Skip to main content
SpringerLink
Log in

Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy

  • Chapter
Book cover Emerging Trends in Visual Computing (ETVC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5416))

Included in the following conference series:

Abstract

Computational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. The goal is not only to model the normal variations among a population, but also discover morphological differences between normal and pathological populations, and possibly to detect, model and classify the pathologies from structural abnormalities. Applications are very important both in neuroscience, to minimize the influence of the anatomical variability in functional group analysis, and in medical imaging, to better drive the adaptation of generic models of the anatomy (atlas) into patient-specific data (personalization).

However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics and computational methods on objects that do not belong to standard Euclidean spaces. We investigate in this chapter the Riemannian metric as a basis for developing generic algorithms to compute on manifolds. We show that few computational tools derived from this structure can be used in practice as the atoms to build more complex generic algorithms such as mean computation, Mahalanobis distance, interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the joint estimation and anisotropic smoothing of diffusion tensor images and with the modeling of the brain variability from sulcal lines.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allassonnière, S., Trouvé, A., Younes, L.: Geodesic shooting and diffeomorphic matching via textured meshes. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 365–381. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  2. Amari, S.-i.: Differential-geometric methods in Statistics, 2nd corr. edn. Lecture Notes in Statistics, vol. 28. Springer, Heidelberg (1990)

    MATH  Google Scholar 

  3. Andrade, A., Kherif, F., Mangin, J.-F., Worsley, K., Paradis, A.-L., Simon, O., Dehaene, S., Poline, J.-B.: Detection of fMRI activation using cortical surface mapping. Human Brain Mapping 12, 79–93 (2001)

    Article  Google Scholar 

  4. Arnaudon, M.: Barycentres convexes et approximations des martingales continues dans les variétés. In: Yor, M., Azema, J., Meyer, P.A. (eds.) Séminaire de probabilités XXIX. Lect. Notes in Math., vol. 1613, pp. 70–85. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  5. Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A log-Euclidean framework for statistics on diffeomorphisms. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4190, pp. 924–931. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A log-Euclidean polyaffine framework for locally rigid or affine registration. In: Pluim, J.P.W., Likar, B., Gerritsen, F.A. (eds.) WBIR 2006. LNCS, vol. 4057, pp. 120–127. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  7. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the log-Euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications 29(1), 328–347 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine 56(2), 411–421 (2006)

    Article  Google Scholar 

  10. Arsigny, V., Pennec, X., Ayache, N.: Bi-invariant means in lie groups. application to left-invariant polyaffine transformations. Research report rr-5885, INRIA Sophia-Antipolis (April 2006)

    Google Scholar 

  11. Ashburner, J., Friston, K.J.: Voxel-based morphometry - the methods. NeuroImage 11(6), 805–821 (2000)

    Google Scholar 

  12. Aubert, G., Kornprobst, P.: Mathematical problems in image processing - Partial differential equations and the calculus of variations. Applied Mathematical Sciences, vol. 147. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  13. Basser, P.J., Mattiello, J., Le Bihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysical Journal 66, 259–267 (1994)

    Article  Google Scholar 

  14. Batchelor, P., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Magnetic Resonance in Medicine 53, 221–225 (2005)

    Article  Google Scholar 

  15. Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. Journal of Computer Vision 61(2), 139–157 (2005)

    Article  Google Scholar 

  16. Bhatia, R.: On the exponential metric increasing property. Linear Algebra and its Applications 375, 211–220 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bhattacharya, R., Patrangenaru, V.: Nonparametric estimation of location and dispersion on Riemannian manifolds. Journal of Statistical Planning and Inference 108, 23–36 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds, I. Annals of Statistics 31(1), 1–29 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds, II. Annals of Statistics 33(3), 1225–1259 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bierkens, G.N.J.C.: Geometric methods in diffusion tensor regularization. Master’s thesis, Technishe Universiteit Eindhoven, Dept. of Math. and Comp. Sci. (2004)

    Google Scholar 

  21. Bingham, C.: An antipodally symmetric distribution on the sphere. Annals of Statistics 2(6), 1201–1225 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bookstein, F.L.: The Measurement of Biological Shape and Shape Change. Lecture Notes in Biomathematics, vol. 24. Springer, Heidelberg (1978)

    MATH  Google Scholar 

  23. Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)

    Article  Google Scholar 

  24. Brun, A.: Manifolds in Image Science and Visualization. PhD thesis, Linköping University, Linköping Studies in Science and Technology Dissertions No. 1157 (2007)

