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aria-label="如何计算线段和圆的交点"><!---->如何计算线段和圆的交点<!----></a></li><li><a class="route-link auto-link vp-sidebar-link" href="/blog/2025/02/Ellipse.html" aria-label="一道数学趣题"><!---->一道数学趣题<!----></a></li><li><a class="route-link auto-link vp-sidebar-link" href="/blog/2025/02/Fibonacci.html" aria-label="斐波那契数列和1/89"><!---->斐波那契数列和1/89<!----></a></li><li><section class="vp-sidebar-group"><button class="vp-sidebar-header clickable" type="button"><!----><span class="vp-sidebar-title">匀速贝塞尔曲线运动的实现</span><span class="vp-arrow end"></span></button><!----></section></li></ul></section></li><li><section class="vp-sidebar-group"><button class="vp-sidebar-header clickable" type="button"><iconify-icon class="vp-icon" icon="fa6-solid:diagram-project" sizing="both" width="1em" height="1em"></iconify-icon><span class="vp-sidebar-title">开源项目</span><span class="vp-arrow end"></span></button><!----></section></li><li><section class="vp-sidebar-group"><button class="vp-sidebar-header clickable" type="button"><iconify-icon class="vp-icon" icon="fa6-solid:lightbulb" sizing="both" width="1em" height="1em"></iconify-icon><span class="vp-sidebar-title">学习笔记</span><span class="vp-arrow end"></span></button><!----></section></li></ul><!----></aside><!--[--><main id="main-content" class="vp-page"><!--[--><!----><!----><nav class="vp-breadcrumb disable"></nav><div class="vp-page-title"><h1><!---->JPEG算法解密（四）</h1><!----><hr></div><!----><div class="" vp-content><!----><div id="markdown-content"><h1 id="jpeg算法解密-四" tabindex="-1"><a class="header-anchor" href="#jpeg算法解密-四"><span>JPEG算法解密（四）</span></a></h1><h3 id="步骤五-哈弗曼编码" tabindex="-1"><a class="header-anchor" href="#步骤五-哈弗曼编码"><span>步骤五：哈弗曼编码</span></a></h3><hr><p>JPEG压缩的最后一步是对数据进行哈弗曼编码(Huffman coding)，哈弗曼几乎是所有压缩算法的基础，它的基本原理是根据数据中元素的使用频率，调整元素的编码长度，以得到更高的压缩比。<br> 举个例子，比如下面这段数据</p><div class="language-" data-highlighter="shiki" data-ext="" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34;"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-"><span class="line"><span>AABCBABBCDBBDDBAABDBBDABBBBDDEDBD</span></span></code></pre></div><p>这段数据里面包含了33个字符，每种字符出现的次数统计如下</p><table class="gridtable" style="width:400px;text-align:center;border-collapse:collapse;"><tbody><tr><th style="width:100px;">字符</th><td style="width:60px;">A</td><td style="width:60px;">B</td><td style="width:60px;">C</td><td style="width:60px;">D</td><td style="width:60px;">E</td></tr><tr><th>次数</th><td>6</td><td>15</td><td>2</td><td>9</td><td>1</td></tr></tbody></table> 如果我们用我们常见的定长编码，每个字符都是3个bit。 <table class="gridtable" style="width:400px;text-align:center;border-collapse:collapse;"><tbody><tr><th style="width:100px;">字符</th><td style="width:60px;">A</td><td style="width:60px;">B</td><td style="width:60px;">C</td><td style="width:60px;">D</td><td style="width:60px;">E</td></tr><tr><th>编码</th><td>001</td><td>010</td><td>011</td><td>100</td><td>101</td></tr></tbody></table><p>那么这段文字共需要<code>3*33=99</code>个bit来保存，但如果我们根据字符出现的概率来编码，也就是出现频率较高的字符，使用较短的编码，如下：</p><table class="gridtable" style="width:400px;text-align:center;border-collapse:collapse;"><tbody><tr><th style="width:100px;">字符</th><td style="width:60px;">A</td><td style="width:60px;">B</td><td style="width:60px;">C</td><td style="width:60px;">D</td><td style="width:60px;">E</td></tr><tr><th>编码</th><td>110</td><td>0</td><td>1110</td><td>10</td><td>1111</td></tr></tbody></table><p>那么这段文字共需要<code>3*6+1*15+4*2+2*9+4*1=63</code>个bit来保存，压缩比为63%，哈弗曼编码一般都是使用二叉树来生成的，这样得到的编码符合前缀规则，也就是较短的编码不能够是较长编码的前缀，比如字符'B'使用的编码是'0'，那么其他字符的编码的第一个字符都不能是‘0’。