Deblurring an image using the heat equation

So I am trying to deblur an image using the heat equation, but when I run the code, I get a black image. So I did this by looking at http://www4.ncsu.edu/eos/users/w/white/www/white/ma325/MVlec3.pdf to write the code:

u0=double(imread('clockmodified.tiff'));

[m,n,k] = size(u0); %k=conductivity
%if k==3
    %u0 = rgb2gray(u0); 
%end;
u0 = double(u0); 
subplot(1,2,1); 
imagesc(u0); 
colormap gray; 
title('Original');

dt = .02; %time step
T = 1000; %stopping time
u = u0;

for t = 0:dt:T
    u_xx = u(:,[2:n n]) - 2*u + u(:,[1 1:n-1]); 
    u_yy = u([2:m m], :) - 2*u + u([1 1:m-1],:); 
    u = u - k*dt*(u_xx+u_yy);
subplot(1,2,2);
imagesc(u);
title(['t=',num2str(t)]);
drawnow;
temp=u;
end;
% imshow(uint8(temp));
imwrite(uint8(temp),'clockdeblur.tiff','TIFF');

Answer 1

[Vidhi Agarwal]

Hi @cee878,

The issue you're experiencing with getting a black image could be due to several factors in implementation of the heat equation for image deblurring.

1. Conductivity Coefficient k: In given code, “k” is being assigned the number of colour channels of the image. This is incorrect for image deblurring. Define “k” as a small positive constant that controls the rate of diffusion (e.g., k = 0.1).
2. Ensure that the pixel values remain within a valid range (0 to 255) during the iterations. If the values exceed this range, they might cause the image to appear completely black or white.
3. When using “imagesc”, the scaling might affect the display. Ensure that the image data is correctly scaled for visualization.

Revised code with above mentioned consideration will be:

% Read and preprocess the image 
u0=imread(‘path_to_your_image.jpeg'); 

% Convert to grayscale if the image is in color 
if size(u0, 3) == 3 
    u0 = rgb2gray(u0); 
end  

% Convert to double for processing 
u0 = double(u0); 

% Get the dimensions of the image 
[m, n] = size(u0); 

% Display the original image 
figure; 
subplot(1,2,1); 
imagesc(u0); 
colormap gray; 
title('Original'); 

% Parameters for the heat equation 
dt = 0.02; % Time step 
T = 100; % Stopping time 
k = 0.1; % Conductivity coefficient a small positive constant 

% Initialize the image for processing 
u = u0; 

% Iterative deblurring using the heat equation 
for t = 0:dt:T 
    % Compute second derivatives  
    u_xx = u(:, [2:n n]) - 2*u + u(:, [1 1:n-1]); 
    u_yy = u([2:m m], :) - 2*u + u([1 1:m-1], :); 
    % Update the image 
    u = u + k*dt*(u_xx + u_yy); 
    % Clip values to ensure they are within valid range 
    u = max(min(u, 255), 0);  
    % Display the updated image 
    subplot(1,2,2); 
    imagesc(u, [0 255]); % Ensure the display range is set 
    colormap gray; 
    title(['t = ', num2str(t)]); 
    drawnow; 
end 

% Save the final deblurred image 
imwrite(uint8(u), 'clockdeblur.tiff', 'TIFF'); 

The output for the following code with sample input is given below:

For better understanding of “imagesc”, refer to the documentation, which you can access by running the following command in MATLAB command window:

• web(fullfile(docroot, "matlab/ref/imagesc.html"))

Hope that helps!