On the Convergence of the IRLS Algorithm in Non-Local Patch Regression

Recently, it was demonstrated in , that the robustness of the classical Non-Local Means (NLM) algorithm can be improved by incorporating
lp
(0 <;
p
≤ 2) regression into the NLM framework. This general optimization framework, called Non-Local Patch Regression (NLPR), contains NLM as a special case. Denoising results on synthetic and natural images show that NLPR consistently performs better than NLM beyond a moderate noise level, and significantly so when
p
is close to zero. An iteratively reweighted least-squares (IRLS) algorithm was proposed for solving the regression problem in NLPR, where the NLM output was used to initialize the iterations. Based on exhaustive numerical experiments, we observe that the IRLS algorithm is globally convergent (for arbitrary initialization) in the convex regime 1 ≤
p
≤ 2, and locally convergent (e.g., fails rarely using NLM initialization) in the non-convex regime 0 <;
p
<; 1. In this letter, we study the cost associated with the IRLS updates, and this, along with the framework of bounded optimization, is used to analyze the convergence of the algorithm.