    Google Scholar 

  25. Burbea, J., Rao, C.R.: Entropy differential metric, distance and divergence measures in probability spaces: a unified approach. Journal of Multivariate Analysis 12, 575–596 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Calvo, M., Oller, J.M.: An explicit solution of information geodesic equations for the multivariate normal model. Statistics and Decisions 9, 119–138 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Chefd’hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 251–265. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  28. Chefd’hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Regularizing flows for constrained matrix-valued images. J. Math. Imaging and Vision 20(1-2), 147–162 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Cohen-Steiner, D., Morvan, J.M.: Restricted delaunay triangulations and normal cycle. In: Proceedings of the nineteenth annual symposium on Computational geometry, pp. 312–321 (2003)

    Google Scholar 

  30. Collins, D.L., Zijdenbos, A.P., Kollokian, V., Sled, J.G., Kabani, N.J., Holmes, C.J., Evans, A.C.: Design and construction of a realistic digital brain phantom. IEEE Transactions on Medical Imaging 17(3), 463–468 (1998)

    Article  Google Scholar 

  31. Darling, R.W.R.: Martingales on non-compact manifolds: maximal inequalities and prescribed limits. Annales de l’institut Poincaré - Probabilités et Statistiques 32(4), 431–454 (1996)

    MathSciNet  MATH  Google Scholar 

  32. Davis, B., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: Proc. of ICCV 2007 (2007)

    Google Scholar 

  33. Dedieu, J.-P., Malajovich, G., Priouret, P.: Newton method on Riemannian manifolds: Covariant alpha-theory. IMA Journal of Numerical Analysis 23, 395–419 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. do Carmo, M.: Riemannian Geometry. Mathematics. Birkhäuser, Boston (1992)

    Book  MATH  Google Scholar 

  35. Dryden, I.L., Koloydenko, A., Zhou, D.: Non-euclidean statistics for covariance matrices with application to diffusion tensor imaging (submitted, 2008)

    Google Scholar 

  36. Dryden, I.L., Mardia, K.V.: Theoretical and distributional aspects of shape analysis. In: Probability Measures on Groups, X (Oberwolfach, 1990), pp. 95–116. Plenum, New York (1991)

    Chapter  Google Scholar 

  37. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. John Wiley, Chichester (1998)

    MATH  Google Scholar 

  38. Durrleman, S., Pennec, X., Trouvé, A., Ayache, N.: A forward model to build unbiased atlases from curves and surfaces. In: Pennec, X., Joshi, S. (eds.) Proc. of the International Workshop on the Mathematical Foundations of Computational Anatomy (MFCA 2008) (September 2008)

    Google Scholar 

  39. Durrleman, S., Pennec, X., Trouvé, A., Ayache, N.: Sparse approximation of currents for statistics on curves and surfaces. In: Metaxas, D., Axel, L., Fichtinger, G., Székely, G. (eds.) MICCAI 2008, Part II. LNCS, vol. 5242. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  40. Durrleman, S., Pennec, X., Trouvé, A., Thompson, P., Ayache, N.: Inferring brain variability from diffeomorphic deformations of currents: an integrative approach. Medical Image Analysis 12(5), 626–637 (2008)

    Article  Google Scholar 

  41. Fisher, R.B., Eggert, D.W., Lorusso, A.: Estimating 3d rigid body transformations: A comparison of four major algorithms. Machine Vision Applications, Special Issue on Performance Characterisitics of Vision Algorithms 9(5/6), 272–290 (1997)

    Google Scholar 

  42. Edelman, A., Arias, T., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM Journal of Matrix Analysis and Applications 20(2), 303–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Emery, M.: Stochastic Calculus in Manifolds. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  44. Emery, M., Mokobodzki, G.: Sur le barycentre d’une probabilité dans une variété. In: Yor, M., Azema, J., Meyer, P.A. (eds.) Séminaire de probabilités XXV. Lect. Notes in Math., vol. 1485, pp. 220–233. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  45. Evans, A.C., Collins, D.L., Mills, S.R., Brown, E.D., Kelly, R.L., Peters, T.M.: 3D statistical neuroanatomical models from 305 MRI volumes. In: Proc. IEEE-Nuclear Science Symposium and Medical Imaging Conference, pp. 1813–1817 (1993)

    Google Scholar 

  46. Facon, D., Ozanne, A., Fillard, P., Lepeintre, J.-F., Tournoux-Facon, C., Ducreux, D.: MR diffusion tensor imaging and fiber tracking in spinal cord compression. American Journal of Neuroradiology (AJNR) 26, 1587–1594 (2005)

    Google Scholar 

  47. Fillard, P., Arsigny, V., Ayache, N., Pennec, X.: A Riemannian framework for the processing of tensor-valued images. In: Fogh Olsen, O., Florack, L.M.J., Kuijper, A. (eds.) DSSCV 2005. LNCS, vol. 3753, pp. 112–123. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  48. Fillard, P., Arsigny, V., Pennec, X., Ayache, N.: Clinical DT-MRI estimation, smoothing and fiber tracking with log-Euclidean metrics. In: Proceedings of the IEEE International Symposium on Biomedical Imaging (ISBI 2006), Crystal Gateway Marriott, Arlington, Virginia, USA, pp. 786–789 (April 2006)

