DOCTORAL SCHOOL
Computational Harmonic Analysis  with Applications to Signal and Image Processing
Dates: 2024 October 2014 at CIRM (Marseille, France)
Computational Harmonic Analysis  with Applications to Signal and Image Processing
Dates: 2024 October 2014 at CIRM (Marseille, France)
ABSTRACTS
Hans Georg Feichtinger (CIRM and University of Vienna)
Mathematical and Numerical Aspects of Frame Theory
Abstract :
Frames are the natural generalization of the concept of a generating system for a Hilbert space, with the additional feature of having some stability. The frame bounds, resp. their quotient which is the condition number of the invertible frame operator provide an indication about the quality of a frame. Equivalently, it tells the user whether the cost (in terms of the l2norm of coefficients) for representing a given vector in a Hilbert space varies with the direction. However, frames (and also the concept or Riesz basic sequences, which generalize the idea of linear independence in a similar way) are often not just frames for the Hilbert space, but for the surrounding function spaces, which are the socalled modulation spaces in this context. Various other methods to describe operators (e.g. using the spreading representation or the KohnNirenberg symbol, or also the kernel theorem for operators) are best described in terms of the socalled Banach Gelfand Triple (S_0,L2,S_0’). This setting is also suitable for a description of approximation (in the w*sense) of continuous problems by finite/discrete models.
Course description:
The course will provide the necessary general background on the function spaces needed in order to describe frames, but not just for the (original) setting of Hilbert spaces, but for families of function spaces. Modulation spaces play here the big role, in particular the Banach Gelfand Triple.
For the numerical side we will give background about the implementation of abstract concepts in a MATLAB setting, and all the needed tools will be provided to the participants.
Among others we will demonstrate how to create in a completely discrete setting a family of Hermitelike functions, forming an orthonormal basis for l2(Z_n), which diagonalizes the FFT.
Discrete Bsplines are important to create bounded partitions of unity, but also for appropriate quasiinterpolation operators, allowing to return from finite sequences back to spaces of continuous functions.
In addition to providing the tools to run experiments we will also explain to the participants which precautions have to be taken when working in the finite, discrete setting.
Mathematical and Numerical Aspects of Frame Theory
Abstract :
Frames are the natural generalization of the concept of a generating system for a Hilbert space, with the additional feature of having some stability. The frame bounds, resp. their quotient which is the condition number of the invertible frame operator provide an indication about the quality of a frame. Equivalently, it tells the user whether the cost (in terms of the l2norm of coefficients) for representing a given vector in a Hilbert space varies with the direction. However, frames (and also the concept or Riesz basic sequences, which generalize the idea of linear independence in a similar way) are often not just frames for the Hilbert space, but for the surrounding function spaces, which are the socalled modulation spaces in this context. Various other methods to describe operators (e.g. using the spreading representation or the KohnNirenberg symbol, or also the kernel theorem for operators) are best described in terms of the socalled Banach Gelfand Triple (S_0,L2,S_0’). This setting is also suitable for a description of approximation (in the w*sense) of continuous problems by finite/discrete models.
Course description:
The course will provide the necessary general background on the function spaces needed in order to describe frames, but not just for the (original) setting of Hilbert spaces, but for families of function spaces. Modulation spaces play here the big role, in particular the Banach Gelfand Triple.
For the numerical side we will give background about the implementation of abstract concepts in a MATLAB setting, and all the needed tools will be provided to the participants.
Among others we will demonstrate how to create in a completely discrete setting a family of Hermitelike functions, forming an orthonormal basis for l2(Z_n), which diagonalizes the FFT.
Discrete Bsplines are important to create bounded partitions of unity, but also for appropriate quasiinterpolation operators, allowing to return from finite sequences back to spaces of continuous functions.
In addition to providing the tools to run experiments we will also explain to the participants which precautions have to be taken when working in the finite, discrete setting.
Some slides related to this part can be downloaded here
Monika Dörfler (NuHAG, Vienna), Nicki Holighaus (ARI, Vienna) and Matthieu Kowalski (L2S, Orsay)
TimeFrequency Frames and Applications to Audio Analysis
Abstract :
Timefrequency (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 shorttime Fourier transform, but also local cosine bases or lapped transforms, in the past years timefrequency frames have increasingly been applied to solve problems in audio signal processing.
