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Concept

Ondelette de Haar

Résumé

L'ondelette de Haar, ou fonction de Rademacher, est une ondelette créée par Alfréd Haar en 1909. On considère que c'est la première ondelette connue. Il s'agit d'une fonction constante par morceaux, ce qui en fait l'ondelette la plus simple à comprendre et à implémenter. L'ondelette de Haar peut être généralisée par ce qu'on appelle le système de Haar. Ondelette de Haar La fonction-mère des ondelettes de Haar est une fonction constante par morceaux : :\psi(t)= \begin{cases} 1 & \quad \textrm{pour} ;; 0\le t < \frac12,\ -1 & \quad \textrm{pour} ;;\frac12\le t

Source officielle

https://fr.wikipedia.org/wiki/Ondelette_de_Haar

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Generalized L-Spline Wavelet Bases

Thierry Blu, Ildar Khalidov, Michaël Unser

We build wavelet-like functions based on a parametrized family of pseudo-differential operators

L _{ v }

that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of

L _{ v }

, generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression a =

2 ^{ i }

. The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm.

SPIE2005

Chargement

, , ,

We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistics across scales within the Haar DWT. The minimization of the MSE estimate is performed independently in each wavelet subband, but this is equivalent to a global image-domain MSE minimization, thanks to the orthogonality of Haar wavelets. This is an important difference with standard Poisson noise-removal methods, in particular those that rely on a non-linear preprocessing of the data to stabilize the variance. Our non-redundant interscale wavelet thresholding outperforms standard variance-stabilizing schemes, even when the latter are applied in a translation-invariant setting (cycle-spinning). It also achieves a quality similar to a state-of-the-art multiscale method that was specially developed for Poisson data. Considering that the computational complexity of our method is orders of magnitude lower, it is a very competitive alternative. The proposed approach is particularly promising in the context of low signal intensities and/or large data sets. This is illustrated experimentally with the denoising of low-count fluorescence micrographs of a biological sample.

Elsevier2010

Chargement

Fast Haar-Wavelet Denoising Of Multidimensional Fluorescence Microscopy Data

Thierry Blu, Florian Luisier, Michaël Unser, Cédric René Jean Vonesch

We propose a novel denoising algorithm to reduce the Poisson noise that is typically dominant in fluorescence microscopy data. To process large datasets at a low computational cost, we use the unnormalized Haar wavelet transform. Thanks to some of its appealing properties, independent unbiased MSE estimates can be derived for each subband. Based on these Poisson unbiased MSE estimates, we then optimize linearly parametrized interscale thresholding. Correlations between adjacent images of the multidimensional data are accounted for through a sliding window approach. Experiments on simulated and real data show that the proposed solution is qualitatively similar to a state-of-the-art multiscale method, while being orders of magnitude faster.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2009

Chargement

Generalized L-Spline Wavelet Bases

Thierry Blu, Ildar Khalidov, Michaël Unser

We build wavelet-like functions based on a parametrized family of pseudo-differential operators

L _{ v }

that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of

L _{ v }

2 ^{ i }

SPIE2005