Time Discrete Geodesic Paths in the Space of Images
Time Discrete Geodesic Paths in the Space of Images
复制标题
DOI：
10.1137/140970719
复制
发表时间：
2015-01-01
影响因子：
2.1
通讯作者：
Rumpf, M.
中科院分区：
文献类型：
--
作者：
Berkels, B.;Effland, A.;Rumpf, M.
在本文中，使用变形方法将图像空间视为黎曼流形（参见 [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61-84；A. Trouve and L. Younes, SIAM J. Math. Anal., 37 (2005), pp. 17-59；以及 A. Trouve and L. Younes, Found. Comput. Math., 5 (2005), pp. 173-198]），其中基础黎曼度量同时测量图像传输和强度变化的成本。提出了一种鲁棒有效的测地线路径变分时间离散化方法。这需要最小化由一组图像强度图和成对匹配变形上的连续图像匹配函数的总和组成的离散路径能量。对于平方可积输入图像，显示了定义为该变分问题的最小化器的离散连接测地路径的存在。此外，证明了底层离散路径能量到连续路径能量的伽玛收敛性。这包括诱导输运的微分同胚性质以及空间和时间中平方可积弱材料导数的存在。提出了通过有限元与图像强度图组和匹配变形组中的交替下降方案相结合的空间离散化，以数字方式近似离散测地线路径。计算结果强调了所提出方法的效率并展示了重要的定性特性。
In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach (see [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61-84; A. Trouve and L. Younes, SIAM J. Math. Anal., 37 (2005), pp. 17-59; and A. Trouve and L. Younes, Found. Comput. Math., 5 (2005), pp. 173-198]), 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 minimizing 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, Gamma-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.