814 research outputs found
Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
This paper concerns an extension of discrete gradient methods to
finite-dimensional Riemannian manifolds termed discrete Riemannian gradients,
and their application to dissipative ordinary differential equations. This
includes Riemannian gradient flow systems which occur naturally in optimization
problems. The Itoh--Abe discrete gradient is formulated and applied to gradient
systems, yielding a derivative-free optimization algorithm. The algorithm is
tested on two eigenvalue problems and two problems from manifold valued
imaging: InSAR denoising and DTI denoising.Comment: Post-revision version. To appear in SIAM Journal on Scientific
Computin
A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images
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
Total Generalized Variation for Manifold-valued Data
In this paper we introduce the notion of second-order total generalized
variation (TGV) regularization for manifold-valued data in a discrete setting.
We provide an axiomatic approach to formalize reasonable generalizations of TGV
to the manifold setting and present two possible concrete instances that
fulfill the proposed axioms. We provide well-posedness results and present
algorithms for a numerical realization of these generalizations to the manifold
setup. Further, we provide experimental results for synthetic and real data to
further underpin the proposed generalization numerically and show its potential
for applications with manifold-valued data
Left-invariant evolutions of wavelet transforms on the Similitude Group
Enhancement of multiple-scale elongated structures in noisy image data is
relevant for many biomedical applications but commonly used PDE-based
enhancement techniques often fail at crossings in an image. To get an overview
of how an image is composed of local multiple-scale elongated structures we
construct a multiple scale orientation score, which is a continuous wavelet
transform on the similitude group, SIM(2). Our unitary transform maps the space
of images onto a reproducing kernel space defined on SIM(2), allowing us to
robustly relate Euclidean (and scaling) invariant operators on images to
left-invariant operators on the corresponding continuous wavelet transform.
Rather than often used wavelet (soft-)thresholding techniques, we employ the
group structure in the wavelet domain to arrive at left-invariant evolutions
and flows (diffusion), for contextual crossing preserving enhancement of
multiple scale elongated structures in noisy images. We present experiments
that display benefits of our work compared to recent PDE techniques acting
directly on the images and to our previous work on left-invariant diffusions on
orientation scores defined on Euclidean motion group.Comment: 40 page
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