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This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate conditions for exact recovery. In particular, we describe how both l 1-minimization and matching pursuit methods can be used to regularize ill-posed problems and moreover, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss extensions of compressed sensing for specific infinite dimensional ill-posed measurement regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the underlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution.

This chapter is devoted to the generalization of the continuous shearlet transform to higher dimensions as well as to the construction of associated smoothness spaces and to the analysis of their structural properties, respectively. To construct canonical scales of smoothness spaces, so-called shearlet coorbit spaces, and associated atomic decompositions and Banach frames we prove that the general coorbit space theory of Feichtinger and Grochenig is applicable for the proposed shearlet setting. For the two-dimensional case we show that for large classes of weights, variants of Sobolev embeddings exist. Furthermore, we prove that for natural subclasses of shearlet coorbit spaces which in a certain sense correspond to “cone-adapted shearlets” there exist embeddings into homogeneous Besov spaces. Moreover, the traces of the same subclasses onto the coordinate axis can again be identified with homogeneous Besov spaces. These results are based on the characterization of Besov spaces by atomic decompositions and rely on the fact that shearlets with compact support can serve as analyzing vectors for shearlet coorbit spaces. Finally, we demonstrate that the proposed multivariate shearlet transform can be used to characterize certain singularities.

The Condensation and the Wavelet Approximated Reduced Vector Machine (W-RVM) approach are joined by the core idea to spend only as much as necessary effort for easy to discriminate regions (Condensation) and measurement locations (W-RVM) of the feature space, but most for regions and locations with high statistical likelihood to contain the object of interest. We unify both approaches by adapting the W-RVM classifier to tracking and refine the Condensation approach. Additionally, we utilize Condensation for abstract multi-dimensional feature vectors and provide a template based tracking of the three-dimensional camera scene. Moreover, we introduce a robust multi-object tracking by extensions to the Condensation approach. The new 3D Cascaded Condensation Tracking (CCT) for multiple objects yields a more than 10 times faster tracking than state-of-art detection methods. In our experiments we compare different tracking approaches using an active dual camera system for face tracking.

Publisher Summary This chapter discusses several iterative strategies for solving inverse problems in the context of signal and image processing. The chapter focuses on problems for which it is reasonable to assume that the solution has a sparse expansion with respect to a wavelet basis or frame. In each case, a variational formulation of the problem is presented, and an iteration scheme for which iterates approximate the solution is constructed. Surrogate functionals are applied; the corresponding strategy is shown to converge in norm and to regularize the problem. The chapter begins with the concrete problem of simultaneously denoising, decomposing, and deblurring a given image. The associated variational formulation of the problem contains terms that promote sparsity and smoothness. The process to transform problem is presented in the chapter, such as the basic method of Daubechies and all. In a second example, a natural extension to vector-valued inverse problems is discussed. Potential applications include seismic or astrophysical data decomposition/reconstruction and color image reconstruction. The illustration presented contains audio data coding. In the linear case, and under fairly general assumptions on the constraint, it is proved that weak convergence of the iterative scheme always holds. In certain cases (i.e, for special families of convex constraints), this weak convergence implies norm convergence. The presented technique covers a wide range of problems. The chapter also discusses image restoration problems in which Besov- or bounded variation (BV) constraints are involved. The chapter concludes with a sketch design of hybrid wavelet-partial differential equation (PDE) image restoration schemes (i.e, with variational problems that contain wavelet and BV constraints).

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted Lp-space. Moreover, we show that a judicious discretization of the representation leads to an atomic decomposition and to Banach frames for these coorbit spaces.

The starting point for this paper is the well known equivalence between convolution filtering with a rescaled Gaussian and the solution of the heat equation. In the first sections we analyze the equivalence between multiscale convolution filtering, linear smoothing methods based on continuous wavelet transforms and the solutions of linear diffusion equations. I.e. we determine a wavelet ψ, resp. a convolution filter φ, which is associated with a given linear diffusion equation ut = Pu and vice versa. This approach has an extension to non-linear smoothing techniques. The main result of this paper is the derivation of a differential equation, whose solution is equivalent to non-linear multi-scale smoothing based on soft shrinkage methods applied to Fourier or continuous wavelet transforms.

