图像时间离散形变模型的图像外推 - 存在性与应用
Image Extrapolation for the Time Discrete Metamorphosis Model - Existence and Applications
AI总结：
本文提出基于时间离散形变模型的图像外推方法，通过离散指数映射在图像空间中计算离散测地线外推路径，并证明了小时间步长下的局部存在性和唯一性，结合空间Galerkin离散化方法，验证了方法的高效与稳定性。
摘要：
图像空间可以配备一个黎曼度量，该度量同时衡量图像强度的运输成本和运动线上强度的变化。由此得到的形变模型由Trouvé和Younes引入并分析，Berkels等人提出了变分时间离散化方法用于测地线插值。本文扩展了该时间离散模型，并通过离散指数映射一致推导出变分时间离散化下的图像外推方法。对于给定的弱可微初始图像和初始图像变化，指数映射允许在图像空间中计算离散测地线外推路径。证明了该射击方法的时间步长可以仅在相关的变形中进行公式化。对于足够小的时间步长，利用合适的不动点公式和隐函数定理证明了局部存在性和唯一性。采用空间Galerkin离散化方法，使用粗网格上的三次样条函数对变形进行离散化，细网格上的分段双线性有限元对图像强度进行离散化，以推导出一个完全实用的算法。不同应用验证了所提出方法的效率和稳定性。
英文摘要：
The space of images can be equipped with a Riemannian metric measuring both the cost of transport of image intensities and the variation of image intensities along motion lines. The resulting metamorphosis model was introduced and analyzed by Trouvé and Younes, and a variational time discretization for the geodesic interpolation was proposed by Berkels et al. In this paper, this time discrete model is expanded and an image extrapolation via a discrete exponential map is consistently derived for the variational time discretization. For a given weakly differentiable initial image and an initial image variation, the exponential map allows to compute a discrete geodesic extrapolation path in the space of images. It is shown that a time step of this shooting method can be formulated in the associated deformations only. For sufficiently small time steps local existence and uniqueness are proved using a suitable fixed point formulation and the implicit function theorem. A spatial Galerkin discretization with cubic splines on coarse meshes for the deformation and piecewise bilinear finite elements on fine meshes for the image intensities are used to derive a fully practical algorithm. Different applications underline the efficiency and stability of the proposed approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
