Image Block Mapping Lemma and Residual Refinement-Enhanced AutoBCS Framework for Block-based Compressive Sensing

This paper presents Block-Based Compressive Sensing (BCS) of images using both traditional and learning-based approaches. An Image Block Mapping Lemma, formulated as a modified version of the Johnson-Lindenstrauss Lemma for image data, is proposed in this work, which justifies distance preservation between image blocks during mapping from the original domain to the compressed domain in block-based compressive sensing. The paper provides both theoretical proof and experimental validation of the proposed lemma. To quantitatively analyse distance preservation between image blocks in the original and compressed domains during block-based compressive sensing using traditional approaches, a new metric termed Distance Ratio (DR) is introduced. The difficulty of obtaining accurate real-time reconstruction using conventional block-based compressive sensing methods has encouraged the transition toward adaptive deep learning approaches. For learning-based sensing and reconstruction, the existing AutoBCS framework is enhanced by introducing an Initial Reconstruction Refinement Network (IRR-Net) between the initial and final reconstruction stages. Using the proposed model, improved reconstruction quality is achieved, with average PSNR gains of 0.69-1.62 dB and SSIM improvements of 0.01-0.02 for sampling rates of 0.30, 0.25, 0.10, and 0.04 across multiple benchmark datasets compared with the baseline architecture. The proposed model introduces an additional 0.05 million parameters and increases the computational cost by 3.64 GFLOPs due to the residual refinement module, while reducing the reconstruction time compared with the baseline AutoBCS framework. Experimental results demonstrate that the proposed model exhibits improved preservation of complex structures, edges, and fine details, particularly for images containing rich textures, dense structures, and high-frequency content.