怎么用mathematica 图像注释作图像频谱分析
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时间：2016-11-16 03:39
如何用Mathematica把一组函数和另外一个函数画在一张图中并显示_数学建模吧_百度贴吧
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如何用Mathematica把一组函数和另外一个函数画在一张图中并显示收藏
如图 我用show 任然只是分开显示两个图像 我想让第一组函数图像和第二个函数图像显示在同一坐标轴中 范围都是0到1 然后从图中标出这一组函数和第二函数的交点坐标 请问该怎么做 Plot[Table[(4 N^2 - N \[Theta]^2 +
N - \[Theta]^4 + \[Theta]^2)/(2 (N - \[Theta] +
1) (N + \[Theta]^2)), {N, 0, 1, 0.1}], {\[Theta], 0, 1}]Plot[1/2*(((\[Theta]^2)/((1 - \[Theta])^2 + \[Theta]^2))*(-\[Theta]) \+ ((\[Theta]^2)/((1 - \[Theta])^2 + \[Theta]^2)) + \[Theta] +
1), {\[Theta], 0, 1}]这是代码 希望各位司机带带我~
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参考：http://cns-alumni.bu.edu/~slehar/fourier/fourier.html#filtering 很棒
如果输入二维图像数据，则显示的图像是输入的灰度分布，傅立叶频谱是输入的频率分布，频谱图中心对称。
图像频谱即二维频谱图通过对原图像进行水平和竖直两个方向的所有扫描线处一维傅立叶变换的叠加得到
频谱图中以图中心为圆心，圆的相位对应原图中频率分量的相位，半径对应频率高低，低频半径小，高频半径大，中心为直流分量，某点亮度对应该频率能量高低。
从测试案例中更清楚的提现以上几点
以下为几个测试案例：
1．水平方向为sin函数，竖直方向没有变化，对应频谱如图为零相位相对低频的一个亮点，而其他位置基本为黑色，说明没有其他频率分量（水平方向其他小亮点是由于原图不是严格的sin函数而产生的噪声）
2.相对1频率变高，频谱图中亮点位置半径变大
3.sin函数方向改变，频谱图的方向也改变，两种解释。可以理解为水平方向和竖直方向单独扫描都是sin函数，对应频谱叠加所得。也可是理解为原图sin函数方向（相位）改变，频谱图相位变化
4.3图叠加1图
5．频率再次升高，频谱图半径再次增大
6.1图叠加2图，频谱图也叠加
7.原图为一条竖直线，每个行扫描需要若干频率（几乎所有频率）正弦曲线去叠加逼近，所以频谱图包含了所有频率，竖直扫描依然没有变化，频谱图只有一条横线。
8.以下两图对比，第一个图变化不明显，对比度低，频谱图中心亮周围暗，主要为低频分量
第二个图相对对比度高，频谱周围部分比第一张图的频谱要亮的多，第二张图高频分量更多，但原图背景仍然是灰色图，所以频谱图也是中心最亮，低频分量最高。
1.图像的傅里叶频谱的意义
之前的博文其实已经归纳过这方面的内容了。我们常用的图像平滑处理，其实就是一个低通滤波，一定程度上去除高频信号，可以使得图像变得柔和（也就是平滑）。但是，在去除周期性噪声时候，空间域内的滤波（卷积）就不是那么好操作了。所以，这里时候，无论是理解起来方便，还是其他原因，都需要在频域内进行滤波。&
详细的叙述还是在下面的博文里面啦！！！！&
2. 傅里叶频谱的计算
这部分的内容，主要就是使用自带的函数&
void cvDFT( const CvArr* src, CvArr* dst, int flags, int nonzero_rows=0 )&
去求取图像的傅里叶变换。这里，输出结果CvArr* dst由两个通道组成，分别代表了实部与虚部。我们再根据如下算式，就可以得到傅里叶频谱了。&
|F(u,v)|=R2(u,v)+F2(u,v)?????????????????√2&
我自己也参考了很多人的代码，然后实现的代码如下。
&code class=&hljs objectivec has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&IplImage* fft2(IplImage* image_input)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//cout&& & Width& &&
image_input-&width && &
&& &\n&;&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//cout&& &Height& && image_input-&height && &
dftHeight && &\n&;&/span&
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&));
IplImage *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Re &--- image_padded &/span&
cvConvertScale(image_padded,image_Re);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Im &--- 0&/span&
cvZero(image_Im);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] &--- image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] &--- image_Im&/span&
cvMerge(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
cvDFT(image_Fourier,image_Fourier,CV_DXT_FORWARD);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] ---& image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] ---& image_Im&/span&
cvSplit(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvAdd(image_Re,image_Im,image_Re);
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
cvAddS(image_Re,cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&),image_Re );
cvLog (image_Re,image_Re);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|&/span&
IplImage *Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvZero(image_Fourier);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& cx = image_Re-&width/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& cy = image_Re-&height/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 1 &/span&
cvSetImageROI( Fourier,cvRect(cx,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 4 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 4 &/span&
cvSetImageROI( Fourier,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 1 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 3 &/span&
cvSetImageROI( Fourier,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 2 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 2 &/span&
cvSetImageROI( Fourier,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 3 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvResetImageROI(image_Re);
cvResetImageROI( Fourier);
