image processing assignment ii(appu)

8/19/2019 Image Processing Assignment II(Appu)

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By

M.Apuroop

110113044



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Explanation:

An image as a function can be expressed as the product of illumination and

reflectance components as follows:

F(x,y) = I(x,y) * R(x,y) (1)

The illumination component

– Slow spatial variations

– Low frequency

The reflectance component

– Vary abruptly, particularly at the junctions of dissimilar objects

–

High frequency

Equation (1) cannot be used directly to operate separately on the frequency

components of illumination and reflectance because the Fourier transform of the

product of two functions is not separable. Instead the function can be represented as

a logarithmic function wherein the product of the Fourier transform can be

represented as the sum of the illumination and reflectance components as shown

below:

ln(x,y) = ln(I(x,y)) + ln(R(x,y)) (2)

The Fourier transform of equation (2) is

Z(u,v) = Fi(u,v) + Fr(u,v) (3)

The fourier transformed signal is processed by means of a filter function H(u,v) and

the resulting function is inverse fourier transformed. Finally, inverse exponential

operation yields an enhanced image. This enhancement approach is termed ashomomorphic filtering .



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The whole operation is expressed as a block diagram below:



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Program :

I = im read('wil lo wt ree.jpg ');

f igure, imsh ow (I);

% the image might be a color im age, so we co nver t i t to greyscale

I = rgb 2gray(I);

% conv ert the image to f loat ing-point typ e from uint8(whic h is default)

I = im2doub le(I);

f igure, imshow (I);

% take the image into the log domain

I = log (1 + I);

f igure, imsh ow (I);

% Now lets co ns tru ct the gaussian fil ter here k = size(I,1) = size(I,2)

M = 2*size(I,1) + 1;

N = 2*si ze(I,2) + 1;

% standard deviat ion of the equiv alent sp atial dom ain gaussian f i l ter

sigm a = 10;

Contd.



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% X and Y are k*k s ize matr ices w ith

% X has co lumn i w ith al l i 's (this is tru e for i = 1,. .. .k

% Y has row i w ith al l i 's (this is true for i = 1,.. ..k

[X, Y] = meshg rid(1:N,1:M);

centerX = ceil(N/2);

centerY = ceil(M/2);

gaus sianNumerator = (X - cen terX).^ 2 + (Y - cent erY).^2;

H = exp(-gaussianNumerator./(2*sigma.^2));

H = 1 - H;

imshow(H,'Ini t ialMagnif icat ion',25)

H = fftshi ft(H);

If = fft2(I, M, N);

Iout = real(ifft 2(H.*If));

Iout = Iou t(1:size(I,1),1:si ze(I,2));

Ihmf = exp(Iout) - 1;

imshow pair( I, Ihm f, 'montage')



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Output

Original Image

Homomorphic Filtered Image

:

image processing assignment ii(appu)

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