The Optimal Weights of Non-local Means for Variance Stabilized Noise Removal

The Non-Local Means (NLM) algorithm is a fundamental denoising technique widely utilized in various domains of image processing. However, further research is essential to gain a comprehensive understanding of its capabilities and limitations. This includes determining the types of noise it can effectively remove, choosing an appropriate kernel, and assessing its convergence behavior. In this study, we optimize the NLM algorithm for all variations of independent and identically distributed (i.i.d.) variance-stabilized noise and conduct a thorough examination of its convergence behavior. We introduce the concept of the optimal oracle NLM, which minimizes the upper bound of pointwise \(L_{1}\) $L_1$ or \(L_{2}\) $L_2$ risk. We demonstrate that the optimal oracle weights comprise triangular kernels with point-adaptive bandwidth, contrasting with the commonly used Gaussian kernel, which has a fixed bandwidth. The computable optimal weighted NLM is derived from this oracle filter by replacing the similarity function with an estimator based on the similarity patch. We present theorems demonstrating that both the oracle filter and the computable filter achieve optimal convergence rates under minimal regularity conditions. Finally, we conduct numerical experiments to validate the performance, accuracy, and convergence of \(L_{1}\) $L_1$ and \(L_{2}\) $L_2$ risk minimization for NLM. These convergence theorems provide a theoretical foundation for further advancing the study of the NLM algorithm and its practical applications.