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<h1>OpenCV基础篇之图像频域</h1>
<h4>2014-09-16 / xiahouzuoxin</h4>
<h4>Tags: OpenCV</h4>
转载请注明出处: <a href="http://xiahouzuoxin.github.io/notes/">http://xiahouzuoxin.github.io/notes/</a>
<div id="TOC">
<ul>
<li><a href="#程序及分析">程序及分析</a></li>
<li><a href="#效果">效果</a></li>
</ul>
</div>
<!---title:OpenCV基础篇之图像频域-->
<!---keywords:OpenCV-->
<!---date:2014-09-16-->
<h2 id="程序及分析">程序及分析</h2>
<pre class="sourceCode c"><code class="sourceCode c"><span class="co">/*</span>
<span class="co"> * FileName : fft2.cpp</span>
<span class="co"> * Author : xiahouzuoxin @163.com</span>
<span class="co"> * Version : v1.0</span>
<span class="co"> * Date : Wed 30 Jul 2014 09:42:12 PM CST</span>
<span class="co"> * Brief : </span>
<span class="co"> * </span>
<span class="co"> * Copyright (C) MICL,USTB</span>
<span class="co"> */</span>
<span class="ot">#include <iostream></span>
<span class="ot">#include <cv.h></span>
<span class="ot">#include <highgui.h></span>
<span class="ot">#include "imgproc/imgproc.hpp"</span>
using namespace std;
using namespace cv;
<span class="dt">int</span> main(<span class="dt">int</span> argc, <span class="dt">char</span> *argv[])
{
<span class="kw">if</span> (argc < <span class="dv">2</span>) {
cout<<<span class="st">"Usage:./fft2 [image name]"</span><<endl;
<span class="kw">return</span> -<span class="dv">1</span>;
}
<span class="co">// Read as grayscale image</span>
Mat image = imread(argv[<span class="dv">1</span>], CV_LOAD_IMAGE_GRAYSCALE);
<span class="kw">if</span> (!image.data) {
cout << <span class="st">"Read image error"</span><<endl;
<span class="kw">return</span> -<span class="dv">1</span>;
}
Mat padded;
<span class="dt">int</span> m = getOptimalDFTSize(image.rows); <span class="co">// Return size of 2^x that suite for FFT</span>
<span class="dt">int</span> n = getOptimalDFTSize(image.cols);
<span class="co">// Padding 0, result is @padded</span>
copyMakeBorder(image, padded, <span class="dv">0</span>, m-image.rows, <span class="dv">0</span>, n-image.cols, BORDER_CONSTANT, Scalar::all(<span class="dv">0</span>));
<span class="co">// Create planes to storage REAL part and IMAGE part, IMAGE part init are 0</span>
Mat planes[] = {Mat_<<span class="dt">float</span>>(padded), Mat::zeros(padded.size(), CV_32F) };
Mat complexI;
merge(planes, <span class="dv">2</span>, complexI);
dft(complexI, complexI);
<span class="co">// compute the magnitude and switch to logarithmic scale</span>
split(complexI, planes);
magnitude(planes[<span class="dv">0</span>], planes[<span class="dv">0</span>], planes[<span class="dv">1</span>]);
Mat magI = planes[<span class="dv">0</span>];
