Image Compression Using Matrix Singular Value Decomposition

The process of image compression using matrix singular value decomposition is implemented as follows: 1. First, partition the image by dividing it into multiple small blocks. In code implementation, this typically involves using block processing functions like `mat2cell()` in MATLAB to create uniform sub-image matrices. 2. For each image block, compute the inner products of corresponding pixels to generate correlation matrix A. This can be computationally achieved through vectorization operations where pixel values are treated as vectors, and their dot products form the correlation matrix elements. 3. Perform singular value decomposition (SVD) on matrix A to obtain eigenvalues and eigenvectors. Using mathematical libraries like NumPy or MATLAB's `svd()` function, this decomposition yields three matrices: U, S, and V, where S contains the singular values (square roots of eigenvalues). 4. Select n eigenvectors corresponding to the n largest eigenvalues, where n is smaller than the total number of eigenvalues N. This threshold selection can be implemented by sorting singular values in descending order and choosing the top n values, often controlled by a compression ratio parameter. 5. Utilize the selected eigenvectors to compute inner products with each image block, enabling both compression and reconstruction. The compression is achieved by storing only the projected coefficients, while reconstruction involves multiplying these coefficients with the eigenvectors to approximate the original blocks. This approach significantly reduces storage requirements while maintaining image quality through optimal linear approximations.