Image Reconstruction in Compressive Sensing Using Daubechies 7 (db7) and Lifting Wavelet Transforms with SP, CoSaMP, and ALISTA Algorithms

This paper proposes an efficient image reconstruction method in compressive sensing (CS) that combines the Lifting Wavelet Transform (LWT) with the Daubechies 7 (db7) wavelet and three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). Unlike the conventional Discrete Wavelet Transform (DWT), whose implementation relies on computationally expensive convolution operations, the LWT enables a faster sparse representation while preserving the sparsity properties essential for reconstruction. The proposed methodology is based on a key observation: among the four subbands generated by the LWT: approximation (CA) and detail coefficients (LH, HL, HH) only the latter three are inherently sparse. Consequently, compression is applied exclusively to these detail components, while the approximation subband is kept intact, thereby preserving critical low-frequency information. Experiments were conducted on two types of images a natural image (Lena) and a medical image (MRI) across various resolutions (from 200×200 to 512×512 pixels) and sampling rates (from 10% to 80%). Performance was evaluated using the Structural Similarity Index (SSIM) and reconstruction time. Results consistently show that ALISTA significantly outperforms SP and CoSaMP in both reconstruction quality and speed. At 80% sampling, ALISTA achieves an SSIM of 0.9936 for Lena and 0.9764 for MRI, compared to approximately 0.975 and 0.934 for the other methods, respectively. Moreover, ALISTA maintains extremely low reconstruction times under 4 seconds even for 512×512-pixel images. This research confirm that the ALISTA + LWT/db7 combination achieves the best quality–speed trade-off and exhibits robustness regardless of image type or size.