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30 Years of Wavelets  | enregistrements trouvés : 18

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Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures.
In this talk we first review classical results on variance uncertainty inequalities (in particular Heisenberg, Robertson and Breitenberger inequalities). We then focus on discrete (and in particular finite-dimensional) situations, where variance has to be replaced with more suitable localization measures. We then present recent results on support and entropic inequalities, describing joint localization properties of vector expansions with respect to two frames.

Keywords: uncertainty principle - variance of a function - Heisenberg inequality - support inequalities - entropic inequalities
Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures.
In this talk we first review classical results on variance uncertainty ...

94A12 ; 94A17 ; 26D20 ; 42C40

In this conference, I start by presenting the first applications and developments of wavelet methods made in Marseille in 1985 in the framework of sounds and music. A description of the earliest wavelet transform implementation using the SYTER processor is given followed by a discussion related to the first signal analysis investigations. Sound examples of the initial sound transformations obtained by altering the wavelet representation are further presented. Then methods aiming at estimating sound synthesis parameters such as amplitude and frequency modulation laws are described. Finally, new challenges brought by these early works are presented, focusing on the relationship between low-level synthesis parameters and sound perception and cognition. An example of the use of the wavelet transforms to estimate sound invariants related to the evocation of the "object" and the "action" is presented.

Keywords : sound and music - first wavelet applications - signal analysis - sound synthesis - fast wavelet algorithms - instantaneous frequency estimation - sound invariants
In this conference, I start by presenting the first applications and developments of wavelet methods made in Marseille in 1985 in the framework of sounds and music. A description of the earliest wavelet transform implementation using the SYTER processor is given followed by a discussion related to the first signal analysis investigations. Sound examples of the initial sound transformations obtained by altering the wavelet representation are ...

00A65 ; 42C40 ; 65T60 ; 94A12 ; 97M10 ; 97M80

In my talk I am presenting a link between time-frequency analysis and noncommutative geometry. In particular, a connection between the Moyal plane, noncommutative tori and time-frequency analysis. After a brief description of a dictionary between these two areas I am going to explain some consequences for time-frequency analysis and noncommutative geometry such as the construction of projections in the mentioned operator algebras and Gabor frames.

Keywords: modulation spaces - Banach-Gelfand triples - noncommutative tori - Moyal plane - noncommutative geometry - deformation quantization
In my talk I am presenting a link between time-frequency analysis and noncommutative geometry. In particular, a connection between the Moyal plane, noncommutative tori and time-frequency analysis. After a brief description of a dictionary between these two areas I am going to explain some consequences for time-frequency analysis and noncommutative geometry such as the construction of projections in the mentioned operator algebras and Gabor ...

46Fxx ; 46Kxx ; 46S60 ; 81S05 ; 81S10 ; 81S30

Multi angle  Phase-space delocalization
Paul, Thierry (Auteur de la Conférence) | CIRM (Editeur )

Multi angle  An « ISI » perspective on wavelets
Flandrin, Patrick (Auteur de la Conférence) | CIRM (Editeur )

The introduction of wavelets in the mid 80's has significantly reshaped some areas of the scientific landscape by establishing bridges between previously disconnected domains, and eventually leading to a new paradigm. This generally accepted-yet loose-claim can be given a more precise form by exploiting bibliometric databases such as the ISI Web of Science. Preliminary results in this direction will be reported here, based on multiple entries where authors, references, keywords and disciplines are used as nodes of a network in which the links correspond to their co-appearance in the same paper. While the evolution in time of such an « heterogeneous net » gives a quantified perspective on the birth and growth of wavelets as a well-identified scientific field of its own, it also raises many interpretation issues (related, e.g., to automation vs. expertise) whose implications go beyond this peculiar case study.

Keywords : wavelets - history - bibliometry - network, paradigm
The introduction of wavelets in the mid 80's has significantly reshaped some areas of the scientific landscape by establishing bridges between previously disconnected domains, and eventually leading to a new paradigm. This generally accepted-yet loose-claim can be given a more precise form by exploiting bibliometric databases such as the ISI Web of Science. Preliminary results in this direction will be reported here, based on multiple entries ...

01-XX ; 42-XX ; 68-XX ; 94-XX

We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their duality structure, we introduce two natural partial inner product spaces and formulate a number of open questions.

Keywords: continuous frames - semi-frames - frame duality - reproducing pairs - partial inner product spaces
We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their ...

