观点一：实际上对二维图像进行傅里叶变化得到的频谱图，就是图像梯度的分布图，当然频谱图上的各点与原图像各点并不存在一一对应关系，即使在不移频的情况下也不是。傅里叶频谱图上看到的明亮不一的亮点，实际上图像上某一点的像素灰度值与它的邻域点差异的强弱，即梯度的大小，也即该点的频率的大小（也可以这么理解，低频部分指低梯度的点，高频部分指高梯度的点）。

观点二： 图像二维频谱图通过对输入图像进行水平和竖直两个方向的所有扫描线的一维傅立叶变换进行叠加得到,用来表示输入图像的频率分布。

频谱图以图像的中心为圆心,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（图像灰度的平均值）,某点的灰度值对应该频率的能量高低。

图像傅立叶频谱分析


参考：http://cns-alumni.bu.edu/~slehar/fourier/fourier.html#filtering 很棒
分析：
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。




代码：




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！！！！ 
[数字图像处理]频域滤波(1)–基础与低通滤波器 
[数字图像处理]频域滤波(2)–高通滤波器，带阻滤波器与陷波滤波器

2. 傅里叶频谱的计算

这部分的内容，主要就是使用OpenCV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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;">IplImage* fft2(IplImage* image_input)
{
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dftWidth  = getOptimalDFTSize(image_input->width);
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dftHeight = getOptimalDFTSize(image_input->height);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//cout<< " Width" <<  image_input->width << "    " <<  dftWidth  << "\n";</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//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-box;">1</span>);
    cvCopyMakeBorder( image_input, image_padded, cvPoint(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>), IPL_BORDER_CONSTANT,cvScalarAll(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)); 

    IplImage *image_Re =<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> , *image_Im = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, *image_Fourier = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; 

    image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
    image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
    image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Re <--- image_padded </span>
    cvConvertScale(image_padded,image_Re);   
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Im <--- 0</span>
    cvZero(image_Im);                 
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Fourier[0] <--- image_Re</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Fourier[1] <--- image_Im</span>
    cvMerge(image_Re,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">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-box;">//image_Fourier[0] ---> image_Re</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//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-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//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-box;">2.0</span>);
    cvPow(image_Im,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">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-box;">0.5</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// log (1 + Mag)</span>
    cvAddS(image_Re,cvScalar(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>),image_Re ); 
    cvLog (image_Re,image_Re); 

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//  |-----|-----|           |-----|-----|   </span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//  |  1  |  3  |           |  4  |  2  |</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//  |-----|-----|   --->    |-----|-----|</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//  |  2  |  4  |           |  3  |  1  |</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//  |-----|-----|           |-----|-----|</span>

    IplImage *Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
    cvZero(image_Fourier);

    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> cx = image_Re->width/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> cy = image_Re->height/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;

    cvSetImageROI(image_Re,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 1 </span>
    cvSetImageROI( Fourier,cvRect(cx,cy,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 4 </span>
    cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,Fourier,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,Fourier);

    cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 4 </span>
    cvSetImageROI( Fourier,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 1 </span>
    cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,Fourier,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,Fourier);

    cvSetImageROI(image_Re,cvRect(cx, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 3 </span>
    cvSetImageROI( Fourier,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cy,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 2 </span>
    cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,Fourier,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,Fourier);

    cvSetImageROI(image_Re,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cy,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 2 </span>
    cvSetImageROI( Fourier,cvRect(cx, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 3 </span>
    cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,Fourier,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,Fourier);

    cvResetImageROI(image_Re);
    cvResetImageROI( Fourier);

    cvNormalize(Fourier,Fourier,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,CV_C,<span class="hljs-literal" style="color: rgb(0, 102, 102); box-sizing: border-box;">NULL</span>);