cvNormalize(Fourier,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,&span class=&hljs-literal& style=&color: rgb(0, 102, 102); box-sizing: border-&&NULL&/span&);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&return&/span&(Fourier);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&li style=&box-sizing: border- padding: 0px 5&&58&/li&&li style=&box-sizing: border- padding: 0px 5&&59&/li&&li style=&box-sizing: border- padding: 0px 5&&60&/li&&li style=&box-sizing: border- padding: 0px 5&&61&/li&&li style=&box-sizing: border- padding: 0px 5&&62&/li&&li style=&box-sizing: border- padding: 0px 5&&63&/li&&li style=&box-sizing: border- padding: 0px 5&&64&/li&&li style=&box-sizing: border- padding: 0px 5&&65&/li&&li style=&box-sizing: border- padding: 0px 5&&66&/li&&li style=&box-sizing: border- padding: 0px 5&&67&/li&&li style=&box-sizing: border- padding: 0px 5&&68&/li&&li style=&box-sizing: border- padding: 0px 5&&69&/li&&li style=&box-sizing: border- padding: 0px 5&&70&/li&&li style=&box-sizing: border- padding: 0px 5&&71&/li&&li style=&box-sizing: border- padding: 0px 5&&72&/li&&li style=&box-sizing: border- padding: 0px 5&&73&/li&&li style=&box-sizing: border- padding: 0px 5&&74&/li&&li style=&box-sizing: border- padding: 0px 5&&75&/li&&li style=&box-sizing: border- padding: 0px 5&&76&/li&&li style=&box-sizing: border- padding: 0px 5&&77&/li&&li style=&box-sizing: border- padding: 0px 5&&78&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&li style=&box-sizing: border- padding: 0px 5&&58&/li&&li style=&box-sizing: border- padding: 0px 5&&59&/li&&li style=&box-sizing: border- padding: 0px 5&&60&/li&&li style=&box-sizing: border- padding: 0px 5&&61&/li&&li style=&box-sizing: border- padding: 0px 5&&62&/li&&li style=&box-sizing: border- padding: 0px 5&&63&/li&&li style=&box-sizing: border- padding: 0px 5&&64&/li&&li style=&box-sizing: border- padding: 0px 5&&65&/li&&li style=&box-sizing: border- padding: 0px 5&&66&/li&&li style=&box-sizing: border- padding: 0px 5&&67&/li&&li style=&box-sizing: border- padding: 0px 5&&68&/li&&li style=&box-sizing: border- padding: 0px 5&&69&/li&&li style=&box-sizing: border- padding: 0px 5&&70&/li&&li style=&box-sizing: border- padding: 0px 5&&71&/li&&li style=&box-sizing: border- padding: 0px 5&&72&/li&&li style=&box-sizing: border- padding: 0px 5&&73&/li&&li style=&box-sizing: border- padding: 0px 5&&74&/li&&li style=&box-sizing: border- padding: 0px 5&&75&/li&&li style=&box-sizing: border- padding: 0px 5&&76&/li&&li style=&box-sizing: border- padding: 0px 5&&77&/li&&li style=&box-sizing: border- padding: 0px 5&&78&/li&&/ul&
从这里开始，还是简单的分析一下代码吧。
&code class=&hljs cs has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)); &/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&/ul&
这里参考了文献[2]中的说法，在尺寸数为2,3,5的倍数的场合，计算的速度是最快的。所以使用函数getOptimalDFTSize()来寻找最匹配的尺寸，然后再同伙cvCopyMakeBorder()进行多余部分的填充，这里选的配置是将图放在从点(0,0)开始的位置，其余不足的地方，用0进行填充。
&code class=&hljs haskell has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IplImage&/span& *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
&span class=&hljs-title& style=&box-sizing: border-&&image_Re&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
&span class=&hljs-title& style=&box-sizing: border-&&image_Im&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
&span class=&hljs-title& style=&box-sizing: border-&&image_Fourier&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
//image_Re &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_padded &/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvConvertScale&/span&(image_padded,image_Re);
//image_Im &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- 0&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvZero&/span&(image_Im);
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&] &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_Re&/span&
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&] &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_Im&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvMerge&/span&(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
&span class=&hljs-title& style=&box-sizing: border-&&cvDFT&/span&(image_Fourier,image_Fourier,&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&CV_DXT_FORWARD&/span&);
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&] &span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&---& image_Re&/span&
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&] &span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&---& image_Im&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvSplit&/span&(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&/ul&
其实这里的很好理解的，将填充到最适尺寸的图像赋值给image_Re，将image_Im赋值为0。让后将这两层图复制到image_Fourier的两个通道里，然后使用函数cvDFT()进行傅里叶变换。得到结果还是存在于image_Fourier的两个通道里，分别代表实部与虚部，然后通过cvSplit()将其抽出到image_Re与image_Im里。
