Mathematical Modeling HW 2 —
A Progressive RPCA Framework for Image Restoration:
From Basic Decomposition to Masked Completion

This paper studies image restoration under a progressive implementation of Robust Principal Component Analysis (RPCA). Five versions are developed in sequence. The Basic version establishes a grayscale low-rank and sparse decomposition pipeline. The Color version extends the same decomposition logic to red-green-blue (RGB) images by processing three channels separately. The GUI_advanced version does not alter the optimization model, but improves the experimental interface through background computation, progress display, interactive inspection, and result export. The TV_Regularization version introduces total variation (TV) regularization on the low-rank component so that local oscillation can be suppressed while large-scale structure is retained. The Masked version further modifies the fidelity constraint by excluding manually selected damaged regions from direct fitting and completing them from the remaining valid observations.