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1. Introduction
Wavelet shrinkage and nonlinear diffusion are two seemingly very different concepts for discontinuity-preserving signal and image denoising. Since they are serving the same purpose, however, it would be desirable to understand if there are intrinsic connections between both worlds. On one hand this may allow to transfer results from one framework to the other, on the other hand it may allow to design hybrid method that combine advantages from both concepts. Although research in this direction is still a relatively young field, already a number of interesting connections between wavelet shrinkage, partial differential equations (PDEs) and related regularisation methods has been esta-blished. Most of them analyse the continuous framework [1–7] or focus on designing methods that use wavelet shrinkage and PDE-based denoising methods in combi-nation [8–13] . Regarding the relations between wavelet shrinkage of discrete signals and PDE-based denoising, not much research has been done so far. One notable exception is a recent paper by Coifman and Sowa [14] where they propose total variation (TV) diminishing flows that act along the direction of Haar wavelets. Bao and Krim [15] addressed the problem of texture loss in diffusion scale-spaces by incorporating ideas from wavelet analysis. An experimental evaluation of the denoising capabilities of 3-D wavelet shrinkage and nonlinear diffusion filters is presented in a paper by Frangakis et al. [16] .

1 引言

小波收缩与非线性扩散是两个看似非常不同的概念的不连续保持信号和图像去噪。因为他们都是服务于同一目的，然而，这将是可取的，了解是否有两个世界之间的内在联系。一方面，这可以允许的结果从一个框架转移到其他的，另一方面它可能允许设计的混合方法，结合两者的优点的概念。虽然在这个方向的研究仍然是一个相对年轻的领域，已经有许多的小波收缩之间有趣的联系，偏微分方程（PDE）和相关的正则化方法已经建立了。他们中的大多数分析框架[ 7 ]连续1–或专注于设计方法，使用小波变换和基于偏微分方程的结合–[ 8 ] 13的去噪方法。对于关系的小波收缩去噪和基于偏微分方程的离散信号，没有太多的研究已经完成，到目前为止。一个值得注意的例外是由Coifman和索瓦[ 14 ]，他们提出了总变异的最近的一篇文章（电视）减少流动行为在Haar小波的方向。宝和克里姆[ 15 ]解决纹理丢失问题在扩散尺度空间将想法从小波分析。的去噪能力的三维小波收缩与非线性扩散过滤器的实验评价是通过frangakis等人提出的[16]。

Also in our recent work we have analysed discrete rela-tions between nonlinear diffusion and wavelet shrinkage in
the one-dimensional setting. By deriving identical analytical solutions we proved equivalence between shift invariant soft Haar wavelet shrinkage on a single scale, space-dis-crete (but time-continuous) nonlinear diffusion with a TV diffusivity, and discrete TV regularisation [17] . Using these ideas on multiple scales and iterating the method comes down to hybrid techniques that aim to combine the efficiency of wavelets with the quality of PDE-based methods. Their performance is evaluated in [18] . Considering space-dis-crete nonlinear diffusion and replacing the time-continuous formulation by an explicit (Euler forward) time discretisa-tion allowed us to find general relations between the diffu-sivity for nonlinear diffusion filtering and the shrinkage function of shift invariant Haar wavelet shrinkage on a sin-gle scale [19] . In this way we identified also nonlinear diffu-sivities for hard, firm and Garrote wavelet shrinkage, and we proposed novel shrinkage functions that were inspired from nonlinear diffusivities and offered competitive performance. Moreover, this connection enabled us to derive novel stability results for single-scale wavelet shrink-age, including monotonic本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。