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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 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)
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 sub-problems 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 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 p-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 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)
a r t i c l e i n f o a b s t r a c t MSC: 42C40 94A12 44A15 Keywords: Wavelet signature Complex wavelets Signal analysis Hilbert transform Phase Feature detection Randomized wavelet coefficients Salient feature We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is(More)
We consider L 1-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)
Mumford–Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even(More)