Directions and projective shapes

Annals of Statistics

Volumn 33, Issue 4, 2005, Pages 1666-1699

  Mardia, Kanti V ^a   Patrangenaru, Vic ^b  

^a UNIVERSITY OF LEEDS   (United Kingdom)

^b TEXAS TECH UNIVERSITY   (United States)

Author keywords

Bootstrapping; Directional statistics; Equivariant embedding; Extrinsic means; Machine vision; Projective frame; Projective shape space; Projective transformations; Shape analysis; Tangent approximation

Indexed keywords

EID: 23744439307     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053605000000273     Document Type: Article

References (29)

1. BERAN, R. and FISHER, N. I. (1998). Nonparametric comparison of mean directions or mean axes. Ann. Statist. 26 472-493.
2. BHATTACHARYA, R. N. and PATRANGENARU, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1-29.
3. BHATTACHARYA, R. N. and PATRANGENARU, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 1225-1259.
4. BOOKSTEIN, F. L. (1991). Morphometric Tools for Landmark Data. Cambridge Univ. Press.
5. DRYDEN, I. L. and MARDIA, K. V. (1998). Statistical Shape Analysis. Wiley, Chichester.
6. FAUGERAS, O. and LUONG, Q.-T. (2001). The Geometry of Multiple Images. MIT Press, Cambridge, MA.
7. FERGUSON, T. (1996). A Course in Large Sample Theory. Chapman and Hall, London.
8. FISHER, N. I., HALL, P., JING, B.-Y. and WOOD, A. T. A. (1996). Improved pivotal methods for constructing confidence regions with directional data. J. Amer. Statist. Assoc. 91 1062-1070.
9. FRÉCHET, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. II. Poincaré 10 215-310.
10. GOODALL, C. R. and MARDIA, K. V. (1993). Multivariate aspects of shape theory. Ann. Statist. 21 848-866.
11. GOODALL, C. R. and MARDIA, K. V. (1999). Projective shape analysis. J. Comput. Graph. Statist. 8 143-168.
12. HARTLEY, R. and ZISSERMAN, A. (2000). Multiple View Geometry in Computer Vision. Cambridge Univ. Press.
13. HEYDEN, A. (1995). Geometry and algebra of multiple projective transformations. Ph.D. dissertation, Univ. Lund, Sweden.
14. KENDALL, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16 81-121.
15. KENDALL, D. G., BARDEN, D., CARNE, T. K. and LE, H. (1999). Shape and Shape Theory. Wiley, New York.
16. KENT, J. T. (1992). New directions in shape analysis. In The Art of Statistical Science. A Tribute to G. S. Watson (K. V. Mardia, ed.) 115-127. Wiley, New York.
17. MARDIA, K. V. (1975). Statistics of directional data (with discussion). J. Roy. Statist. Soc. Ser. B 37 349-393.
18. MARDIA, K. V. and JUPP, P. E. (2000). Directional Statistics. Wiley, New York.
19. MARDIA, K. V. and PATRANGENARU, V. (2002). Directions and projective shapes. Technical Report No. 02/04, Dept. Statistics, Univ. Leeds.
20. MAYBANK, S. J. and BEARDSLEY, P. A. (1994). Classification based on the cross ratio. Applications of Invariance in Computer Vision. Lecture Notes in Comput. Sci. 825 (J. L. Mundy, A. Zisserman and D. Forsyth, eds.) 433-472. Springer, Berlin.
21. PRENTICE, M. J. (1984). A distribution-free method of interval estimation for unsigned directional data. Biometrika 71 147-154.
22. PRENTICE, M. J. and MARDIA, K. V. (1995). Shape changes in the plane for landmark data. Ann. Statist. 23 1960-1974.
23. SINGH, H., HNIZDO, V. and DEMCHUK, E. (2002). Probabilistic model for two dependent circular variables. Biometrika 89 719-723.
24. SPARR, G. (1992). Depth computations from polyhedral images. Image and Vision Computing 10 683-688.
25. SPIVAK, M. (1970). A Comprehensive Introduction to Differential Geometry. Publish or Perish, Boston.
26. SRIVASTAVA, A. and KLASSEN, E. (2002). Monte Carlo extrinsic estimators of manifold-valued parameters. IEEE Trans. Signal Process. 50 299-308.
27. TRICERRI, F. and VANHECKE, L. (1983). Homogeneous Structures on Riemannian Manifolds. Cambridge Univ. Press.
28. WATSON, G. S. and WILLIAMS, E. (1956). On the construction of significance tests on the circle and the sphere. Biometrika 43 344-352.
29. ZIEZOLD, H. (1977). On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Trans. Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians A 591-602. Czechoslovak Academy of Sciences, Prague.