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In \cite{AV99}, Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in \cite{AV99} by providing a complete and detailed proof.

The increasing demand for renewable energy encourages the installation of wind turbine sites for power generation across Europe, thus supporting the important energy transition, but also having a negative impact on weather radar measurements near wind turbine sites. In recent years, the fast construction, expansion and repowering of wind parks have been a major source of concern for the weather radar community and meteorological services. Among others because wind turbines are extremely tall, reflective, and moving objects, which make them a source of interference that is hard to distinguish from meteorological echoes and therefore difficult to filter and even more difficult to correct. Polarimetric C-Band Doppler weather radar measurements enable us to analyse and understand the impact of wind turbine interference on meteorological weather radar echoes and to build up knowledge.The main idea is to analyse the raw IQ-data in order to quantify the behaviour of wind turbine interference with meteorological scattering. As a first step in this direction, this paper will focus on the derivation and analysis of radar moments such as Reflectivity (Z), Differential Reflectivity (ZDR), Differential Phase (PHIDP), and Mean Doppler Velocity (V). We will consider two cases: (i) events with precipitation, and (ii) events without precipitation, in order to understand and model the impact of wind turbine interference.For this purpose, weather radar measurements from Deutscher Wetterdienst (DWD), recorded under the aegis of the RIWER project (Removing the Influence of Wind Park Echoes in Weather Radar Measurements), are presented, analysed and discussed in detail.

The overall goal consists in understanding and overcoming the influence of weather radar returns that stem from wind energy turbines. We tackle this problem on two different levels. First, we model and simulate radar raw data. The goal is to understand the echo characteristics from wind turbine returns and to use this knowledge to construct classifiers that are able to discriminate between “clean” and “contaminated” radar echoes. If possible, raw data-based “decontamination”- routines will be applied. The second approach relies on the analysis of higher moment data. In this approach we are faced with incomplete spatial-temporal data, namely with data gaps that are caused by strong wind turbine backscatter. The goal is to develop and to apply so-called “gap-infilling” routines that deliver physically feasible higher moment recoveries for the missing parts of the data. Mathematically the first approach relies on sophisticated Fourier analysis and classification/learning theory. The second approach involves techniques from inverse problems, sparse recovery principles, partial differential equation based infilling and optical flow.

This paper deals with the analysis of the sampling setup for Doppler profilers aiming at the determination of vertical profiles of the wind. An explicit solution for the retrieval of mean wind vectors under the assumption of local homogeneity is presented for the case of a symmetric velocity–azimuth display sampling, and a stability analysis is performed. Furthermore, the explicit solution allows a detailed investigation of the propagation of radial wind measurement errors on the retrieved wind vector.

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Frechet space $$\mathcal T$$ of functions on G, which generalizes the classical choice $$\mathcal T=L_w^1(G)$$ . Our basic example is $$ \mathcal T=\bigcap _{p\in (1,+\infty )} L^p(G)$$ , or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrodingerlets.

This chapter is concerned with recent progress in the context of coorbit space theory. Based on a square-integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decompositions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this chapter, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.

In this paper, we apply wavelet thresholding for removing automatically ground and intermittent clutter (airplane echoes) from wind profiler radar data. Using the concept of discrete multi-resolution analysis and non-parametric estimation theory, we develop wavelet domain thresholding rules, which allow us to identify the coefficients relevant for clutter and to suppress them in order to obtain filtered reconstructions.Key words. Meteorology and atmospheric dynamics (instruments and techniques) – Radio science (remote sensing; signal processing)

This paper deals with the analysis of the sampling setup for Doppler profilers aiming at the determination of vertical profiles of the wind. An explicit solution for the retrieval of mean wind vectors under the assumption of local homogeneity is presented for the case of a symmetric Velocity-Azimuth-Display sampling and a stability analysis is performed. Furthermore, the explicit solution allows a detailed investigation of the propagation of radial wind measurement errors on the retrieved wind vector.

