Probabilistic models for shapes as continuous curves

Journal of Mathematical Imaging and Vision

Volumn 33, Issue 1, 2009, Pages 39-65

  Kim, Jeong Gyoo ^a   Noble, J Alison ^a   Brady, J Michael ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Average over a function space; Fr chet mean; Shape space; Wiener measure space

Indexed keywords

FUNCTIONAL ANALYSIS; FUNCTIONS; IMAGE PROCESSING; PROBABILITY DENSITY FUNCTION;

AVERAGE OVER A FUNCTION SPACE; DATA SETS; EIGENSTRUCTURE; EUCLIDEAN SPACES; GAUSSIANS; PLANAR SHAPES; PROBABILISTIC MODELS; SAMPLE SETS; SHAPE DEFORMATIONS; SHAPE MODELS; SHAPE SPACE; SHAPE VARIATIONS; SPARSE SETS; WIENER MEASURE SPACE;

DEFORMATION;

EID: 58149484996     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-008-0104-3     Document Type: Article

References (54)

1. Bartle, R.G.: The Elements of Real Analysis. Wiley, New York (1976)
2. R. Beichel H. Bischof F. Leberl M. Sonka 2005 Robust active appearance models and their application to medical image analysis IEEE Trans. Med. Imaging 24 9 1151 1169
3. R. Berthilsson K. Åström 1999 Extension of affine shape J. Math. Imaging Vis. 11 119 136
4. F. Bookstein 1986 Size and shape spaces for landmark data in two dimensions Stat. Sci. 1 2 181 221
5. C. Brechbühler G. Gerig O. Kübler 1995 Parameterisation of closed surfaces for 3-D shape description CVGIP: Image Underst. 61 154 170
6. Cootes, T.F., Taylor, C.J.: Statistical models of appearance for computer vision. Technical Report, University of Manchester, Manchester M13 9PT, UK (2004)
7. T.F. Cootes C.J. Taylor D.H. Cooper J. Graham 1995 Active shape models-their training and application Comput. Vis. Image Underst. 61 1 38 59
8. Davies, R.H., Cootes, T.F., Taylor, C.J.: A minimum description length approach to statistical shape modelling. In: IPMI 2001. LNCS, vol. 2082, pp. 50-63. Springer, Berlin (2001)
9. R.H. Davies C. Twining T.F. Cootes C.J. Taylor 2002 A minimum description length approach to statistical shape modelling IEEE Trans. Med. Imaging 21 525 537
10. I.L. Dryden K.V. Mardia 1991 General shape distributions in a plane Adv. Appl. Probab. (SGSA) 23 259 276
11. Ericsson, A., Åström, K.: An affine invariant deformable shape representation for general curves. In: ICCV 2003 (2003)
12. Fletcher, P.T., Joshi, S., Lu, C., Pizer, S.M.: Gaussian distributions on Lie groups and their application to statistical shape analysis. In: Taylor, C.J., Noble, J.A. (eds.) IPMI 2003. LNCS, vol. 2732. Springer, Berlin (2003)
13. P.T. Fletcher C. Lu S.M. Pizer S. Joshi 2004 Principal geodesic analysis for the study of nonlinear statistics of shape IEEE Trans. Med. Imaging 23 8 995 1005
14. M. Fréchet 1948 Les éléments aléatoires de nature quelconque dans un espace distancié Ann. Inst. Henri Poincaré 10 215 310
15. Gerig, G., Styner, M.: Shape vs size: improved understanding of the morphology of brain structures. In: MICCAI 2001. LNCS, vol. 2208. Springer, Berlin (2001)
16. J.C. Gower 1975 Generalised Procrustes analysis Psychometrika 40 1 33 51
17. Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman's Operational Calculus. Oxford University Press, London (2000)
18. H. Karcher 1977 Riemannian center of mass and mollifier smoothing Commun. Pure Appl. Math. 30 509 541
19. A. Kelemen G. Székely G. Gerig 1999 Elastic model-based segmentation of 3-D neuroradiological data sets IEEE Trans. Med. Imaging 18 10 828 839
20. D.G. Kendall 1984 Shape-manifolds, procrustes metrics, and complex projective spaces Bull. Lond. Math. Soc. 16 81 121
21. Kim, J.-G.: Probabilistic shape models: application to medical images. Ph.D. thesis, Oxford University (2005)
22. E. Klassen A. Srivastava W. Mio S.H. Joshi 2004 Analysis of planar shapes using geodesic paths on shape spaces IEEE Trans. PAMI 26 3 372 383
23. A.C.W. Kotcheff C.J. Taylor 1988 Automatic construction of eigenshape models by direct optimization Med. Image Anal. 2 4 303 314
