Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

Abstract

In Functional Data Analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called Functions on Surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries.A set of FoSs, such as the ones in Figure 1, can be mathematically formulated as a collection of pairs tpM i , Y i q : i " 1, . . . , nu. The collection tM i : i " 1, . . . , nu is a set of topologically equivalent smooth two-dimensional manifolds, embedded in R 3 , representing the geometry of the data. The functional aspect of the data is represented by the collection tY i : i " 1, . . . , nu, where Y i is an element of the function space L 2 pM i q, i.e. the Hilbert space of square integrable functions on M i with respect to the area measure.Here, we propose a statistical generative model for FoSs, modelled in terms of mathematically more tractable objects. To this end, we define a deformation operator ϕ, such that ϕ v : R 3 Ñ R 3 is parametrized by the elements of a Hilbert space tv : v P Vu. Moreover, we assume ϕ v is an homomorphism of R 3 for all v P V and that ϕ 0 pxq " x for all x P R 3 . For each v P V, ϕ v : R 3 Ñ R 3 represents a deformation of the space R 3 , which means that when ϕ v is applied to a point x P R 3 this is relocated to the location ϕ v pxq P R 3 . In addition, ϕ v being a homomorphism of R 3 implies that, for a fixed v P V, there is a one-to-one correspondence between each element x P R 3 and the relocated element ϕ v pxq P R 3 .Moreover, we introduce M 0 , a smooth two-dimensional manifold topologically equivalent to tM i u, which represents a fixed template geometric object. Given a FoS, the geometric template together with the deformation operator offers an alternative representation of the geometry of the FoS in hand as: ϕ v˝M0 , for a particular choice of v P V.Here, ϕ v˝M0 is the geometric object obtained by deforming M 0 through the map ϕ v , and specifically, by relocating each point x P M 0 to the new location ϕ v pxq, to resemble the target manifold. For this reason, we will informally say that the element v P V encodes the geometry, or the shape, of a FoS, as in fact v defines the deformation ϕ v , which defines the geometry ϕ v˝M0 . The choice of the deformation operator is driven by the particular problem in hand. We first introduce the generative model and subsequently discuss different choices of this operator.

Modeling Time-Varying Random Objects and Dynamic Networks

“…Eigenfunctions φ j can be interpreted as coordinate directions, thereby providing the basis for principal modes of variation of the subject specific oracle Fréchet variance trajectories around the population Fréchet variance trajectory. Modes of variation (Castro et al 1986;Lila and Aston 2019) are useful to quantify the departure of a random object trajectory from the Fréchet mean function. The eigenfunctions can be viewed as modes of outlyingness of the subject-specific trajectories.…”

Functional random effects modeling of brain shape and connectivity

“…In our application setting, this step is generally performed by first inflating the surfaces to a spherical domain, and then aligning the maps projected onto the spherical domain in order to increase a measure of structural/functional 'coherence' across subjects, while minimizing the amount of distortion introduced by the registration (Fischl et al, 1999;Yeo et al, 2010;Robinson et al, 2014. A registration model that instead is able to deal with maps mapped on general manifold domains has been proposed in Lila and Aston (2019).…”

Technology for High-Sensitivity Analysis of Medical Diagnostic Images

“…The use of statistical models [6][7][8] to identify and analyze the trends of small changes in the images of anatomical objects is unacceptable for the following reasons:…”