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IPCV >> Image Enhancement and Restoration > imdeconvsobolev

imdeconvsobolev

Deconvolution by Sobolev Regularization

Syntax

imout = imdeconvsobolev(imin,psf,lambda)

Parameters

imin :

Source Image

f :

Blur function

lamda :

Regularization parameter

imout :

Deblurred Image

Description

L2 regularization did not perform any denoising. To remove some noise, we can penalize high frequencies using Sobolev regularization (quadratic grow).

The Sobolev prior reads (note the conversion from spacial domain to Fourier domain)

$[J(f) = \sum_x \|\nabla f(x)\|^2 = \sum_{\omega} S(\omega) \|\hat f(\omega)\|^2 ] where S(\omega)=\|\omega\|^2)$

Since this prior can be written over the Fourier domain, one can compute the solution to the deblurring with Sobolev prior simply with the Fourier coefficients:

$[\hat f^\star(\omega) = \frac{\hat y(\omega) \hat h(\omega)}{ \|\hat h(\omega)\|^2 + \lambda S(\omega) }]$

Compute the Sobolev prior penalty S (rescale to [0,1]).

Examples

S = imread(fullpath(getIPCVpath() + "/images/balloons_gray.png"));
S = im2double(S);
h = fspecial('motion',25,45);
S2 = imfilter(S,h,'circular');
imshow(S2);
S3 = imdeconvsobolev(S2,h,0);
figure;imshow(S3);

See also

Authors

Bibliography

1. Advanced Signal, Image and Surface Processing, Ceremade, Université Paris-Dauphine


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