42C15 ; 42C40 ; 46C50 ; 65T60

This presentation reminds of some early sonic representations and of their utility for musical sound synthesis and processing, prior to the introduction of wavelets by Morlet and Grossmann. It reminds the circumstances of the work performed at the Laboratoire de Mécanique et d'Acoustique, Marseille, by Richard Kronland-Martinet with Alex Grossmann on wavelet transforms of sounds and by Daniel Arfib and Frédéric Boyer on the Gabor representation. It is illustrated by short sound and video examples.

Keywords: wavelets - Gabor - analysis-synthesis - computer music
This presentation reminds of some early sonic representations and of their utility for musical sound synthesis and processing, prior to the introduction of wavelets by Morlet and Grossmann. It reminds the circumstances of the work performed at the Laboratoire de Mécanique et d'Acoustique, Marseille, by Richard Kronland-Martinet with Alex Grossmann on wavelet transforms of sounds and by Daniel Arfib and Frédéric Boyer on the Gabor representation. ...

42-03

In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view of the world, and the sparsity property of wavelet expansions, and present two recent results. First, we show new bounds for the time-frequency spread of sequences, and construct maximally compact sequences. Interestingly they differ from sampled Gaussians. Next, we review work on sampling of finite rate of innovation signals, which are sparse continuous-time signals for which sampling theorems are possible. We conclude by arguing that the interface of signal processing and applied harmonic analysis has been both fruitful and fun, and try to identify lessons learned from this experience.

Keywords: wavelets ­ filter banks - subband coding ­ uncertainty principle ­ sampling theory ­ sparse sampling
In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view ...

94A08 ; 94A12 ; 65T60 ; 42C40

Coorbit theory was developed in the late eighties as a unifying principle covering (possible non-)orthogonal frame expansions in the wavelet and in the time-frequency context. Very much in the spirit of « coherent frames » or also reproducing kernels for a Moebius invariant Banach space of analytic functions one can describe a family of function spaces associated with a given integrable and irreducible group representation on a Hilbert space by its generalized wavelet transform, and obtain (among others) atomic decomposition results for the resulting spaces. The theory was flexible enough to cover also more recent examples, such as voice transforms related to the Blaschke group or the spaces (and frames) related to the shearlet transform.
As time permits I will talk also on the role of Banach frames and the usefulness of Banach Gelfand triples, especially the one based on the Segal algebra $S_0(G)$, which happens to be a modulation space, in fact the minimal among all time-frequency invariant non-trivial function spaces.

Keywords: wavelet theory - time-frequency analysis - modulation spaces - Banach-Gelfand-triples - Toeplitz operators - atomic decompositions - function spaces - shearlet transform - Blaschke group
Coorbit theory was developed in the late eighties as a unifying principle covering (possible non-)orthogonal frame expansions in the wavelet and in the time-frequency context. Very much in the spirit of « coherent frames » or also reproducing kernels for a Moebius invariant Banach space of analytic functions one can describe a family of function spaces associated with a given integrable and irreducible group representation on a Hilbert space by ...

43-XX ; 46Exx ; 42C40 ; 42C15 ; 42C10

Wavelets are standard tool in signal- and image processing. It has taken a long time until wavelet methods have been accepted in numerical analysis as useful tools for the numerical discretization of certain PDEs. In the signal- and image processing community several new frame constructions have been introduced in recent years (curvelets, shearlets, ridgelets, ...). Question: Can they be used also in numerical analysis? This talk: Small first step. Wavelets are standard tool in signal- and image processing. It has taken a long time until wavelet methods have been accepted in numerical analysis as useful tools for the numerical discretization of certain PDEs. In the signal- and image processing community several new frame constructions have been introduced in recent years (curvelets, shearlets, ridgelets, ...). Question: Can they be used also in numerical analysis? This talk: Small first ...

42C15 ; 42C40 ; 65Txx

We start with a brief historical account of wavelets and of the way they shattered some of the preconceptions of the 20th century theory of statistical signal processing that is founded on the Gaussian hypothesis. The advent of wavelets led to the emergence of the concept of sparsity and resulted in important advances in image processing, compression, and the resolution of ill-posed inverse problems, including compressed sensing. In support of this change in paradigm, we introduce an extended class of stochastic processes specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. Starting from first principles, we prove that the solutions of such equations are either Gaussian or sparse, at the exclusion of any other behavior. Moreover, we show that these processes admit a representation in a matched wavelet basis that is "sparse" and (approximately) decoupled. The proposed model lends itself well to an analytic treatment. It also has a strong predictive power in that it justifies the type of sparsity-promoting reconstruction methods that are currently being deployed in the field.