In this course, we will introduce the basic concepts of timefrequency frames, keeping their connection to audio applications as a guideline. 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 nonstationary Gabor transform. Applications such as the realization of an invertible constantQ 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.
Course description:
Download Slides (zip files contains the sounds)
TimeFrequency Frames and Applications to Audio Analysis
Abstract :
Timefrequency (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 shorttime Fourier transform, but also local cosine bases or lapped transforms, in the past years timefrequency frames have increasingly been applied to solve problems in audio signal processing.
In this course, we will introduce the basic concepts of timefrequency frames, keeping their connection to audio applications as a guideline. 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 nonstationary Gabor transform. Applications such as the realization of an invertible constantQ 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.
Course description:
 Timefrequency (Gabor) frames: definition and structure of Gabor frames, dual frames and inversion, relation to shorttime Fourier transform, flexible time or frequency resolution, coefficient priors
 Applications in Audio processing: extraction of signal components, adaptive transforms, denoising, declipping
 Implementation issues
Download Slides (zip files contains the sounds)
TFTheory_part1.pdf  
File Size:  5546 kb 
File Type: 
TFTheory_part1_sounds.zip  
File Size:  8166 kb 
File Type:  zip 
TFTheory_part2.pdf  
File Size:  4863 kb 
File Type: 
Download ltfat for the practical session and the following files
tf_practical_session_1.zip  
File Size:  9379 kb 
File Type:  zip 
tf_practical_session2.zip  
File Size:  751 kb 
File Type:  zip 
Philipp Grohs (ETH, Zurich) and Axel Obermeier (ETH, Zurich)
Wavelets, Shearlets and Geometric Frames
Abstract :
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 wellestablished 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.
Course description:
Wavelets, Shearlets and Geometric Frames
Abstract :
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 wellestablished 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.
Course description:
 Basics of nonlinear approximation
 Short introduction to wavelets and their approximation properties
 Construction of curvelets and shearlets and similar systems and their approximation properties
 Implementation aspects and applications
geometrical_part1.pdf  
File Size:  4376 kb 
File Type: 
geometrical_part2.pdf  
File Size:  1766 kb 
File Type: 
geometrical_session_codes.zip  
File Size:  3 kb 
File Type:  zip 
Have a look at the slides of the practical session here
shearlet_demo.zip  
File Size:  485 kb 
File Type:  zip 
Sandrine Anthoine (I2M, Marseille), Caroline Chaux (I2M, Marseille), Clothilde Melot (I2M, Marseille) and Pierre Weiss (Toulouse)
Inverse Problems and Optimization
Abstract :
Inverse problems are the art of estimating quantities that are not directly observable from observed measurements, e.g. estimating the depth of a scene from 2D images of it. This essentially boils to inverting the direct operator, which difficulty is linked to the illposedness of the problem.
In this course, we introduce basic notions of inverse problems, and present a general optimization framework to solve it, as well as some classical algorithms. Then, the notion of sparsity on a frame is explored for two different aspects of the problem: 1) the regularization of an inverse problem, 2) the approximation of the operator at stake.
Course description:
Inverse Problems and Optimization
Abstract :
Inverse problems are the art of estimating quantities that are not directly observable from observed measurements, e.g. estimating the depth of a scene from 2D images of it. This essentially boils to inverting the direct operator, which difficulty is linked to the illposedness of the problem.
In this course, we introduce basic notions of inverse problems, and present a general optimization framework to solve it, as well as some classical algorithms. Then, the notion of sparsity on a frame is explored for two different aspects of the problem: 1) the regularization of an inverse problem, 2) the approximation of the operator at stake.
Course description:
 Inverse problems: introduction, examples, illposedness, direct solution versus solution of an optimization problem.
 The art of regularizing: (joint)sparsity on a frame, nondifferentiable convex optimization and algorithms.
 Operator approximation using a wavelet basis.
InverseProblems_part1.pdf  
File Size:  3294 kb 
File Type: 
InverseProblems_part2.pdf  
File Size:  829 kb 
File Type: 
Practical session
During the practical session you received a .zip file. Below is an updated version where you find



Matlab: some basic commands: An option is a first look at the website Basic commands instructions Another possibility is the guided tutorial 