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target="_blank"><!--[--><!--[--><!--]--><!--]-->Cyclone<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading collapsible">学习笔记 <span class="right arrow"></span></p><ul style="display:none;" 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 collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" 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/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><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 collapsible">环境光渲染 <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/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 auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></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="jpeg算法解密-二" tabindex="-1"><a class="header-anchor" href="#jpeg算法解密-二"><span>JPEG算法解密（二）</span></a></h1><h3 id="步骤三-离散余弦变换" tabindex="-1"><a class="header-anchor" href="#步骤三-离散余弦变换"><span>步骤三：离散余弦变换</span></a></h3><hr><p>这次我们来介绍JPEG算法中的核心内容，离散余弦变换（Discrete cosine transform），简称DCT。<br> 离散余弦变换属于傅里叶变换的另外一种形式，没错，就是大名鼎鼎的傅里叶变换。傅里叶是法国著名的数学家和物理学家，1807年，39岁的傅里叶在他的一篇论文里提出了一个想法，他认为<em>任何周期性的函数，都可以分解为为一系列的三角函数的组合</em>，这个想法一开始并没有得到当时科学界的承认，比如当时著名的数学家拉格朗日提出质疑，三角函数无论如何组合，都无法表达带有“尖角”的函数，一直到1822年拉格朗日死后，傅里叶的想法才正式在他的著作《热的解析理论》一书中正式发表。</p><figure><img src="/images/2014/08/jpeg_20.jpg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>金子总会闪光，傅里叶变换如今广泛应用于数学、物理、信号处理等等领域，变换除了它在数学上的意义外，还有其哲学上的伟大意义，那就是，世上任何复杂的事物，都可以分解为简单的事物的组合，而这个过程只需要借助数学工具就可以了。但是当年拉格朗日的质疑是正确的，三角函数的确无法表达出尖角形状的函数，不过只要三角函数足够多，可以无限逼近最终结果。比如下面这张动图，就动态描述了一个矩形方波，是如何做傅里叶分析的。</p><p align="center"><img src="/images/2014/08/jpeg_22.jpg" width="45%"><img src="/images/2014/08/jpeg_21.gif" width="45%"></p><p>当我们要处理的不再是函数，而是一堆离散的数据时，并且这些数据是对称的话，那么傅里叶变化出来的函数只含有余弦项，这种变换称为离散余弦变换。举个例子，有一组一维数据<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><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-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 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-c1D45B TEX-I"></mjx-c></mjx-mi><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-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>,那么可以通过DCT变换得到<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-c1D45B 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>n</mi></math></mjx-assistive-mml></mjx-container>个变换级数<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-c1D456 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><mi>F</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:19.133em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:19.133em;" side="right"><mjx-table style="width:auto;min-width:13.421em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mtable style="min-width:13.421em;"><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-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-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><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-munderover space="4"><mjx-over style="padding-bottom:0.142em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-c30"></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-c1D465 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-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-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A 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-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></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-c1D45A 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-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:19.133em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:4.287em;"><mjx-mtd id="mjx-eqn:2.1"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:4.287em;vertical-align:-1.893em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></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"><mlabeledtr><mtd><mtext>(2.1)</mtext></mtd><mtd><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>F</mi><mi>m</mi></msub></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi>x</mi><mi>k</mi></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mi>π</mi><mi>n</mi></mfrac><mi>m</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>此时原始数据<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-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45A 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><mi>x</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container>可以通过离散余弦变换变化的逆变换（IDCT)表达出来</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:23.705em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:23.705em;" side="right"><mjx-table