Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View
Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View
复制标题
形状样条和随机形状演化：二阶观点
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Franccois
中科院分区：
文献类型：
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作者：
A. Trouv'e;Franccois
本文提出了一种新的数学框架，对二维或三维形状的时间索引序列进行统计分析。这种统计分析的核心是对这些数据进行时间插值。当前使用的模型可以与一维数据的线性插值进行比较。本文提出了一种与黎曼流形上的三次样条直接相关的样条插值方法。我们的策略包括引入一个控制变量的哈密顿方程的测地线。受时空数据统计建模的启发，我们还设计了一个随机模型来处理随机形状演变。该模型是密切相关的样条模型，因为先前引入的控制变量被设置为一个随机力扰动的演变。虽然我们专注于有限维的情况下的地标，我们的模型可以扩展到无限维的形状空间，他们提供了第一步的非参数增长模型的形状，利用广泛发展的框架大变形的仿射。
This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian equations of the geodesics. Motivated by statistical modeling of spatiotemporal data, we also design a stochastic model to deal with random shape evolutions. This model is closely related to the spline model since the control variable previously introduced is set as a random force perturbing the evolution. Although we focus on the finite dimensional case of landmarks, our models can be extended to infinite dimensional shape spaces, and they provide a first step for a non parametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms.