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Professor David Larson - A New Generalization of Frame Theory

Texas A & M University,

Abstract: We show that there are some natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of normal linear mappings between von Neumann algebras. This leads to a natural generalization of frame theory. In fact two natural generalizations, one subordinate to the other.

Date: Thursday, March 24, 2011, 12:30

Location: Room 03.04.011, TU München, Boltzmannstr. 3, 85748 Garching

Prof. Ole Christensen - Gabor frames in L2 and the duality principle in general Hilbert spaces

Technical University of Denmark

Date: 08.07.2010, 11:15 am

Room: 03.04.011

Prof. Dr. Manfred Tasche - Nichtlineare Approximation durch anharmonische Fouriersummen

Universität Rostock


Date: March 17 2010, 10:30 am

Location: Zentrum Mathematik der TUM in Garching, Raum 03.04.011.

Ulrich Rührmair - Physical Cryptography: A new approach in crypto and security

Lehrstuhl für Effiziente Algorithmen, Institut für Informatik, Technische Universität München

Abstract: Security and privacy are amongst the most relevant topics in computer science. Nevertheless, standard approaches in the field are based on unproven mathematical assumptions and on the supposed secrecy of binary keys. Unfortunately, such keys can be cloned, extracted from mobile environments by invasive attacks, or transferred from computer systems by viruses. These facts restrict both the security and applicability of existing cryptographic concepts. In our talk, we will describe a recent alternative approach known as physical cryptography. It is based on the inherent complexity of nanoscale electronic and photonic systems. Its key idea is to use disordered, random nanostructures in order to replace or complement standard binary keys. This can lead to new applications and/or better security properties. Concrete examples for systems considered by us are optical systems such as photonic crystals or random arrays of quantum dots, or electronic circuits (for complex cellular non-linear networks). Our approach is complementary to quantum cryptography, but may be easier to implement, and covers a yet broader spectrum of applications. In order to implement physical cryptography successfully in practice, the stable read-out of the analog signals of said disordered structures must be guaranteed. This establishes interesting relations to the fields of image transformation and image analysis, with physical cryptographer being one potential customer for image transformations with specific, non-standard properties.

Date: Tuesday, Februar 23, 2010, 13:30 h

Location: Zentrum Mathematik, TU München, MI 03.04.011

Ildar Khalidov - Activelets: From Hemodynamics To Wavelets For Sparse Representations in fMRI

EPF Lausanne

Abstract: Functional magnetic resonance imaging (fMRI) is a widely used technique in brain activity mapping. Its popularity is due to good spatial resolution compared to other non-invasive methods. In fMRI, neural activity is observed through the blood-oxygenation level-dependent (BOLD) changes in the magnetic resonance signal. Once the dataset is acquired, voxels that expose task-related activity need to be identified. In the traditional framework, precise knowledge of the event timing is assumed. Given the timing, the BOLD response is modeled, and a linear fit is performed on the measured data. In this work, we start from a hemodynamics-based model of the BOLD signal and design wavelets (activelets) that mimic the behaviour of the underlying differential operator. These wavelets provide a sparse representation of the activity-related component of the fMRI time signal. Without any knowledge on the timing, the activity-related signal is extracted from the measurements by means of a sparse-solution search algorithm. We demonstrate the performance of the algorithm on synthetic and real fMRI datasets. As a generalization of our approach, we design wavelets that behave like a given arbitrary differential operator. Thanks to their favourable decorrelation and energy compaction properties, our new wavelets could be useful in treating biomedical signals from imaging devices that admit a description by a system of differential equations.

Date: 26. Juni 2008, 16:45 Uhr

Location: Institut für Biomathematik und Biometrie, Helmholtz Zentrum München, Ingolstädter Landstr. 1, 85764 Neuherberg

Prof. Ole Christensen - Frames, Gabor-System und Wavelets

Technical University of Denmark, Lyngby, July 11-12, 2007.

Abstract: In der Signalverarbeitung verwendet man oft Darstellungen von komplizierten Signalen via Orthonormalbasen. Allerdings sind die ONBBedingungen sehr restriktiv, und so findet man manchmal keine ONB, die eventuelle zusätzliche Bedingungen erfüllt. In diesem Vortrag werden wir Frames betrachten und konkrete Beispiele studieren, in welchen die Flexibilität (im Vergleich zu den Orthonormalbasen) nützlich ist. Zum Beispiel werden wir Gabor-Frames konstruieren, für die der Frame-Generator und auch der duale Generator beide explizit gegebene Splines mit kompakten Träger sind.

Zeit: am Mittwoch, den 11. Juli 2007 11:00 Uhr

Ort: GSF-Forschungszentrum für Umwelt und Gesundheit Ingolstädter Landstr. 1 85746 Neuherberg/Oberschleißheim Raum : 121/Gebäude 58a , Etage : 1

Zeit: am Donnerstag, den 12. Juli 2007 10:15 Uhr

Ort: Zentrum Mathematik, TU München, MI 03.06.011

Dr. Akira Hirabayashi - Sampling and reconstruction of signals in generalized under-sampling scenarios

Yamaguchi University, Japan, Feb. 27, 2007

Abstract: Sampling and reconstruction of signals are basic tasks that provide interface between analog and digital signals. A classical example is Shannon's sampling theorem, which is not practical in many aspects. A number of substitutions have been proposed to date. Most of them discussed so-called over-sampling or normal sampling scenarios. On the other hand, under-sampling is a scenario which is difficult to deal with, but easy to encounter. In this talk, we focus on the latter case. We first formulate sampling and reconstruction problem, and generalize the definition of over-sampling and under-sampling within the formulation. Then, we know that there are two kinds of under-sampling cases. Note that the second case has never been pointed out so far. We show some results obtained for the first case based on the consistency criterion. We also provide some perspectives on the second case.

Datum: Feb. 27, 2007

Ort: Zentrum Mathematik, TU München

Dr. Dimitri Van De Ville - Wavelets in Neuro-Imaging: Applications to Functional MRI

Swiss Federal Institute of Technology Lausanne, EPF Lausanne, 17. January 2007

Abstract: Imaging the functioning of human brain is an important task in neurosciences and neurology. One of the most popular modalities is functional magnetic resonance imaging (fMRI), known for its versatility and good spatial resolution. Unfortunately, the blood- oxygenation-level-depedent (BOLD) signal that is indirectly related to the neuronal activity, is weak and deteriorated by noise and background activity. Successful analysis methods for this type of signals mostly rely on a suitable transformation, allowing to better represent the signals' key properties, and a statistical framework, providing a significance level of the results (e.g., probability of a false positive in a detection task). The wavelet decomposition is an exquisite transformation for this application due to the following properties: (1) non-redundant multi-resolution representation; (2) sparse representation of piecewise smooth signals; (3) decorrelation of wavelet coefficients. In this talk, we focus on two applications of wavelets to neuro-imaging: (1) detection of coordinated stimuli; (2) revealing functional connectivity. So-called fixed-effects analysis of fMRI data refers to checking the data for a hypothetical response that is modeled as the stimulus function convolved with the hemodynamic response function. Recently, we proposed a framework [1,2] to detect such coordinated stimuli that relies on the spatial discrete wavelet transform and the temporal linear model. Activation patterns are detected by exploiting the spatial correlation and at the same time properly controlling the multiple-hypothesis-testing problem. The data is processed in the wavelet domain (by adaptive thresholding of the wavelet coefficients), and a suitable statistical testing procedure is applied afterwards in the spatial domain. This method is based on conservative assumptions only and has a strong type-I error control by construction; i.e., this allows us to detect activated regions with a prescribed level of confidence. We show experimental results for the analysis of brain activity. Several extensions of our technique are highlighted; e.g., to obtain a quasi-shift-invariant result. Evaluation and comparison of our technique is done by determining receiver-operator-characteristics (ROC) curves from a reproducibility study.

Another active area of neuro-imaging research involves examining functional relationships between spatially remote brain regions. However, one must account for non-neurophysiological background spatial correlation inherent to fMRI data. We developed a non- parametric approach based on statistical resampling in the wavelet domain to generate surrogate data that preserves average background spatial correlation in 3D+T. Interestingly, the polyharmonic wavelet transform with quincunx subsampling matrix is playing a crucial role to compose surrogate datasets with isotropic spectral density in the axial slices. As an example, we apply our resampling technique to determine significant functional connectivity from resting state and motor task fMRI datasets.

