Hans-Georg Stark

THE UNCERTAINTY PRINCIPLE ASSOCIATED WITH THE CONTINUOUS SHEARLET TRANSFORM

Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

Continuous wavelet transform and continuous multiscale analysis

Abstract We study a generalization to the continuous wavelet transform of the concept of dyadic multiscale analysis, useful in the analysis and application of orthonormal wavelet decompositions. The resulting notion of scaling and its limitation is discussed and a possible modification is introduced. With this modified notion of scaling the “information loss” which occurs, when analyzing a signal with orthonormal wavelets (i.e., using a discrete subset of the shift dilation plane only) instead of performing the same analysis with continuously varying scale factors and translations, is illustrated by analyzing an example signal.

ON IMAGE RETRIEVAL WITH WAVELETS

In this article a multiscale scheme for image classification is reviewed. It requires a wavelet decomposition of the image and a multiresolution decomposition of object contours. Both procedures are needed for building multiscale feature vectors, which are used for image classification. We describe a classification experiment and deduce from it a simple wavelet based method for retrieving images from image data bases.

Variance based uncertainty principles and minimum uncertainty samplings

Abstract It has been shown recently that the uncertainty measure defined by variance products of subgroup generators, belonging to representations of popular signal transformation groups like, e.g., the wavelet group, can be arbitrarily small such that the search for a single function minimizing this measure becomes obsolete. Following an analysis of “local minima” of such an uncertainty measure, in this letter we introduce minimum uncertainty samplings. Denoting the transformation group with G , these samplings lead to lattices g m ( m ∈ M ) with g m ∈ G such that, starting from a function f , the total uncertainty of the function system { f g m } m ∈ M has the lowest possible value under some given constraints (here f g m denotes the function transformed with g m ). We apply this construction to, e.g., wavelet transforms and related techniques like shearlets.

Fractal graphs and wavelet series

Abstract The potential usefulness of wavelets as basic building blocks for constructing fractal objects by self-similar series is illustrated by two examples.

Adjoint translation, adjoint observable and uncertainty principles

The motivation to this paper stems from signal/image processing where it is desired to measure various attributes or physical quantities such as position, scale, direction and frequency of a signal or an image. These physical quantities are measured via a signal transform, for example, the short time Fourier transform measures the content of a signal at different times and frequencies. There are well known obstructions for completely accurate measurements formulated as “uncertainty principles”. It has been shown recently that “conventional” localization notions, based on variances associated with Lie-group generators and their corresponding uncertainty inequality might be misleading, if they are applied to transformation groups which differ from the Heisenberg group, the latter being prevailing in signal analysis and quantum mechanics. In this paper we describe a generic signal transform as a procedure of measuring the content of a signal at different values of a set of given physical quantities. This viewpoint sheds a light on the relationship between signal transforms and uncertainty principles.In particular we introduce the concepts of “adjoint translations” and “adjoint observables”, respectively. We show that the fundamental issue of interest is the measurement of physical quantities via the appropriate localization operators termed “adjoint observables”. It is shown how one can define, for each localization operator, a family of related “adjoint translation” operators that translate the spectrum of that localization operator. The adjoint translations in the examples of this paper correspond to well-known transformations in signal processing such as the short time Fourier transform (STFT), the continuous wavelet transform (CWT) and the shearlet transform. We show how the means and variances of states transform appropriately under the translation action and compute associated minimizers and equalizers for the uncertainty criterion. Finally, the concept of adjoint observables is used to estimate concentration properties of ambiguity functions, the latter being an alternative localization concept frequently used in signal analysis.

A stationary wavelet transform and a time-frequency based spike detection algorithm for extracellular recorded data

OBJECTIVE Spike detection from extracellular recordings is a crucial preprocessing step when analyzing neuronal activity. The decision whether a specific part of the signal is a spike or not is important for any kind of other subsequent preprocessing steps, like spike sorting or burst detection in order to reduce the classification of erroneously identified spikes. Many spike detection algorithms have already been suggested, all working reasonably well whenever the signal-to-noise ratio is large enough. When the noise level is high, however, these algorithms have a poor performance. APPROACH In this paper we present two new spike detection algorithms. The first is based on a stationary wavelet energy operator and the second is based on the time-frequency representation of spikes. Both algorithms are more reliable than all of the most commonly used methods. MAIN RESULTS The performance of the algorithms is confirmed by using simulated data, resembling original data recorded from cortical neurons with multielectrode arrays. In order to demonstrate that the performance of the algorithms is not restricted to only one specific set of data, we also verify the performance using a simulated publicly available data set. We show that both proposed algorithms have the best performance under all tested methods, regardless of the signal-to-noise ratio in both data sets. SIGNIFICANCE This contribution will redound to the benefit of electrophysiological investigations of human cells. Especially the spatial and temporal analysis of neural network communications is improved by using the proposed spike detection algorithms.

Signal representation, uncertainty principles and localization measures

The following collection of articles addresses one of the most basic problems in signal and image processing, namly the search for function systems (basis, frames, dictionaries) which allow efficient representations of certain classes of signals/images. Such representations are essential for decomposition and synthesis of signals, hence they are at the core of almost any application (coding, compression, pattern matching, feature extraction, classification, etc.) in this field. Accordingly, this is one of the best-studied topics in data analysis and a multitude of different concepts also addressing discretization/algorithmic issues has been investigated in this context. The starting point for reviving activities in this field was a recently rediscovered inconsistency in the concept of constructing optimally localized basis functions by minimizing uncertainty principles. In this short introductory note, we shortly sketch the basic dilemma, which was the starting point for this research approximately three years ago. However, the subsequent investigations presented in this collection of papers cover a much wider range of more general localization measures, discretization concepts as well as discussing algorithmic efficiency and stability.

Bilddatenkompression mit Wavelet—Methoden

The limited capacity of transmission Channels poses a severe restriction to many applications of modern communication technology. In particular sequences of digital images cannot be efhciently transmitted or stored without compression. The aim of this project is to develop, analyse and implement new wavelet techniques for image compression.

Wavelets und Bildarchive

Mit der Steigerung der Leistungsfähigkeit von Personal-Computern und dem Vordringen von digitalem Video und Multimedia-Diensten wie dem World Wide Web ist der Umgang mit multimedialen Daten, insbesondere mit Bildern, auf den Consumer-Markt vorgedrungen. Dies hat dazu geführt, daß die Bildverarbeitung zu einem hochaktuellen und mittlerweile auch kommerziell bedeutungsvollen Thema geworden ist. Wir werden uns in dieser Arbeit insbesondere mit zwei Themen beschäftigen, die für das Management digitaler Bildarchive von entscheidender Bedeutung sind: Ohne Bilddatenkompression ist weder an die Erstellung großer Bildarchive zu denken, noch an die Echtzeitübertragung von Bildsequenzen über konventionelle Computernetze. Andererseits bietet die Bildverarbeitung bei der Erschließung digitaler Bildarchive neue Möglichkeiten, die über Standard-Datenbanktechniken hinausgehen. Wir werden beide Problemkreise und die Beiträge einer neuen und unter dem Namen Wavelet Analysis bekannten Technik der Signalanalyse zu ihrer Lösung illustrieren.