In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider TV and TGV regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results for synthetic and real data to show the potential of the proposed schemes for applications.
The first definition of the Riemannian center of mass can be traced back to Fr 'echet; a lot of work has been done by Karcher @cite_74 . For details including an historic overview we refer to @cite_75 ; see also @cite_55 . Due to its use as a means of averaging in a manifold, it is employed as a basic building block in various works; e.g., @cite_37 @cite_16 @cite_9 as well as many references in the paragraph above. Various contributions deal with the numerical computation of the Riemannian center of mass; see, for instance @cite_82 @cite_13 @cite_47 .
@cite_13 @cite_37 @cite_55 @cite_9 @cite_74 @cite_47 @cite_16 @cite_75 @cite_82
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Let be a complete Riemannian manifold and a probability measure on . Assume . We derive a new bound (in terms of , the injectivity radius of and an upper bound on the sectional curvatures of ) on the radius of a ball containing the support of which ensures existence and uniqueness of the global Riemannian center of mass with respect to . A significant consequence of our result is that under the best available existence and uniqueness conditions for the so-called local'' center of mass, the global and local centers coincide. In our derivation we also give an alternative proof for a uniqueness result by W. S. Kendall. As another contribution, we show that for a discrete probability measure on , under the existence and uniqueness conditions, the (global) center of mass belongs to the closure of the convex hull of the masses. We also give a refined result when is of constant curvature. We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere ( S ^ n-1 R ^ n ) with ( n ), and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Cote, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Cote, Commun. Pure Appl. Math. 69(4):609–612, 2016; Cote, Am. J. Math. 137(1):139–207, 2015; , Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Riviere (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.