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public:image_convolution_in_fourier_domain [2015/12/02 17:30] fangfufu [Computing the Laplacian of an image via deconvolution] |
public:image_convolution_in_fourier_domain [2018/03/31 01:38] (current) |
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+ | ====== Image convolution and deconvolution in the frequency domain ====== | ||

+ | Image convolution and deconvolution can be performed quickly in the frequency domain. In the frequency domain, pointwise multiplication is convolution, while pointwise division is deconvolution. The pseudo-code of the procedure is: | ||

+ | - Convert your input image into the frequency domain, via fast Fourier transform. | ||

+ | - Pad your convolution kernel with zeros, until it reaches the same size as your input image. | ||

+ | - Convert your convolution kernel into the frequency domain, via fast Fourier transform. | ||

+ | - Pointwise multiply the frequency domain input image and the frequency domain convolution kernel together. | ||

+ | - Convert the product from pointwise multiplication back into the spatial domain via inverse fast Fourier transform. | ||

+ | If you want to perform deconvolution, rather than perform pointwise multiplication in step 4, you do pointwise division. You have to be aware divide-by-zero problem, when you are performing deconvolution. You might want to replace all the zeros with eps(). | ||

+ | |||

+ | Below are some Matlab examples. Note that Matlab takes care of the zero-padding inside the fft2() function call. | ||

+ | |||

+ | ===== Computing the Laplacian of an image via convolution ===== | ||

+ | <code matlab> | ||

+ | function [ out ] = get_laplacian( in ) | ||

+ | %GET_LAPLACIAN Obtain the Laplacian of an image | ||

+ | % This is an implementation of the algorithm described in page 32-35 of | ||

+ | % Lightness and Brightness Computation by Retinex-like Algorithms by Eran | ||

+ | % Borenstein | ||

+ | |||

+ | if size(in, 3) == 3 | ||

+ | for i = 1:3 | ||

+ | tmp = in(:,:,i); | ||

+ | out(:,:,i) = get_laplacian_real(tmp); | ||

+ | end | ||

+ | else | ||

+ | out = get_laplacian_real(in); | ||

+ | end | ||

+ | |||

+ | end | ||

+ | |||

+ | function [ out ] = get_laplacian_real(in) | ||

+ | %% Fourier transform input image | ||

+ | a = fft2(in); | ||

+ | b = fft2(fspecial('laplacian'), size(in, 1), size(in, 2)); | ||

+ | c = a.*b; | ||

+ | out = real(ifft2(c)); | ||

+ | end | ||

+ | |||

+ | </code> | ||

+ | |||

+ | ===== Computing the Laplacian of an image via deconvolution ===== | ||

+ | <code matlab> | ||

+ | function [ out ] = inv_laplacian(in) | ||

+ | %INV_LAPLACIAN Convert a Laplacian image back to its original | ||

+ | |||

+ | if size(in, 3) == 3 | ||

+ | for i = 1:3 | ||

+ | tmp = in(:,:,i); | ||

+ | out(:,:,i) = inv_laplacian_real(tmp); | ||

+ | end | ||

+ | else | ||

+ | out = inv_laplacian_real(in); | ||

+ | end | ||

+ | |||

+ | end | ||

+ | |||

+ | function [out] = inv_laplacian_real(in) | ||

+ | %% Fourier transform input image | ||

+ | a = fft2(in); | ||

+ | b = fft2(fspecial('laplacian'), size(in, 1), size(in, 2)); | ||

+ | b(b == 0) = eps; | ||

+ | c = a./b; | ||

+ | out = real(ifft2(c)); | ||

+ | out = out + abs(min(out(:))); | ||

+ | end | ||

+ | |||

+ | |||

+ | </code> |