TY - JOUR
T1 - Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds
AU - Effland, Alexander
AU - Neumayer, Sebastian
AU - Rumpf, Martin
PY - 2020/4/6
Y1 - 2020/4/6
N2 - Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouv\'e, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouv\'e and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198] 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 diffusion tensor imaging (DTI), 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 [S. Neumayer, J. Persch, and G. Steidl, SIAM J. Imaging Sci., 11 (2018), pp. 1898--1930] based on the general variational time discretization proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. 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 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.
AB - Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouv\'e, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouv\'e and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198] 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 diffusion tensor imaging (DTI), 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 [S. Neumayer, J. Persch, and G. Steidl, SIAM J. Imaging Sci., 11 (2018), pp. 1898--1930] based on the general variational time discretization proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. 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 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.
KW - Hadamard manifolds
KW - Image morphing
KW - Manifold-valued images
KW - Metamorphosis
KW - Shape space
KW - Variational time discretization
UR - http://www.scopus.com/inward/record.url?scp=85087411279&partnerID=8YFLogxK
U2 - 10.1137/19M1247073
DO - 10.1137/19M1247073
M3 - Article
SN - 1936-4954
VL - 13
SP - 557
EP - 588
JO - SIAM Journal on Imaging Sciences
JF - SIAM Journal on Imaging Sciences
IS - 2
ER -