Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View

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 non parametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms.

[1]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[2]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[3]  Aaron Strauss.  Introduction to Optimal Control Theory , 1968 .

[4]  J. Gallego,et al.  INTRODUCTION TO OPTIMAL CONTROL THEORY , 1985 .

[5]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[6]  J. N. Kapur.  Maximum-entropy models in science and engineering , 1992 .

[7]  Lyle Noakes,et al.  Cubic Splines on Curved Spaces , 1989 .

[8]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[9]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[10]  Peter E. Crouch,et al.  Dynamic interpolation and application to flight control , 1991 .

[11]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[12]  P. Crouch,et al.  The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces , 1995 .

[13]  P. Crouch,et al.  Splines of class Ck on non-euclidean spaces , 1995 .

[14]  K. Mardia,et al.  Statistical Shape Analysis , 1998 .

[15]  Michael I. Miller,et al.  Landmark matching via large deformation diffeomorphisms , 2000, IEEE Trans. Image Process..

[16]  L. Younes,et al.  On the metrics and euler-lagrange equations of computational anatomy. , 2002, Annual review of biomedical engineering.

[17]  R. Giambò,et al.  An analytical theory for Riemannian cubic polynomials , 2002 .

[18]  L. Younes,et al.  Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching , 2004, CVPR 2004.

[19]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[20]  Alain Trouvé,et al.  Local Geometry of Deformable Templates , 2005, SIAM J. Math. Anal..

[21]  Alain Trouvé,et al.  Geodesic Shooting for Computational Anatomy , 2006, Journal of Mathematical Imaging and Vision.

[22]  Anuj Srivastava,et al.  Maximum-Likelihood Estimation of Biological Growth Variables , 2005, EMMCVPR.

[23]  Alain Trouvé,et al.  Geodesic Shooting and Diffeomorphic Matching Via Textured Meshes , 2005, EMMCVPR.

[24]  D. Doman,et al.  Optimal Control Problems on Parallelizable Riemannian Manifolds: Theory and Applications , 2006 .

[25]  Anuj Srivastava,et al.  Characterization of biological growth using iterated diffeomorphisms , 2006, 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006..

[26]  D. Mumford,et al.  An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach , 2006, math/0605009.

[27]  Alain Trouvé,et al.  Modeling Planar Shape Variation via Hamiltonian Flows of Curves , 2006, Statistics and Analysis of Shapes.

[28]  P. Thomas Fletcher,et al.  Population Shape Regression from Random Design Data , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[29]  Y. Amit,et al.  Towards a coherent statistical framework for dense deformable template estimation , 2007 .

[30]  S. Vadlamani.  On the Diffusion of Shape , 2007 .

[31]  Anuj Srivastava,et al.  A Pattern-Theoretic Characterization of Biological Growth , 2007, IEEE Transactions on Medical Imaging.

[32]  Ali R. Khan,et al.  Representation of time-varying shapes in the large deformation diffeomorphic framework , 2008, 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro.

[33]  Alain Trouvé,et al.  The Euler-Poincare theory of Metamorphosis , 2008, ArXiv.

[34]  Guido Gerig,et al.  Spatiotemporal Atlas Estimation for Developmental Delay Detection in Longitudinal Datasets , 2009, MICCAI.

[35]  Edward R. Vrscay,et al.  Direct Estimation of Biological Growth Properties from Image Data Using the "GRID" Model , 2009, ICIAR.

[36]  Darryl D. Holm,et al.  Hamiltonian Approach to Shape Spaces in a Diffeomorphic Framework : from the Discontinuous Image Matching Problem to a Stochastic Growth Model , 2009 .

[37]  Michael I. Miller,et al.  Evolutions equations in computational anatomy , 2009, NeuroImage.