图像空间中的时间离散测地路径
Time Discrete Geodesic Paths in the Space of Images
AI总结：
本文基于metamorphosis方法将图像空间视为黎曼流形，提出了一种稳健有效的变分时间离散测地路径方法，证明了离散测地路径的存在性及Γ收敛性，并通过有限元和交替下降法进行数值模拟。
摘要：
本文考虑将图像空间作为黎曼流形，利用metamorphosis方法，其中底层黎曼度量同时测量图像运输和强度变化的成本。提出了一种稳健有效的变分时间离散测地路径方法，要求最小化由连续图像匹配函数组成离散路径能量。对于平方可积输入图像，证明了离散测地路径的存在性，即作为该变分问题的最小化器。进一步证明了底层离散路径能量对连续路径能量的Γ收敛性，包括诱导运输的微分同胚性质以及在空间和时间中的平方可积弱材料导数的存在性。通过有限元空间离散化结合图像强度映射和匹配变形的交替下降方案，提出近似离散测地路径的数值方法。计算结果验证了所提方法的有效性，并展示了重要的定性性质。
英文摘要：
In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach, where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $Γ$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.
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