    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span>(Fourier);
}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li><li style="box-sizing: border-box; padding: 0px 5px;">35</li><li style="box-sizing: border-box; padding: 0px 5px;">36</li><li style="box-sizing: border-box; padding: 0px 5px;">37</li><li style="box-sizing: border-box; padding: 0px 5px;">38</li><li style="box-sizing: border-box; padding: 0px 5px;">39</li><li style="box-sizing: border-box; padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li><li style="box-sizing: border-box; padding: 0px 5px;">44</li><li style="box-sizing: border-box; padding: 0px 5px;">45</li><li style="box-sizing: border-box; padding: 0px 5px;">46</li><li style="box-sizing: border-box; padding: 0px 5px;">47</li><li style="box-sizing: border-box; padding: 0px 5px;">48</li><li style="box-sizing: border-box; padding: 0px 5px;">49</li><li style="box-sizing: border-box; padding: 0px 5px;">50</li><li style="box-sizing: border-box; padding: 0px 5px;">51</li><li style="box-sizing: border-box; padding: 0px 5px;">52</li><li style="box-sizing: border-box; padding: 0px 5px;">53</li><li style="box-sizing: border-box; padding: 0px 5px;">54</li><li style="box-sizing: border-box; padding: 0px 5px;">55</li><li style="box-sizing: border-box; padding: 0px 5px;">56</li><li style="box-sizing: border-box; padding: 0px 5px;">57</li><li style="box-sizing: border-box; padding: 0px 5px;">58</li><li style="box-sizing: border-box; padding: 0px 5px;">59</li><li style="box-sizing: border-box; padding: 0px 5px;">60</li><li style="box-sizing: border-box; padding: 0px 5px;">61</li><li style="box-sizing: border-box; padding: 0px 5px;">62</li><li style="box-sizing: border-box; padding: 0px 5px;">63</li><li style="box-sizing: border-box; padding: 0px 5px;">64</li><li style="box-sizing: border-box; padding: 0px 5px;">65</li><li style="box-sizing: border-box; padding: 0px 5px;">66</li><li style="box-sizing: border-box; padding: 0px 5px;">67</li><li style="box-sizing: border-box; padding: 0px 5px;">68</li><li style="box-sizing: border-box; padding: 0px 5px;">69</li><li style="box-sizing: border-box; padding: 0px 5px;">70</li><li style="box-sizing: border-box; padding: 0px 5px;">71</li><li style="box-sizing: border-box; padding: 0px 5px;">72</li><li style="box-sizing: border-box; padding: 0px 5px;">73</li><li style="box-sizing: border-box; padding: 0px 5px;">74</li><li style="box-sizing: border-box; padding: 0px 5px;">75</li><li style="box-sizing: border-box; padding: 0px 5px;">76</li><li style="box-sizing: border-box; padding: 0px 5px;">77</li><li style="box-sizing: border-box; padding: 0px 5px;">78</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li><li style="box-sizing: border-box; padding: 0px 5px;">35</li><li style="box-sizing: border-box; padding: 0px 5px;">36</li><li style="box-sizing: border-box; padding: 0px 5px;">37</li><li style="box-sizing: border-box; padding: 0px 5px;">38</li><li style="box-sizing: border-box; padding: 0px 5px;">39</li><li style="box-sizing: border-box; padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li><li style="box-sizing: border-box; padding: 0px 5px;">44</li><li style="box-sizing: border-box; padding: 0px 5px;">45</li><li style="box-sizing: border-box; padding: 0px 5px;">46</li><li style="box-sizing: border-box; padding: 0px 5px;">47</li><li style="box-sizing: border-box; padding: 0px 5px;">48</li><li style="box-sizing: border-box; padding: 0px 5px;">49</li><li style="box-sizing: border-box; padding: 0px 5px;">50</li><li style="box-sizing: border-box; padding: 0px 5px;">51</li><li style="box-sizing: border-box; padding: 0px 5px;">52</li><li style="box-sizing: border-box; padding: 0px 5px;">53</li><li style="box-sizing: border-box; padding: 0px 5px;">54</li><li style="box-sizing: border-box; padding: 0px 5px;">55</li><li style="box-sizing: border-box; padding: 0px 5px;">56</li><li style="box-sizing: border-box; padding: 0px 5px;">57</li><li style="box-sizing: border-box; padding: 0px 5px;">58</li><li style="box-sizing: border-box; padding: 0px 5px;">59</li><li style="box-sizing: border-box; padding: 0px 5px;">60</li><li style="box-sizing: border-box; padding: 0px 5px;">61</li><li style="box-sizing: border-box; padding: 0px 5px;">62</li><li style="box-sizing: border-box; padding: 0px 5px;">63</li><li style="box-sizing: border-box; padding: 0px 5px;">64</li><li style="box-sizing: border-box; padding: 0px 5px;">65</li><li style="box-sizing: border-box; padding: 0px 5px;">66</li><li style="box-sizing: border-box; padding: 0px 5px;">67</li><li style="box-sizing: border-box; padding: 0px 5px;">68</li><li style="box-sizing: border-box; padding: 0px 5px;">69</li><li style="box-sizing: border-box; padding: 0px 5px;">70</li><li style="box-sizing: border-box; padding: 0px 5px;">71</li><li style="box-sizing: border-box; padding: 0px 5px;">72</li><li style="box-sizing: border-box; padding: 0px 5px;">73</li><li style="box-sizing: border-box; padding: 0px 5px;">74</li><li style="box-sizing: border-box; padding: 0px 5px;">75</li><li style="box-sizing: border-box; padding: 0px 5px;">76</li><li style="box-sizing: border-box; padding: 0px 5px;">77</li><li style="box-sizing: border-box; padding: 0px 5px;">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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dftWidth  = getOptimalDFTSize(image_input->width);
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">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-box;">1</span>);
cvCopyMakeBorder( image_input, image_padded, cvPoint(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>), IPL_BORDER_CONSTANT,cvScalarAll(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)); </code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;"><span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">IplImage</span> *image_Re =<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> , *image_Im = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, *image_Fourier = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; 

