Hartmut Führ

Abstract harmonic analysis of continuous wavelet transforms

Introduction.- Wavelet Transforms and Group Representations.- The Plancherel Transform for Locally Compact Groups.- Plancherel Inversion and Wavelet Transforms.- Admissible Vectors for Group Extension.- Sampling Theorems for the Heisenberg Group.- References.- Index.

Wavelet frames and admissibility in higher dimensions

This paper is concerned with the relations between discrete and continuous wavelet transforms on k‐dimensional Euclidean space. We start with the construction of continuous wavelet transforms with the help of square‐integrable representations of certain semidirect products, thereby generalizing results of Bernier and Taylor. We then turn to frames of L2(Rk) and to the question, when the functions occurring in a given frame are admissible for a given continuous wavelet transform. For certain frames we give a characterization which generalizes a result of Daubechies to higher dimensions.

Simultaneous estimates for vector-valued Gabor frames of Hermite functions

We derive frame bound estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides estimates for windows composed of Hermite functions. The proof is based on a recently established sampling theorem for the simply connected Heisenberg group, which is translated to a family of frame bound estimates via a direct integral decomposition.

An adaptive algorithm for the detection of microcalcifications in simulated low-dose mammography

This paper uses the task of microcalcification detection as a benchmark problem to assess the potential for dose reduction in x-ray mammography. We present the results of a newly developed algorithm for detection of microcalcifications as a case study for a typical commercial film-screen system (Kodak Min-R 2000/2190). The first part of the paper deals with the simulation of dose reduction for film-screen mammography based on a physical model of the imaging process. Use of a more sensitive film-screen system is expected to result in additional smoothing of the image. We introduce two different models of that behaviour, called moderate and strong smoothing. We then present an adaptive, model-based microcalcification detection algorithm. Comparing detection results with ground-truth images obtained under the supervision of an expert radiologist allows us to establish the soundness of the detection algorithm. We measure the performance on the dose-reduced images in order to assess the loss of information due to dose reduction. It turns out that the smoothing behaviour has a strong influence on detection rates. For moderate smoothing. a dose reduction by 25% has no serious influence on the detection results. whereas a dose reduction by 50% already entails a marked deterioration of the performance. Strong smoothing generally leads to an unacceptable loss of image quality. The test results emphasize the impact of the more sensitive film-screen system and its characteristics on the problem of assessing the potential for dose reduction in film-screen mammography. The general approach presented in the paper can be adapted to fully digital mammography.

Admissible vectors for the regular representation

It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admissible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group G with type I regular representation λ G , we show that λ G itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff G is nonunimodular.

Heart rate estimation on a beat-to-beat basis via ballistocardiography - a hybrid approach

We present an algorithm for obtaining the heart rate from the signal of a single, contact-less sensor recording the mechanical activity of the heart. This vital parameter is required on a beat-to-beat basis for applications in sleep analysis and heart failure disease management. Our approach bundles information from various sources for first robust estimates. These estimates are further refined in a second step. An unambiguous comparison with the ECG RR-intervals taken as reference is possible for 98.5% of the heart beats. In these cases, a mean absolute error of 17 ms for the inter-beat interval lengths has been achieved, over a test corpus of 20 whole nights.

Continuous wavelet transforms with Abelian dilation groups

This paper is devoted to the construction of wavelet (or coherent state) systems arising from the action of certain semidirect products G=Rk⋊H on L2(Rk). For this purpose previous results which guarantee the existence of inversion formulas are applied to the special case where H is Abelian. The questions of systematic construction and conjugacy of such groups are completely resolved by setting up a correspondence to unit groups of commutative associative algebras. As an application the numbers of conjugacy classes of possible Abelian groups are computed for k=2,3,4. For k⩾7, there are uncountably many conjugacy classes. We then compute the admissibility conditions belonging to Abelian groups. The final section contains a characterization of Abelian matrix groups acting ergodically on some subset of Rk. This result ensures that the approach via associative algebras yields all possible groups.

Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space 𝐵 𝑠 𝑝 , 𝑞 in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces 𝐵 𝑠 𝑝 , 𝑞 with 1 ≤ 𝑝 , 𝑞 ∞ and 𝑠 ∈ ℝ .

Coorbit spaces and wavelet coefficient decay over general dilation groups

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L 2 (R d ). It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbitspace norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted L p,q -estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

Poincaré and Plancherel--Polya Inequalities in Harmonic Analysis on Weighted Combinatorial Graphs

We prove Poincare and Plancherel--Polya inequalities for weighted