This paper addresses the morphing of manifold-valued images based on the timediscrete geodesic paths model of Berkels, Effland and Rumpf 2015. Although forour manifold-valued setting such an interpretation of the energy functional isnot available so far, the model is interesting on its own. We prove theexistence of a minimizing sequence within the set of $L^2(\Omega,\mathcal{H})$images having values in a finite dimensional Hadamard manifold $\mathcal{H}$together with a minimizing sequence of admissible diffeomorphism. To this end,we show that the continuous manifold-valued functions are dense in$L^2(\Omega,\mathcal{H})$. We propose a space discrete model based on a finitedifference approach on a staggered grid, where we focus on the linearizedelastic potential in the regularizing term. The numerical minimizationalternates between i) the computation of a deformation sequence between givenimages via the parallel solution of certain registration problems formanifold-valued images, and ii) the computation of an image sequence with fixedfirst (template) and last (reference) frame based on a given sequence ofdeformations. Numerical examples give a proof of the concept of our ideas.
展开▼
机译：本文基于Berkels，Effland和Rumpf 2015的时间离散测地路径模型，对流形值图像的变形进行了处理。尽管到目前为止，尚无关于能量函数的流形值设置这样的解释，但该模型本身很有趣。我们证明了在具有有限维Hadamard流形$ \ mathcal {H} $的值的$ L ^ 2（\ Omega，\ mathcal {H}）$ images集合中最小化序列的存在以及可容许的亚纯性的最小化序列。为此，我们证明了连续流形值函数在$ L ^ 2（\ Omega，\ mathcal {H}）$中是密集的。我们在交错网格上基于有限差分方法提出了一个空间离散模型，其中我们关注正则化项中的线性弹性势。数值最小化在以下两者之间进行：i）通过对形成配准值图像的某些配准问题的并行求解来计算给定图像之间的变形序列，以及ii）基于a来计算固定第一个（模板）和最后一个（参考）帧的图像序列给定变形顺序。数值示例证明了我们思想的概念。
展开▼
