A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the s of as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley. A multiresolution analysis of the Lebesgue space consists of a sequence of nested subspaces that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations. Self-similarity in time demands that each subspace Vk is invariant under shifts by integer multiples of 2k. That is, for each the function g defined as also contained in . Self-similarity in scale demands that all subspaces are time-scaled versions of each other, with scaling respectively dilation factor 2k-l. I.e., for each there is a with . In the sequence of subspaces, for k>l the space resolution 2l of the l-th subspace is higher than the resolution 2k of the k-th subspace. Regularity demands that the model subspace V0 be generated as the linear hull (algebraically or even topologically closed) of the integer shifts of one or a finite number of generating functions or . Those integer shifts should at least form a frame for the subspace , which imposes certain conditions on the decay at infinity. The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support. Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in , and that they are not too redundant, i.e., their intersection should only contain the zero element. In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Cours associés (1)
COM-406: Foundations of Data Science
We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas an
Séances de cours associées (8)
Traitement d'image II: Haar Filterbank et Wavelet Transforms
Explore la banque de filtres Haar, les transformées en ondelettes et le filtre Wiener dans le traitement d'image.
Représentations du signal
Couvre les représentations de signaux en utilisant des concepts tels que les ondelettes Haar et les filtres FIR.
Représentations du signal
Couvre le concept de représentations de signaux à l'aide d'ondelettes et de l'ondelette de Haar.
Afficher plus
Publications associées (70)

Systematic analysis of wavelet denoising methods for neural signal processing

Silvestro Micera

Objective. Among the different approaches for denoising neural signals, wavelet-based methods are widely used due to their ability to reduce in-band noise. All wavelet denoising algorithms have a common structure, but their effectiveness strongly depends o ...
IOP PUBLISHING LTD2020

Multiresolution Subdivision Snakes

Michaël Unser, Daniel Andreas Schmitter, Virginie Sophie Uhlmann, Anaïs Laure Marie-Thérèse Badoual

We present a new family of snakes that satisfy the property of multiresolution by exploiting subdivision schemes. We show in a generic way how to construct such snakes based on an admissible subdivision mask. We derive the necessary energy formulations and ...
2017

Optimized Wavelet Denoising for Self-Similar alpha-Stable Processes

Michaël Unser, Pedram Pad

We investigate the performance of wavelet shrinkage methods for the denoising of symmetric-a-stable (S alpha S) self-similar stochastic processes corrupted by additive white Gaussian noise (AWGN), where a is tied to the sparsity of the process. The wavelet ...
2017
Afficher plus
Concepts associés (3)
Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. A function is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space of square integrable functions.
Ondelette
thumb|Ondelette de Daubechies d'ordre 2. Une ondelette est une fonction à la base de la décomposition en ondelettes, décomposition similaire à la transformée de Fourier à court terme, utilisée dans le traitement du signal. Elle correspond à l'idée intuitive d'une fonction correspondant à une petite oscillation, d'où son nom. Cependant, elle comporte deux différences majeures avec la transformée de Fourier à court terme : elle peut mettre en œuvre une base différente, non forcément sinusoïdale ; il existe une relation entre la largeur de l'enveloppe et la fréquence des oscillations : on effectue ainsi une homothétie de l'ondelette, et non seulement de l'oscillation.
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time). Haar wavelet The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.