A Geometric Framework for Stochastic Shape Analysis

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-018-9394-z A Geometric Framework for Stochastic Shape Analysis Alexis Arnaudon1 · Darryl D. Holm1 · Stefan Sommer2 Received: 30 March 2017 / Revised: 17 January 2018 / Accepted: 5 April 2018 © The Author(s) 2018 Abstract We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the large deformation diffeomorphic metric mapping framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finitedimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker–Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker– Planck equation to sample moments of the data, while the second approach employs an expectation-maximization based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise. Keywords Shape analysis · Stochastic flows of diffeomorphisms · Stochastic landmark dynamics · Stochastic geometric mechanics Mathematics Subject Classification 60G99 · 70H99 · 65C30 Communicated by Endre Suli. B Alexis Arnaudon B Stefan Sommer 1 Department of Mathematics, Imperial College, London SW7 2AZ, UK 2 Department of Computer Science (DIKU), University of Copenhagen, 2100 Copenhagen E, Denmark 123 Foundations of Computational Mathematics 1 Introduction In this work, we aim at modelling variability of shapes using a theory of stochastic perturbations consistent with the action of the diffeomorphism group underlying the large deformation diffeomorphic metric mapping framework (LDDMM, see [65]). In applications, such variability arises and can be observed, for example, when human organs are influenced by disease processes, as analysed in computational anatomy [66]. Spatially independent white noise contains insufficient information to describe these large-scale variabilities of shapes. In addition, the coupling of the spatial correlations of the noise must be adapted to a variety of transformation properties of the shape spaces. The theory developed here addresses this problem by introducing spatially correlated transport noise which respects the geometric structure of the data. This method provides a new way of characterizing stochastic variability of shapes using spatially correlated noise in the context of the standard LDDMM framework. We will show that this specific type of noise can be used for all the data structures to which the LDDMM framework applies. The LDDMM theory was initiated by [6,12,19,46,60] based on the pattern theory of [23]. LDDMM models the dynamics of shapes by the action of diffeomorphisms (smooth invertible transformations) on shape spaces. It gives a unified approach to shape modelling and shape analysis that is valid for a range of structures such as landmarks, curves, surfaces, images, densities or even tensor-valued images. For any such data structure, the optimal shape deformations are described via the Euler–Poincaré equation of the diffeomorphism group, usually referred to as the EPDiff equation [26,27,66]. In this work, we will show how to obtain a stochastic EPDiff equation valid for any data structure, and in particular for the finite-dimensional spaces of landmarks. For this, we will follow the LDDMM derivation in [8] based on geometric mechanics [24,43]. This view is based on the existence of momentum maps, which are characterized by the transformation properties of the data structures for images and shapes. These momentum maps persist in the process of introducing noise into the EPDiff equation, and they thereby preserve most of the technology developed for shape analysis in the deterministic context and in computational anatomy. This work is not the first to consider stochastic evolutions in LDDMM. Indeed, [61,64] and more recently [44] have already investigated the possibility of stochastic perturbations of landmark dynamics. In these works, the noise is introduced into the momentum equation, as though it was an external random force acting on each landmark independently. In [44], an extra dissipative force was added to balance the energy input from the noise and to make the dynamics correspond to a certain type of heat bath used in statistical physics. Refs. [55,56] considered evolutions on the landmark manifold with stochastic parts being Brownian motion with respect to a Riemannian metric and estimated parameters of the models from observed data. Here, we will introduce Eulerian noise directly into the reconstruction relation used to find the deformation flows from the velocity fields, which are solutions of the EPDiff equation [26,65]. As we will see, this derivation of stochastic models is compatible with variational principles, preserves the momentum map structure and yields a stochastic EPDiff equation with a novel type of multiplicative noise, depending on the gradient of the solution, as well as its magnitude. This model is based on the previous works [2,25], where, 123 Foundations of Computational Mathematics (a) (b) (c) Fig. 1 In this figure, we compare the deterministic evolution of landmarks arranged in an ellipse (black line) with a translated ellipse as final position (black dashed line), to two different stochastically perturbed evolutions. The radius for the landmark kernel is twice their average initial distances. In blue is the stochastic perturbation developed in this paper. The black dots represent the J Eulerian noise fields arranged in a grid configuration. In magenta is the evolution resulting from additive noise in the momentum equation, different for each landmark but with the same amplitude as the Eulerian noise. We run three initial value simulations to compare the limit of a large number of landmarks and small noise correlation. The Eulerian noise model (blue) is robust to the continuum limit and can reproduce the general behaviour of the additive noise model. Furthermore, the choice of the noise fields provides an additional freedom in parameterization which will be studied and exploited in this work. a Low resolution and large noise correlation (100 landmarks, 6 × 6 noise fields), b high resolution and large noise correlation (200 landmarks, 6 × 6 noise fields), c high resolution and small noise correlation (200 landmarks, 12 × 12 noise fields) (Color figure online) respectively, stochastic perturbations of infinite- and fi (...truncated)