Gerald A. Kaiser · Birkhäuser Boston eBooks · 2010
In all of the frames studied so far, the analysis (computation of ƒ (Ψ, t) or ƒ ( s , t) or their discrete samples) must be made directly by computing the relevant integrals for all the necessary values of the time-frequency or timescale parameters. Around 1986, a radically new method for performing discrete wavelet analysis and synthesis was born, known as multiresolution analysis. This method is completely recursive and therefore ideal for computations.
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2018
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Georges‐Pierre Bonneau, Gershon Elber, Stéfanie Hahmann et al. · Mathematics and visualization · 2007
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Patrick L. Combettes, J.-C. Pesquet · IEEE Transactions on Pattern Analysis and Machine Intelligence · 1998
A standard wavelet multiresolution analysis can be defined via a sequence of projectors onto a monotone sequence of closed vector subspaces possessing certain properties. We propose a nonlinear extension of this framework in which the vector subspaces are replaced by convex subsets. These sets are chosen so as to provide a recursive, monotone approximation scheme that allows for various signal and image features to be investigated. Several classes of convex multiresolution analyses are discussed and numerical applications to signal and image-processing problems are demonstrated.
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Elisabetta Grecchi · Research Portal (King's College London) · 2016
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S. Pitchai Murugan, G. P. Youvaraj · Journal of the Australian Mathematical Society · 2021
Abstract Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158 (1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.
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Laurent Navarro, Michel Jourlin, Guy Courbebaisse · HAL (Le Centre pour la Communication Scientifique Directe) · 2015
International audience
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Beacom, J., R. Wilson · Warwick Research Archive Portal (University of Warwick) · 1996
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M. A. Fortes, M. L. Rodríguez · 2013
Beam and Warming proved in [1] that the supercompact wavelets can exactly represent any piecewise polynomial function in one variable, generalizing the fact that Haar wavelet can exactly represent any piecewise constant function. Higher level of accuracy is attained by higher order polynomials of supercompact wavelets. Later, in [2], the authors developed an extension of the work [1] to the case of surfaces defined over uniform meshes of the domain of the surface. Such construction keeps the same advantages attained by [1] in relation with orthogonality, short support, approximation of surfaces with no border effects, detection of discontinuities, higher degree of accuracy and compressibility. The approach in [2] allows transfer information of a function between different resolution levels by means of reconstruction and decomposition algorithms stated in a multiresolution context. Actually, we are working in generalizations of [2] in two different ways: First of all, we extend the multiresolution scheme for surfaces to the case of non-uniform meshes. On the other hand, we handle the problem of generalizing the multiresolution analysis with the aim of transfering other kinds of information, like derivative or integral values, curvatures,...between different resolution levels.
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Carl Rohwer · International series of numerical mathematics · 2005
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Manisha Gupta, Andrew Gilbert · 2002
We explore the use of multiresolution analysis for vector signals, such as multispectral images or stock market portfolio time series. These signals often contain local correlations among components that are overlooked in a component-by-component analysis. We show that a coarse signal defined by taking local arithmetic averages is equivalent to analyzing the signal component by component, but by using the average that minimizes the L/sup 2/ distance to the local points results in a non-separable vector multiresolution analysis. We propose using the vector multiresolution representation for signal processing tasks such as compression and denoising. We prove some results in denoising and present color image examples.
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Usha Desai, C. Gurudasnayak · 2015
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Abdeldjalil Ouahabi · 2012
This chapter introduces the “wavelet” tool via multiresolution analysis (MRA) to make it accessible to the widest possible range of users from the academic world as well as from the socioeconomic world. It then focuses on the concept of MRA and the equivalence of orthonormal wavelet bases and filter banks, leading to the rapid, linearly complex algorithms used for the calculation of a wavelet transform. The multiresolution analysis and the criteria for choosing a wavelet provided underpin the chapter, highlighting the take-home message of the problems discussed. Finally, informative illustrations help the uninitiated reader to better conceptualize certain ideas, manifested in particular applications and frequently accompanied by the corresponding Matlab® code. Controlled Vocabulary Terms audio watermarking; curvelet transforms
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Laurent Simons · Open Repository and Bibliography (University of Liège) · 2010
An orthonormal basis of wavelets of $L^2(\R)$ is an orthonormal basis of $L^2(\R)$ of type \[ \psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\Z. \] A classical method to obtain such bases consists in constructing a multiresolution analysis. When the mother wavelet $\psi$ depends on the scale (i.e. the index $j$), a non stationary version of multiresolution analysis is then used. It is for example the case in the general context of Sobolev spaces. We generalize different characterizations in the standard theory of wavelets to the case of multi-scales wavelets and non stationary multiresolution analyses.
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Christensen, Ole · Technical University of Denmark, DTU Orbit (Technical University of Denmark, DTU) · 2005
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Brigitte Forster · 2014
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Hubert Konik, Alain Trémeau, Bernard Laget · Color and Imaging Conference · 1995
Because of the color space size, a color image contains too much information. Consequently, before analyzing it, its information has to be reduced without a loss of relevant data. In order to reach that, the multiresolution process is well adapted. Indeed, it enables a spatiocolor approach considering color and spatial distribution at the same time. More precisely, the construction of a color pyramid is presented. Finally, its utilisation during a segmentation step is discussed.
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Tom Lyche, Marie‐Laurence Mazure · HAL (Le Centre pour la Communication Scientifique Directe) · 2012
International audience
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P. A. Andrianov · Journal of Mathematical Sciences · 2023
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Paolo Zatelli · 2004
The ability of gerarchically manage data available in a Geographical Information System is becoming increasingly important, given the huge data quantity provided by the modern surveying techniques. In particular, the spatial features of an element must be framed in an intrensically multiresolution structure with an efficient management of the information at different resolutions. Four new modules have been created for the multiresolution analysis with GRASS. These modules use the discrete wavelet transform to perform a MultiResolution Analysis (MRA) by the Mallat algorithm, which has been extended for the discrete bidimensional case. Same applications exploiting the potentiality of the technique have been carried out, among these the filtering of LIDAR data and the automatich shape recognition. 1.
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