&code class=&hljs scss has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvAdd(image_Re,image_Im,image_Re)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&)&/span&;
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
&span class=&hljs-function& style=&box-sizing: border-&&cvAddS(image_Re,&span class=&hljs-function& style=&box-sizing: border-&&cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)&/span&,image_Re )&/span&;
cvLog (image_Re,image_Re); &/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&
以上代码，实现了以下计算。&
|F(u,v)|=R2(u,v)+F2(u,v)?????????????????√2&
还有就是进行了一个对数变换，这个也没的说，看傅里叶频谱的标配操作。
&code class=&hljs mathematica has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&//
|-----|-----|
|-----|-----|
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
|-----|-----|
|-----|-----|
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
|-----|-----|
|-----|-----|
IplImage *&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvZero(image_Fourier);
int cx = image_Re-&width/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
int cy = image_Re-&height/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect(cx,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvResetImageROI(image_Re);
cvResetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvNormalize(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,NULL);
return(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&/ul&
其实重头戏在这里，这里需要一个交换操作。至于为何所求得的傅里叶频谱为什么需要交换的原因是，这个代码求得的结果其实是范围[0,2π]内的傅里叶变换。而我们所需要的是[-π,π]内的结果。详细的原因，还是在我以前的博客里有说明。&
这里，我使用了ROI操作与cvAddWeighted()函数进行了实现。其运行的结果如下所示。&
恩，基本可以看出来，直流分量也被我移动到了中心，以上代码实现了傅里叶频谱的计算与显示。
3. 不用交换操作的代码
使用MATLAB去求取尺寸为M×N图像f的傅里叶频谱时候，通常会用fft2(f,2*M,2*N)。使用此函数求得的福利叶变换，其实还是[0,2π]范围内的傅里叶变换。要想使得傅里叶频谱的DC分量位于中心的话，其实还是需要一些别的操作的。冈萨雷斯的《数字图像处理》一书中，对于这个问题，是利用了傅里叶变换的评议特性，即&
f(x,y)=f(x,y)×(-1)x+y&
然后再对函数f(x,y)进行福利叶变换，所得到结果，就是所需要的是[-π,π]内傅里叶变换的结果。具体的原理与推导，还是参看冈萨雷斯的《数字图像处理》英文原本的p258页左右，中文译本的p148左右。当然，嫌麻烦的话，还可以看我的博文，博文中也简明推导了平移特性。&
为此，实现的代码变为了如下形式。
&code class=&hljs cpp has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&IplImage* fft2_New(IplImage* image_input)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
&span class=&hljs-built_in& style=&color: rgb(102, 0, 102); box-sizing: border-&&cout&/span&&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&& Width&&/span& &&
image_input-&width && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&
&&/span& &&
&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&\n&&/span&;
&span class=&hljs-built_in& style=&color: rgb(102, 0, 102); box-sizing: border-&&cout&/span&&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&Height&&/span& && image_input-&height && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&
&&/span& &&
dftHeight && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&\n&&/span&;
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&));
IplImage *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Re = image_padded .* (-1)^(x+y);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&double&/span&
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& y=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;y&image_padded-&y++)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& x=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;x&image_padded-&x++)
pixel = cvGetReal2D(image_padded,x,y);
pixel = ((x+y)%&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span& == &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)?(pixel):((-&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)*pixel);
cvSetReal2D(image_Re,x,y,pixel);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Im &--- 0&/span&
cvZero(image_Im);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] &--- image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] &--- image_Im&/span&
cvMerge(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
cvDFT(image_Fourier,image_Fourier,CV_DXT_FORWARD);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] ---& image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] ---& image_Im&/span&
cvSplit(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvAdd(image_Re,image_Im,image_Re);
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
cvAddS(image_Re,cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&),image_Re );
cvLog (image_Re,image_Re);
