Martin Storath

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We consider an extension of the 1-D concept of analytical wavelet to <i>n</i>-D which is by construction compatible with rotations. This extension, called a monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined(More)
We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals(More)
We propose a fast splitting approach to the classical variational formulation of the image partitioning problem, which is frequently referred to as the Potts or piecewise constant Mumford–Shah model. For vector-valued images, our approach is significantly faster than the methods based on graph cuts and convex relaxations of the Potts model which are(More)
We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford–Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments(More)
We give a methodology-oriented perspective on directional image analysis and rotation-invariant processing. We review the state of the art in the field and make connections with recent mathematical developments in functional analysis and wavelet theory. We unify our perspective within a common framework using operators. The intent is to provide(More)
This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notion of(More)
We consider total variation (TV) minimization for manifold-valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with -type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds.(More)
We reconsider the continuous curvelet transform from a signal processing point of view. We show that the analyzing elements of the curvelet transform, the curvelets, can be understood as analytic signals in the sense of the partial Hilbert transform. We then generalize the usual curvelets by the monogenic curvelets, which are analytic signals in the sense(More)
We consider L1-TV regularization of univariate signals with values on the real line or on the unit circle. While the real data space leads to a convex optimization problem, the problem is nonconvex for circle-valued data. In this paper, we derive exact algorithms for both data spaces. A key ingredient is the reduction of the infinite search spaces to a(More)
Magnetic particle imaging (MPI) is an emerging medical imaging modality which is based on the non-linear response of magnetic nanoparticles to an applied magnetic field. It is an important feature of MPI that even fast dynamic processes can be captured for 3D volumes. The high temporal resolution in turn leads to large amounts of data which have to be(More)