A Novel Riemannian Framework for Shape Analysis of Annotated Surfaces

Kurtek, Sebastian

doi:10.5244/c.29.diffcv.3

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…Almost all studies of these kind presume a preprocessing registration step and so do not consider the inherent variability effects that might be induced by the registration step on the functional measurements. Indeed, this issue goes beyond neuroimaging, as the same techniques are often used in a wide variety of medical imaging settings (Audette et al, 2000), as well as computer vision applications (Zaetz and Kurtek, 2015).…”

Section: Motivating Applicationmentioning

confidence: 99%

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

Lila

Aston

2019

Journal of the American Statistical Association

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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.

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Section: Motivating Applicationmentioning

confidence: 99%

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

Lila

Aston

2019

Journal of the American Statistical Association

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Interpretable discriminant analysis for functional data supported on random nonlinear domains with an application to Alzheimer’s disease

Lila,

Zhang,

Rane Levendovszky

et al. 2024

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We introduce a novel framework for the classification of functional data supported on nonlinear, and possibly random, manifold domains. The motivating application is the identification of subjects with Alzheimer’s disease from their cortical surface geometry and associated cortical thickness map. The proposed model is based upon a reformulation of the classification problem as a regularized multivariate functional linear regression model. This allows us to adopt a direct approach to the estimation of the most discriminant direction while controlling for its complexity with appropriate differential regularization. Our approach does not require prior estimation of the covariance structure of the functional predictors, which is computationally prohibitive in our application setting. We provide a theoretical analysis of the out-of-sample prediction error of the proposed model and explore the finite sample performance in a simulation setting. We apply the proposed method to a pooled dataset from Alzheimer’s Disease Neuroimaging Initiative and Parkinson’s Progression Markers Initiative. Through this application, we identify discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer’s disease that are consistent with the existing neuroscience literature.

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Bayesian Model-Based Automatic Landmark Detection for Planar Curves

Strait

Kurtek

2016

2016 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)

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A Novel Riemannian Framework for Shape Analysis of Annotated Surfaces

Cited by 3 publications

References 21 publications

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

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

Interpretable discriminant analysis for functional data supported on random nonlinear domains with an application to Alzheimer’s disease

Bayesian Model-Based Automatic Landmark Detection for Planar Curves

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