Discrete Fourier Transform

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.

Example of 1-D Fourier Transform

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')

2-D Discrete Fourier Transform

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,[ ])

2-D FFT with CenterShift

2-D FFT on Texture Images

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

Discrete Cosine Transform (DCT) X: a block of 8x8 pixels A=Q 8 : 8x8 DCT matrix as shown above Y=AXA t

DCT on a 8x8 Block

Quantized DCT Coefficients

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 i=1:8 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 end fprintf(fout,'Quantized DCT coefficients\n'); for i=1:8 for j=1:8 fprintf(fout,'%4d',Y(i,j)); end; fprintf(fout,'\n'); end fclose(fout);

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

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

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

未採用流程

壓縮結果差異

壓縮數據比較 原始檔案 CameraManPluspluslenaboats 原始大小 ( 壓縮比 ) 49.3 KB468 KB257 KB188 KB 壓縮格式壓縮大小 zip KB (82.61%) 111KB (23.72%) 219KB (85.21%) 137KB (72.87%) rar KB (74.24%) 97.3KB (20.79%) 164KB (63.81%) 116KB (61.7%) jpg KB (26.17%) 26.5KB (5.662%) 62.0KB (24.12%) 46.6KB (24.79) project KB (22.75%) 9.48KB (2.024%) 28.1KB (10.95%) 24.3KB (12.93%)

壓縮數據比較