References: [1] D. Van De Ville, T. Blu, M. Unser, 'Surfing the Brain—An Overview of Wavelet-Based Techniques for fMRI Data Analysis,' IEEE Engineering in Medicine and Biology Magazine, vol. 25, no. 2, pp. 65-78, March-April 2006. [2] D. Van De Ville, T. Blu, M. Unser, 'Integrated Wavelet Processing and Spatial Statistical Testing of fMRI Data,' NeuroImage, vol. 23, no. 4, pp. 1472-1485, December 2004. [3] R.S. Patel, D. Van De Ville, F. DuBois Bowman, 'Determining Significant Connectivity by 4D Spatiotemporal Wavelet Packet Resampling of Functional Neuroimaging Data,' NeuroImage, vol. 31, no. 3, pp. 1142-1155, July 2006.

Zeit: am Mittwoch, den 17. Januar 2007 16:00 Uhr

Ort: GSF-Forschungszentrum für Umwelt und Gesundheit Ingolstädter Landstr. 1 85746 Neuherberg/Oberschleißheim Raum : 121/Gebäude 58a , Etage : 1

Prof. V. Andriyevskyy

Department of Mathematical Sciences, Kent State University, Ohio, 17.- 28. July 2006

Prof. Daryl N.Geller - Smooth Compactly Supported Nearly Tight Frames on Stratified Lie Groups

Mathematics Department, Stony Brook University, USA, 25. May/ 08. June 2006,

Workshop: Harmonic Analysis and Applications, June 1-2, 2006, IBB/GSF and TU Munich.

Laurent Condat - Reconstruction of uniform signals and images in shift-invariant spaces: overview and new asymptotically optimal quasi-projection methods

Polytechnical University of Grenoble (INPG), France, 8./9. May 2006

Abstract: The analysis and processing of signals and images inherently rely on the existence of methods for reconstructing a continuous-time signal from a sequence of discrete-time samples. The wide range of applications includes edge detection and all resampling problems. The most common setting considered in the one advocated by Shannon, in which samples are ideal point-values, and the input signal and reconstructed function are bandlimited. To overcome the limitations ot this traditional setting, we consider the more recent and realistic framework in which a continuously defined process is prefiltered prior to sampling at uniform infinitely-many locations, and the samples corrupted by additive noise. This modelizes the output of a linear acquisition device. The reconstruction problem can then be formulated as the estimation of the unknown underlying process given these discrete measurements on it. Looking for a linear reconstruction process amounts to seek a solution lying in a linear shift-invariant functional space. The expansion coefficients that parameterize the solution are obtained by processing the data with a digital prefilter. Once the reconstruction space has been chosen, several strategies are available for designing the prefilter, that differ in their assumptions on the signal and noise. We will present a panorama of these methods, including consistent and minimax reconstruction. We will then investigate new solutions that are asymptotically optimal, in the sense that the approximation error has the maximum decay when the sampling step tends to zero. This approximation-theoretic approach leads to efficient quasi-projecting schemes, that are particularly well-suited for lowpass signals like natural images. We will present the 1D and 2D problems, emphasizing the versatility our our approach, through the case study of reconstruction with multi-dimensional splines on the hexagonal lattice. We will illustrate our approach by practical resampling and image rotation experiments.

Dr. Peter Massopust - Mathematische Probleme verbunden mit einer Klasse zerstörungsfreier Prüfverfahren

Tuboscope Inc., Houston,Texas, 16./17. March 2006

Abstract: Defekte in magnetisierbaren Materialen können durch Messung magnetischer Felddaten gefunden und ausgewertet werden. Die Auswertung dieser Daten für eine Klasse zerstörungsfreier Prüfverfahren beruht zum größten Teil auf einer effektiven Entrauschung. Zum Teil müssen auch unvollständige oder lückenhafte Meßdaten einer approximativen Verbesserung unterzogen werden. Im Vortrag werden auf Wavelets und Curvelets basierende Entrauschungsverfahren präsentiert und anhand experimenteller Meßdaten validiert. Eine Approximativmethode zur Verbesserung inkompletter Datenkollektionen wird vorgestellt.

Stefan Held - Faktorisierung des Gaborframeoperators in C^L

Munich University of Technology, 15. March 2006

Abstract: Gaborframes erlauben die Darstellung der Elemente des L^2(R) durch Limearkombinationen transliertere und modulierter Versionen eines Operatos. Um diese Zerlegung für numerische Berechnungen zu nutzen ist ein endliches diskretes Modell nötig. Dieser Vortrag befasst sich mit Faktorisierungen des Framesoperators von Frames auf CL, die Gaborstruktur haben. Es wird gezeigt, wie diese Faktorisierungen dazu genutzt werden können, den dualen Frame zu finden und somit eine Darstellung der Elemente des CL durch translierte und mudulierte Versionen eines Elemenets ermöglichen.