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Metric Spaces Of Non Positive Curvature

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.

/pdf/a-second-order-non-smooth-variational-model-for-restoring-221k3joheh.pdf

A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

In this paper, an optimization method for nonsmooth locally Lipschitz functions on complete Riemannian manifolds is presented. The method is based on approximating the subdifferential of the cost function at every iteration by the convex hull of transported gradients from tangent spaces at randomly generated nearby points to the tangent space of the current iterate and can hence be seen as a generalization of the well known gradient sampling algorithm to a Riemannian setting. A convergence result is obtained under the assumption that the cost function is bounded below and continuously differentiable on an open set of full measure and that the employed vector transport and retraction satisfy certain conditions, which hold, for instance, for the exponential map and parallel transport. Then with probability one the algorithm produces iterates at which the cost function is differentiable, and each cluster point of the iterates is a Clarke stationary point. Modifications yielding only $\varepsilon$-stationary ...

A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds

We introduce a new nonsmooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization, and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machin...

A Second Order Nonsmooth Variational Model for Restoring Manifold-Valued Images

Over the last decades, the total variation (TV) evolved to one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation (TGV), and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging (MRI), computed tomography (CT), magnetic-resonance positron emission tomography (MR-PET), and electron tomography.

Higher-order total variation approaches and generalisations

We consider the problem of reconstructing an image from noisy and/or incomplete data, where the image/data take values in a metric space X (e.g. R for grayscale, S for the chromaticity component of RGB-images or SPD(3), the set of positive definite 3× 3-Matrices, for Diffusion Tensor Magnetic Resonance Imaging (DT-MRI)). We use the common technique of minimizing a total variation (TV) functional J. After having defined J for arbitrary metric spaces X we will propose an adaption of the Iteratively Reweighted Least Squares (IRLS) algorithm to minimize J. For the case of X being a Hadamard space, such as SPD(n), we prove existence and uniqueness of a minimizer of a regularized functional Jε where ε > 0 and show that these minimizers convergence to a minimizer of J when the regularization parameter ε tends to zero. We show that IRLS can also be applied for X being a half-sphere. For the case of X being a Riemannian manifold we propose to use Newton’s method on Manifolds to numerically compute the minimizer of Jε . To demonstrate our algorithm we present some numerical experiments where we denoise and/or inpaint sphere-valued and SPD-valued images.

Total Variation Regularization by Iteratively Reweighted Least Squares on Hadamard Spaces and the Sphere

Counting unique-sink orientations

We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embe...

pdf/projection-based-finite-elements-for-nonlinear-function-5g4etmhn1y.pdf

Projection-Based Finite Elements for Nonlinear Function Spaces

We give a polynomial-time dynamic programming algorithm for solving the linear complementarity problem with tridiagonal or, more generally, Hessenberg P-matrices. We briefly review three known tractable matrix classes and show that none of them contains all tridiagonal P-matrices.

A Polynomial-Time Algorithm for the Tridiagonal and Hessenberg P-Matrix Linear Complementarity Problem

A polynomial-time algorithm for the tridiagonal and Hessenberg P-matrix linear complementarity problem

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