Nonparametric statistics on manifolds with applications to shape spaces

This article presents certain recent methodologies and some new results for the statistical analysis of probability distributions on manifolds. An important example considered in some detail here is the 2-D shape space of k-ads, comprising all configurations of k planar landmarks (k > 2)-modulo translation, scaling and rotation.

[1]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[2]  H. Hopf,et al.  Ueber den Begriff der vollständigen differentialgeometrischen Fläche , 1931 .

[3]  M. Fréchet.  Les éléments aléatoires de nature quelconque dans un espace distancié , 1948 .

[4]  F. Smithies.  Linear Operators , 2019, Nature.

[5]  K. Mardia.  Statistics of Directional Data , 1972 .

[6]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[7]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[8]  H. Ziezold.  On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces , 1977 .

[9]  D. Kendall.  The diffusion of shape , 1977, Advances in Applied Probability.

[10]  J. Ghosh,et al.  ON THE VALIDITY OF THE FORMAL EDGEWORTH EXPANSION , 1978 .

[11]  F. Bookstein,et al.  The Measurement of Biological Shape and Shape Change. , 1980 .

[12]  G. S. Watson.  Statistics on Spheres , 1983 .

[13]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

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

[15]  M. J. Prentice.  A distribution-free method of interval estimation for unsigned directional data , 1984 .

[16]  F. Bookstein.  Size and Shape Spaces for Landmark Data in Two Dimensions , 1986 .

[17]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[18]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[19]  D. Kendall.  A Survey of the Statistical Theory of Shape , 1989 .

[20]  Second Order and $L^p$-Comparisons between the Bootstrap and Empirical Edgeworth Expansion Methodologies , 1989 .

[21]  W. Kendall.  Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence , 1990 .

[22]  P. Hall.  The Bootstrap and Edgeworth Expansion , 1992 .

[23]  Jayanta K. Ghosh,et al.  Higher Order Asymptotics , 1994 .

[24]  D. Kendall.  MORPHOMETRIC TOOLS FOR LANDMARK DATA: GEOMETRY AND BIOLOGY , 1994 .

[25]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[26]  H. Hendriks,et al.  Mean Location and Sample Mean Location on Manifolds , 1998 .

[27]  Nicholas I. Fisher,et al.  Improved pivotal methods for constructing confidence regions with directional data , 1996 .

[28]  C. Small.  The statistical theory of shape , 1996 .

[29]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[30]  N. Fisher,et al.  Nonparametric comparison of mean directions or mean axes , 1998 .

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

[32]  David G. Kendall,et al.  Shape & Shape Theory , 1999 .

[33]  Anders Heyden,et al.  Recognition of Planar Objects Using the Density of Affine Shape , 1999, Comput. Vis. Image Underst..

[34]  T. K. Carne,et al.  Shape and Shape Theory , 1999 .

[35]  H. Le,et al.  Locating Fréchet means with application to shape spaces , 2001, Advances in Applied Probability.

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

[37]  José M. F. Moura,et al.  Affine-permutation symmetry: invariance and shape space , 2003, IEEE Workshop on Statistical Signal Processing, 2003.

[38]  Kanti V. Mardia,et al.  A Shape-based Glaucoma Index for Tomographic Images , 2004 .

[39]  Kalle Åström,et al.  Extension of Affine Shape , 2004, Journal of Mathematical Imaging and Vision.

[40]  Bede Liu,et al.  Image registration by "Super-curves" , 2004, IEEE Transactions on Image Processing.

[41]  Kanti V. Mardia,et al.  Directions and projective shapes , 2005, math/0508280.

[42]  R. Bhattacharya,et al.  Large sample theory of intrinsic and extrinsic sample means on manifolds--II , 2005, math/0507423.

[43]  Jérémie Bigot.  Landmark-Based Registration of Curves via the Continuous Wavelet Transform , 2006 .

[44]  Anuj Srivastava,et al.  Riemannian Structures on Shape Spaces: A Framework for Statistical Inferences , 2006, Statistics and Analysis of Shapes.

[45]  Hamid Krim,et al.  Statistics and Analysis of Shapes , 2006, Modeling and Simulation in Science, Engineering and Technology.

[46]  H. Hendriks,et al.  Asymptotic data analysis on manifolds , 2007, 0708.0474.

[47]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[48]  R. Bhattacharya,et al.  Statistics on Riemannian manifolds: asymptotic distribution and curvature , 2008 .

[49]  Abhishek Bhattacharya.  Nonparametric Inference on Shape Spaces , 2008 .

[50]  Ian L. Dryden,et al.  A multi-dimensional scaling approach to shape analysis , 2008 .

[51]  Vic Patrangenaru,et al.  Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging , 2009, J. Multivar. Anal..

[52]  Abhishek Bhattacharya.  Statistical Analysis on Manifolds : A Nonparametric Approach for Inference on Shape Spaces , 2009 .

[53]  Abhishek Bhattacharya,et al.  Statistics on Manifolds with Applications to Shape Spaces , 2009 .