This papers examines structural properties of the recently developed shearlet coorbit spaces in higher dimensions. We prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone in three dimensions into Besov spaces. The results are based on general atomic decompositions of Besov spaces. Furthermore, we establish trace results for these subspaces with respect to the coordinate planes. It turns out that in many cases these traces are contained in lower dimensional shearlet coorbit spaces.

The inversion of the one-dimensional Radon transform on the rotation group SO ( 3 ) is an ill-posed inverse problem that can be applied to X-ray tomography with polycrystalline materials. This paper is concerned with the development of a method to stably approximate the inverse of the noisy Radon transform on SO ( 3 ) . The proposed approach is composed by basic building blocks of the coorbit theory on homogeneous spaces, Gabor frame constructions and variational principles for sparse recovery. The performance of the finally obtained iterative approximation is studied through several experiments.

We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real axes.

This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full $n$-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.

In this paper, a novel method for reducing the runtime complexity of a support vector machine classifier is presented. The new training algorithm is fast and simple. This is achieved by an over-complete wavelet transform that finds the optimal approximation of the support vectors. The presented derivation shows that the wavelet theory provides an upper bound on the distance between the decision function of the support vector machine and our classifier. The obtained classifier is fast, since a Haar wavelet approximation of the support vectors is used, enabling efficient integral image-based kernel evaluations. This provides a set of cascaded classifiers of increasing complexity for an early rejection of vectors easy to discriminate. This excellent runtime performance is achieved by using a hierarchical evaluation over the number of incorporated and additional over the approximation accuracy of the reduced set vectors. Here, this algorithm is applied to the problem of face detection, but it can also be used for other image-based classifications. The algorithm presented, provides a 530-fold speedup over the support vector machine, enabling face detection at more than 25 fps on a standard PC.

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.

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.

This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness spaces that can be identified with α-modulation spaces.

This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different channels of the solution exhibit a common or so-called joint sparsity pattern encoding special characteristics of the function under consideration (e.g. a coupling of non-vanishing channel components). Quite recently, an iterative strategy for linear inverse problems with such joint sparsity constraints was presented. Here, we develop an iterative approach for nonlinear inverse problems for which we show norm convergence and regularization properties. The focus throughout the paper is in the context of color image inpainting/recolorization.

We develop a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel-based methods. Copyright © 2007 John Wiley & Sons, Ltd.

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted Lp space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S1 we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.

We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algo- rithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively mod- erate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex con- straints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov- or BV-restoration for which we present also numerical experiments in the context of image processing.

This paper is concerned with linear inverse problems where the solution is assumed to have a sparse expansion with respect to several bases or frames. We were mainly motivated by the following two different approaches: (1) Jaillet and Torresani [F. Jaillet, B. Torresani, Time–frequency jigsaw puzzle: Adaptive multi-window and multi-layered Gabor expansions, preprint, 2005] and Molla and Torresani [S. Molla, B. Torresani, A hybrid audio scheme using hidden Markov models of waveforms, Appl. Comput. Harmon. Anal. (2005), in press] have suggested to represent audio signals by means of at least a wavelet for transient and a local cosine dictionary for tonal components. The suggested technology produces sparse representations of audio signals that are very efficient in audio coding. (2) Also quite recently, Daubechies et al. [I. Daubechies, M. Defrise, C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004) 1413–1541] have developed an iterative method for linear inverse problems that promote a sparse representation for the solution to be reconstructed. Here in this paper, we bring both ideas together and construct schemes for linear inverse problems where the solution might then have a sparse representation (we also allow smoothness constraints) with respect to several bases or frames. By a few numerical examples in the field of audio and image processing we show that the resulting method works quite nicely.

We shall be concerned with the construction of Tikhonov-based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On the basis of the minimizers of the replacement functional, we introduce an iterative algorithm that converges towards a critical point of the Tikhonov functional, and under additional assumptions for the nonlinear operator F, to a global minimizer. Combining our iterative strategy with an appropriate parameter selection rule, we obtain convergence and convergence rates. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (single photon emission computerized tomography) problem.