cvNormalize(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,NULL);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&return&/span&(image_Re);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&/ul&
在这里，由于考虑到计算的原因，我将f(x,y)=f(x,y)×(-1)x+y这个计算，用下面代码去进行了实现。
&code class=&hljs perl has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&&image_padded-&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&++)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&&image_padded-&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&++)
pixel = cvGetReal2D(image_padded,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&);
pixel = ((&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&+&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&)&span class=&hljs-variable& style=&color: rgb(102, 0, 102); box-sizing: border-&&%2&/span& == &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)?(pixel):((-&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)&span class=&hljs-variable& style=&color: rgb(102, 0, 102); box-sizing: border-&&*pixel&/span&);
cvSetReal2D(image_Re,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&,pixel);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&
其实也就相当于，x+y是偶数的话f(x,y)不变，x+y是奇数的话f(x,y)变为负。用这样一个简单的判断去实现。其运行的结果，如下所示。
原图的傅里叶频谱&
使用11×11高斯滤波器后的傅里叶频谱&
从实验结果看来，可以看出以下两点
对于原图而言两个函数的结果基本一致，两个函数都得到了正确的结果。使用平滑处理后，频谱的高频成分明显变小，对于空间域的图像而言，图像变得模糊了。
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如何用Mathematica把一组函数和另外一个函数画在一张图中并显示收藏
如图 我用show 任然只是分开显示两个图像 我想让第一组函数图像和第二个函数图像显示在同一坐标轴中 范围都是0到1 然后从图中标出这一组函数和第二函数的交点坐标 请问该怎么做 Plot[Table[(4 N^2 - N \[Theta]^2 +
N - \[Theta]^4 + \[Theta]^2)/(2 (N - \[Theta] +
1) (N + \[Theta]^2)), {N, 0, 1, 0.1}], {\[Theta], 0, 1}]Plot[1/2*(((\[Theta]^2)/((1 - \[Theta])^2 + \[Theta]^2))*(-\[Theta]) \+ ((\[Theta]^2)/((1 - \[Theta])^2 + \[Theta]^2)) + \[Theta] +
1), {\[Theta], 0, 1}]这是代码 希望各位司机带带我~
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为兴趣而生，贴吧更懂你。或更多公众号：gh_a百纳知识致力于求真、创新的知识精神，立志要做各类知识的分享者。最新文章相关作者文章搜狗：感谢您阅读[分享]Mathematica在图像处理中的简单应用，本文由网友投稿产生，如果侵犯了您的相关权益，请联系管理员。图像傅立叶频谱分析
参考：http://cns-alumni.bu.edu/~slehar/fourier/fourier.html#filtering 很棒
如果输入二维图像数据，则显示的图像是输入的灰度分布，傅立叶频谱是输入的频率分布，频谱图中心对称。
图像频谱即二维频谱图通过对原图像进行水平和竖直两个方向的所有扫描线处一维傅立叶变换的叠加得到
频谱图中以图中心为圆心，圆的相位对应原图中频率分量的相位，半径对应频率高低，低频半径小，高频半径大，中心为直流分量，某点亮度对应该频率能量高低。
从测试案例中更清楚的提现以上几点
以下为几个测试案例：
1．水平方向为sin函数，竖直方向没有变化，对应频谱如图为零相位相对低频的一个亮点，而其他位置基本为黑色，说明没有其他频率分量（水平方向其他小亮点是由于原图不是严格的sin函数而产生的噪声）
2.相对1频率变高，频谱图中亮点位置半径变大
3.sin函数方向改变，频谱图的方向也改变，两种解释。可以理解为水平方向和竖直方向单独扫描都是sin函数，对应频谱叠加所得。也可是理解为原图sin函数方向（相位）改变，频谱图相位变化
4.3图叠加1图
5．频率再次升高，频谱图半径再次增大
6.1图叠加2图，频谱图也叠加
7.原图为一条竖直线，每个行扫描需要若干频率（几乎所有频率）正弦曲线去叠加逼近，所以频谱图包含了所有频率，竖直扫描依然没有变化，频谱图只有一条横线。
8.以下两图对比，第一个图变化不明显，对比度低，频谱图中心亮周围暗，主要为低频分量
第二个图相对对比度高，频谱周围部分比第一张图的频谱要亮的多，第二张图高频分量更多，但原图背景仍然是灰色图，所以频谱图也是中心最亮，低频分量最高。
1.图像的傅里叶频谱的意义
之前的博文其实已经归纳过这方面的内容了。我们常用的图像平滑处理，其实就是一个低通滤波，一定程度上去除高频信号，可以使得图像变得柔和（也就是平滑）。但是，在去除周期性噪声时候，空间域内的滤波（卷积）就不是那么好操作了。所以，这里时候，无论是理解起来方便，还是其他原因，都需要在频域内进行滤波。&
详细的叙述还是在下面的博文里面啦！！！！&
2. 傅里叶频谱的计算
这部分的内容，主要就是使用自带的函数&
void cvDFT( const CvArr* src, CvArr* dst, int flags, int nonzero_rows=0 )&
去求取图像的傅里叶变换。这里，输出结果CvArr* dst由两个通道组成，分别代表了实部与虚部。我们再根据如下算式，就可以得到傅里叶频谱了。&
|F(u,v)|=R2(u,v)+F2(u,v)?????????????????√2&
我自己也参考了很多人的代码，然后实现的代码如下。
&code class=&hljs objectivec has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&IplImage* fft2(IplImage* image_input)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//cout&& & Width& &&
image_input-&width && &
&& &\n&;&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//cout&& &Height& && image_input-&height && &
dftHeight && &\n&;&/span&
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&));
IplImage *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Re &--- image_padded &/span&
cvConvertScale(image_padded,image_Re);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Im &--- 0&/span&
cvZero(image_Im);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] &--- image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] &--- image_Im&/span&
cvMerge(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
cvDFT(image_Fourier,image_Fourier,CV_DXT_FORWARD);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] ---& image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] ---& image_Im&/span&
cvSplit(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvAdd(image_Re,image_Im,image_Re);
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
cvAddS(image_Re,cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&),image_Re );
