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class="vp-sidebar-children"><!--[--><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/turbolink" aria-label="TurboLink" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->TurboLink<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading active collapsible">学习笔记 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学相关 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">图形学 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">环境光渲染 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-高光部分" aria-label="3. 高光部分"><!--[--><!--[--><!--]--><!--]-->3. 高光部分<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-1-原理" aria-label="3.1 原理"><!--[--><!--[--><!--]--><!--]-->3.1 原理<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-2-预渲染环境光贴图-pre-filtered-environment-map" aria-label="3.2 预渲染环境光贴图（pre-filtered environment map)"><!--[--><!--[--><!--]--><!--]-->3.2 预渲染环境光贴图（pre-filtered environment map)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-3-预渲染贴图的生成" aria-label="3.3 预渲染贴图的生成"><!--[--><!--[--><!--]--><!--]-->3.3 预渲染贴图的生成<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-3-brdf部分" aria-label="3.3 BRDF部分"><!--[--><!--[--><!--]--><!--]-->3.3 BRDF部分<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-4-brdf查找贴图" aria-label="3.4 BRDF查找贴图"><!--[--><!--[--><!--]--><!--]-->3.4 BRDF查找贴图<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-4-1-brdf贴图的生成" aria-label="3.4.1 BRDF贴图的生成"><!--[--><!--[--><!--]--><!--]-->3.4.1 BRDF贴图的生成<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="环境光渲染-二" tabindex="-1"><a class="header-anchor" href="#环境光渲染-二"><span>环境光渲染(二）</span></a></h1><hr><h2 id="_3-高光部分" tabindex="-1"><a class="header-anchor" href="#_3-高光部分"><span>3. 高光部分</span></a></h2><h3 id="_3-1-原理" tabindex="-1"><a class="header-anchor" href="#_3-1-原理"><span>3.1 原理</span></a></h3><p>光照方程中的高光部分</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.474em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.474em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>L</mi><mrow data-mjx-texclass="ORD"><mi>o</mi><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi></mrow></msub><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><p>同样使用蒙特卡洛进行近似积分</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:14.605em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-msub space="2"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.474em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.474em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2248"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi></mi><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi></mrow></msub><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd><mo>≈</mo></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>EPIC在此基础上提出了一种近似<a href="http://blog.selfshadow.com/publications/s2013-shading-course/karis/s2013_pbs_epic_notes_v2.pdf" target="_blank" rel="noopener noreferrer">计算方法</a></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:21.045em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c 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stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>≈</mo></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>这种方法的好处是可以通过分开预计算，尽量减少实际渲染时的计算量，这个公式可以理解为“Env. lighting x BRDF”，左边的第一部分是小平面受到的所有环境光的采样值，第二部分只跟材质的BRDF属性相关。</p><figure><img src="/assets/ibl_02-M9fcYT_u.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h3 id="_3-2-预渲染环境光贴图-pre-filtered-environment-map" tabindex="-1"><a class="header-anchor" href="#_3-2-预渲染环境光贴图-pre-filtered-environment-map"><span>3.2 预渲染环境光贴图（pre-filtered environment map)</span></a></h3><p>第一个拆分项只和环境光照以及采样点有关，首先采样点集中在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>周围（<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>相对于法线<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的反射向量），并且粗糙度越大，采样点越分散。 <img src="/assets/ibl_specular_lobe-B1VhXs6g.png" alt="" loading="lazy"> 实际应用中，这个拆分项是通过把环境光贴图预先渲染到一个环境贴图中实现的，称为为"pre-filter map"或者"radiance map"，最终渲染时根据<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>，以及把粗糙度(0~1)转换为mip获得该方向的数值</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c78"></mjx-c><mjx-c class="mjx-c43"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c62"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c4C"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c50"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c4D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mtext>TexCubeLod</mtext><mo stretchy="false">(</mo><mtext>PrefilterMap</mtext><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>r</mi></msub><mo>,</mo><mtext>mip</mtext><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><figure><img src="/assets/ibl_prefilter_map-DNm9qyCg.png" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h3 id="_3-3-预渲染贴图的生成" tabindex="-1"><a class="header-anchor" href="#_3-3-预渲染贴图的生成"><span>3.3 预渲染贴图的生成</span></a></h3><p>生成这张贴图时，首先假定<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.512em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，也就是摄像机从法线方向垂直观察小平面，然后开始采样</p><figure><img src="/assets/ibl_03-HvKFbkIi.