Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.