cvLog (image_Re,image_Re);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//
|-----|-----|
|-----|-----|&/span&
IplImage *Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvZero(image_Fourier);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& cx = image_Re-&width/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& cy = image_Re-&height/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 1 &/span&
cvSetImageROI( Fourier,cvRect(cx,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 4 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 4 &/span&
cvSetImageROI( Fourier,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 1 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 3 &/span&
cvSetImageROI( Fourier,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 2 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 2 &/span&
cvSetImageROI( Fourier,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// 3 &/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,Fourier);
cvResetImageROI(image_Re);
cvResetImageROI( Fourier);
cvNormalize(Fourier,Fourier,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,&span class=&hljs-literal& style=&color: rgb(0, 102, 102); box-sizing: border-&&NULL&/span&);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&return&/span&(Fourier);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&li style=&box-sizing: border- padding: 0px 5&&58&/li&&li style=&box-sizing: border- padding: 0px 5&&59&/li&&li style=&box-sizing: border- padding: 0px 5&&60&/li&&li style=&box-sizing: border- padding: 0px 5&&61&/li&&li style=&box-sizing: border- padding: 0px 5&&62&/li&&li style=&box-sizing: border- padding: 0px 5&&63&/li&&li style=&box-sizing: border- padding: 0px 5&&64&/li&&li style=&box-sizing: border- padding: 0px 5&&65&/li&&li style=&box-sizing: border- padding: 0px 5&&66&/li&&li style=&box-sizing: border- padding: 0px 5&&67&/li&&li style=&box-sizing: border- padding: 0px 5&&68&/li&&li style=&box-sizing: border- padding: 0px 5&&69&/li&&li style=&box-sizing: border- padding: 0px 5&&70&/li&&li style=&box-sizing: border- padding: 0px 5&&71&/li&&li style=&box-sizing: border- padding: 0px 5&&72&/li&&li style=&box-sizing: border- padding: 0px 5&&73&/li&&li style=&box-sizing: border- padding: 0px 5&&74&/li&&li style=&box-sizing: border- padding: 0px 5&&75&/li&&li style=&box-sizing: border- padding: 0px 5&&76&/li&&li style=&box-sizing: border- padding: 0px 5&&77&/li&&li style=&box-sizing: border- padding: 0px 5&&78&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&li style=&box-sizing: border- padding: 0px 5&&58&/li&&li style=&box-sizing: border- padding: 0px 5&&59&/li&&li style=&box-sizing: border- padding: 0px 5&&60&/li&&li style=&box-sizing: border- padding: 0px 5&&61&/li&&li style=&box-sizing: border- padding: 0px 5&&62&/li&&li style=&box-sizing: border- padding: 0px 5&&63&/li&&li style=&box-sizing: border- padding: 0px 5&&64&/li&&li style=&box-sizing: border- padding: 0px 5&&65&/li&&li style=&box-sizing: border- padding: 0px 5&&66&/li&&li style=&box-sizing: border- padding: 0px 5&&67&/li&&li style=&box-sizing: border- padding: 0px 5&&68&/li&&li style=&box-sizing: border- padding: 0px 5&&69&/li&&li style=&box-sizing: border- padding: 0px 5&&70&/li&&li style=&box-sizing: border- padding: 0px 5&&71&/li&&li style=&box-sizing: border- padding: 0px 5&&72&/li&&li style=&box-sizing: border- padding: 0px 5&&73&/li&&li style=&box-sizing: border- padding: 0px 5&&74&/li&&li style=&box-sizing: border- padding: 0px 5&&75&/li&&li style=&box-sizing: border- padding: 0px 5&&76&/li&&li style=&box-sizing: border- padding: 0px 5&&77&/li&&li style=&box-sizing: border- padding: 0px 5&&78&/li&&/ul&
从这里开始，还是简单的分析一下代码吧。
&code class=&hljs cs has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)); &/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&/ul&
这里参考了文献[2]中的说法，在尺寸数为2,3,5的倍数的场合，计算的速度是最快的。所以使用函数getOptimalDFTSize()来寻找最匹配的尺寸，然后再同伙cvCopyMakeBorder()进行多余部分的填充，这里选的配置是将图放在从点(0,0)开始的位置，其余不足的地方，用0进行填充。
&code class=&hljs haskell has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IplImage&/span& *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
&span class=&hljs-title& style=&box-sizing: border-&&image_Re&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
&span class=&hljs-title& style=&box-sizing: border-&&image_Im&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