<span class="co">// => log(1+sqrt(Re(DFT(I))^2+Im(DFT(I))^2))</span>
magI += Scalar::all(<span class="dv">1</span>);
log(magI, magI);
<span class="co">// crop the spectrum</span>
magI = magI(Rect(<span class="dv">0</span>, <span class="dv">0</span>, magI.cols & (-<span class="dv">2</span>), magI.rows & (-<span class="dv">2</span>)));
Mat _magI = magI.clone();
normalize(_magI, _magI, <span class="dv">0</span>, <span class="dv">1</span>, CV_MINMAX);
<span class="co">// rearrange the quadrants of Fourier image so that the origin is at the image center</span>
<span class="dt">int</span> cx = magI.cols/<span class="dv">2</span>;
<span class="dt">int</span> cy = magI.rows/<span class="dv">2</span>;
Mat q0(magI, Rect(<span class="dv">0</span>,<span class="dv">0</span>,cx,cy)); <span class="co">// Top-Left</span>
Mat q1(magI, Rect(cx,<span class="dv">0</span>,cx,cy)); <span class="co">// Top-Right</span>
Mat q2(magI, Rect(<span class="dv">0</span>,cy,cx,cy)); <span class="co">// Bottom-Left</span>
Mat q3(magI, Rect(cx,cy,cx,cy)); <span class="co">// Bottom-Right</span>
<span class="co">// exchange Top-Left and Bottom-Right</span>
Mat tmp;
q0.copyTo(tmp);
q3.copyTo(q0);
tmp.copyTo(q3);
<span class="co">// exchange Top-Right and Bottom-Left</span>
q1.copyTo(tmp);
q2.copyTo(q1);
tmp.copyTo(q2);
normalize(magI, magI, <span class="dv">0</span>, <span class="dv">1</span>, CV_MINMAX);
imshow(<span class="st">"Input image"</span>, image);
imshow(<span class="st">"Spectrum magnitude before shift frequency"</span>, _magI);
imshow(<span class="st">"Spectrum magnitude after shift frequency"</span>, magI);
waitKey();
<span class="kw">return</span> <span class="dv">0</span>;
}</code></pre>
<p>本程序的作用是：将图像从空间域转换到频率域，并绘制频域图像。</p>
<ol style="list-style-type: decimal">
<li><p>二维图像的DFT（离散傅里叶变换），</p>
<p><img src="https://latex.codecogs.com/png.latex? \Large F(k,l)={\sum_{i=0}^{i=N-1}}{\sum_{j=0}^{j=N-1}}f(i,j)e^{-j2\pi{(\frac{ki}{N}+\frac{lj}{N}})}"></p>
<p>图像的频域表示的是什么含义呢？又有什么用途呢？图像的频率是表征图像中灰度变化剧烈程度的指标，是灰度在平面空间上的梯度。图像的边缘部分是突变部分，变化较快，因此反应在频域上是高频分量；图像的噪声大部分情况下是高频部分；图像大部分平缓的灰度变化部分则为低频分量。也就是说，傅立叶变换提供另外一个角度来观察图像，可以将图像从灰度分布转化到频率分布上来观察图像的特征。</p>
<p>频域在图像处理中，就我所知的用途主要在两方面：图像压缩和图像去噪。关于这两点将在下面给出图片DFT的变换结果后说明。</p>
<p>有关DFT的更多性质请参考胡广书教授的《数字信号处理》教材。</p></li>
<li><p>请注意读图片的函数与之前有所不同：</p>
<pre class="sourceCode c"><code class="sourceCode c">Mat image = imread(argv[<span class="dv">1</span>], CV_LOAD_IMAGE_GRAYSCALE);</code></pre>
<p><code>CV_LOAD_IMAGE_GRAYSCALE</code>参数表示将原图像转换为灰度图后读入，这是因为后面的DFT变换都是基于二维信号的，而彩色图像是三维信号。当然，也可以对RGB每一通道都进行DFT运算。</p></li>
<li><p>DFT算法的原理要求输入信号的长度最好为2^n，这样可以使用快速傅里叶变换算法（FFT算法）进行加速。所以程序中使用</p>
