Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines (original) (raw)
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Abstract. In medical imaging analysis and computer vision, there is a growing interest in analyzing various manifold-valued data including 3D rotations, planar shapes, oriented or directed directions, the Grassmann manifold, deformation field, symmetric positive definite (SPD) matrices and medial shape representations (m-rep) of subcortical structures.
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Shape regression is emerging as an important tool for the statistical analysis of time dependent shapes. In this paper, we develop a new generative model which describes shape change over time, by extending simple linear regression to the space of shapes represented as currents in the large deformation diffeomorphic metric mapping (LDDMM) framework. By analogy with linear regression, we estimate a baseline shape (intercept) and initial momenta (slope) which fully parameterize the geodesic shape evolution. This is in contrast to previous shape regression methods which assume the baseline shape is fixed. We further leverage a control point formulation, which provides a discrete and low dimensional parameterization of large diffeomorphic transformations. This flexible system decouples the parameterization of deformations from the specific shape representation, allowing the user to define the dimensionality of the deformation parameters. We present an optimization scheme that estimates the baseline shape, location of the control points, and initial momenta simultaneously via a single gradient descent algorithm. Finally, we demonstrate our proposed method on synthetic data as well as real anatomical shape complexes.
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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 nonparametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms. Contents
Lecture Notes in Computer Science, 2010
Manifolds are widely used to model non-linearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the non-linear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton end-effectors acquired using a stereo camera and tracking software.
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In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a high order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval.
Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape
IEEE Transactions on Medical Imaging, 2004
A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.