MATLAB Implementation of PCA Algorithm with Projection Visualization

In this document, we present a comprehensive implementation of Principal Component Analysis (PCA) using MATLAB, with detailed explanations on computing image projections onto the first three principal components. PCA is a fundamental dimensionality reduction technique widely used for data compression, feature extraction, and pattern recognition. Using image processing as our application example, we demonstrate PCA's practical implementation and interpretation. First, let's understand the mathematical foundation of PCA. The algorithm's core objective is to transform original data into a new coordinate system where the maximum variance directions become the principal components. These components represent orthogonal directions capturing the greatest data variability, allowing us to preserve essential features while reducing dimensionality by retaining only the most significant components. Our MATLAB implementation follows a structured pipeline. We begin by loading the image and converting it into a matrix format using imread() and im2double() functions. The data then undergoes mean-centering - a crucial preprocessing step where we subtract the mean vector from each data point using mean() and bsxfun() operations. This ensures the data is centered around the origin for proper covariance calculation. The computational core involves calculating the covariance matrix through cov() function or matrix multiplication, followed by eigenvalue decomposition using eig() or svd() functions. The resulting eigenvectors form our new principal component basis, while eigenvalues indicate their relative importance in explaining data variance. We sort components in descending eigenvalue order and select the top k components for dimensionality reduction. Finally, we project the centered data onto the principal component subspace through matrix multiplication (X_centered * eigenvectors), generating reconstructed images using inverse transformation. The projection results are visualized using subplot() and imshow() functions, demonstrating how each principal component contributes to feature representation. This implementation provides practical insights into PCA's mechanism while building proficiency in MATLAB's numerical computing capabilities for multivariate analysis.