A Dimension-Keeping Semi-Tensor Product Framework for Compressed Sensing
Qi Qi, Abdelhamid Tayebi, Daizhan Cheng, Jun-e Feng
TL;DR
The paper addresses sub-Nyquist image reconstruction by introducing the dimension-keeping semi-tensor product (DK-STP) to design sensing matrices that preserve dimensionality while exploiting intra-group correlations. The proposed DK-STP-CS framework links the recovered signal to grouped sums via $y = (1/\sqrt{\gamma}) A x^\gamma$, and provides theoretical guarantees through spark, coherence, and RIP analyses along with an error decomposition into original, CS, and distribution components. Empirical results on standard 256×256 images show consistent PSNR gains (e.g., up to several dB) over conventional CS and STP-CS across varying sampling rates and noise levels, with memory efficiency and faster convergence highlighted. The method offers practical benefits for resource-constrained imaging, guiding the design of adaptive, dimensionality-preserving measurement matrices for robust reconstruction.
Abstract
In compressed sensing (CS), sparse signals can be reconstructed from significantly fewer samples than required by the Nyquist-Shannon sampling theorem. While non-sparse signals can be sparsely represented in appropriate transformation domains, conventional CS frameworks rely on the incoherence of the measurement matrix columns to guarantee reconstruction performance. This paper proposes a novel method termed Dimension-Keeping Semi-Tensor Product Compressed Sensing (DK-STP-CS), which leverages intra-group correlations while maintaining inter-group incoherence to enhance the measurement matrix design. Specifically, the DK-STP algorithm is integrated into the design of the sensing matrix, enabling dimensionality reduction while preserving signal recovery capability. For image compression and reconstruction tasks, the proposed method achieves notable noise suppression and improves visual fidelity. Experimental results demonstrate that DK-STP-CS significantly outperforms traditional CS and STP-CS approaches, as evidenced by higher Peak Signal-to-Noise Ratio (PSNR) values between the reconstructed and original images. The robustness of DK-STP-CS is further validated under noisy conditions and varying sampling rates, highlighting its potential for practical applications in resource-constrained environments.