19 research outputs found
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
Mazur intersection property for Asplund spaces
The main result of the present note states that it is consistent with the ZFC
axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund
space of density character has a renorming with the Mazur
intersection property. Combined with the previous result of Jim\' enez and
Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain
that the MIP normability of Asplund spaces of density is undecidable
in ZFC.Comment: 6 page
Point Simpliciality in Choquet's Theory
Katedra matematické analýzyDepartment of Mathematical AnalysisFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult
Point Simpliciality in Choquet's Theory
Katedra matematické analýzyDepartment of Mathematical AnalysisFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult
Alternating projections in CAT(0) spaces
By using recently developed theory which extends the idea of weak convergence into CAT(0) space we prove the convergence of the alternating projection method for convex closed subsets of a CAT(0) space. Given the right notion of weak convergence it turns out that the generalization of the well-known results in Hilbert spaces is straightforward and allows the use of the method in a nonlinear setting. As an application, we use the alternating projection method to minimize convex functionals on a CAT(0) space