Display Fisherface after reducing dimensions using PCA

Hi,

I'm learning to implement the Fisherface algorithm using Matlab. I understand that to deal with the singularity issue of Sw, I can project Sb and Sw onto the PCA space of r dimensions, with r ≤ rank(Sw) and compute the Fisherfaces in this PCA space. If I set r = 1000, what are my Fisherfaces' dimensions, and how do I display them as images? Do I need to reconstruct them back to their original dimensions? If so, how do I do this?

Thank you.

Answer 1

[Aditya]

Hi Rayne,

The Fisherface algorithm involves two main steps: applying Principal Component Analysis (PCA) to reduce dimensionality and then performing Linear Discriminant Analysis (LDA) on the reduced data to find the Fisherfaces. Here's a step-by-step guide to implementing this in MATLAB, along with how to handle the dimensions and display the Fisherfaces.

% Assume X is your data matrix with each column as a flattened image
% labels is a vector of class labels

% Step 1: Compute PCA
[coeff, score, ~, ~, explained] = pca(X');

% Choose r dimensions based on cumulative variance explained or a fixed number
r = 1000; % or choose based on explained variance
pcaData = score(:, 1:r)'; % Transpose to have each column as a data point

% Step 2: Compute LDA
% Compute the within-class and between-class scatter matrices in the PCA space
uniqueClasses = unique(labels);
numClasses = numel(uniqueClasses);
meanTotal = mean(pcaData, 2);

Sw = zeros(r, r);
Sb = zeros(r, r);

for i = 1:numClasses
    classData = pcaData(:, labels == uniqueClasses(i));
    meanClass = mean(classData, 2);
    Sw = Sw + cov(classData');
    Sb = Sb + size(classData, 2) * (meanClass - meanTotal) * (meanClass - meanTotal)';
end

% Solve the generalized eigenvalue problem for LDA
[eigVecs, eigVals] = eig(Sb, Sw);

% Sort eigenvectors by eigenvalues in descending order
[~, idx] = sort(diag(eigVals), 'descend');
fisherfaces = eigVecs(:, idx);

% Step 3: Display Fisherfaces
% Fisherfaces are in the PCA space, reconstruct them to original dimensions
fisherfacesOriginal = coeff(:, 1:r) * fisherfaces;

% Choose how many Fisherfaces to display
numFisherfacesToDisplay = 5;
figure;
for i = 1:numFisherfacesToDisplay
    % Reshape each Fisherface back to image dimensions
    fisherfaceImage = reshape(fisherfacesOriginal(:, i), [imageHeight, imageWidth]);
    subplot(1, numFisherfacesToDisplay, i);
    imshow(mat2gray(fisherfaceImage)); % Normalize and display
    title(['Fisherface ', num2str(i)]);
end

This implementation assumes your images are vectorized and that you know their original dimensions (imageHeight and imageWidth). Adjust the number of Fisherfaces to display based on your needs.