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[-]

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[-]

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[-]

In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.
In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (Slides in attachment).[-]

In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.
In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (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 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 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).[-]

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[-]

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[-]

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[-]