Speakers
- Swanhild Bernstein, Freiberg University of Mining and Technology, Germany
Wavelets and generalized splines for Radon transform on compact Lie groups with applications to crystallography - Paula Cerejeiras, University of Aveiro, Portugal
A short-time Fourier transform for quaternionic signals
(Abstract) - Lasse Hjuler Christiansen, Technical University of Denmark
Construction of smooth windows generating dual pairs of Gabor - and wavelet frames
(Abstract)
- Laurent Demaret, Helmholtz Zentrum München, Germany
The L1-Potts functional. Fast algorithm and application to deconvolution
(Abstract)
- Frank Eckhardt, Universität Marburg, Germany
Besov-regularity for the Stokes system in polyhedral cones
(Abstract) - Martin Ehler, Helmholtz Zentrum München, Germany
Signal reconstruction from the magnitudes of subspace components
We consider signal reconstruction from the norms of its subspace components. If the weighted linear subspaces form a tight p-fusion frame and a cubature formula of strength 4, then even in case that p of the subspace norms are erased, we find a finite list of potential signals, one of which is the correct one. Moreover, we present a computationally feasible algorithm to determine this list. Alternatively, we use semi-definite programming and random subspaces to decrease the required number of subspace components for reconstruction. - Wolfgang Erb, Universität zu Lübeck, Germany
Localized polynomials and a time-frequency analysis on the unit sphere
The aim of this talk is to present a time-frequency theory for spherical harmonics on the unit sphere that runs parallel to the time-frequency analysis of bandlimited functions developed by Landau, Pollak and Slepian. For this purpose, the spectral decomposition of a particular band-time-band limiting operator
is studied. The spectral decomposition and the eigenvalues of this operator are closely related to the theory of orthogonal polynomials. Results from both theories, the theory of orthogonal polynomials and the Landau-Pollak-Slepian theory, can be used to prove localization and approximation properties of the corresponding eigenfunctions. Finally, an uncertainty principle is presented that reflects the limitation of coupled space and frequency locatability on the unit sphere.
- Frank Filbir, Helmholtz Zentrum München, Germany
Data driven harmonic analysis
In the classical scenario in harmonic analysis we assume that the manifold is fixed beforehand and one has to approximate functions defined on this manifold. Many modern applications require an analysis of huge, unstructured data set which belongs to a very high dimensional ambient Euclidean space. The basic idea is to assume that the data lies on an unknown low dimensional manifold. The desire to take advantage of this low intrinsic dimensionality has recently prompted a great deal of research on diffusion geometry techniques. We will present some new results in the direction of a data driven harmonic analysis.
This is joint work with Hrushikesh N. Mhaskar (Caltech, Pasadena, USA). - Stefan Held, Technische Universität München, Germany
Wavelet frames of higher Riesz transforms
(Abstract) - Bettina Heise, Johannes Kepler Universität Linz, Austria
Fourier plane filtering, Riesz transforms, and singularities in optics
(Abstract)
- Mads Sielemann Jacobsen,Technical University of Denmark
Dual pairs of Gabor frames generated by trigonometric functions without the partition of unity property
(Abstract) - Uwe Kähler, University of Aveiro, Portugal
Discrete Dirac operators and harmonic analysis
(Abstract) - Ofer Levi, Ben-Gurion University of the Negev, Israel
Direct and exact PPFFT and radon transforms using orthogonalizing weights
The Pseudo-Polar FFT (PPFFT), which was developed and presented by Averbuch et. al, computes Discrete Fourier Transform coecients on a nearly polar grid. The special structural properties of the Pseudo-Polar grid allow a computational complexity of O(n^2*log(n)) for an n by n image (as opposed to O(n^4) in the polar case). Averbuch et. al. also presented a weighted version of the PPFFT that is nearly orthogonal and can be used for the application of an extremely fast iterative inverse solver. The PPFFT is directly related to a highly accurate and fast version of the Discrete Radon Transform that possesses the same desirable computational properties as the PPFFT, the Fast Slant-Stack Transform. In many instances and applications the PPFFT can be a very good substitute for the Polar FT and its superior computational properties can speed up many related algorithms by several orders of magnitude. Classical applications of the Polar FFT include rotational registration of images and reconstruction in medical and biological imaging. The preliminary part of this talk will introduce the basics of 2D DFTs in Cartesian, Polar and PP grids using matrices and vectors notation. Later, a new direct an exact inverse PPFFT will be presented, the algorithm is based on a preprocessing step in which an optimal set of weights is computed for the given image size, these weights perfectly orthogonalize the columns of the transform's matrix so that the inverse problem can be solved exactly by a single application of the Adjoint PPFFT which can be computed as well in O(n^2*log(n)) complexity. - Clothilde Melot, LATP, Marseille, France
Study of an example of multifractal and "sparse" signal
We consider here a slight modification of a family of multifractal signals by Jaffard (1992). They can be considered somehow as "sparse" signals since at each scale only few wavelet coefficients of these signals do not vanish.