&span class=&hljs-title& style=&box-sizing: border-&&image_Fourier&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&IPL_DEPTH_64F&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
//image_Re &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_padded &/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvConvertScale&/span&(image_padded,image_Re);
//image_Im &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- 0&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvZero&/span&(image_Im);
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&] &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_Re&/span&
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&] &-&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&-- image_Im&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvMerge&/span&(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
&span class=&hljs-title& style=&box-sizing: border-&&cvDFT&/span&(image_Fourier,image_Fourier,&span class=&hljs-type& style=&box-sizing: border- color: rgb(102, 0, 102);&&CV_DXT_FORWARD&/span&);
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&] &span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&---& image_Re&/span&
//image_Fourier[&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&] &span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&---& image_Im&/span&
&span class=&hljs-title& style=&box-sizing: border-&&cvSplit&/span&(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&/ul&
其实这里的很好理解的，将填充到最适尺寸的图像赋值给image_Re，将image_Im赋值为0。让后将这两层图复制到image_Fourier的两个通道里，然后使用函数cvDFT()进行傅里叶变换。得到结果还是存在于image_Fourier的两个通道里，分别代表实部与虚部，然后通过cvSplit()将其抽出到image_Re与image_Im里。
&code class=&hljs scss has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvAdd(image_Re,image_Im,image_Re)&/span&;
&span class=&hljs-function& style=&box-sizing: border-&&cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&)&/span&;
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
&span class=&hljs-function& style=&box-sizing: border-&&cvAddS(image_Re,&span class=&hljs-function& style=&box-sizing: border-&&cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)&/span&,image_Re )&/span&;
cvLog (image_Re,image_Re); &/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&
以上代码，实现了以下计算。&
|F(u,v)|=R2(u,v)+F2(u,v)?????????????????√2&
还有就是进行了一个对数变换，这个也没的说，看傅里叶频谱的标配操作。
&code class=&hljs mathematica has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&//
|-----|-----|
|-----|-----|
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
|-----|-----|
|-----|-----|
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
|-----|-----|
|-----|-----|
IplImage *&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span& = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvZero(image_Fourier);
int cx = image_Re-&width/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
int cy = image_Re-&height/&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&;
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect(cx,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&4&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvSetImageROI(image_Re,cvRect( &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cy,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&
cvSetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,cvRect(cx, &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,cx,cy));
// &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&3&/span&
cvAddWeighted(image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvResetImageROI(image_Re);
cvResetImageROI( &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);
cvNormalize(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,NULL);
return(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&Fourier&/span&);&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&/ul&
其实重头戏在这里，这里需要一个交换操作。至于为何所求得的傅里叶频谱为什么需要交换的原因是，这个代码求得的结果其实是范围[0,2π]内的傅里叶变换。而我们所需要的是[-π,π]内的结果。详细的原因，还是在我以前的博客里有说明。&
这里，我使用了ROI操作与cvAddWeighted()函数进行了实现。其运行的结果如下所示。&
恩，基本可以看出来，直流分量也被我移动到了中心，以上代码实现了傅里叶频谱的计算与显示。
3. 不用交换操作的代码
使用MATLAB去求取尺寸为M×N图像f的傅里叶频谱时候，通常会用fft2(f,2*M,2*N)。使用此函数求得的福利叶变换，其实还是[0,2π]范围内的傅里叶变换。要想使得傅里叶频谱的DC分量位于中心的话，其实还是需要一些别的操作的。冈萨雷斯的《数字图像处理》一书中，对于这个问题，是利用了傅里叶变换的评议特性，即&
f(x,y)=f(x,y)×(-1)x+y&
然后再对函数f(x,y)进行福利叶变换，所得到结果，就是所需要的是[-π,π]内傅里叶变换的结果。具体的原理与推导，还是参看冈萨雷斯的《数字图像处理》英文原本的p258页左右，中文译本的p148左右。当然，嫌麻烦的话，还可以看我的博文，博文中也简明推导了平移特性。&
为此，实现的代码变为了如下形式。
&code class=&hljs cpp has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&IplImage* fft2_New(IplImage* image_input)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftWidth
