Regularized Orthogonal Matching Pursuit (ROMP) - A Compressive Sensing Reconstruction Algorithm

Regularized Orthogonal Matching Pursuit (ROMP) is an efficient signal reconstruction algorithm in the field of compressive sensing. This algorithm primarily reconstructs sparse or compressible signals from a small number of linear measurements, addressing the high sampling rate requirements of traditional sampling theorems. The implementation typically involves iterative support set construction with regularization constraints for improved stability.

The core concept of ROMP involves progressively building the signal's support set through an iterative process. Unlike traditional matching pursuit algorithms, ROMP selects multiple atoms satisfying regularization conditions in each iteration, providing better stability and convergence. The algorithm workflow consists of several key steps that can be implemented through systematic matrix operations:

The initialization phase begins with the residual signal, where the algorithm computes correlations between all atoms and the current residual. In code implementation, this involves calculating the inner products between the measurement matrix columns and the residual vector. The candidate set selection phase then filters a group of atoms with similar correlations based on regularization conditions - this step effectively addresses the instability issues of single-atom selection in traditional algorithms through proper thresholding mechanisms.

The support set expansion phase merges the selected candidate atoms into the current support set. The signal estimate is then updated using least squares minimization, typically implemented through pseudoinverse calculations (e.g., using pinv() in MATLAB or lstsq() in Python). The residual is recalculated based on the updated estimate. This iterative process continues until meeting preset stopping conditions, such as reaching the specified sparsity level or achieving a sufficiently small residual error threshold.

ROMP's advantage lies in its balance between theoretical guarantees and practical performance. Compared to basic Orthogonal Matching Pursuit (OMP), ROMP better handles measurement noise and provides more stable reconstruction results. The computational complexity remains relatively moderate, making it suitable for medium-scale problems. This algorithm finds important applications in medical imaging, wireless communications, and radar systems, where efficient signal recovery from limited measurements is crucial.