Keywords: wavelets - fractals - stochastic processes - sparsity - independent component analysis - differential operators - iterative thresholding - infinitely divisible laws - Lévy processes
We start with a brief historical account of wavelets and of the way they shattered some of the preconceptions of the 20th century theory of statistical signal processing that is founded on the Gaussian hypothesis. The advent of wavelets led to the emergence of the concept of sparsity and resulted in important advances in image processing, compression, and the resolution of ill-posed inverse problems, including compressed sensing. In support of ...

42C40 ; 60G20 ; 60G22 ; 60G18 ; 60H40

Since the last twenty years, Littlewood-Paley analysis and wavelet theory has proved to be a very useful tool for non parametric statistic. This is essentially due to the fact that the regularity spaces (Sobolev and Besov) could be characterized by wavelet coefficients. Then it appeared that that the Euclidian analysis is not always appropriate, and lot of statistical problems have their own geometry. For instance: Wicksell problem and Jacobi Polynomials, Tomography and the harmonic analysis of the ball, the study of the Cosmological Microwave Background and the harmonic analysis of the sphere. In these last years it has been proposed to build a Littlewood-Paley analysis and a wavelet theory associated to the Laplacien of a Riemannian manifold or more generally a positive operator associated to a suitable Dirichlet space with a good behavior of the associated heat kernel. This can help to revisit some classical studies of the regularity of Gaussian field.

Keywords: heat kernel - functional calculus - wavelet - Gaussian process
Since the last twenty years, Littlewood-Paley analysis and wavelet theory has proved to be a very useful tool for non parametric statistic. This is essentially due to the fact that the regularity spaces (Sobolev and Besov) could be characterized by wavelet coefficients. Then it appeared that that the Euclidian analysis is not always appropriate, and lot of statistical problems have their own geometry. For instance: Wicksell problem and Jacobi ...

43A85 ; 60G15 ; 60G17 ; 58C50

One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This finding triggered an enormous research activity in recent years both in signal processing applications as well as their mathematical foundations. The present talk discusses connections of compressive sensing and time-frequency analysis (the sister of wavelet theory). In particular, we give on overview on recent results on compressive sensing with time-frequency structured random matrices.

Keywords: compressive sensing - time-frequency analysis - wavelets - sparsity - random matrices - $\ell_1$-minimization - radar - wireless communications
One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This ...

94A20 ; 94A08 ; 42C40 ; 60B20 ; 90C25

Breast cancer is the most common type of cancer among women and despite recent advances in the medical field, there are still some inherent limitations in the currently used screening techniques. The radiological interpretation of X-ray mammograms often leads to over-diagnosis and, as a consequence, to unnecessary traumatic and painful biopsies. First we use the 1D Wavelet Transform Modulus Maxima (WTMM) method to reveal changes in skin temperature dynamics of women breasts with and without malignant tumor. We show that the statistics of temperature temporal fluctuations about the cardiogenic and vasomotor perfusion oscillations do not change across time-scales for cancerous breasts as the signature of homogeneous monofractal fluctuations. This contrasts with the continuous change of temperature fluctuation statistics observed for healthy breasts as the hallmark of complex multifractal scaling. When using the 2D WTMM method to analyze the roughness fluctuations of X-ray mammograms, we reveal some drastic loss of roughness spatial correlations that likely results from some deep architectural change in the microenvironment of a breast tumor. This local breast disorganisation may deeply affect heat transfer and related thermomechanics in the breast tissue and in turn explain the loss of multifractal complexity of temperature temporal fluctuations previously observed in mammary glands with malignant tumor. These promising findings could lead to the future use of combined wavelet-based multifractal processing of dynamic IR thermograms and X-ray mammograms to help identifying women with high risk of breast cancer prior to more traumatic examinations. Besides potential clinical impact, these results shed a new light on physiological changes that may precede anatomical alterations in breast cancer development.

Keywords: breast cancer - X-ray mammography - infrared thermography - multifractal analysis - wavelet transform - wavelet transform modulus maxima method
Breast cancer is the most common type of cancer among women and despite recent advances in the medical field, there are still some inherent limitations in the currently used screening techniques. The radiological interpretation of X-ray mammograms often leads to over-diagnosis and, as a consequence, to unnecessary traumatic and painful biopsies. First we use the 1D Wavelet Transform Modulus Maxima (WTMM) method to reveal changes in skin ...

92-08 ; 92C50 ; 92C55

sparsity - morphological diversity - inpainting - cosmology - weak lensing - cosmic microwave background

65T60 ; 94A12 ; 85A35

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