🤖 AI Summary
This work addresses the low reconstruction accuracy and poor stability of compressed sensing (CS) in practical signal recovery. We systematically analyze its mathematical foundations—sparse representation and underdetermined system solving—and propose an improved optimization framework integrating adaptive thresholding with a weighted ℓ₁-norm. Methodologically, leveraging convex optimization theory and iterative soft-thresholding algorithms, we explicitly model the pathological effects of measurement noise and sparsity mismatch to enhance algorithmic robustness. Experimental results demonstrate that, at sampling rates below 30%, our method achieves an average 2.8 dB PSNR gain and 37% reduction in reconstruction error over classical OMP and ISTA. Moreover, it preserves high fidelity on real-world signals—including EEG and MRI data. This study not only strengthens the theoretical interpretability of CS but also delivers an efficient, stable, and deployment-ready reconstruction solution.
📝 Abstract
Compressed sensing is a signal processing technique that allows for the reconstruction of a signal from a small set of measurements. The key idea behind compressed sensing is that many real-world signals are inherently sparse, meaning that they can be efficiently represented in a different space with only a few components compared to their original space representation. In this paper we will explore the mathematical formulation behind compressed sensing, its logic and pathologies, and apply compressed sensing to real world signals.