24. A. Kume H. Le 2000 Estimating Fréchet means in Bookstein's shape space Adv. Appl. Probab. (SGSA) 32 663 674
25. H. Le 1995 Mean size-and-shapes and mean shapes: a geometric point of view Adv. Appl. Probab. (SGSA) 27 44 55
26. H. Le 1998 On the consistency of Procrustean mean shape Adv. Appl. Probab. (SGSA) 30 53 63
27. H. Le 2001 Locating Fréchet means with application to shape spaces Adv. Appl. Probab. (SGSA) 33 324 338
28. H. Le A. Kume 2000 The Fréchet mean shape and the shape of the means Adv. Appl. Probab. (SGSA) 32 101 113
29. K.V. Mardia I.L. Dryden 1989 Shape distributions for landmark data Adv. Appl. Probab. (SGSA) 21 742 755
30. D. Meier E. Fisher 2002 Parameter space warping: shape-based correspondence between morphologically different objects IEEE Trans. Med. Imaging 21 1 31 47
31. Mumford, D.: The problem of robust shape descriptors. In: 1st ICCV 1987, pp. 602-606. IEEE (1987)
32. X. Pennec 2006 Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements J. Math. Imaging Vis. 25 1 127 154
33. X. Pennec N. Ayache 1998 Uniform distribution, distance and expectation problems for geometric features processing J. Math. Imaging Vis. 9 46 67
34. X. Pennec J.-P. Thirion 1997 A framework for uncertainty and validation of 3D registration methods based on points and frames Int. J. Comput. Vis. 25 3 203 229
35. S.M. Pizer D. Eberly D.S. Fritsch 1998 Zoom-invariant vision of figural shape: the mathematics of cores Comput. Vis. Image Underst. 69 1 55 71
36. S.M. Pizer D.S. Fritsch P.A. Yushkevich V.E. Johnson E.L. Chaney 1999 Segmentation, registration and measurement of shape variation via image object shape IEEE Trans. Med. Imaging 18 10 851 865
37. Reed, M., Simon, B.: Functional Analysis. Methods of Modern Mathematical Physics, vol. 1. Academic Press, New York (1980)
38. Royden, H.L.: Real Analysis, 3rd edn. Macmillan, New York (1988)
39. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)
40. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
41. Small, C.G.: The Statistical Theory of Shape. Springer, Berlin (1996)
42. G. Sparr 1992 Depth computations from polyhedral images Image Vis. Comput. 10 10 683 688
43. Sparr, G.: A common framework for kinetic depth, reconstruction and motion for deformable objects. In: European Conf. on Computer Vision. LNCS, vol. 801, pp. 471-482. Springer, Berlin (1994)
44. Sparr, G.: Euclidean and affine structure/motion for uncalibrated cameras from affine shape and subsidiary information. In: SMILE Workshop (1998)
45. L.H. Staib J.S. Duncan 1992 Boundary finding with parametrically deformable models IEEE Trans. PAMI 14 11 1061 1075
46. Styner, M., Gerig, G.: Medial models incorporating object variability for 3D shape analysis. In: IPMI 2001. LNCS, vol. 2082. Springer, Berlin (2001)
47. G. Székely A. Kelemen C. Brechbühler G. Gerig 1996 Segmentation of 2D and 3D object from MRI volume data using constrained elastic deformations of flexible Fourier contour and surface models Med. Image Anal. 1 1 19 34
48. D. Terzopoulos D. Metaxas 1991 Dynamic 3D models with local and global deformations: deformable superquadrics IEEE Trans. in PAMI 13 7 703 714
49. Thodberg, H.H.: Minimum description length shape and appearance models. In: IPMI 2003. LNCS, vol. 2732, pp. 51-62. Springer, Berlin (2003)
50. Thompson, D.: On Growth and Form. Cambridge University Press, Cambridge (1961)
51. Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. SIAM (1997)
52. Troutman, J.L.: Variational Calculus and Optimal Control, 2nd edn. Springer, New York (1996)
53. Twining, C., Marsland, S.: Constructing diffeomorphic representations of non-rigid registrations of medical images. In: Taylor, C.J., Noble, J.A. (eds.) IPMI 2003. LNCS, vol. 2732, pp. 413-425. Springer, Berlin (2003)
54. Yeh, J.: Stochastic Processes and the Wiener Integral. Pure and Applied Mathematics, vol. 13. Marcel Dekker, New York (1973)