MATLAB Code Implementation in Compressed Sensing Domain

The Subspace Pursuit method is an efficient reconstruction algorithm in the compressed sensing field, representing a significant improvement over the Orthogonal Matching Pursuit (OMP) method, particularly suitable for precise recovery of sparse signals. Similar to OMP, Subspace Pursuit iteratively selects the most correlated atoms to approximate the original signal, but it dynamically adjusts the selected support set during each iteration, thereby enhancing reconstruction accuracy and convergence speed.

The core concept of this algorithm lies in its ability to not only select atoms sequentially but also to backtrack and optimize the chosen atom set during the iteration process. Specifically, in each iteration, Subspace Pursuit calculates the inner products between the current residual and each column of the measurement matrix, selects several atoms with the highest correlation, and recalculates the optimal linear combination to minimize the residual. From a code implementation perspective, this involves computing correlations using matrix-vector multiplications, maintaining an active set of indices, and solving least-squares problems using MATLAB's backslash operator or pinv() function for pseudoinverse calculations. This strategy avoids local optimum issues that OMP might encounter and provides better approximation to the true sparse signal.

Compared to OMP, Subspace Pursuit has slightly higher computational complexity but offers superior reconstruction accuracy, especially under conditions of low signal-to-noise ratio or limited measurement samples. The algorithm typically implements an iterative loop that includes correlation computation, candidate set expansion, pruning, and projection steps. This method finds widespread applications in radar imaging, medical image reconstruction, and other fields, serving as an important practical algorithm in compressed sensing theory. MATLAB implementations often leverage built-in functions for matrix operations and include parameters for controlling the sparsity level and convergence criteria.