Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds

Abstract

Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes and coworkers [39,53] casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g. in DTI imaging the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented in [42] based on the general variational time discretization proposed in [13]. Here, we prove the Mosco-convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function -which characterizes Hadamard manifolds -is a crucial ingredient to establish existence of the metamorphosis model.

Image Morphing in Deep Feature Spaces: Theory and Applications

“…is identical to the corresponding reasoning in [13]. At this point, we also observe that the velocity field w K with…”

“…is uniformly bounded in L 2 ((0, 1), V) and thus converges weakly in L 2 ((0, 1), V) to some limit velocity field v. This fact is again proved following the corresponding reasoning as in [13].…”

“…holds true, which follows from Korn's inequality and the Poincaré inequality. Thus, the lemma follows by combining (11), (12), (13) and (14).…”

“…In [13,Section 3], the equivalence of this energy functional and (3) in the isotropic case has been shown. Let ψ as above denote the Lagrangian flow map induced by the Eulerian motion field withψ…”

“…(ii) Recovery sequence. Before constructing the recovery sequence, we note that the infimum in (6) is actually attained with an associated pair (v, z) ∈ C( f ), which follows from [13,Proposition 5.3] together with Remark 1.…”