<span class="hljs-title" style="box-sizing: border-box;">image_Re</span> = cvCreateImage(cvSize(dftWidth,dftHeight),<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">IPL_DEPTH_64F</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
<span class="hljs-title" style="box-sizing: border-box;">image_Im</span> = cvCreateImage(cvSize(dftWidth,dftHeight),<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">IPL_DEPTH_64F</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
<span class="hljs-title" style="box-sizing: border-box;">image_Fourier</span> = cvCreateImage(cvSize(dftWidth,dftHeight),<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">IPL_DEPTH_64F</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>);

//image_Re <-<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">-- image_padded </span>
<span class="hljs-title" style="box-sizing: border-box;">cvConvertScale</span>(image_padded,image_Re);   
//image_Im <-<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">-- 0</span>
<span class="hljs-title" style="box-sizing: border-box;">cvZero</span>(image_Im);                 
//image_Fourier[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>] <-<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">-- image_Re</span>
//image_Fourier[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>] <-<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">-- image_Im</span>
<span class="hljs-title" style="box-sizing: border-box;">cvMerge</span>(image_Re,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,image_Fourier); 

<span class="hljs-title" style="box-sizing: border-box;">cvDFT</span>(image_Fourier,image_Fourier,<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">CV_DXT_FORWARD</span>);

//image_Fourier[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>] <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">---> image_Re</span>
//image_Fourier[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>] <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">---> image_Im</span>
<span class="hljs-title" style="box-sizing: border-box;">cvSplit</span>(image_Fourier,image_Re,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>);</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;"><span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//Mag = sqrt(Re^2 + Im^2)</span>
<span class="hljs-function" style="box-sizing: border-box;">cvPow(image_Re,image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2.0</span>)</span>;
<span class="hljs-function" style="box-sizing: border-box;">cvPow(image_Im,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2.0</span>)</span>;
<span class="hljs-function" style="box-sizing: border-box;">cvAdd(image_Re,image_Im,image_Re)</span>;
<span class="hljs-function" style="box-sizing: border-box;">cvPow(image_Re,image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0.5</span>)</span>;