= getOptimalDFTSize(image_input-&width);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& dftHeight = getOptimalDFTSize(image_input-&height);
&span class=&hljs-built_in& style=&color: rgb(102, 0, 102); box-sizing: border-&&cout&/span&&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&& Width&&/span& &&
image_input-&width && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&
&&/span& &&
&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&\n&&/span&;
&span class=&hljs-built_in& style=&color: rgb(102, 0, 102); box-sizing: border-&&cout&/span&&& &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&Height&&/span& && image_input-&height && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&
&&/span& &&
dftHeight && &span class=&hljs-string& style=&color: rgb(0, 136, 0); box-sizing: border-&&&\n&&/span&;
IplImage* image_padded = cvCreateImage(cvSize(dftWidth,dftHeight),
IPL_DEPTH_8U,
&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
cvCopyMakeBorder( image_input, image_padded, cvPoint(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&), IPL_BORDER_CONSTANT,cvScalarAll(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&));
IplImage *image_Re =&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span& , *image_Im = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&, *image_Fourier = &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;
image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&);
image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Re = image_padded .* (-1)^(x+y);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&double&/span&
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& y=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;y&image_padded-&y++)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& x=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;x&image_padded-&x++)
pixel = cvGetReal2D(image_padded,x,y);
pixel = ((x+y)%&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2&/span& == &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)?(pixel):((-&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)*pixel);
cvSetReal2D(image_Re,x,y,pixel);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Im &--- 0&/span&
cvZero(image_Im);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] &--- image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] &--- image_Im&/span&
cvMerge(image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,image_Fourier);
cvDFT(image_Fourier,image_Fourier,CV_DXT_FORWARD);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[0] ---& image_Re&/span&
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//image_Fourier[1] ---& image_Im&/span&
cvSplit(image_Fourier,image_Re,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&//Mag = sqrt(Re^2 + Im^2)&/span&
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvPow(image_Im,image_Im,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&2.0&/span&);
cvAdd(image_Re,image_Im,image_Re);
cvPow(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0.5&/span&);
&span class=&hljs-comment& style=&color: rgb(136, 0, 0); box-sizing: border-&&// log (1 + Mag)&/span&
cvAddS(image_Re,cvScalar(&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&),image_Re );
cvLog (image_Re,image_Re);
cvNormalize(image_Re,image_Re,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&,&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&,CV_C,NULL);
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&return&/span&(image_Re);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&li style=&box-sizing: border- padding: 0px 5&&10&/li&&li style=&box-sizing: border- padding: 0px 5&&11&/li&&li style=&box-sizing: border- padding: 0px 5&&12&/li&&li style=&box-sizing: border- padding: 0px 5&&13&/li&&li style=&box-sizing: border- padding: 0px 5&&14&/li&&li style=&box-sizing: border- padding: 0px 5&&15&/li&&li style=&box-sizing: border- padding: 0px 5&&16&/li&&li style=&box-sizing: border- padding: 0px 5&&17&/li&&li style=&box-sizing: border- padding: 0px 5&&18&/li&&li style=&box-sizing: border- padding: 0px 5&&19&/li&&li style=&box-sizing: border- padding: 0px 5&&20&/li&&li style=&box-sizing: border- padding: 0px 5&&21&/li&&li style=&box-sizing: border- padding: 0px 5&&22&/li&&li style=&box-sizing: border- padding: 0px 5&&23&/li&&li style=&box-sizing: border- padding: 0px 5&&24&/li&&li style=&box-sizing: border- padding: 