Suboptimality of Nonlocal Means for Images with Sharp Edges

TL;DR

This paper analyzes the nonlocal means algorithm for image denoising with sharp edges, showing it performs better than some traditional filters but still falls short of the optimal minimax rate.

Contribution

It provides an asymptotic risk analysis of nonlocal means for piecewise constant images, quantifying its decay rate and comparing it to other methods and the optimal rate.

Findings

Risk decays as $n^{-1} ext{log}^{1/2+ ext{epsilon}} n$ with optimal tuning.

Outperforms linear convolution, median, and SUSAN filters.

Still below the minimax optimal rate of $n^{-4/3}$.

Abstract

We conduct an asymptotic risk analysis of the nonlocal means image denoising algorithm for the Horizon class of images that are piecewise constant with a sharp edge discontinuity. We prove that the mean square risk of an optimally tuned nonlocal means algorithm decays according to $n^{-1}\log^{1/2+\epsilon} n$, for an $n$-pixel image with $\epsilon>0$. This decay rate is an improvement over some of the predecessors of this algorithm, including the linear convolution filter, median filter, and the SUSAN filter, each of which provides a rate of only $n^{-2/3}$. It is also within a logarithmic factor from optimally tuned wavelet thresholding. However, it is still substantially lower than the the optimal minimax rate of $n^{-4/3}$.