Implementing Matching Pursuit Algorithm

Matching Pursuit (MP) algorithm is an iterative method for sparse signal representation that greedily selects the best-matching atoms from an overcomplete dictionary to gradually approximate the target signal. This algorithm has broad applications in signal processing, image analysis, and pattern recognition domains. When implementing Matching Pursuit in MATLAB, Gabor atoms are commonly used as dictionary elements due to their excellent time-frequency localization properties, which effectively capture local signal characteristics. The core implementation follows this algorithmic flow: Initialization Phase: First construct an overcomplete Gabor dictionary and initialize the residual signal as the original input signal. In MATLAB implementation, this typically involves generating Gabor atoms with varying time-frequency parameters using vectorized operations for efficiency. Atom Selection Phase: During each iteration, the algorithm computes inner products between all Gabor atoms and the current residual signal, selecting the atom with maximum inner product as the best match. The MATLAB implementation can optimize this computationally intensive step using matrix multiplication (e.g., residual_signals' * dictionary_matrix) rather than loop-based calculations. Update Phase: Subtract the contribution of the selected best-matching atom from the residual signal. This update is mathematically expressed as residual = residual - (atom' * residual) * atom, where the projection coefficient is calculated through inner product operations. Termination Condition: The iteration stops when the residual becomes sufficiently small or reaches a predefined number of iterations. In practical MATLAB implementation, common termination criteria include residual energy thresholds (e.g., norm(residual) < tolerance) or maximum iteration counts. The MATLAB implementation benefits significantly from matrix operations optimization, particularly in the inner product calculations. Gabor atom parameters (center position, scale, frequency) can be flexibly adjusted according to signal characteristics using parameter sweeping techniques. Key implementation considerations include efficient dictionary storage using sparse matrices and optimized atom selection through vectorized comparisons. The algorithm's strength lies in its straightforward implementation approach combined with Gabor atoms' ability to effectively capture local time-frequency features. This combination performs exceptionally well in non-stationary signal analysis and image processing applications, with MATLAB providing built-in functions for visualization and performance validation of the reconstruction results.