<br> 上面这个编码实例，就是由下面的这颗二叉树生成的。</p><figure><img src="/images/2014/08/hufman7.gif" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>我们回到JPEG压缩上，回顾上一节的内容，经过数据量化，我们现在要处理的数据是一串一维数组，举例如下：</p><table class="gridtable" style="width:600px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:500px;"><!----></td></tr></tbody></table> 在实际的压缩过程中，数据中的0出现的概率非常高，所以首先要做的事情，是使用RLE编码对其中的0进行处理，把数据中的非零的数据，以及数据前面0的个数作为一个处理单元。 <table class="gridtable" align="center" style="width:600px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:500px;" colspan="8"><!----></td></tr><tr><th style="text-align:left;"><strong>②RLE编码</strong></th><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td>0,0,…,0,8</td><td>0,0,…,0</td></tr></tbody></table> 如果其中某个单元的0的个数超过16，则需要分成每16个一组，如果最后一个单元全都是0，则使用特殊字符“EOB”表示，EOB意思就是“后面的数据全都是0”, <table class="gridtable" align="center" style="width:600px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:500px;" colspan="8"><!----></td></tr><tr><th style="text-align:left;" rowspan="3"><strong>②RLE编码</strong></th><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td colspan="2">0,0,…,0,8</td><td>0,0,…,0</td></tr><tr><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td>0,0,…,0</td><td>0,0,8</td><td>0,0,…,0</td></tr><tr><td>(0,35)</td><td>(0,7)</td><td>(3,-6)</td><td>(0,-2)</td><td>(2,-9)</td><td>(15,0)</td><td>(2,8)</td><td>EOB</td></tr></tbody></table> 其中（15,0）表示15+1也就是16个0，接下来我们要处理的是括号里右面的数字，这个数字的取值范围在-2047~2047之间，JPEG提供了一张标准的码表用于对这些数字编码： <table class="gridtable" style="width:550px;text-align:center;"><tbody><tr><th style="width:250px;" colspan="2">Value</th><th style="width:50px;">Size</th><th style="width:250px;" colspan="2">Bits </th></tr><tr><td align="center" colspan="2">0</td><td>0</td><td align="center" colspan="2">–</td></tr><tr><td align="right">-1</td><td align="left">1</td><td>1</td><td align="right">0</td><td align="left">1</td></tr><tr><td align="right">-3,-2</td><td align="left">2,3</td><td>2</td><td align="right">00,01</td><td align="left">10,11</td></tr><tr><td align="right">-7,-6,-5,-4</td><td align="left">4,5,6,7</td><td>3</td><td align="right">000,001,010,011</td><td align="left">100,101,110,111</td></tr><tr><td align="right">-15,…,-8</td><td align="left">8,…,15</td><td>4</td><td align="right">0000,…,0111</td><td align="left">1000,…,1111</td></tr><tr><td align="right">-31,…,-16</td><td align="left">16,…,31</td><td>5</td><td align="right">0 0000,…,0 1111</td><td align="left">1 0000,…,1 1111 </td></tr><tr><td align="right">-63,…,-32</td><td align="left">32,…,63</td><td>6</td><td align="right">00 0000,…</td><td align="left">…,11 1111 </td></tr><tr><td align="right">-127,…,-64</td><td align="left">64,…,127</td><td>7</td><td align="right">000 0000,…</td><td align="left">…,111 1111 </td></tr><tr><td align="right">-255,…,-128</td><td align="left">128,…,255</td><td>8</td><td align="right">0000 0000,…</td><td align="left">…,1111 1111 </td></tr><tr><td align="right">-511,…,-256</td><td align="left">256,…,511</td><td>9</td><td align="right">0 0000 0000,…</td><td align="left">…,1 1111 1111 </td></tr><tr><td align="right">-1023,…,-512</td><td align="left">512,…,1023</td><td>10</td><td align="right">00 0000 0000,…</td><td align="left">…,11 1111 1111 </td></tr><tr><td align="right">-2047,…,-1024</td><td align="left">1024,…,2047</td><td>11</td><td align="right">000 0000 0000,…</td><td align="left">…,111 1111 1111</td></tr></tbody></table> 举例来说，第一个单元中的“35”这个数字，在表中的位置是长度为6的那组，所对应的bit码是“100011”，而“-6”的编码是”001″，由于这种编码附带长度信息，所以我们的数据变成了如下的格式。 <table class="gridtable" align="center" style="width:800px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:700px;" colspan="8"><!----></td></tr><tr><th style="text-align:left;" rowspan="3"><strong>②RLE编码</strong></th><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td colspan="2">0,0,…,0,8</td><td>0,0,…,0</td></tr><tr style="padding:3px 3px;"><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td>0,0,…,0</td><td>0,0,8</td><td>0,0,…,0</td></tr><tr><td>(0,35)</td><td>(0,7)</td><td>(3,-6)</td><td>(0,-2)</td><td>(2,-9)</td><td>(15,0)</td><td>(2,8)</td><td>EOB</td></tr><tr><th style="text-align:left;"><strong>③BIT编码</strong></th><td>(0,6, <em><b>100011</b></em>)</td><td>(0,3, <em><b>111</b></em>)</td><td>(3,3, <em><b>001</b></em>)</td><td>(0,2, <em><b>01</b></em>)</td><td>(2,4, <em><b>0110</b></em>)</td><td>(15,-)</td><td>(2,4, <em><b>1000</b></em>)</td><td>EOB</td></tr></tbody></table> 括号中前两个数字分都在0~15之间，所以这两个数可以合并成一个byte，高四位是前面0的个数，后四位是后面数字的位数。 <table class="gridtable" align="center" style="width:800px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:700px;" colspan="8"><!----></td></tr><tr><th style="text-align:left;" rowspan="3"><strong>②RLE编码</strong></th><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td colspan="2">0,0,…,0,8</td><td>0,0,…,0</td></tr><tr style="padding:3px 3px;"><td>35</td><td>7</td><td>0,0,0,-6</td><td>-2</td><td>0,0,-9</td><td>0,0,…,0</td><td>0,0,8</td><td>0,0,…,0</td></tr><tr><td>(0,35)</td><td>(0,7)</td><td>(3,-6)</td><td>(0,-2)</td><td>(2,-9)</td><td>(15,0)</td><td>(2,8)</td><td>EOB</td></tr><tr><th style="text-align:left;" rowspan="2"><strong>③BIT编码</strong></th><td>(0,6, <em><b>100011</b></em>)</td><td>(0,3, <em><b>111</b></em>)</td><td>(3,3, <em><b>001</b></em>)</td><td>(0,2, <em><b>01</b></em>)</td><td>(2,4, <em><b>0110</b></em>)</td><td>(15,-)</td><td>(2,4, <em><b>1000</b></em>)</td><td>EOB</td></tr><tr><td>(0x6,<em><b>100011</b></em>)</td><td>(0x3,<em><b>111</b></em>)</td><td>(0x33,<em><b>001</b></em>)</td><td>(0x2,<em><b>01</b></em>)</td><td>(0x24,<em><b>0110</b></em>)</td><td>(0xF0,-)</td><td>(0x24,<em><b>1000</b></em>)</td><td>EOB</td></tr></tbody></table> 对于括号前面的数字的编码，就要使用到我们提到的哈弗曼编码了，比如下面这张表，就是一张针对数据中的第一个单元，也就是直流(DC)部分的哈弗曼表，由于直流部分没有前置的0，所以取值范围在0~15之间。 <table class="gridtable" style="width:300px;text-align:left;"><tbody><tr><th style="width:50px;">Length</th><th style="width:80px;">Value</th><th style="width:170px;">Bits</th></tr><tr><td>3 bits </td><td>04<br>05<br>03<br>02<br>06<br>01<br>00 (EOB) </td><td>000<br>001<br>010<br>011<br>100<br>101<br>110</td></tr><tr><td>4 bits </td><td>07</td><td>1110</td></tr><tr><td>5 bits </td><td>08</td><td>1111 0</td></tr><tr><td>6 bits </td><td>09</td><td>1111 10</td></tr><tr><td>7 bits </td><td>0A</td><td>1111 110 </td></tr><tr><td>8 bits </td><td>0B</td><td>1111 1110 </td></tr></tbody></table> 举例来说，示例中的DC部分的数据是0x06，对应的二进制编码是“100”，而对于后面的交流部分，取值范围在0~255之间，所以对应的哈弗曼表会更大一些 <table class="gridtable" style="width:300px;text-align:left;"><tbody><tr><th style="width:50px;">Length</th><th style="width:80px;">Value</th><th style="width:170px;">Bits</th></tr><tr><td>2 bits </td><td>01<br>02</td><td>00<br>01</td></tr><tr><td>3 bits </td><td>03</td><td>100</td></tr><tr><td>4 bits </td><td>00 (EOB)<br>04<br>11</td><td>1010<br>1011<br>1100</td></tr><tr><td>5 bits </td><td>05<br>12<br>21</td><td>1101 0<br>1101 1<br>1110 0 </td></tr><tr><td>6 bits </td><td>31<br>41</td><td>1110 10<br>1110 11 </td></tr><tr><td>…</td><td>…</td><td>…</td></tr><tr><td>12 bits </td><td>24<br>33<br>62<br>72</td><td>1111 1111 0100<br>1111 1111 0101<br>1111 1111 0110<br>1111 1111 0111</td></tr><tr><td>15 bits</td><td>82</td><td>1111 1111 1000 000</td></tr><tr><td>16 bits </td><td>09<br>…<br>FA</td><td>1111 1111 1000 0010<br>…<br>1111 1111 1111 1110</td></tr></tbody></table> 这样经过哈弗曼编码，并且序列化后，最终数据成为如下形式 <table class="gridtable" style="width:1000px;text-align:center;"><tbody><tr><th style="width:100px;text-align:left;">①原始数据</th><td style="width:900px;" colspan="14"><!----></td></tr><tr><th rowspan="3" style="text-align:left;"><strong>②RLE编码</strong></th><td colspan="2">35</td><td colspan="2">7</td><td colspan="2">0,0,0,-6</td><td colspan="2">-2</td><td colspan="2">0,0,-9</td><td colspan="3">0,0,…,0,8</td><td>0,0,…,0</td></tr><tr style="padding:3px 3px;"><td colspan="2">35</td><td colspan="2">7</td><td colspan="2">0,0,0,-6</td><td colspan="2">-2</td><td colspan="2">0,0,-9</td><td>0,0,…,0</td><td colspan="2">0,0,8</td><td>0,0,…,0</td></tr><tr><td colspan="2">(0,35)</td><td colspan="2">(0,7)</td><td colspan="2">(3,-6)</td><td colspan="2">(0,-2)</td><td colspan="2">(2,-9)</td><td>(15,0)</td><td colspan="2">(2,8)</td><td>EOB</td></tr><tr><th style="text-align:left;" rowspan="2"><strong>③BIT编码</strong></th><td colspan="2">(0,6, <em><b>100011</b></em>)</td><td colspan="2">(0,3, <em><b>111</b></em>)</td><td colspan="2">(3,3, <em><b>001</b></em>)</td><td colspan="2">(0,2, <em><b>01</b></em>)</td><td colspan="2">(2,4, <em><b>0110</b></em>)</td><td>(15,-)</td><td colspan="2">(2,4, <em><b>1000</b></em>)</td><td>EOB</td></tr><tr><td colspan="2">(0x6,<em><b>100011</b></em>)</td><td colspan="2">(0x3,<em><b>111</b></em>)</td><td colspan="2">(0x33,<em><b>001</b></em>)</td><td colspan="2">(0x2,<em><b>01</b></em>)</td><td colspan="2">(0x24,<em><b>0110</b></em>)</td><td>0xF0</td><td colspan="2">(0x24,<em><b>1000</b></em>)</td><td>EOB</td></tr><tr><th style="text-align:left;"><strong>④哈弗曼编码</strong></th><td><em><b>100</b></em></td><td><em><b>100011</b></em></td><td><em><b>100</b></em></td><td><em><b>111</b></em></td><td><em><b>1111 1111 0101</b></em></td><td><em><b>001</b></em></td><td><em><b>01</b></em></td><td><em><b>01</b></em></td><td><em><b>1111 1111 0100</b></em></td><td><em><b>0110</b></em></td><td><em><b>1111 1111 001</b></em></td><td><em><b>1111 1111 0100</b></em></td><td><em><b>1000</b></em></td><td><em><b>1010</b></em></td></tr><tr><th style="text-align:left;" rowspan="2">⑤序列化</th><td colspan="14"><!----></td></tr><tr><td colspan="14"><!----></td></tr></tbody></table></div><!----><!----><!----></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><!----><!----></div></footer><nav class="vp-page-nav"><a class="route-link 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