<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// log (1 + Mag)</span>
<span class="hljs-function" style="box-sizing: border-box;">cvAddS(image_Re,<span class="hljs-function" style="box-sizing: border-box;">cvScalar(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)</span>,image_Re )</span>; 
cvLog (image_Re,image_Re); </code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;">//  |-----|-----|           |-----|-----|   
//  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>  |           |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>  |
//  |-----|-----|   --->    |-----|-----|
//  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>  |           |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>  |  <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>  |
//  |-----|-----|           |-----|-----|

IplImage *<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span> = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
cvZero(image_Fourier);

int cx = image_Re->width/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;
int cy = image_Re->height/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;

cvSetImageROI(image_Re,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span> 
cvSetImageROI( <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,cvRect(cx,cy,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span> 
cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);

cvSetImageROI(image_Re,cvRect(cx,cy,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span> 
cvSetImageROI( <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span> 
cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);

cvSetImageROI(image_Re,cvRect(cx, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span> 
cvSetImageROI( <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cy,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span> 
cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);

cvSetImageROI(image_Re,cvRect( <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cy,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span> 
cvSetImageROI( <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,cvRect(cx, <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,cx,cy));  // <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span> 
cvAddWeighted(image_Re,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);

cvResetImageROI(image_Re);
cvResetImageROI( <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);

cvNormalize(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,CV_C,NULL);

return(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Fourier</span>);</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">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。 
[数字图像处理]频域滤波(1)–基础与低通滤波器

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。 
[数字图像处理]频域滤波(1)–基础与低通滤波器 

为此，实现的代码变为了如下形式。
<code class="hljs cpp has-numbering" style="display: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;">IplImage* fft2_New(IplImage* image_input)
{
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dftWidth  = getOptimalDFTSize(image_input->width);
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dftHeight = getOptimalDFTSize(image_input->height);

    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">cout</span><< <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">" Width"</span> <<  image_input->width << <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"    "</span> <<  dftWidth  << <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"\n"</span>;
    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">cout</span><< <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"Height"</span> << image_input->height << <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"    "</span> <<  dftHeight << <span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"\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-box;">1</span>);
    cvCopyMakeBorder( image_input, image_padded, cvPoint(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>), IPL_BORDER_CONSTANT,cvScalarAll(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)); 

    IplImage *image_Re =<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> , *image_Im = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, *image_Fourier = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; 

    image_Re = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
    image_Im = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>);
    image_Fourier = cvCreateImage(cvSize(dftWidth,dftHeight),IPL_DEPTH_64F,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Re = image_padded .* (-1)^(x+y);  </span>
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">double</span> pixel;
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> y=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;y<image_padded->height;y++)
    {
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> x=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;x<image_padded->width;x++)
        {
            pixel = cvGetReal2D(image_padded,x,y);
            pixel = ((x+y)%<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span> == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)?(pixel):((-<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)*pixel);
            cvSetReal2D(image_Re,x,y,pixel);
        }
    }

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Im <--- 0</span>
    cvZero(image_Im);                 
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Fourier[0] <--- image_Re</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//image_Fourier[1] <--- image_Im</span>
    cvMerge(image_Re,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">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-box;">//image_Fourier[0] ---> image_Re</span>
    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//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-box;">0</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//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-box;">2.0</span>);
    cvPow(image_Im,image_Im,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">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-box;">0.5</span>);

    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// log (1 + Mag)</span>
    cvAddS(image_Re,cvScalar(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">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-box;">1</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,CV_C,NULL);

