Discrete Fourier Transform

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# Discrete Fourier Transform - PowerPoint PPT Presentation

Discrete Fourier Transform. FFT and Its Applications.

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Presentation Transcript
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.

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')
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,[ ])
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=Q8: 8x8 DCT matrix as

shown above

Y=AXAt

Matlab Code for 2d DCT
• 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
• 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

write to file

huffman encoding

shift 128

DCT

run-length encoding

convert 2D matrix to 1D array

round

quantize with quantize matrix

Standard Quantization Table

run-length encoding

-26,-3,0,……,-1,-1,0,0,0,0…….

JPEG Decoding

image result