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Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds
SIAM Journal on Imaging Sciences ( IF 2.1 ) Pub Date : 2020-04-06 , DOI: 10.1137/19m1247073 Alexander Effland , Sebastian Neumayer , Martin Rumpf
SIAM Journal on Imaging Sciences ( IF 2.1 ) Pub Date : 2020-04-06 , DOI: 10.1137/19m1247073 Alexander Effland , Sebastian Neumayer , Martin Rumpf
SIAM Journal on Imaging Sciences, Volume 13, Issue 2, Page 557-588, January 2020.
Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé 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.
中文翻译：
Hadamard流形上时间离散变态模型的收敛性
SIAM影像科学杂志，第13卷，第2期，第557-588页，2020年1月。
连续图像变形是图像处理中的经典任务。Trouvé，Younes和同事提出的变态模型[MI Miller和L.Younes，Int。J.计算机 Vis.41（2001），第61--84页; A.Trouvé和L. Younes，找到。计算 Math。（5）（2005），第173--198页]在黎曼几何和图像之间的测地线框架中提出了这个问题。图像空间中的相关度量结合了由粘性流传输图像强度及其沿运动路径的变化所引起的耗散。在许多应用中，图像是从图像域到流形的映射（例如，在扩散张量成像（DTI）中，具有合适的黎曼度量的对称正定矩阵的流形）。在本文中，我们为流形值图像提出了一个广义变形模型，范围空间是有限维Hadamard流形。相应的时间离散版本在[S. Neumayer，J.Persch，and G.Steidl，SIAM J.Imaging Sci。，11（2018），pp.1898--1930]是基于[B. Berkels，A。Effland和M. Rumpf，SIAM J. Imaging Sci。，第8卷，2015年，第1457--1488页]。在这里，我们证明了时间离散变态的Mosco收敛对拟议的流形值变态模型具有功能性，这意味着时间离散测地路径到（时间连续）变态模型中的测地路径的收敛。特别地，建立了测地路径的存在。特别地，建立了测地路径的存在。实际上，映射到Hadamard流形中的图像不仅与应用相关，
更新日期：2020-04-06
Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé 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.
中文翻译：
Hadamard流形上时间离散变态模型的收敛性
SIAM影像科学杂志，第13卷，第2期，第557-588页，2020年1月。
连续图像变形是图像处理中的经典任务。Trouvé，Younes和同事提出的变态模型[MI Miller和L.Younes，Int。J.计算机 Vis.41（2001），第61--84页; A.Trouvé和L. Younes，找到。计算 Math。（5）（2005），第173--198页]在黎曼几何和图像之间的测地线框架中提出了这个问题。图像空间中的相关度量结合了由粘性流传输图像强度及其沿运动路径的变化所引起的耗散。在许多应用中，图像是从图像域到流形的映射（例如，在扩散张量成像（DTI）中，具有合适的黎曼度量的对称正定矩阵的流形）。在本文中，我们为流形值图像提出了一个广义变形模型，范围空间是有限维Hadamard流形。相应的时间离散版本在[S. Neumayer，J.Persch，and G.Steidl，SIAM J.Imaging Sci。，11（2018），pp.1898--1930]是基于[B. Berkels，A。Effland和M. Rumpf，SIAM J. Imaging Sci。，第8卷，2015年，第1457--1488页]。在这里，我们证明了时间离散变态的Mosco收敛对拟议的流形值变态模型具有功能性，这意味着时间离散测地路径到（时间连续）变态模型中的测地路径的收敛。特别地，建立了测地路径的存在。特别地，建立了测地路径的存在。实际上，映射到Hadamard流形中的图像不仅与应用相关，
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