## 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}log1 ^{/2+ε}n, for an n×n-pixel image with ε>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 optimal minimax rate of n- ^{4/3}.

Original language | English (US) |
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Pages (from-to) | 370-387 |

Number of pages | 18 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 33 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2012 |

## Keywords

- Denoising
- Horizon class
- Linear filter
- Minimax risk
- Nonlocal means
- SUSAN filter
- Wavelet thresholding

## ASJC Scopus subject areas

- Applied Mathematics