how to represent an image with the autoregressive model

how to represent an image with the autoregressive model matlab code

Answer 1

[Zuber Khan]

Hi,

Firstly, it's essential to grasp the concept of an Autoregressive Model.

Many observed time series exhibit serial autocorrelation; that is, linear association between lagged observations. This suggests past observations might predict current observations. The autoregressive (AR) process models the conditional mean of 'yt' as a function of past observations, 'yt−1','yt−2',…,'yt−p'. An AR process that depends on 'p' past observations is called an AR model of degree 'p', denoted by AR(p).

Autoregressive Image Modeling (AIM) uses the same approach but on image pixels as data points. The approach divides the image into segments and treats the segments as a sequence of data points. The model learns to predict the next image segment given the previous data point. Mathematically, AR models treat an image as a sequence of pixels and represent its probability as the product of the conditional probabilities of all pixels.

Now, in order to apply Autoregression on an image, you can follow the steps as I have illustrated in an example code snippet below:

• Load image.

image = imread('cameraman.tif'); % Load a sample image
image = double(image); 

• Define AR parameters and estimate AR coefficients, and use it to construct a prediction image.

% Define AR model parameters
order = 1; % Order of the AR model
[rows, cols] = size(image);
predicted_image = zeros(rows, cols);

% Simple 2D AR model with a 3x3 neighborhood
neighborhood_size = 3;
half_window = floor(neighborhood_size / 2);

% Loop through each pixel in the image
for i = (1 + half_window):(rows - half_window)
    for j = (1 + half_window):(cols - half_window)
        % Extract the neighborhood
        neighborhood = image((i-half_window):(i+half_window), (j-half_window):(j+half_window));
        neighborhood_vector = neighborhood(:); 
        
        X = neighborhood_vector([1:4, 6:9]); % Exclude center pixel 
        y = neighborhood_vector(5); % Center pixel
        
        % Estimate AR coefficients (using pseudo-inverse)
        coeff = pinv(X) * y;
        
        % Predict the center pixel
        predicted_pixel = X' * coeff';
        predicted_image(i, j) = predicted_pixel;
    end
end

• Visualize and compare the original image with reconstructed image.

% Calculate prediction error
error_image = image - predicted_image;

% Display original and predicted images
figure;
subplot(1, 3, 1);
imshow(uint8(image));
title('Original Image');

subplot(1, 3, 2);
imshow(uint8(predicted_image));
title('Predicted Image');

subplot(1, 3, 3);
imshow(uint8(error_image));
title('Prediction Error');

I hope this will address your query.

Regards,

Zuber