Question 1: How does the spectrum change after simultaneous translation in the x and y directions?
Answer: The Fourier transform amplitude map after translation is basically the same as the Fourier transform amplitude map obtained from the original image, and translation does not change the amplitude of the spectrum.
Code running result:
Code:
clc;
clear all;
I=imread('C:\Users\Ch04\4.bmp');
fftI=fft2(I);
sfftI=fftshift(fftI); %求离散傅里叶频谱
%对原始图像进行二维离散傅里叶变换，并将其坐标原点移到频谱图中央位置
RRfdp1=real(sfftI);
IIfdp1=imag(sfftI); a=sqrt(RRfdp1.^2+IIfdp1.^2);
a=(a-min(min(a)))/(max(max(a))-min(min(a)))*225;
figure(4);
subplot(1,2,1),imshow(real(I)),title('同时沿X轴和Y轴平移图');
subplot(1,2,2),imshow(real(a)),title('同时沿X轴和Y轴的傅里叶谱');
Question 2: Programming to verify that after a picture is rotated 45 degrees, its Fourier spectrum is also rotated 45 degrees.
Code:
%构造原始图像
clc;clear all;
I = zeros(256,256); I(88:168,124:132) = 1;
J = fft2(I); %求原始图像的傅里叶频谱
F = abs(J);
J1 = fftshift(F);
figure;
subplot(2,2,1),imshow(I),title('原图');
subplot(2,2,2),imshow(J1,[5 50]),title('原图的傅里叶谱');
%对原始图像进行旋转
J = imrotate(I,45,'bilinear','crop');
%求旋转后图像的傅里叶频谱
J1 = fft2(J);
F = abs(J1);
J2 = fftshift(F);
%figure;
subplot(2,2,3),imshow(J),title('旋转45度后的图');
subplot(2,2,4),imshow(J2,[5 50]),title('旋转45度后的图的傅里叶谱');
Question 3: The impact of the number size of 10 in line 8 on the result;
Answer: The number 10 in line 8 means replacing the values less than 10 in the transformation matrix with 0. The larger the number, the worse the recovery effect of the inverse cosine transform when using idct2 to reconstruct the image.
Code running results:
Code:
% 选取一幅图像，进行离散余弦变换，并对其进行离散余弦反变换，观察其结果。
clc;clear all;close all;
%对 cameraman.tif 文件计算二维 DCT 变换
RGB = imread('C:\Users\...\lenna_rgb.tif');
I = rgb2gray(RGB); %真彩色图像转换成灰度图像
J = dct2(I);%计算二维 DCT 变换
% figure(2); imshow(log(abs(J)),[]); %图像大部分能量集中在上左角处
J(abs(J) < 10) = 0; %%：第8行10的数字大小对结果的影响；
%把变换矩阵中小于 10 的值置换为 0，然后用 idct2 重构图像
K = idct2(J)/255;
figure;
subplot(1,3,1),imshow(I),title('灰度图');
% subplot(1,3,2),imshow(abs(J),[]),title('余弦变换系数图');
subplot(1,3,2),imshow(log(abs(J)),[]),title('余弦变换系数图');
JJ=log(abs(J));
subplot(1,3,3),imshow(K),title('余弦反变换恢复图');
%
Question 4: Change the number of 1’s in mask1 and study its impact on the quality of the reconstructed image
Answer: mask represents a binary mask matrix, used to compress the coefficients of DCT. The original code indicates that the matrix only retains the 10 coefficients in the upper left corner of the DCT transformation matrix for data compression, and discards the high coefficients in the lower right corner. Frequency data.
Through experiments, it can be seen that when 1 in the upper left corner is taken, the number of 1s has little impact on the image quality, and the picture can be restored well. When the upper left corner takes 0 (as shown in the figure below, the upper left corner takes 3 zeros), only very few edges of the image can be restored.
According to experiments and the principles of DCT, it can be concluded that by retaining less data in the upper left corner, a relatively complete picture can be obtained. However, if all other data except the data in the upper left corner are retained, only Can get some edge detail information of the image.
