Image Compression and Reconstruction Using PCA Method in MATLAB

In this project, we will utilize Principal Component Analysis (PCA) method in MATLAB to compress and reconstruct images. PCA is a widely-used dimensionality reduction technique that transforms high-dimensional data into lower-dimensional representations. In image processing applications, PCA is extensively employed for image compression. The fundamental principle involves performing linear transformations on image pixel values to eliminate redundant information, thereby achieving image compression. Additionally, PCA finds applications in image denoising and enhancement, making it an essential component in image processing workflows. The implementation typically involves these key steps: First, the image matrix is normalized by subtracting the mean from each pixel value. Then, we compute the covariance matrix and perform eigenvalue decomposition to identify principal components. The core MATLAB functions used include 'mean()' for centralization, 'cov()' for covariance calculation, and 'eig()' for eigenvalue decomposition. By retaining only the top k eigenvectors corresponding to the largest eigenvalues, we create a projection matrix for dimensionality reduction. The compression ratio can be controlled by adjusting the number of principal components preserved. For reconstruction, we reverse the process by projecting the compressed data back to the original space using the transpose of the projection matrix and adding the mean back. This approach effectively demonstrates how PCA balances compression efficiency with reconstruction quality. We will provide detailed explanations of PCA's mathematical foundations and its practical implementation in image processing, offering comprehensive insights into this powerful technique.