thejinchao.github.io/assets/JPEG002.html-DgUoCOgr.js

import{_ as j,c,a as x,b as m,d as l,o as t}from"./app-DCgfI_Ar.js";const n="/images/2014/08/jpeg_20.jpg",a="/images/2014/08/jpeg_22.jpg",e="/images/2014/08/jpeg_21.gif",d="/images/2014/08/jpeg_25.gif",u="/images/2014/08/jpeg_26.gif",i="/images/2014/08/jpeg_27.gif",o="/images/2014/08/jpeg_28.gif",r="/images/2014/08/jpeg_29.gif",p="/images/2014/08/jpeg_30.gif",g="/images/2014/08/jpeg_31.gif",y="/images/2014/08/jpeg_32.gif",h="/images/2014/08/jpeg_33.gif",f={};function b(E,s){return t(),c("div",null,s[0]||(s[0]=[x('<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="'+n+'" 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="'+a+'" width="45%"><img src="'+e+'" width="45%"></p>',7),m("p",null,[l("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"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c5B"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 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src="'+d+'" 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="'+u+'" 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="'+i+'" 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="'+o+'" 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="'+r+'" 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="'+p+'" 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="'+g+'" 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="'+y+'" 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="'+h+'" 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>',12),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",width:"full",style:{"min-width":"31.745em",position:"relative"}},[m("mjx-math",{width:"full",display:"true",class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{width:"full",style:{"min-width":"31.745em"},side:"right"},[m("mjx-table",{style:{width:"auto","min-width":"26.033em",margin:"0 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