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*Outreach;Mathematics Education and Popularization of Mathematics*

The Jean Morlet Chair is a scientific collaboration between CIRM -CNRS-SMF-, Aix-Marseille Université and the City of Marseille. Two international calls are launched every year to attract innovative researchers in an area of mathematical sciences. Selected candidates who must come from a foreign institution can spend a semester in residence at CIRM, where they run a full program of mathematical events in collaboration with a local project holder. Hans-Georg Feichtinger (University of Vienna) and Bruno Torresani (I2M Marseille) have been in charge of the second semester 2014 which will end in January 2015. The focus is on 'Computational Time-Frequency and Coorbit Theory'. Starting with a Research in Pairs event at the end of August, then three larger events-a School for young scientists, a main Conference and Small group- rather close in dates to enable participants to stay for more than one event, their semester will end on a second Research in Pairs in January 2015 and a celebratory event at the very end of the semester to celebrate 30 years of wavelets.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
The Jean Morlet Chair is a scientific collaboration between CIRM -CNRS-SMF-, Aix-Marseille Université and the City of Marseille. Two international calls are launched every year to attract innovative researchers in an area of mathematical sciences. Selected candidates who must come from a foreign institution can spend a semester in residence at CIRM, where they run a full program of mathematical events in collaboration with a local project ...

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*Research talks;Analysis and its Applications;Mathematics in Science and Technology*

Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

CIRM - ...

15-XX ; 41-XX ; 42-XX ; 46-XX

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*Special events;30 Years of Wavelets;Analysis and its Applications;Computer Science;History of Mathematics;Mathematics in Science and Technology*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications;Probability and Statistics;Numerical Analysis and Scientific Computing*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications;Mathematical Physics;Mathematics in Science and Technology*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications;Mathematics in Science and Technology*

94A12 ; 65T60 ; 42C40

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*Special events;30 Years of Wavelets;Analysis and its Applications;Mathematics in Science and Technology*

94A12 ; 94A08 ; 42C40

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*Research talks;Analysis and its Applications;Mathematics in Science and Technology*

Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

CIRM - ...

15-XX ; 41-XX ; 42-XX ; 46-XX

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*Special events;30 Years of Wavelets;Analysis and its Applications;Mathematical Physics;Mathematics in Science and Technology*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications*

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

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*Special events;30 Years of Wavelets;Mathematics in Science and Technology*

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

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*Special events;30 Years of Wavelets;Mathematics in Science and Technology*

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

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*Research talks;Analysis and its Applications*

Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal processing.

In this course, we will introduce the basic concepts of time-frequency frames, keeping their connection to audio applications as a guide-line. We will show how standard mathematical tools such as the Walnut representations can be used to obtain convenient reconstruction methods and also generalizations such the non-stationary Gabor transform. Applications such as the realization of an invertible constant-Q transform will be presented. Finally, we will introduce the basic notions of transform domain modelling, in particular those based on sparsity and structured sparsity, and their applications to denoising, multilayer decomposition and declipping. (Slides in attachment).
Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal ...

42C15

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*Research talks;Analysis and its Applications*

Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal processing.

In this course, we will introduce the basic concepts of time-frequency frames, keeping their connection to audio applications as a guide-line. We will show how standard mathematical tools such as the Walnut representations can be used to obtain convenient reconstruction methods and also generalizations such the non-stationary Gabor transform. Applications such as the realization of an invertible constant-Q transform will be presented. Finally, we will introduce the basic notions of transform domain modelling, in particular those based on sparsity and structured sparsity, and their applications to denoising, multilayer decomposition and declipping. (Slides in attachment).
Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal ...

94A12

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*Special events;30 Years of Wavelets;Analysis and its Applications;Numerical Analysis and Scientific Computing*

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

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*Special events;30 Years of Wavelets;Mathematical Physics;Mathematics in Science and Technology*

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

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*Special events;30 Years of Wavelets;Analysis and its Applications;Probability and Statistics*

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

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