Code execution result:
Code:
% %对图像进行离散余弦变换，做图像压缩解压，取不同的 DCT 系数，并观察其结果。
% clc;clear all;close all;
% %对图像文件计算二维 DCT 变换
% RGB = imread('C:\Users\Ch04\lenna_rgb.tif');
% I = rgb2gray(RGB); %真彩色图像转换成灰度图像
% I = im2double(I); %转换图像矩阵为双精度型
% T = dctmtx(8); %产生二维 DCT 变换矩阵T，矩阵 T 及其转置 T'是 DCT 函数 P1*X*P2 的参数
% B = blkproc(I,[8 8],'P1*x*P2',T,T'); %块状操作函数，B实际上是DCT系数矩阵
% mask1= [ 1 1 1 1 0 0 0 0
% 1 1 1 0 0 0 0 0
% 1 1 0 0 0 0 0 0
% 1 0 0 0 0 0 0 0
% 0 0 0 0 0 0 0 0
% 0 0 0 0 0 0 0 0
% 0 0 0 0 0 0 0 0
% 0 0 0 0 0 0 0 0]; %二值掩模，用来压缩DCT系数
% % 定义一个二值掩模矩阵，用来压缩DCT的系数
% % 该矩阵只保留DCT变换矩阵的最左上角的10个系数
% % 数据压缩，丢弃右下角高频数据
% TB = blkproc(B,[8 8],'P1*x*P2',T',T); %重构图像
% B2 = blkproc(B,[8 8],'P1.*x',mask1); %只保留 DCT 变换的 10 个系数，即压缩DCT系数矩阵
% I2 = blkproc(B2,[8 8],'P1*x*P2',T',T); %压缩DCT系数后重构图像
% figure;
% subplot(2,3,1),imshow(I),title('灰度图');
% subplot(2,3,2);imshow(B),title('DCT系数矩阵图');
% subplot(2,3,3);imshow(B2),title('压缩后的DCT系数矩阵图');
% subplot(2,3,4),imshow(TB),title('DCT系数解压图像');
% subplot(2,3,5),imshow(I2),title('压缩DCT系数mask1解压图像');
% %%%%
%对图像进行离散余弦变换，做图像压缩解压，取不同的 DCT 系数，并观察其结果。
clc;clear all;close all;
%对图像文件计算二维 DCT 变换
RGB = imread('C:\Users\ark\Desktop\newwork\DIP\Ch04\lenna_rgb.tif');
I = rgb2gray(RGB); %真彩色图像转换成灰度图像
I = im2double(I); %转换图像矩阵为双精度型
T = dctmtx(8); %产生二维 DCT 变换矩阵T，矩阵 T 及其转置 T'是 DCT 函数 P1*X*P2 的参数
B = blkproc(I,[8 8],'P1*x*P2',T,T'); %块状操作函数，B实际上是DCT系数矩阵
mask1= [ 0 0 1 1 1 1 1 1
0 1 1 1 1 1 1 0
1 1 1 1 1 1 0 0
1 1 1 1 1 0 0 0
1 1 1 1 0 0 0 0
1 1 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0]; %二值掩模，用来压缩DCT系数
TB = blkproc(B,[8 8],'P1*x*P2',T',T); %重构图像
B2 = blkproc(B,[8 8],'P1.*x',mask1); %只保留 DCT 变换的 10 个系数，即压缩DCT系数矩阵
I2 = blkproc(B2,[8 8],'P1*x*P2',T',T); %压缩DCT系数后重构图像
figure;
subplot(2,3,1),imshow(I),title('灰度图');
subplot(2,3,2);imshow(B),title('DCT系数矩阵图');
subplot(2,3,3);imshow(B2),title('压缩后的DCT系数矩阵图');
subplot(2,3,4),imshow(TB),title('DCT系数解压图像');
subplot(2,3,5),imshow(I2),title('压缩DCT系数mask1解压图像');
%%%%%end homework
figure;
subplot(1,2,1),imshow(abs(TB-I)),title('');
subplot(1,2,2),imshow(abs(I2-I)),title('');
RGB = imread('C:\Users\ark\Desktop\newwork\DIP\Ch04\lenna_rgb.tif'); I=rgb2gray(RGB);
I = im2double(I); %转换图像矩阵为双精度型
T = dctmtx(8); %产生二维 DCT 变换矩阵， returns the N-by-N DCT transform matrix
%矩阵 T 及其转置 T'是 DCT 函数 P1*X*P2 的参数
B = blkproc(I,[8 8],'P1*x*P2',T,T'); %DCT变换
mask2= [ 1 1 1 1 0 0 0 0
1 1 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0]; %二值掩模，用来压缩 DCT 系数
B2 = blkproc(B,[8 8],'P1.*x',mask2); %只保留 DCT 变换的 8 个系数
I2 = blkproc(B2,[8 8],'P1*x*P2',T',T); %重构图像figure,imshow(I);
figure;
subplot(1,3,1),imshow(I),title('灰度图');
subplot(1,3,2);imshow(B2),title('');
subplot(1,3,3),imshow(I2),title('压缩DCT 系数 mask2 解压图像');
figure;
subplot(2,3,1),imshow(I),title('灰度图');
subplot(2,3,2),imshow(T),title('DCT变换矩阵');
subplot(2,3,3),imshow(B),title('DCT变换');
subplot(2,3,4),imshow(B2),title('压缩后的DCT变换');
subplot(2,3,5),imshow(I2),title('重构图像');
%改变mask1中1的个数，研究其对重构图像的质量影响