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>再最终渲染时，当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>不等时，采样点的分布会有一定差异，但基本上可以忽略</p><h4 id="_3-3-1-采样点的生成" tabindex="-1"><a class="header-anchor" href="#_3-3-1-采样点的生成"><span>3.3.1 采样点的生成</span></a></h4><p>根据前面章节的分析，微表面的法线分布可以用法线分布函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>描述，并且符合</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi></mrow></msub><mi>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub><mi>d</mi><mi>ω</mi><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><p>由于微表面法线的概率密度函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>符合以下公式</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi></mrow></msub><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><p>所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><p>用极坐标表示</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></math></mjx-assistive-mml></mjx-container><p>对于各向同性系统，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container>无关，所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:17.799em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.506em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="3"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></msubsup><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>ϕ</mi><mo>=</mo><mn>2</mn><mi>π</mi><mo>⋅</mo><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>D函数使用GGX函数，也就是</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msup><mi>α</mi><mn>2</mn></msup><mrow><mi>π</mi><mo stretchy="false">(</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mtext><mjx-script style="vertical-align:0.421em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><msup><mtext>roughness</mtext><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container>，此时</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="3"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>π</mi><mo>⋅</mo><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>2</mn><msup><mi>α</mi><mn>2</mn></msup><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>计算<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container>的概率分布函数</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:16.671em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-spacer style="margin-top:1.402em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-spacer style="margin-top:1.402em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mtext>cdf</mtext><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>h</mi></msub></mrow></msubsup><mtext>pdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>θ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>h</mi></msub></mrow></msubsup><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>2</mn><msup><mi>α</mi><mn>2</mn></msup><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mstyle><mi>d</mi><mi>θ</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>使用Mathmatica计算可得</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:12.036em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.396em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mtext>cdf</mtext><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></mrow><mrow><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></mrow><mrow><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>根据前面的概率知识，可以得到微表面的法线的球坐标<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container>的采样函数为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:19.223em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-script style="vertical-align:0.432em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="4"><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mrow space="2"><mjx-mo class="mjx-s4"><mjx-c class="mjx-c28 TEX-S4"></mjx-c></mjx-mo><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-s4"><mjx-c class="mjx-c221A TEX-S4"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.38em;"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-box></mjx-sqrt></mjx-msqrt><mjx-mo class="mjx-s4"><mjx-c class="mjx-c29 TEX-S4"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>θ</mi></mtd><mtd><mi></mi><mo>=</mo><msup><mtext>cdf</mtext><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>θ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>arccos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msqrt><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mi>θ</mi></msub></mrow><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>ξ</mi><mi>θ</mi></msub><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></msqrt><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd><mi>ϕ</mi></mtd><mtd><mi></mi><mo>=</mo><mn>2</mn><mi>π</mi><msub><mi>ξ</mi><mi>ϕ</mi></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>然后即可计算出光线采样点的入射向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></p><h4 id="_3-3-2-代码" tabindex="-1"><a class="header-anchor" href="#_3-3-2-代码"><span>3.3.2 代码</span></a></h4><p>实际运算时，每个采样点的权重是不一样的，采用和宏法线的点积作为权重</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2248"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-munderover limits="false"><mjx-mo class="mjx-sop"><mjx-c class="mjx-c2211 TEX-S1"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.285em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:0.272em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-munderover><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≈</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></math></mjx-assistive-mml></mjx-container><div class="language-cpp" data-highlighter="prismjs" data-ext="cpp" data-title="cpp"><pre><code><span class="line"><span class="token comment">// make the simplyfying assumption that V equals R equals the normal </span></span>
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<span class="line">vec3 R <span class="token operator">=</span> N<span class="token punctuation">;</span></span>
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<span class="line">vec3 V <span class="token operator">=</span> R<span class="token punctuation">;</span></span>
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<span class="line"></span>
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<span class="line"><span class="token keyword">const</span> uint SAMPLE_COUNT <span class="token operator">=</span> <span class="token number">1024u</span><span class="token punctuation">;</span></span>
