Comparison of Detection Probability Among Various Compressed Sensing Algorithms

Compressed sensing is a widely used technique in signal processing that reconstructs original signals from sampling data collected at rates significantly below the Nyquist rate. In compressed sensing, signal reconstruction algorithms play a critical role, and detection probability serves as one of the key metrics for evaluating algorithm performance. Among common algorithms (M-OMP, MP, OMP), their detection probability characteristics vary significantly.

MP (Matching Pursuit) MP is a fundamental greedy algorithm that reconstructs signals by iteratively selecting atoms most correlated with the residual. Its primary advantages include low computational complexity and straightforward implementation. However, since MP selects only one atom per iteration without backward optimization, its detection probability remains relatively low. Under conditions of low signal-to-noise ratio or high signal sparsity, the reconstruction performance may be suboptimal. In code implementation, MP typically involves calculating inner products between the residual and dictionary atoms, then updating the residual by subtracting the contribution of the selected atom.

OMP (Orthogonal Matching Pursuit) OMP is an improved version of MP that incorporates orthogonalization of selected atoms after each iteration, thereby enhancing reconstruction accuracy. Compared to MP, OMP achieves higher detection probability, particularly with moderately sparse signals. The trade-off is slightly higher computational complexity due to the orthogonalization process required in each iteration. Algorithm implementation typically involves maintaining an orthogonal basis for selected atoms using Gram-Schmidt orthogonalization or QR decomposition, ensuring that subsequent selections are orthogonal to previously chosen atoms.

M-OMP (Modified OMP) M-OMP represents further optimization of OMP, often incorporating threshold adjustment mechanisms or adaptive strategies to improve detection probability in high-noise or high-sparsity environments. Enhancements may include dynamic adjustment of atom selection criteria or optimized residual update mechanisms, enabling robust reconstruction performance under challenging conditions. Code implementation might feature adaptive thresholding based on noise estimation or modified stopping criteria that consider both residual energy and sparsity constraints.

Detection Probability Comparison Overall, MP demonstrates the lowest detection probability, making it suitable for scenarios with limited computational resources where precision requirements are moderate. OMP performs well in most cases and is ideal for reconstructing signals with medium sparsity levels. M-OMP excels under conditions of high noise or elevated sparsity, though it may incur greater computational overhead. When selecting an algorithm, practitioners must balance detection probability requirements against computational costs based on specific application scenarios.