A new extrinsic sample mean in the shape space with applications to the boundaries of anatomical structures

Shape analysis is of great importance in many fields of medical imaging and computational biology. In this paper, we consider the shape space as the set of smooth planar immersed curves in R2 (parameterized curves) and, using the property of being isometric to a classical manifold immersed in a Euclidean space, we introduce a new extrinsic sample mean and a new extrinsic variance for a finite set of shapes, which are not necessarily star shaped. This is a fundamental tool in medical image analysis, for instance, to assess uncertainties that arise in locating anatomical structures such as the prostate and the bladder. We apply it to a dataset consisting of parallel planar axial CT sections of human prostate, in order to study the variability between boundaries that have been manually delineated by several observers.

[1]  J. Miller.  Numerical Analysis , 1966, Nature.

[2]  Ralph Roskies,et al.  Fourier Descriptors for Plane Closed Curves , 1972, IEEE Transactions on Computers.

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

[4]  J. Koplowitz,et al.  Design Of Perimeter Estimators For Digitized Planar Shapes , 1988, Other Conferences.

[5]  Alfred M. Bruckstein,et al.  Design of Perimeter Estimators for Digitized Planar Shapes , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

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

[7]  C. Goodall.  Procrustes methods in the statistical analysis of shape , 1991 .

[8]  Robert Azencott,et al.  A distance for elastic matching in object recognition , 1996, Proceedings of 13th International Conference on Pattern Recognition.

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

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

[11]  Mariano Alcañiz Raya,et al.  Outlining of the prostate using snakes with shape restrictions based on the wavelet transform , 1999, Pattern Recognit..

[12]  Anil K. Jain,et al.  Automatic Construction of 2D Shape Models , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

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

[14]  D. Mumford,et al.  Riemannian Geometries on Spaces of Plane Curves , 2003, math/0312384.

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

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

[17]  P. Thomas Fletcher,et al.  Principal geodesic analysis for the study of nonlinear statistics of shape , 2004, IEEE Transactions on Medical Imaging.

[18]  Florence H. Sheehan,et al.  Performance of User Independent Echocardiographic Border Detection Algorithm: Comparison with Human Observer Variability , 2005, The International Journal of Cardiovascular Imaging.

[19]  F Lliso,et al.  Organ contouring for prostate cancer: interobserver and internal organ motion variability. , 2005, Computerized medical imaging and graphics : the official journal of the Computerized Medical Imaging Society.

[20]  Aaron Fenster,et al.  Prostate contouring uncertainty in megavoltage computed tomography images acquired with a helical tomotherapy unit during image-guided radiation therapy. , 2006, International journal of radiation oncology, biology, physics.

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

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

[23]  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..

[24]  J Steve Marron,et al.  Overview of object oriented data analysis , 2014, Biometrical journal. Biometrische Zeitschrift.

[25]  Anuj Srivastava,et al.  Statistical Shape Analysis , 2014, Computer Vision, A Reference Guide.

[26]  Amelia Simó,et al.  The $$k$$k-means algorithm for 3D shapes with an application to apparel design , 2016, Adv. Data Anal. Classif..