Intrinsic sample mean in the space of planar shapes

In this paper we consider the shape space as the set of smooth simple closed curves in R 2 (parameterized curves), modulo translations, rotations and scale changes. An algorithm to obtain the intrinsic average of a sample data (set of planar shape realizations), from the identification of the shape space with an infinite dimensional Grassmannian is proposed using a gradient descent type algorithm. A simulation study is carried out to check the performance of the algorithm. HighlightsWe propose an algorithm to obtain the intrinsic mean of sample data (planar shapes).Shape space is identified with a Grassmann manifold of infinite dimension.The Frechet mean for manifolds and the gradient descent algorithm are used.A simulation study is carried out to check the performance of the methodology.

[1]  Cong Zhao,et al.  Plant identification using leaf shapes - A pattern counting approach , 2015, Pattern Recognit..

[2]  Anuj Srivastava,et al.  Analysis of planar shapes using geodesic paths on shape spaces , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Stefano Soatto,et al.  A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering , 2011, SIAM J. Imaging Sci..

[4]  H. Hendriks.  A Crame´r-Rao–type lower bound for estimators with values in a manifold , 1991 .

[5]  Paul Suetens,et al.  Isometric deformation invariant 3D shape recognition , 2012, Pattern Recognit..

[6]  Rama Chellappa,et al.  Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  D. Mumford,et al.  A Metric on Shape Space with Explicit Geodesics , 2007, 0706.4299.

[8]  Ralph R. Martin,et al.  Euclidean-distance-based canonical forms for non-rigid 3D shape retrieval , 2015, Pattern Recognit..

[9]  Ulf Grenander,et al.  On the Shape of Plane Images , 1993, SIAM J. Appl. Math..

[10]  Ilya Molchanov,et al.  Expectations of Random Sets , 2017 .

[11]  Sergio Escalera,et al.  Continuous Generalized Procrustes analysis , 2014, Pattern Recognit..

[12]  R. Bhattacharya,et al.  Nonparametic estimation of location and dispersion on Riemannian manifolds , 2002 .

[13]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[14]  Xavier Pennec,et al.  Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements , 2006, Journal of Mathematical Imaging and Vision.

[15]  Yurii A. Neretin.  On Jordan Angles and the Triangle Inequality in Grassmann Manifolds , 2001 .

[16]  Amelia Simó,et al.  A new extrinsic sample mean in the shape space with applications to the boundaries of anatomical structures , 2015, Biometrical journal. Biometrische Zeitschrift.

[17]  Anuj Srivastava,et al.  On Shape of Plane Elastic Curves , 2007, International Journal of Computer Vision.

[18]  D. Kendall,et al.  The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics , 1993 .

[19]  R. Bhattacharya,et al.  LARGE SAMPLE THEORY OF INTRINSIC AND EXTRINSIC SAMPLE MEANS ON MANIFOLDS—II , 2003 .

[20]  D. Kendall.  SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES , 1984 .

[21]  Andrea C. G. Mennucci,et al.  Geodesics in infinite dimensional Stiefel and Grassmann manifolds , 2012 .

[22]  PennecXavier.  Intrinsic Statistics on Riemannian Manifolds , 2006 .

[23]  Laurent Younes,et al.  Optimal matching between shapes via elastic deformations , 1999, Image Vis. Comput..

[24]  Laurent Younes,et al.  Computable Elastic Distances Between Shapes , 1998, SIAM J. Appl. Math..