<pre class="sourceCode c"><code class="sourceCode c">copyMakeBorder(image, padded, <span class="dv">0</span>, m-image.rows, <span class="dv">0</span>, n-image.cols, BORDER_CONSTANT, Scalar::all(<span class="dv">0</span>));</code></pre>
<p>填充0使横纵长度都为2^n。</p>
<p>对于一维信号，原DFT直接运算的复杂度是O(N^2)，而快速傅里叶变换的复杂度降低到O(Nlog2(N))，假设N为512，足足提高了512/9≈57倍。</p></li>
<li><p>由DFT的性质知，输入为实信号（图像）的时候，频域输出为复数，因此将频域信息分为幅值和相位。频域的幅值高的代表高频分量，幅值低的地方代表低频分量，因此程序中使用</p>
<pre class="sourceCode c"><code class="sourceCode c"><span class="co">// => log(1+sqrt(Re(DFT(I))^2+Im(DFT(I))^2))</span>
magI += Scalar::all(<span class="dv">1</span>);
log(magI, magI);
<span class="co">// crop the spectrum</span>
magI = magI(Rect(<span class="dv">0</span>, <span class="dv">0</span>, magI.cols & (-<span class="dv">2</span>), magI.rows & (-<span class="dv">2</span>)));
Mat _magI = magI.clone();
normalize(_magI, _magI, <span class="dv">0</span>, <span class="dv">1</span>, CV_MINMAX);</code></pre>
<p>进行log幅值计算及归一化幅值（归一化目的主要是方便将频域通过图像的形式进行显示）。</p></li>
<li><p>关于频域中心平移：将图像的高频分量平移到图像的中心，便于观测。</p>
<pre><code>int cx = magI.cols/2;
int cy = magI.rows/2;
Mat q0(magI, Rect(0,0,cx,cy)); // Top-Left
Mat q1(magI, Rect(cx,0,cx,cy)); // Top-Right
Mat q2(magI, Rect(0,cy,cx,cy)); // Bottom-Left
Mat q3(magI, Rect(cx,cy,cx,cy)); // Bottom-Right
// exchange Top-Left and Bottom-Right
Mat tmp;
q0.copyTo(tmp);
q3.copyTo(q0);
tmp.copyTo(q3);
// exchange Top-Right and Bottom-Left
q1.copyTo(tmp);
q2.copyTo(q1);
tmp.copyTo(q2);</code></pre>
<p>其原理就是将左上角的频域和右下角的互换，右上角和左下角互换。</p>
<p>请注意：频域点和空域点的坐标没有一一对应的关系，两者的关系只是上面的DFT公式所见到的。</p></li>
<li><p>本程序因为使用到图像处理相关的函数，所以包含了头文件<code>imgproc/imgproc.hpp</code>，该文件位于opencv安装目录的include/opencv2/目录下，在编写Makefile时也要增加相关的头文件路径和库，本程序使用的Makefile如下：</p>
<pre class="sourceCode c"><code class="sourceCode c">TARG=fft2
SRC=fft2.cpp
LIB=-L/usr/local/lib/
INC=-I/usr/local/include/opencv/ -I/usr/local/include/opencv2
CFLAGS=
$(TARG):$(SRC)
g++ -g -o $@ ${CFLAGS} $(LIB) $(INC) \
-lopencv_core -lopencv_highgui -lopencv_imgproc \
$^
.PHONY:clean
clean:
-rm $(TARG) tags -f </code></pre>
<p>其中Makefile中的（反斜杠后不能再有任何字符，包括空格），如上库增加了<code>-lopencv_imgproc</code>，头文件路径增加了<code>-I/usr/local/include/opencv2</code>。</p></li>
</ol>
<h2 id="效果">效果</h2>
<div class="figure">
<img src="../images/OpenCV基础篇之图像频域/dft.png" alt="dft" /><p class="caption">dft</p>
</div>
<ol style="list-style-type: decimal">
<li><p>上图从左到右分别是：原始灰度图（我大爱的杨过啊）、频域平移前的频域图像、频域中心平移后的频域图像。</p></li>
<li><p>提到图像频域变换的用途：压缩和去噪。压缩的原理就是在频域中，大部分频域的值为0（或接近0，可以进行有损压缩，如jpeg图像），只要压缩频域中的少数非0值即可达到图片压缩的目的。去噪则是通过频域的滤波实现，因为噪声大部分情况下体现为高频信号，使用低通滤波器即可滤除高频噪声（当然，也会带来损失，那就是边缘会变得模糊（之前说过，边缘也是高频信号））。</p></li>
</ol>
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