一、序言：
该教程基于之前的图像处理类MYCV，是对其的补充。
二、设计目标
对图像进行简单的离散傅里叶变换，并输出生成的频谱图。
三、需要提前掌握的知识
二维傅里叶变换公式：
四、详细步骤
1．首先定义一个方法，该方法对输入的图像进行傅里叶变换
输入：MyImage 源图像
输出：ComplexNu 进行离散傅里叶变换后的复数数组
定义：
static ComplexNumber* Dft2(MyImage const &Scr);
实现：
1 ComplexNumber* MyCV::Dft2(MyImage const &Scr)
2 {
3 int width = Scr.m_width;
4 int height = Scr.m_height;
5
6 // 将 scr_data 转化为灰度
7 MyImage *grayimage = Gray(Scr);
8 unsigned char* gray_data = grayimage->m_data;
9 int gray_bytesPerLine = grayimage->m_bytesPerLine;
10
11 // 将 gray_data 转化为 double 型，并去掉用于填充的多余空间
12 double *double_data = new double[width*height];
13
14 for(int i=0;i<height;i++)
15 for(int j=0;j<width;j++)
16 {
17 double_data[i*width+j]=(double)gray_data[i*gray_bytesPerLine+j];
18 }
19
20 // 对 double_data 进行傅里叶变换
21 ComplexNumber *dft2_data = new ComplexNumber[width*height];
22 double fixed_factor_for_axisX = (-2 * PI) / height;
23 // evaluate -i2π/N of -i2πux/N, and store the value for computing efficiency
24 double fixed_factor_for_axisY = (-2 * PI) / width;
25 // evaluate -i2π/N of -i2πux/N, and store the value for computing efficiency
26
27 for (int u = 0; u<height; u++) {
28 for (int v = 0; v<width; v++) {
29 for (int x = 0; x<height; x++) {
30 for (int y = 0; y<width; y++) {
31 double powerX = u * x * fixed_factor_for_axisX; // evaluate -i2πux/N
32 double powerY = v * y * fixed_factor_for_axisY; // evaluate -i2πux/N
33 ComplexNumber cplTemp;
34 cplTemp.m_rl = double_data[y + x*width] * cos(powerX + powerY);
35 // evaluate f(x) * e^(-i2πux/N), which is equal to f(x) * (cos(-i2πux/N)+sin(-i2πux/N)i) according to Euler's formula
36 cplTemp.m_im = double_data[y + x*width] * sin(powerX + powerY);
37 dft2_data[v + u*width] = dft2_data[v + u*width] + cplTemp;
38 }
39 }
40 }
41 }
42
43 // 返回傅里叶数组
44 return dft2_data;
45 }
2．为了让傅里叶变换可视化，旭阳对其进行标准化和中性化
输入：ComplexNumber 离散傅里叶变换生成的复数数组
输出：MyImage 可视化后的图像
定义：
static MyImage* Dft22MyImage(ComplexNumber *Scr,int width,int height);
实现：
1 MyImage* MyCV::Dft22MyImage(ComplexNumber *Scr, int const width, int const height)
2 {
3 // 将傅里叶数组归一化
4 // 取模
5 double mold[width*height];
6 for(int i = 0 ;i<width*height;i++)
7 {
8 mold[i] = Scr[i].get_mold();
9 }
10
11 // 获取最小值
12 double min = mold[0];
13 for(int i = 0;i<width*height;i++)
14 {
15 if(mold[i]<min)
16 min = mold[i];
17 }
18
19 // 获取去掉前几大值的最大值
20 double maxqueue[20] = {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},max;
21
22 for(int i = 0;i<width*height;i++){
23 if(mold[i]>maxqueue[0])
24 maxqueue[0] = mold[i];
25 }
26
27 for(int j =1;j<20;j++){
28 for(int i = 0;i<width*height;i++){
29 if(mold[i]>maxqueue[j]&&mold[i]<maxqueue[j-1])
30 maxqueue[j] = mold[i];
31 }
32 }
33
34 max = maxqueue[19];
35
36 unsigned char *normalized_data = new unsigned char[width*height];
37
38 for(int i=0;i<height;i++)
39 for(int j=0;j<width;j++)
40 {
41 unsigned char t = (unsigned char)((mold[i*width+j]-min)/(max-min)*255);
42 if(t>255)
43 t = 255;
44 normalized_data[i*width+j]=t;
45 }
46
47 // 将图像中心化
48 unsigned char* center_data = new unsigned char[width*height];
49
50 for (int u = 0; u<height; u++){
51 for (int v = 0; v<width; v++) {
52 if ((u<(height / 2)) && (v<(width / 2))) {
53 center_data[v + u*width] =
54 normalized_data[width / 2 + v + (height / 2 + u)*width];
55 }
56 else if ((u<(height / 2)) && (v >= (width / 2))) {
57 center_data[v + u*width] =
58 normalized_data[(v - width / 2) + (height / 2 + u)*width];
59 }
60 else if ((u >= (height / 2)) && (v<(width / 2))) {
61 center_data[v + u*width] =
62 normalized_data[(width / 2 + v) + (u - height / 2)*width];
63 }
64 else if ((u >= (height / 2)) && (v >= (width / 2))) {
65 center_data[v + u*width] =
66 normalized_data[(v - width / 2) + (u - height / 2)*width];
67 }
68 }
69 }
70
71 // 向中心化的数组填充空间
72 int bytesPerLine = (width*8+31)/32*4;
73 unsigned char *dst_data = new unsigned char[bytesPerLine*height];
74
75 for(int i=0;i<height;i++)
76 for(int j=0;j<width;j++)
77 {
78 dst_data[i*bytesPerLine+j] = center_data[i*width+j];
79 }
80
81 return new MyImage(dst_data,width,height,MyImage::format::GRAY8);
82 }
至此，离散傅里叶变换的方法实现完成，效果图如下：
