Activity 19 - Restoration of Blurred image

This activity will show how to restore blurred and noisy images from an a priori knowledge of degradation function and noise. The main idea of image degradation and restoration is presented using the model below:

General model in spatial domain:

where g(x,y) is the degraded image, f(x,y) is the original image, h(x,y) is the degradation function and n(x,y) is the noise model.

General model in frequency domain:

The variables presented are the Fourier transforms of the above equation.

From the above equations, we are to find the degrading function and the noise model. The noise model is described in the previous activity, so we are left with the degrading function, or the motion blurring effect. It is given by:

The image restoration procedure is summarized in the following way:

One filter that is shown in this activity is Weiner filter which is also known as the minimum mean-square error (MMSE) filter by modeling the error through statistical characterization of the noise. It minimizes the average error giving the Weiner filter:

or

where , , and K is a specified constant.

Results

The result of filtering for different types of images are presented. The first image is the original, the second is the blurred and noisy image, the last two are the two types of Weiner filter the last being specified by K = 1.

As seen, the Weiner filter restores the blurred and noisy image but it can't recover the original image itself. The phrase can still be discerned from the restoration but not fully. The last image is not a good restoration using the constant K. Even if K value is varied, no visible change in the restored image is observed.

I give myself 10 pts for understanding and finishing the activity. :)

References
M. Soriano. A19 – Restoration of blurred image. pdf file
Digital Image Processing. Gonzales and Woods.

Scilab Code

// Degradation

chdir('E:\Documents and Settings\vergara\My Documents\acads\1st sem 09-10\186\act19\');
img = gray_imread('k.bmp');
//img = gray_imread('phrase_ei.bmp');
//img = gray_imread('people1.bmp');
FTimg = fft2(img);
FT = fftshift(abs(FTimg));

a = .05;
b = .05;
T = 1;

// noise
y = grand(size(img,1), size(img,2), 'nor', 0.1, 0.1); // gauss

FTy = fft2(y);

for i = 1:size(FT,1)
for j = 1:size(FT,2)
H(i,j) = T/(%pi*(i*a + j*b)) * sin(%pi*(i*a + j*b)) * exp(-%i*(%pi*(i*a + j*b)));
end
end

G = H.*FTimg + FTy;
Sn = FTy.*conj(FTy);
Sf = FTimg.*conj(FTimg);

a = (H.^(-1));
b = ((H.*conj(H))./((H.*conj(H)) + (Sn./Sf)));
Fh1 = a.*b.*G;

K = 10000000000;
b = ((H.*conj(H))./((H.*conj(H)) + K));
Fh2 = a.*b.*G;

//scf(), imshow(FTimg, [])
scf(), imshow(abs(ifft(G)), [])
scf(), imshow(abs(ifft(Fh1)), [])
scf(), imshow(abs(ifft(Fh2)), [])
//scf(), imshow(img, [])