The common way to process radar wind profiler (RWP) data by moments estimation of the Fourier power spectrum fails in presence of transient intermittent clutter contributions. Wavelets are especially suitable for detecting and removing transient components because of their high localization in time and frequency domain. We give an overview on the wavelet filtering of contaminated discrete RWP signals and introduce a new technique involving the wavelet packet decomposition and a splitting in progressive and regressive signal components. This technique has been successfully tested on severely real-data sets where classical wavelet routines fail.

We discuss a wavelet based treatment of variational problems arising in the context of image processing, inspired by papers of Vese-Osher and Osher-Sole-Vese, in particular, we introduce a special class of variational functionals, that induce a decomposition of images in oscillating and cartoon components. Cartoons are often modeled by BV functions. In the setting of Vese et.el. and Osher et.al. the incorporation of BV penalty terms leads to PDE schemes that are numerically intensive. We propose to embed the problem in a wavelet framework. This provides us with elegant and numerically efficient schemes even though a basic requirement, the involvement of the space BV , has to be softened slightly. We show results on test images of our wavelet algorithm with a B11 (L1) penalty term, and we compare them with the BV restorations of Osher-Sole-Vese.

Generalized sampling is a new framework for sampling and reconstruction in infinite-dimensional Hilbert spaces. Given measurements (inner products) of an element with respect to one basis, it allows one to reconstruct in another, arbitrary basis, in a way that is both convergent and numerically stable. However, generalized sampling is thus far only valid for sampling and reconstruction in systems that comprise bases. Thus, in the first part of this paper we extend this framework from bases to frames, and provide fundamental sampling theorems for this more general case. The second part of the paper is concerned with extending the idea of generalized sampling to the solution of inverse and ill-posed problems. In particular, we introduce two generalized sampling frameworks for such problems, based on regularized and non-regularized approaches. We furnish evidence of the usefulness of the proposed theories by providing a number of numerical experiments.

In this paper, we are concerned with compressive sampling strategies and sparse recovery principles for linear inverse and ill-posed problems. As the main result, we provide compressed measurement models for ill-posed problems and recovery accuracy estimates for sparse approximations of the solution of the underlying inverse problem. The main ingredients are variational formulations that allow the treatment of ill-posed operator equations in the context of compressively sampled data. In particular, we rely on Tikhonov variational and constrained optimization formulations. One essential difference to the classical compressed sensing framework is the incorporation of joint sparsity measures allowing the treatment of infinite-dimensional reconstruction spaces. The theoretical results are furnished with a number of numerical experiments.

This paper is concerned with the construction of an iterative algorithm to solve nonlinear inverse problems with an l1 constraint on x. One extensively studied method to obtain a solution of such an l1 penalized problem is iterative soft-thresholding. Regrettably, such iteration schemes are computationally very intensive. A subtle alternative to iterative soft-thresholding is the projected gradient method that was quite recently proposed by Daubechies et al (2008 J. Fourier Anal. Appl. 14 764–92). The authors have shown that the proposed scheme is indeed numerically much thriftier. However, its current applicability is limited to linear inverse problems. In this paper we provide an extension of this approach to nonlinear problems. Adequately adapting the conditions on the (variable) thresholding parameter to the nonlinear nature, we can prove convergence in norm for this projected gradient method, with and without acceleration. A numerical verification is given in the context of nonlinear and non-ideal sensing. For this particular recovery problem we can achieve an impressive numerical performance (when comparing it to non-accelerated procedures).

In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear ill-posed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber's method is numerically very intensive. Therefore we propose an adaptive variant of Landweber's iteration that may reduce the computational expense significantly, i.e. leading to a compressed version of Landweber's iteration. We borrow the concept of adaptivity that was primarily developed for well-posed operator equations (in particular, for elliptic PDE's) essentially exploiting the concept of wavelets (frames), Besov regularity, best N-term approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber's iteration. In combination with an adequate refinement/stopping rule (a priori as well as a posteriori principles) we prove that the proposed procedure is a regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.

This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider "generalized" localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.