1. Discrete Fourier Transform

2. FFT and Its Applications
FFTSHIFT Shift zero-frequency component to the center of spectrum. For vectors, FFTSHIFT(X) swaps the left and right halves of X. For matrices, FFTSHIFT(X) swaps the first and third quadrants and the second and fourth quadrants. For N-D arrays, FFTSHIFT(X) swaps "half-spaces" of X along each dimension.

3. Example of 1-D Fourier Transform

4. fftBox.m – Plot Fourier Spectrum%
% Script file: fftBox.m
% Fourier Spectrum Plot of Box function
X1=linspace(0,1,17);
Y1=ones(1,length(X1));
X2=linspace(1,16,241);
Y2=zeros(1,length(X2));
X=[X1 X2]; Y=[Y1 Y2];
W=abs(fftshift(fft(Y)));
subplot(2,1,1)
plot(X,Y,'r'); axis([0 16, 0,1.2]); title('Box function')
subplot(2,1,2)
plot(W,'b-');
title('Fourier Spectrum of Box function')

5. 2-D Discrete Fourier Transform

6. Example of 2-D FFT Matlab Code
% Script file: fourier.m - 2D Fourier Transform
% Pictures on P.113 of Gonzalez, Woods, Eddins
m=128; n=128;
f=zeros(m,n);
f(56:71,48:79)=255;
F0=fft2(f); S0=abs(F0);
Fc=fftshift(fft2(f)); Sc=abs(Fc);
Fd=fft2(fftshift(f)); Sd=log(1+abs(Fc));
subplot(2,2,1)
imshow(f,[])
subplot(2,2,2)
imshow(S0,[])
subplot(2,2,3)
imshow(Sc,[ ])
subplot(2,2,4)
imshow(Sd,[ ])

7. 2-D FFT with CenterShift

8. 2-D FFT on Texture Images

9. Discrete Cosine Transform
Partition an image into nonoverlapping 8 by 8 blocks, and apply a 2d DCT on each block to get DC and AC coefficients.
Most of the high frequency coefficients become insignificant, only the DC term and some low frequency AC coefficients are significant.
Fundamental for JPEG Image Compression

10. Discrete Cosine Transform (DCT)
X: a block of 8x8 pixels
A=Q8: 8x8 DCT matrix as
shown above
Y=AXAt

11. DCT on a 8x8 Block

12. Quantized DCT Coefficients

13. Matlab Code for 2d DCT Q=xlsread('Qtable.xls','A2:H9');
fin=fopen('block8x8.txt','r');
fout=fopen('dctO.txt','w');
fgetl(fin); X=fscanf(fin,'%f',[8,8]); fclose(fin); X=X';
Y=dct2(X-128,[8,8]);
fprintf(fout,'DCT coefficients\n');
for i=1:8
for j=1:8 fprintf(fout,'%6.1f',Y(i,j)); end; fprintf(fout,'\n');
end
Y=Y./Q; % Y=fix(Y+0.5*(Y>0));
for j=1:8
if (Y(i,j)>0) Y(i,j)=fix(Y(i,j)+0.5); else Y(i,j)=fix(Y(i,j)-0.5); end
fprintf(fout,'Quantized DCT coefficients\n');
for j=1:8 fprintf(fout,'%4d',Y(i,j)); end; fprintf(fout,'\n');
fclose(fout);

14. DCT-Based JPEG Conversion
Input image
write to file
huffman encoding
shift 128
DCT
run-length encoding
convert 2D matrix to 1D array
round
quantize with quantize matrix

15. Standard Quantization Table
run-length encoding
產生一維結果:
-26,-3,0,……,-1,-1,0,0,0,0…….
後皆為零，簡化可以減少資料儲存量

16. JPEG Decoding image result read compression file huffman decoding
shift 128
IDCT
run-length decoding
quantize with quantize matrix
convert 1D array to 2D matrix

17. 未採用流程

18. 壓縮結果差異

19. 壓縮數據比較 原始檔案 CameraMan Plusplus lena boats 原始大小 (壓縮比) 49.3 KB 468 KB
壓縮格式
壓縮大小
zip
42.5KB
(82.61%)
111KB
(23.72%)
219KB
(85.21%)
137KB
(72.87%)
rar
36.6KB
(74.24%)
97.3KB
(20.79%)
164KB
(63.81%)
116KB
(61.7%)
jpg
12.9KB
(26.17%)
26.5KB
(5.662%)
62.0KB
(24.12%)
46.6KB
(24.79)
project
11.2KB
(22.75%)
9.48KB
(2.024%)
28.1KB
(10.95%)
24.3KB
(12.93%)

20. 壓縮數據比較