    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span>(image_Re);
}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li><li style="box-sizing: border-box; padding: 0px 5px;">35</li><li style="box-sizing: border-box; padding: 0px 5px;">36</li><li style="box-sizing: border-box; padding: 0px 5px;">37</li><li style="box-sizing: border-box; padding: 0px 5px;">38</li><li style="box-sizing: border-box; padding: 0px 5px;">39</li><li style="box-sizing: border-box; padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li><li style="box-sizing: border-box; padding: 0px 5px;">44</li><li style="box-sizing: border-box; padding: 0px 5px;">45</li><li style="box-sizing: border-box; padding: 0px 5px;">46</li><li style="box-sizing: border-box; padding: 0px 5px;">47</li><li style="box-sizing: border-box; padding: 0px 5px;">48</li><li style="box-sizing: border-box; padding: 0px 5px;">49</li><li style="box-sizing: border-box; padding: 0px 5px;">50</li><li style="box-sizing: border-box; padding: 0px 5px;">51</li><li style="box-sizing: border-box; padding: 0px 5px;">52</li><li style="box-sizing: border-box; padding: 0px 5px;">53</li><li style="box-sizing: border-box; padding: 0px 5px;">54</li><li style="box-sizing: border-box; padding: 0px 5px;">55</li><li style="box-sizing: border-box; padding: 0px 5px;">56</li><li style="box-sizing: border-box; padding: 0px 5px;">57</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li><li style="box-sizing: border-box; padding: 0px 5px;">35</li><li style="box-sizing: border-box; padding: 0px 5px;">36</li><li style="box-sizing: border-box; padding: 0px 5px;">37</li><li style="box-sizing: border-box; padding: 0px 5px;">38</li><li style="box-sizing: border-box; padding: 0px 5px;">39</li><li style="box-sizing: border-box; padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li><li style="box-sizing: border-box; padding: 0px 5px;">44</li><li style="box-sizing: border-box; padding: 0px 5px;">45</li><li style="box-sizing: border-box; padding: 0px 5px;">46</li><li style="box-sizing: border-box; padding: 0px 5px;">47</li><li style="box-sizing: border-box; padding: 0px 5px;">48</li><li style="box-sizing: border-box; padding: 0px 5px;">49</li><li style="box-sizing: border-box; padding: 0px 5px;">50</li><li style="box-sizing: border-box; padding: 0px 5px;">51</li><li style="box-sizing: border-box; padding: 0px 5px;">52</li><li style="box-sizing: border-box; padding: 0px 5px;">53</li><li style="box-sizing: border-box; padding: 0px 5px;">54</li><li style="box-sizing: border-box; padding: 0px 5px;">55</li><li style="box-sizing: border-box; padding: 0px 5px;">56</li><li style="box-sizing: border-box; padding: 0px 5px;">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: block; padding: 0px; background-color: transparent; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background-position: initial initial; background-repeat: initial initial;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span>=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span><image_padded->height;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span>++)
{
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span>=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span><image_padded->width;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span>++)
    {
        pixel = cvGetReal2D(image_padded,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span>);
        pixel = ((<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span>+<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span>)<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%2</span> == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)?(pixel):((-<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*pixel</span>);
        cvSetReal2D(image_Re,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">x</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">y</span>,pixel);
    }
}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; background-color: rgb(238, 238, 238); top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right;"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">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。

在数字图像处理中，经常需要进行傅立叶变换，分析数字图像的频谱，用python简单方便，如下：

import numpy as np

import matplotlib.pyplot as plt 

import cv2 

import sys 

 

img = cv2.imread('%s.png' % sys.argv[1],0)

 

dft = cv2.dft(np.float32(img),flags = cv2.DFT_COMPLEX_OUTPUT)

dft_shift = np.fft.fftshift(dft)

 

magnitude_spectrum = 20*np.log(cv2.magnitude(dft_shift[:,:,0],dft_shift[:,:,1]))

 

plt.subplot(121),plt.imshow(img, cmap = 'gray')

plt.title('Input Image'), plt.xticks([]), plt.yticks([])

plt.subplot(122),plt.imshow(magnitude_spectrum, cmap = 'gray')

plt.title('Magnitude Spectrum'), plt.xticks([]), plt.yticks([])

plt.show()

