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In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regu-larization term is replaced by a one–homogeneous (typically weighted p) penalty on the coefficients (or(More)
Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since it might be difficult to obtain directional information by means of wavelets, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently,(More)
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(More)
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 di-lations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group,(More)
On March 11, 1944, the famous Eremitani Church in Padua (Italy) was destroyed in an Allied bombing along with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60 years, several attempts have been made to restore the fresco fragments by traditional methods, but without much success. One of the authors(More)
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(More)
In this paper, we present a novel algorithm for reducing the runtime computational complexity of a Support Vector Machine classifier. This is achieved by approximating the Support Vector Machine decision function by an over-complete Haar wavelet transformation. This provides a set of classifiers of increasing complexity that can be used in a cascaded(More)
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(More)