style="width:auto;min-width:17.993em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mtable style="min-width:17.993em;"><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-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><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-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><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-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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-munderover space="3"><mjx-over style="padding-bottom:0.142em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><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-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-c1D458 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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></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-c1D45A 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-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:23.705em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:4.287em;"><mjx-mtd id="mjx-eqn:2.2"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:4.287em;vertical-align:-1.893em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></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"><mlabeledtr><mtd><mtext>(2.2)</mtext></mtd><mtd><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>x</mi><mi>m</mi></msub></mtd><mtd><mi></mi><mo>=</mo><mfrac><msub><mi>F</mi><mn>0</mn></msub><mi>n</mi></mfrac><mo>+</mo><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><mo>−</mo><mn>1</mn></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mi>k</mi></msub></mrow><mi>n</mi></mfrac><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mi>π</mi><mi>n</mi></mfrac><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo><mi>k</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>也就是说，经过DCT变换，可以把一个数组分解成数个数组的和，如果我们数组视为一个一维矩阵，那么可以把结果看做是一系列矩阵的和</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:35.995em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:35.995em;" side="right"><mjx-table style="width:auto;min-width:30.283em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mtable style="min-width:30.283em;"><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-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><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-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c 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class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></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-TeXAtom space="2" texclass="ORD"><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-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-mi class="mjx-i"><mjx-c class="mjx-c1D70B 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></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-TeXAtom space="2" texclass="ORD"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-mi class="mjx-i"><mjx-c class="mjx-c1D70B 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:35.995em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:12.745em;"><mjx-mtd id="mjx-eqn:2.3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:12.745em;vertical-align:-6.123em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></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"><mlabeledtr><mtd><mtext>(2.3)</mtext></mtd><mtd><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>0</mn></msub><mi>n</mi></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mn>1</mn></msub></mrow><mi>n</mi></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mi>π</mi><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mn>2</mn></msub></mrow><mi>n</mi></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>6</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>10</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mn>3</mn></msub></mrow><mi>n</mi></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>9</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>15</mn><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>3</mn><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mo>…</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>n</mi></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mn>3</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>π</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>举个例子，我们有一个长度为8的数字，内容为[50,55,67,80,-10,-5,20,30]，在数轴上展示的话，没有任何明显的规律</p><figure><img src="/images/2014/08/jpeg_25.gif" alt="" tabindex="0" loading="lazy"><figcaption>[50,55,67,80,-10,-5,20,30]</figcaption></figure><p>经过DCT转换，得到8个级数为[287.0,106.3,14.2,-110.8,9.2,65.7,-8.2,-43.9]，根据公式2.3把这个数组转换为8个新的数组的和,再使用图像来表达的话，就可以发现DCT转换的有趣之处了</p><figure><img src="/images/2014/08/jpeg_26.