 

 

频谱图的物理意义是什么呢，频率的振幅能够反映什么物理意义呢？(例如下图)请哪位大师指点，不胜感激！
图不太看得清，我2113没猜错的话横坐标是频率，纵坐5261标是振幅，4102例如频谱图可以用来表示声音频率1653与能量的关系，就像一个声音一般由各种不同频率声音信号组成，每个频率的信号幅值都不一样，就形成了频谱图，一个频谱图就可以表示一个复合信号(例如声音)，不知道说清楚了没~不清楚就hi我
为什么要频谱分析？即频谱分析的意义何在，跪求答案。
推断粒子的性质好像是光谱图吧 频谱一般是研究信号的吧 将信号源发出的信号强度按频率顺序展开，使其成为频率的函数，并考察变化规律，称为频谱分析常见的有振幅频谱图和相位频谱图.
对信号的频域分析有什么意义？
对信号进行时域分析时，2113有时一些信号的5261时域参数相同，但并不能说4102明信号就完全相同。因为1653信号不仅随时间变化，还与频率、相位等信息有关，这就需要进一步分析信号的频率结构，并在频率域中对信号进行描述。动态信号从时间域变换到频率域主要通过傅立叶级数和傅立叶变换实现。周期信号靠傅立叶级数，非周期信号靠傅立叶变换。
如何理解实时频谱分析的含义
对信号进行频谱抄分析的原因袭：
在看似杂乱无bai章的信号中，找出一定du振幅、相位、频率的zhi基本的正弦(余dao弦)信号中，振幅较大(能量较高)信号对应的频率，从而找出信号的主要振动频率特点。如减速机故障时，通过频谱分析，根据各级齿轮转速，齿数与杂音频谱中振幅大的对比，可以快速判断哪级齿轮损伤。
信号谱分析是数字信号处理的重要内容，对确定的信号其时 域表示是确定的，其频谱可以通 过傅立叶变换得到。但在实际应用中，携带信息的信号本质上都是随机的，随机信号不能用 确定的时间函数表示，只能用概率分布函数、概率密度函数或统计平均特性来描述。通常把 随机信号看作无限长度和无限能量的功率信号，由于不满足绝对可积，其傅立叶变换不存在 ，因此只能研究其功率在频域的分布，即功率谱或功率谱密度。
实际应用中人们所能得到的 随机信号的样本函数总是有限长序列，根据有限长度的信号所得的功率谱只是随机信号真实 功率谱的估计，称为功率谱估计。功率谱是平稳随机信号在频域上，描述各频率分量功率分 布情况的基本特征量，由于功率谱与相关函数之间是一对傅立叶变换，经典功率谱估计都依 据DFT，而采用FFT算法，故称之为非参数方法。
频谱图有什么作用？
亲爱的2113楼主：
做频谱特性分析5261用，最简单的就是瀑布图，频谱反4102应的是频域上的能量，引入一个时间的分1653量可以研究分析频域对时间的变化。还可以做多域联合分析，看看频域、时域、调制域上的变化。例如可以剥离分析 相位对时间 频率对时间 上的特性，雷达设计中常常用到
祝您步步高升
期望你的采纳，谢谢
频域图像各点的意义
频域的图像只是(图像)2113信号变换到频域后的5261函数以图4102像形式表示出来，如果你原来的1653是与信号是图像信号，可以这样理解，低频部分极接近原点部分对应于图像平滑变化不大的地方，如背景之类，高频部分对应于图像变化剧烈的地方，如边缘。不知道你变换时的显示形式是怎样，一般0表示暗，1或255表示亮，亮的那块就是你的频域值大罗，如果高频亮就是边缘比较多的地方，但这种变换关系是近似的，不是严格的数学表达，一般图像的书都有相关的介绍。冈萨雷斯的比较好懂
本文标签：
频谱图分析的意义
频谱
意义
物理