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<span class="line">vec3 prefilteredColor <span class="token operator">=</span> <span class="token function">vec3_splat</span><span class="token punctuation">(</span><span class="token number">0.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"><span class="token keyword">float</span> totalWeight <span class="token operator">=</span> <span class="token number">0.0</span><span class="token punctuation">;</span></span>
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<span class="line"></span>
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<span class="line"><span class="token keyword">for</span><span class="token punctuation">(</span>uint i <span class="token operator">=</span> <span class="token number">0u</span><span class="token punctuation">;</span> i <span class="token operator"><</span> SAMPLE_COUNT<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span></span>
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<span class="line"><span class="token punctuation">{</span></span>
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<span class="line"> <span class="token comment">// generates a sample vector that's biased towards the preferred </span></span>
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<span class="line"> <span class="token comment">//alignment direction (importance sampling).</span></span>
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<span class="line"> vec2 Xi <span class="token operator">=</span> <span class="token function">Hammersley</span><span class="token punctuation">(</span>i<span class="token punctuation">,</span> SAMPLE_COUNT<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> vec3 H <span class="token operator">=</span> <span class="token function">ImportanceSampleGGX</span><span class="token punctuation">(</span>Xi<span class="token punctuation">,</span> N<span class="token punctuation">,</span> roughness<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> vec3 L <span class="token operator">=</span> <span class="token function">normalize</span><span class="token punctuation">(</span><span class="token number">2.0</span> <span class="token operator">*</span> <span class="token function">dot</span><span class="token punctuation">(</span>V<span class="token punctuation">,</span> H<span class="token punctuation">)</span> <span class="token operator">*</span> H <span class="token operator">-</span> V<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"></span>
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<span class="line"> <span class="token keyword">float</span> NdotL <span class="token operator">=</span> <span class="token function">max</span><span class="token punctuation">(</span><span class="token function">dot</span><span class="token punctuation">(</span>N<span class="token punctuation">,</span> L<span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token number">0.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> <span class="token keyword">if</span><span class="token punctuation">(</span>NdotL <span class="token operator">></span> <span class="token number">0.0</span><span class="token punctuation">)</span></span>
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<span class="line"> <span class="token punctuation">{</span></span>
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<span class="line"> prefilteredColor <span class="token operator">+=</span> <span class="token function">texture2D</span><span class="token punctuation">(</span>s_texSkybox<span class="token punctuation">,</span> <span class="token function">SampleSphericalMap</span><span class="token punctuation">(</span>L<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">.</span>rgb <span class="token operator">*</span> NdotL<span class="token punctuation">;</span></span>
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<span class="line"> totalWeight <span class="token operator">+=</span> NdotL<span class="token punctuation">;</span></span>
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<span class="line"> <span class="token punctuation">}</span></span>
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<span class="line"><span class="token punctuation">}</span></span>
|
||
<span class="line"></span>
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||
<span class="line">prefilteredColor <span class="token operator">=</span> prefilteredColor <span class="token operator">/</span> totalWeight<span class="token punctuation">;</span></span>
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<span class="line"></span></code></pre></div><h2 id="_3-3-brdf部分" tabindex="-1"><a class="header-anchor" href="#_3-3-brdf部分"><span>3.3 BRDF部分</span></a></h2><p>观察公式的第二个拆分项</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>其中</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:17.887em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c 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class="mjx-c35"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>D</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><msup><mo stretchy="false">)</mo><mn>5</mn></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>设<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>c</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><msup><mo stretchy="false">)</mo><mn>5</mn></msup></math></mjx-assistive-mml></mjx-container>，那么</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>F</mi><mi>c</mi></msub></math></mjx-assistive-mml></mjx-container><p>带入上面的式中，可得</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:38.144em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mstyle