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组0：[35.9,35.9,35.9,35.9,35.9,35.9,35.9,35.9]</figcaption></figure><figure><img src="/images/2014/08/jpeg_27.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组1：[26.0,22.1,14.8,5.2,-5.2,-14.8,-22.1,-26.1]</figcaption></figure><figure><img src="/images/2014/08/jpeg_28.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组2：[3.3,1.4,-1.4,-3.3,-3.3,-1.4,1.4,3.3]</figcaption></figure><figure><img src="/images/2014/08/jpeg_29.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组3：[-23.0,5.4,27.2,15.4,-15.4,-27.2,-5.4,23.0]</figcaption></figure><figure><img src="/images/2014/08/jpeg_30.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组4：[1.6,-1.6,-1.6,1.6,1.6,-1.6,-1.6,1.6]</figcaption></figure><figure><img src="/images/2014/08/jpeg_31.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组5：[9.1,-16.1,3.2,13.6,-13.6,-3.2,16.1,-9.1]</figcaption></figure><figure><img src="/images/2014/08/jpeg_32.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组6：[-0.8,1.9,-1.9,0.8,0.8,-1.9,1.9,-0.8]</figcaption></figure><figure><img src="/images/2014/08/jpeg_33.gif" alt="" tabindex="0" loading="lazy"><figcaption>数组7：[-2.1,6.1,-9.1,10.8,-10.8,9.1,-6.1,2.1]</figcaption></figure><p>奥妙之处在于，经过DCT，数据中隐藏的规律被发掘了出来,杂乱的数据被转换成几个工整变化的数据。DCT转换后的数组中第一个是一个直线数据，因此又被称为“直流数据”，简称DC，后面的数据被称为“交流数据”，简称AC，这个称呼起源于信号分析中的术语。<br> 在JPEG压缩过程中，经过颜色空间的转换，每一个8X8的图像块，在数据上表现为3个8X8的矩阵，紧接着我们对这三个矩阵做一个二维的DCT转换，二维的DCT转换公式为</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:31.745em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" 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style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c75"></mjx-c></mjx-mtext><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-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c75"></mjx-c></mjx-mtext><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2260"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:31.745em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:6.001em;"><mjx-mtd id="mjx-eqn:2.4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:6.001em;vertical-align:-2.788em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></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"><mlabeledtr><mtd><mtext>(2.4)</mtext></mtd><mtd><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>α</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>⋅</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>x</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mn>7</mn></mrow></munderover><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>y</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mn>7</mn></mrow></munderover><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mn>16</mn></mfrac><mi>u</mi><mi>π</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mn>16</mn></mfrac><mi>v</mi><mi>π</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd><mi>u</mi><mo>,</mo><mi>v</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>7</mn></mtd></mtr><mtr><mtd><mi>α</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msqrt><mn>8</mn></msqrt><mo>,</mo></mtd><mtd><mtext>when u</mtext><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo></mtd><mtd><mtext>when u</mtext><mo>≠</mo><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd></mtr></mtable></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>DCT的威力究竟有多大，我们可以做一个实际的测试，比如一个所有数值都一样的矩阵，经过DCT转换后，将所有级数组合成一个新的矩阵</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:20.334em;"><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-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:10.8em;vertical-align:-5.15em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:19em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn 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style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:10.8em;vertical-align:-5.15em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr><mtr><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd><mtd><mn>100</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mstyle mathsize="1.2em"><mrow data-mjx-texclass="ORD"><mo stretchy="false">⇓</mo></mrow><mtext>DCT</mtext></mstyle></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>800</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>可以看到，经过DCT转换，矩阵的“能量”被全部集中在左上角上的直流分量F(0,0）上，其他位置都变成了0。