Inspired by papers of Vese–Osher [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and Osher–Sole–Vese [Image decomposition and restoration using total variation minimization and the H −1 norm, Technical Report 02-57, 2002] we present a wavelet-based treatment of variational problems arising in the field of image processing. In particular, we follow their approach and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an appropriate ‘noise’ component. In the setting of [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H −1 norm, Technical Report 02-57, 2002], the cartoon component of an image is modeled by a BV function; the corresponding incorporation of BV penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing the BV penalty term by a B 1 1 ( L 1 ) term (which amounts to a slightly stronger constraint on the minimizer), and writing the problem in a wavelet framework, we obtain elegant and numerically efficient schemes with results very similar to those obtained in [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H −1 norm, Technical Report 02-57, 2002]. This approach allows us, moreover, to incorporate general bounded linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, deblurring and denoising.

This paper is concerned with the construction of a data-driven parameter rule which is essential in the context of regularization methods for inverse problems. In particular, we consider the in- version problem of Sobolev embedding operators. For a Fourier- and wavelet-based treatment of this problem we refer the reader to our previous paper Regularization of Sobolev Embedding Operators and Applications Part I: Fourier and Wavelet based Methods1. Here we focus on developing a related data-driven scheme for an optimal choice of the regularization parameter. We prove order optimal convergence of the resulting regularization scheme, and, moreover, show by two examples the applicability of the proposed method.

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be defined. Moreover, we can construct a specific reproducing kernel which, after a judicious discretization, gives rise to atomic decompositions for these coorbit spaces. Furthermore, we show that under certain additional conditions our discretization method generates Banach frames. We also discuss nonlinear approximation schemes based on the atomic decomposition. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.

This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the narrowband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). Following the approach of Naparst and Rebolla-Neira et al., we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula can be derived, provided that the set of outgoing signals forms a frame.

In this paper we present a smoothing technique for noisy data by the inversion of Sobolev embedding operators. We assume that the true data x belong to a Sobolev space Hs with known smoothness index s, and that the measurements xδ of x belong to L2 (or to Ht with t< s) only. Our main goal is to compute a Hs approximation of the true data. We will use Tikhonov regularization or the stable inversion of the embedding operator, and show how wavelet and Fourier methods can be effectively used to compute a pointwise approximation to the inverse embedding operator. In the forthcoming paper Regularization of Sobolev Embedding Operators and Applications Part II: Data Driven Regularization and Applications we will propose a data driven rule for the regularization parameter and we prove order optimal convergence of the resulting regularization (smoothing) scheme.

This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the general wideband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). In this case, the echo is given by an inverse wavelet transform of the density D where the role of the analyzing wavelet is played by the transmitted signal. However, the null space of an inverse wavelet transform is nontrivial, it is described by the corresponding reproducing kernel. Following the approach of Naparst [14] and Rebolla-Neira et al. [16], we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula for one- and 2-dimensional reflectivity densities can be derived, provided that the set of outgoing signals forms an orthogonal basis or – more general – a frame. We also present some rigorous error estimates for these reconstruction formulas. The theoretical results are confirmed by some numerical examples.

In this chapter, we will provide a comprehensive overview of shearlet coorbit theory. We will present an almost self-contained introduction into coorbit theory which is the basis for all our investigations. We also discuss the group theoretical background of the continuous shearlet transform, and we explain how the shearlet transform can be combined with coorbit theory. By proceeding this way we can establish new smoothness spaces, the shearlet coorbit spaces. The structure of these spaces will be discussed in detail. In particular, we derive density, embedding, and trace results for the shearlet coorbit spaces.

The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized sampling to inverse and ill-posed problems. And third, the combination of generalized sampling with sparse recovery techniques. This final contribution leads to a theory and set of methods for infinite-dimensional compressed sensing, or as we shall also refer to it, compressed sensing over the continuum.