We study here their pointwise behavior, computing several pointwise exponents (Hölder exponent, oscillating singularity exponent, p-exponent defined with the help of local Lp norms) at every point and prove that from the point of view of each of these exponents, the signals are multifractal. Furthermore we will prove that one can compute the Hausdorff dimension of the set of points at which the signal has a given pointwise regularity with the help of a formula, the so-called multifractal formalism, which uses global quantities for the computation.
This work is a joint work with Claire Coiffard and Thomas Willer, LATP, Marseille, France.
- Sebastian Penka, Universität zu Lübeck, Germany
Frequency detection with orthogonal polynomials on the bitorus
Abstract
- Martin Storath, Helmholtz Zentrum München, Germany
Signal analysis via complex wavelet signatures
We propose a new analysis tool for signals, called signature, based on wavelet signs. The complex-valued signature at some spatial point $b$ is defined as the limit of the wavelet signs,
$\sign\langle f,\kappa_{a,b}\rangle,$
provided the limit exists for $a\to 0$ and is independent of $\kappa$ belonging to a certain class of complex wavelets; otherwise the signature is set equal to zero. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature is invariant under fractional differentiation and rotates in the complex plane under fractional Hilbert transforms. Thus, the signature may be regarded as the dual object to the local Sobolev regularity index, which is altered by fractional differentiations but not affected by fractional Hilbert transforms.
We validate the theoretical results with several numerical experiments. These experiments show that wavelet signatures can be explicitly computed and therefore be used in signal and possibly image analysis.
This is joint work with L. Demaret and P. Massopust.
- Myung-Sin Song, Southern Illinois University, USA:
Haar wavelet-like analysis with MRA method extended to fractals
Extending the Haar wavelet-like analysis with MRA method to both fractals and to discrete hierarchical models to study two computational features:
(a) Approximation of the father or mother functions by subdivision schemes, and
(b) matrix formulas for the wavelet coefficients where a variety of data will be considered;
typically for fractals, - convergence is more restrictive than is the case for wavelets. This makes wavelets closely related to fractals and fractal processes. Investigation of the relation between wavelets and fractals and fractal processes has theoretical and practical potential. It has been recently shown (by Palle Jorgensen, Ola Bratteli, David Larson, X. Dai and others) that a unifying approach to wavelets, dynamical systems, iterated function systems, self-similarity and fractals may be based on the systematic use of operator analysis and representation theory. Motivated by hierarchical models and multiscaling, operators of multiplication, and dilations, and more general weighted composition operators are studied. In these models, scaling is implemented by non-linear and non-invertible transformations. This in turn generalizes affine transformations of variables from wavelet analysis and analysis on affine fractals.
- Pat Van Fleet, University of St. Thomas, USA
Nonnegative scaling vectors on the interval
In this talk, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors on the interval have been constructed by Dahmen and Micchelli; Goh, Jiang, and Xia; and Lakey and Pereyra, but our approach is different in that we start with an existing scaling vector $\Phi$ that generates a multiresolution analysis for $L^2(\mathbb{R})$ to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. The talk will conclude with some examples of constructions in the case where the given scaling vectors have length $A=2$.
- Jürgen Frikel, Helmholtz Zentrum München, Germany
Curvelet reconstructions in limited angle x-ray tomography
We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations.
To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this presentation, we discuss the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will present the practical relevance of these results. - Regina de Almeida, Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal
- Min Ku, University of Aveiro, Portugal
- Johannes Nagler, Universität Passau, Germany
- Elnaz Osgooei, Technical University of Denmark
- Xing Liu, Imperial College London, United Kingdom
Share this page