<br> 在实际的JPEG压缩过程中，由于图像本身的连贯性，一个8X8的图像中的数值一般不会出现大的跳跃，经过DCT转换会有类似的效果，左上角的直流分量保存了一个大的数值，其他分量都接近于0，我们以Lenna左上角第一块图像的Y分量为例，经过变换的矩阵为</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:21.253em;"><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-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:10.8em;vertical-align:-5.15em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:15em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c 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rowspacing="4pt"><mtr><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>32</mn></mtd></mtr><mtr><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>32</mn></mtd></mtr><mtr><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>32</mn></mtd></mtr><mtr><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>32</mn></mtd></mtr><mtr><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>34</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>32</mn></mtd></mtr><mtr><mtd><mn>36</mn></mtd><mtd><mn>36</mn></mtd><mtd><mn>29</mn></mtd><mtd><mn>27</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>31</mn></mtd><mtd><mn>30</mn></mtd><mtd><mn>31</mn></mtd></mtr><mtr><mtd><mn>32</mn></mtd><mtd><mn>32</mn></mtd><mtd><mn>35</mn></mtd><mtd><mn>30</mn></mtd><mtd><mn>32</mn></mtd><mtd><mn>33</mn></mtd><mtd><mn>31</mn></mtd><mtd><mn>27</mn></mtd></mtr><mtr><mtd><mn>30</mn></mtd><mtd><mn>30</mn></mtd><mtd><mn>27</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>30</mn></mtd><mtd><mn>30</mn></mtd><mtd><mn>28</mn></mtd><mtd><mn>29</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mstyle mathsize="1.2em"><mrow data-mjx-texclass="ORD"><mo stretchy="false">⇓</mo></mrow><mtext>DCT</mtext></mstyle></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mstyle mathsize="0.85em"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>257.1</mn></mtd><mtd><mn>6.4</mn></mtd><mtd><mn>2.5</mn></mtd><mtd><mo>−</mo><mn>0.3</mn></mtd><mtd><mn>0.4</mn></mtd><mtd><mn>0.1</mn></mtd><mtd><mo>−</mo><mn>6.0</mn></mtd><mtd><mn>6.9</mn></mtd></mtr><mtr><mtd><mn>8.4</mn></mtd><mtd><mn>0.0</mn></mtd><mtd><mn>0.5</mn></mtd><mtd><mo>−</mo><mn>0.5</mn></mtd><mtd><mn>1.9</mn></mtd><mtd><mn>3.4</mn></mtd><mtd><mo>−</mo><mn>4.2</mn></mtd><mtd><mn>3.3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>5.3</mn></mtd><mtd><mo>−</mo><mn>1.0</mn></mtd><mtd><mo>−</mo><mn>1.4</mn></mtd><mtd><mn>1.3</mn></mtd><mtd><mo>−</mo><mn>0.7</mn></mtd><mtd><mo>−</mo><mn>0.5</mn></mtd><mtd><mn>2.1</mn></mtd><mtd><mo>−</mo><mn>1.7</mn></mtd></mtr><mtr><mtd><mn>2.4</mn></mtd><mtd><mn>1.7</mn></mtd><mtd><mn>1.5</mn></mtd><mtd><mn>1.5</mn></mtd><mtd><mo>−</mo><mn>0.6</mn></mtd><mtd><mo>−</mo><mn>1.5</mn></mtd><mtd><mn>0.2</mn></mtd><mtd><mn>0.4</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>1.1</mn></mtd><mtd><mo>−</mo><mn>1.6</mn></mtd><mtd><mo>−</mo><mn>0.2</mn></mtd><mtd><mo>−</mo><mn>1.8</mn></mtd><mtd><mn>1.6</mn></mtd><mtd><mn>1.2</mn></mtd><mtd><mo>−</mo><mn>1.4</mn></mtd><mtd><mo>−</mo><mn>0.1</mn></mtd></mtr><mtr><mtd><mn>1.4</mn></mtd><mtd><mn>0.9</mn></mtd><mtd><mo>−</mo><mn>1.9</mn></mtd><mtd><mo>−</mo><mn>0.1</mn></mtd><mtd><mo>−</mo><mn>2.0</mn></mtd><mtd><mn>0.9</mn></mtd><mtd><mn>1.5</mn></mtd><mtd><mn>0.7</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>2.0</mn></mtd><mtd><mo>−</mo><mn>0.1</mn></mtd><mtd><mn>3.1</mn></mtd><mtd><mn>2.0</mn></mtd><mtd><mn>1.8</mn></mtd><mtd><mo>−</mo><mn>2.7</mn></mtd><mtd><mo>−</mo><mn>0.9</mn></mtd><mtd><mo>−</mo><mn>1.3</mn></mtd></mtr><mtr><mtd><mn>1.5</mn></mtd><mtd><mo>−</mo><mn>0.2</mn></mtd><mtd><mo>−</mo><mn>2.3</mn></mtd><mtd><mo>−</mo><mn>1.9</mn></mtd><mtd><mo>−</mo><mn>1.0</mn></mtd><mtd><mn>2.3</mn></mtd><mtd><mn>0.3</mn></mtd><mtd><mn>1.1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>可以看到，数据经过DCT变化后，被明显分成了直流分量和交流分量两部分，为后面的进一步压缩起到了充分的铺垫作用，可以说是整个JPEG中最重要的一步，后面我们会介绍数据量化。</p></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="/blog/2025/02/JPEG001.html" aria-label="Part1"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>Part1</span></div><!--]--></a><a class="route-link auto-link next" 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