Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling stochastic evolution of human organ shapes, and in geometric mechanics for modelling turbulence parts of multi-scale fluid flows. Recently introduced models involve stochastic differential equations that govern the dynamics of a diffusion process $X$ where, in applications, $X$ is only partially observed at times $0$ and $T>0$. Conditional on these observations, the challenge is to infer parameters of the dynamics of the diffusion and to reconstruct the path $(X_t,\, t\in [0,T])$. The latter problem is known as bridge simulation. We develop a general scheme for bridge simulation in the case of finite dimensional systems of shape landmarks and singular solutions in fluid dynamics. The scheme allows for subsequent statistical inference of properties of the evolution of the shapes. We show how the approach covers stochastic landmark models for which no suitable simulation method has been proposed in the literature; that it removes restrictions of earlier approaches; that it improves the handling of the nonlinearity of the configuration space leading to more effective sampling schemes; and that it allows to generalise the common inexact matching scheme to the stochastic setting.
翻译:用于模拟人体器官形状变化的形状分析和计算解剖以及用于模拟多尺度流流中动荡部分的几何力学研究。最近引入的模型涉及控制扩散过程动态的随机分方程式,在应用中,仅部分观测到X美元,时间为0美元和0.美元。这些观测的前提条件是,如何推导扩散动态参数,并重建路径$(X_t,\\,t\in,[0,T]),后者被称为桥梁模拟。我们开发了一个通用的桥架模拟方案,在形状标志的有限维系系统和流体动态的独有解决办法的情况下,我们开发了一个通用的桥架式模拟方案。这个方案允许随后对形状演变特性进行统计推导。我们展示了该方法如何涵盖没有在文献中提出适当模拟方法的随机里程碑模型;它消除了早先方法的限制;它改进了对配置空间的非线性处理方法的处理,从而能够形成更有效的模拟方案。