space="3"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd><mtd><mi></mi><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><msub><mi>F</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>+</mo><msub><mi>F</mi><mi>c</mi></msub><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>设</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>G</mi><mi>v</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>那么可得</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c66"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c28 TEX-S3"></mjx-c></mjx-mo><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-lop" space="2"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-mo class="mjx-s3"><mjx-c class="mjx-c29 TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mstyle space="3"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-lop" space="2"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>c</mi></msub><mo stretchy="false">)</mo><msub><mi>G</mi><mi>v</mi></msub></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><mo data-mjx-texclass="OP">∑</mo><msub><mi>F</mi><mi>c</mi></msub><msub><mi>G</mi><mi>v</mi></msub></math></mjx-assistive-mml></mjx-container><h2 id="_3-4-brdf查找贴图" tabindex="-1"><a class="header-anchor" href="#_3-4-brdf查找贴图"><span>3.4 BRDF查找贴图</span></a></h2><p>上面的公式中，除了<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>和运行时的材质相关，其他数值都只取决于所使用的微表面的函数类型，所以可以预先计算在一张二维贴图中</p><figure><img src="/assets/env_brdf-HRbq0kY7.png" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>在运行时可以使用查表得方式从这张贴图上获取数据</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.086em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.524em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.537em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c 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class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>f</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pdf</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>k</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mo>∗</mo><mtext>Tex2D</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mtext>Roughness</mtext><mo stretchy="false">)</mo><mo>.</mo><mi>r</mi><mo>+</mo><mtext>Tex2D</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mtext>Roughness</mtext><mo stretchy="false">)</mo><mo>.</mo><mi>g</mi></math></mjx-assistive-mml></mjx-container><h3 id="_3-4-1-brdf贴图的生成" tabindex="-1"><a class="header-anchor" href="#_3-4-1-brdf贴图的生成"><span>3.4.1 BRDF贴图的生成</span></a></h3><div class="language-cpp" data-highlighter="prismjs" data-ext="cpp" data-title="cpp"><pre><code><span class="line">vec2 <span class="token function">IntegrateBRDF</span><span class="token punctuation">(</span><span class="token keyword">float</span> NdotV<span class="token punctuation">,</span> <span class="token keyword">float</span> roughness<span class="token punctuation">)</span></span>
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<span class="line"><span class="token punctuation">{</span></span>
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<span class="line"> vec3 V<span class="token punctuation">;</span></span>
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<span class="line"> V<span class="token punctuation">.</span>x <span class="token operator">=</span> <span class="token function">sqrt</span><span class="token punctuation">(</span><span class="token number">1.0</span> <span class="token operator">-</span> NdotV<span class="token operator">*</span>NdotV<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> V<span class="token punctuation">.</span>y <span class="token operator">=</span> <span class="token number">0.0</span><span class="token punctuation">;</span></span>
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||
<span class="line"> V<span class="token punctuation">.</span>z <span class="token operator">=</span> NdotV<span class="token punctuation">;</span></span>
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<span class="line"></span>
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<span class="line"> <span class="token keyword">float</span> A <span class="token operator">=</span> <span class="token number">0.0</span><span class="token punctuation">;</span></span>
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||
<span class="line"> <span class="token keyword">float</span> B <span class="token operator">=</span> <span class="token number">0.0</span><span class="token punctuation">;</span> </span>
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||
<span class="line"></span>
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||
<span class="line"> vec3 N <span class="token operator">=</span> <span class="token function">vec3</span><span class="token punctuation">(</span><span class="token number">0.0</span><span class="token punctuation">,</span> <span class="token number">0.0</span><span class="token punctuation">,</span> <span class="token number">1.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> </span>
|
||
<span class="line"> <span class="token keyword">const</span> uint SAMPLE_COUNT <span class="token operator">=</span> <span class="token number">1024u</span><span class="token punctuation">;</span></span>
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||
<span class="line"> <span class="token keyword">for</span><span class="token punctuation">(</span>uint i <span class="token operator">=</span> <span class="token number">0u</span><span class="token punctuation">;</span> i <span class="token operator"><</span> SAMPLE_COUNT<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span></span>
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||
<span class="line"> <span class="token punctuation">{</span></span>
|
||
<span class="line"> <span class="token comment">// generates a sample vector that's biased towards the</span></span>
|
||
<span class="line"> <span class="token comment">// preferred alignment direction (importance sampling).</span></span>
|
||
<span class="line"> vec2 Xi <span class="token operator">=</span> <span class="token function">Hammersley</span><span class="token punctuation">(</span>i<span class="token punctuation">,</span> SAMPLE_COUNT<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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<span class="line"> vec3 H <span class="token operator">=</span> <span class="token function">ImportanceSampleGGX</span><span class="token punctuation">(</span>Xi<span class="token punctuation">,</span> N<span class="token punctuation">,</span> roughness<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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||
<span class="line"> vec3 L <span class="token operator">=</span> <span class="token function">normalize</span><span class="token punctuation">(</span><span class="token number">2.0</span> <span class="token operator">*</span> <span class="token function">dot</span><span class="token punctuation">(</span>V<span class="token punctuation">,</span> H<span class="token punctuation">)</span> <span class="token operator">*</span> H <span class="token operator">-</span> V<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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||
<span class="line"></span>
|
||
<span class="line"> <span class="token keyword">float</span> NdotL <span class="token operator">=</span> <span class="token function">max</span><span class="token punctuation">(</span>L<span class="token punctuation">.</span>z<span class="token punctuation">,</span> <span class="token number">0.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"> <span class="token keyword">float</span> NdotH <span class="token operator">=</span> <span class="token function">max</span><span class="token punctuation">(</span>H<span class="token punctuation">.</span>z<span class="token punctuation">,</span> <span class="token number">0.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"> <span class="token keyword">float</span> VdotH <span class="token operator">=</span> <span class="token function">max</span><span class="token punctuation">(</span><span class="token function">dot</span><span class="token punctuation">(</span>V<span class="token punctuation">,</span> H<span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token number">0.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"> <span class="token keyword">if</span><span class="token punctuation">(</span>NdotL <span class="token operator">></span> <span class="token number">0.0</span><span class="token punctuation">)</span></span>
|
||
<span class="line"> <span class="token punctuation">{</span></span>
|
||
<span class="line"> <span class="token keyword">float</span> G <span class="token operator">=</span> <span class="token function">GeometrySmith</span><span class="token punctuation">(</span>N<span class="token punctuation">,</span> V<span class="token punctuation">,</span> L<span class="token punctuation">,</span> roughness<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"> <span class="token keyword">float</span> G_Vis <span class="token operator">=</span> <span class="token punctuation">(</span>G <span class="token operator">*</span> VdotH<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span>NdotH <span class="token operator">*</span> NdotV<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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||
<span class="line"> <span class="token keyword">float</span> Fc <span class="token operator">=</span> <span class="token function">pow</span><span class="token punctuation">(</span><span class="token number">1.0</span> <span class="token operator">-</span> VdotH<span class="token punctuation">,</span> <span class="token number">5.0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"> A <span class="token operator">+=</span> <span class="token punctuation">(</span><span class="token number">1.0</span> <span class="token operator">-</span> Fc<span class="token punctuation">)</span> <span class="token operator">*</span> G_Vis<span class="token punctuation">;</span></span>
|
||
<span class="line"> B <span class="token operator">+=</span> Fc <span class="token operator">*</span> G_Vis<span class="token punctuation">;</span></span>
|
||
<span class="line"> <span class="token punctuation">}</span></span>
|
||
<span class="line"> <span class="token punctuation">}</span></span>
|
||
<span class="line"> A <span class="token operator">/=</span> <span class="token keyword">float</span><span class="token punctuation">(</span>SAMPLE_COUNT<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"> B <span class="token operator">/=</span> <span class="token keyword">float</span><span class="token punctuation">(</span>SAMPLE_COUNT<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
|
||
<span class="line"> <span class="token keyword">return</span> <span class="token function">vec2</span><span class="token punctuation">(</span>A<span class="token punctuation">,</span> B<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
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||
<span class="line"><span class="token punctuation">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"></span></code></pre></div></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>环境光渲染(一)</span></div><!--]--></a><!----></nav><!--[--><!--]--></main><!--]--></div><!--[--><!----><!--]--><!--]--></div>
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