Suboptimality of Nonlocal Means for Images with Sharp Edges

Details Suboptimality of Nonlocal Means for Images with Sharp Edges Suboptimality of Nonlocal Means for Images with Sharp Edges

2011-11-24

arxiv.org/abs/1111.5867v1

Mathematics

Statistics Theory

33 pages, 3 figures

Scientific paper

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 $n^{-4/3}$.

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