0px 5&&25&/li&&li style=&box-sizing: border- padding: 0px 5&&26&/li&&li style=&box-sizing: border- padding: 0px 5&&27&/li&&li style=&box-sizing: border- padding: 0px 5&&28&/li&&li style=&box-sizing: border- padding: 0px 5&&29&/li&&li style=&box-sizing: border- padding: 0px 5&&30&/li&&li style=&box-sizing: border- padding: 0px 5&&31&/li&&li style=&box-sizing: border- padding: 0px 5&&32&/li&&li style=&box-sizing: border- padding: 0px 5&&33&/li&&li style=&box-sizing: border- padding: 0px 5&&34&/li&&li style=&box-sizing: border- padding: 0px 5&&35&/li&&li style=&box-sizing: border- padding: 0px 5&&36&/li&&li style=&box-sizing: border- padding: 0px 5&&37&/li&&li style=&box-sizing: border- padding: 0px 5&&38&/li&&li style=&box-sizing: border- padding: 0px 5&&39&/li&&li style=&box-sizing: border- padding: 0px 5&&40&/li&&li style=&box-sizing: border- padding: 0px 5&&41&/li&&li style=&box-sizing: border- padding: 0px 5&&42&/li&&li style=&box-sizing: border- padding: 0px 5&&43&/li&&li style=&box-sizing: border- padding: 0px 5&&44&/li&&li style=&box-sizing: border- padding: 0px 5&&45&/li&&li style=&box-sizing: border- padding: 0px 5&&46&/li&&li style=&box-sizing: border- padding: 0px 5&&47&/li&&li style=&box-sizing: border- padding: 0px 5&&48&/li&&li style=&box-sizing: border- padding: 0px 5&&49&/li&&li style=&box-sizing: border- padding: 0px 5&&50&/li&&li style=&box-sizing: border- padding: 0px 5&&51&/li&&li style=&box-sizing: border- padding: 0px 5&&52&/li&&li style=&box-sizing: border- padding: 0px 5&&53&/li&&li style=&box-sizing: border- padding: 0px 5&&54&/li&&li style=&box-sizing: border- padding: 0px 5&&55&/li&&li style=&box-sizing: border- padding: 0px 5&&56&/li&&li style=&box-sizing: border- padding: 0px 5&&57&/li&&/ul&
在这里，由于考虑到计算的原因，我将f(x,y)=f(x,y)×(-1)x+y这个计算，用下面代码去进行了实现。
&code class=&hljs perl has-numbering& style=&display: padding: 0 background-color: color: box-sizing: border- font-family: 'Source Code Pro',font-size: white-space: border-top-left-radius: 0 border-top-right-radius: 0 border-bottom-right-radius: 0 border-bottom-left-radius: 0 word-wrap: background-position: background-repeat:&&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&&image_padded-&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&++)
&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&for&/span&(&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&int&/span& &span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&=&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&;&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&&image_padded-&&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&++)
pixel = cvGetReal2D(image_padded,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&);
pixel = ((&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&+&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&)&span class=&hljs-variable& style=&color: rgb(102, 0, 102); box-sizing: border-&&%2&/span& == &span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&0&/span&)?(pixel):((-&span class=&hljs-number& style=&color: rgb(0, 102, 102); box-sizing: border-&&1&/span&)&span class=&hljs-variable& style=&color: rgb(102, 0, 102); box-sizing: border-&&*pixel&/span&);
cvSetReal2D(image_Re,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&x&/span&,&span class=&hljs-keyword& style=&color: rgb(0, 0, 136); box-sizing: border-&&y&/span&,pixel);
}&/code&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&&ul class=&pre-numbering& style=&box-sizing: border- position: width: 50 background-color: rgb(238, 238, 238); top: 0 left: 0 margin: 0 padding: 6px 0px 40 border-right-width: 1 border-right-style: border-right-color: rgb(221, 221, 221); list-style: text-align:&&&li style=&box-sizing: border- padding: 0px 5&&1&/li&&li style=&box-sizing: border- padding: 0px 5&&2&/li&&li style=&box-sizing: border- padding: 0px 5&&3&/li&&li style=&box-sizing: border- padding: 0px 5&&4&/li&&li style=&box-sizing: border- padding: 0px 5&&5&/li&&li style=&box-sizing: border- padding: 0px 5&&6&/li&&li style=&box-sizing: border- padding: 0px 5&&7&/li&&li style=&box-sizing: border- padding: 0px 5&&8&/li&&li style=&box-sizing: border- padding: 0px 5&&9&/li&&/ul&
其实也就相当于，x+y是偶数的话f(x,y)不变，x+y是奇数的话f(x,y)变为负。用这样一个简单的判断去实现。其运行的结果，如下所示。
原图的傅里叶频谱&
使用11×11高斯滤波器后的傅里叶频谱&
从实验结果看来，可以看出以下两点
对于原图而言两个函数的结果基本一致，两个函数都得到了正确的结果。使用平滑处理后，频谱的高频成分明显变小，对于空间域的图像而言，图像变得模糊了。
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