Question

Matlab question:

1. Implement image filtering in frequency domain on the "testtpattern512.tif" image. You will design the filter, and filter the image in the frequency domain, then use inverse DFT to display the filtered image in the spatial domain. Assume zero padding for filtering process.

a- Design a rectangular ideal low pass filter with a cutoff frequency of (Uo, Vo), filtering the image so that the vertical lines are blurred together, horizontal lines are basically intact and the letters are still readable. Uo, and Vo represent the cutoff frequencies in two dimensions. You may need to try different Uo, Vo, to find the best settings.

b- Design a Gaussian low pass filter with cutoff frequency of radius (sigma) Do, so that in the filtered image the large letter ‘a’ is barely readable, the other letters are not. You may need to try different Uo, to find the best settings. Please compute mean square error (MSE) between the original image and resulted blurred image for five Uo settings, sort them and explain why MSE decreases along with the increase of Do.

c- Design a Fifth-order Butterworth high pass filter, use circular cutoff, Do, so that only edges of shapes are clearly shown. You may need to try different Uo, to find the best settings. Please compute mean square error (MSE) between the original image and resulted image for five Uo settings, sort them and explain why MSE increases along with the increase of Do.

Answer 1

% Demo to filter an image, with periodic ripple, in the Fourier domain.
clc; % Clear the command window.
close all; % Close all figures (except those of imtool.)
clear; % Erase all existing variables. Or clearvars if you want.
workspace; % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 14;

% Read in a standard MATLAB gray scale demo image.
folder = fullfile(matlabroot, '\toolbox\images\imdemos');
baseFileName = 'cameraman.tif';
% Get the full filename, with path prepended.
fullFileName = fullfile(folder, baseFileName);
% Check if file exists.
if ~exist(fullFileName, 'file')
% File doesn't exist -- didn't find it there. Check the search path for it.
fullFileName = baseFileName; % No path this time.
if ~exist(fullFileName, 'file')
% Still didn't find it. Alert user.
errorMessage = sprintf('Error: %s does not exist in the search path folders.', fullFileName);
uiwait(warndlg(errorMessage));
return;
end
end
grayImage = imread(fullFileName);
[rows columns numberOfColorBands] = size(grayImage);
if numberOfColorBands > 1
grayImage = rgb2gray(grayImage);
end
subplot(2, 3, 1);
imshow(grayImage, [0 255]);
set(gcf, 'Name', ['Results for ' fullFileName]);
title('Original Image', 'FontSize', fontSize);
set(gcf, 'units','normalized','outerposition',[0 0 1 1]); % Maximize figure.

% Add big ripples to it.
rowVector = (1 : rows)';
period = 10; % 20 rows
amplitude = 0.5; % Magnitude of the ripples.
offset = 1 - amplitude; % How much the cosine is raised above 0.
cosVector = amplitude * (1 + cos(2 * pi * rowVector / period))/2 + offset;
ripplesImage = repmat(cosVector, [1, columns]);
subplot(2, 3, 2);
minValue = min(min(ripplesImage))
maxValue = max(max(ripplesImage))
imshow(ripplesImage, [0 maxValue]);
axis on;
title('Ripples to multiply the image by', 'FontSize', fontSize);
% Multiply the ripples by the image to get an image with periodic "noise" in it.
grayImage = ripplesImage .* double(grayImage);
minValue = min(min(grayImage))
maxValue = max(max(grayImage))
subplot(2, 3, 3);
imshow(grayImage, [0 255]);
axis on;
title('Original Image with Periodic "Noise" ripples', 'FontSize', fontSize);

% Compute the 2D fft.
frequencyImage = fftshift(fft2(grayImage));
% Take log magnitude so we can see it better in the display.
amplitudeImage = log(abs(frequencyImage));
minValue = min(min(amplitudeImage))
maxValue = max(max(amplitudeImage))
subplot(2, 3, 4);
imshow(amplitudeImage, []);
caption = sprintf('Notice the two spikes\nperpendicular to the periodic frequency');
title(caption, 'FontSize', fontSize);
axis on;
% zoom(10)

% Find the location of the big spikes.
amplitudeThreshold = 10.9;
brightSpikes = amplitudeImage > amplitudeThreshold; % Binary image.
subplot(2, 3, 5);
imshow(brightSpikes);
axis on;
title('Bright Spikes', 'FontSize', fontSize);
% Let user see the image.
promptMessage = sprintf('The image below shows the bright spikes.\nNow we will exclude the central spike.');
titleBarCaption = 'Continue?';
button = questdlg(promptMessage, titleBarCaption, 'Continue', 'Cancel', 'Continue');
if strcmpi(button, 'Cancel')
return;
end
% Exclude the central DC spike. Everything from row 115 to 143.
brightSpikes(115:143, :) = 0;
imshow(brightSpikes);
title('Bright spikes other than central spike', 'FontSize', fontSize);

promptMessage = sprintf('Now we will use these bright spikes to filter (mask) the spectrum.');
button = questdlg(promptMessage, titleBarCaption, 'Continue', 'Cancel', 'Continue');
if strcmpi(button, 'Cancel')
return;
end
% Filter/mask the spectrum.
frequencyImage(brightSpikes) = 0;
% Take log magnitude so we can see it better in the display.
amplitudeImage2 = log(abs(frequencyImage));
minValue = min(min(amplitudeImage2))
maxValue = max(max(amplitudeImage2))
subplot(2, 3, 5);
imshow(amplitudeImage2, [minValue maxValue]);
axis on;
title('Spikes zeroed out', 'FontSize', fontSize);
% zoom(10)

filteredImage = ifft2(fftshift(frequencyImage));
amplitudeImage3 = abs(filteredImage);
minValue = min(min(amplitudeImage3))
maxValue = max(max(amplitudeImage3))
subplot(2, 3, 6);
imshow(amplitudeImage3, [minValue maxValue]);
title('Filtered Image', 'FontSize', fontSize);
% set(gcf, 'units','normalized','outerposition',[0 0 1 1]); % Maximize figure.
% %