    Google Scholar 

  49. Fillard, P., Arsigny, V., Pennec, X., Ayache, N.: Clinical DT-MRI estimation, smoothing and fiber tracking with log-Euclidean metrics. IEEE Transactions on Medical Imaging 26(11), 1472–1482 (2007)

    Article  Google Scholar 

  50. Fillard, P., Arsigny, V., Pennec, X., Hayashi, K.M., Thompson, P.M., Ayache, N.: Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines. Neuroimage 34(2), 639–650 (2007)

    Article  Google Scholar 

  51. Fillard, P., Arsigny, V., Pennec, X., Thompson, P.M., Ayache, N.: Extrapolation of sparse tensor fields: Application to the modeling of brain variability. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 27–38. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  52. Fillard, P., Pennec, X., Thompson, P.M., Ayache, N.: Evaluating brain anatomical correlations via canonical correlation analysis of sulcal lines. NeuroImage (accepted for publication, 2008)

    Google Scholar 

  53. Fletcher, P.T., Joshi, S. C.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 87–98. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  54. Fletcher, P.T., Joshi, S., Lu, C., Pizer, S.: 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, pp. 450–462. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  55. Fleute, M., Lavallée, S.: Building a complete surface model from sparse data using statistical shape models: Application to computer assisted knee surgery. In: Wells, W.M., Colchester, A.C.F., Delp, S.L. (eds.) MICCAI 1998. LNCS, vol. 1496, pp. 879–887. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  56. Förstner, W., Moonen, B.: A metric for covariance matrices. In: Krumm, F., Schwarze, V.S. (eds.). Qua vadis geodesia..? Festschrift for Erik W. Grafarend on the occasion of his 60th birthday, number 1999.6 in Tech. Report of the Dpt of Geodesy and Geoinformatics, pp. 113–128. Stuttgart University (1999)

    Google Scholar 

  57. Fréchet, M.: L’intégrale abstraite d’une fonction abstraite d’une variable abstraite et son application à la moyenne d’un élément aléatoire de nature quelconque. Revue Scientifique, pp. 483–512 (1944)

    Google Scholar 

  58. Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’Institut Henri Poincaré 10, 215–310 (1948)

    MATH  Google Scholar 

  59. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 2nd edn. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  60. Gamkrelidze, R.V. (ed.): Geometry I. Encyclopaedia of Mathematical Sciences, vol. 28. Springer, Heidelberg (1991)

    MATH  Google Scholar 

  61. Gerig, G., Kikinis, R., Kübler, O., Jolesz, F.A.: Nonlinear anisotropic filtering of MRI data. IEEE Transactions on Medical Imaging 11(2), 221–232 (1992)

    Article  Google Scholar 

  62. Gramkow, C.: On averaging rotations. International Journal of Computer Vision 42(1-2), 7–16 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  63. Granger, S., Pennec, X.: Statistiques exactes et approchées sur les normales aléatoires. Research report RR-4533, INRIA (2002)

    Google Scholar 

  64. Grenander, U.: Probabilities on Algebraic Structures. Wiley, Chichester (1963)

    MATH  Google Scholar 

  65. Grenander, U., Miller, M.I., Srivastava, A.: Hilbert-schmidt lower bounds for estimators on matrix Lie groups for ATR. IEEE Transations on Pattern Analysis and Machine Intelligence (PAMI) 20(8), 790–802 (1998)

    Article  Google Scholar 

  66. Grigor’yan, A.: Heat kernels on weighted manifolds and applications. In: Jorgenson, J., Walling, L. (eds.) The Ubiquitous Heat Kernel. Contemporary Mathematics, vol. 398, pp. 91–190. AMS (2006)

    Google Scholar 

  67. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structure preserving algorithm for ordinary differential equations. Springer series in computational mathematics, vol. 31. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  68. Helgason, S.: Differential Geometry, Lie groups, and Symmetric Spaces. Academic Press, London (1978)

    MATH  Google Scholar 

  69. Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Communication and Control Engineering Series. Springer, Heidelberg (1994)

    Book  MATH  Google Scholar 

  70. Hendricks, H.: A Cramer-Rao type lower bound for estimators with values in a manifold. Journal of Multivariate Analysis 38, 245–261 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  71. Huber, P.: Robust Statistics. John Wiley, New York (1981)

    Book  MATH  Google Scholar 

  72. Amari, S.-i., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs. American Mathematical Society (2000)

    Google Scholar 

  73. Joshi, S.C., Miller, M.I.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Processing 9(8), 1357–1370 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  74. Jupp, P.E., Mardia, K.V.: A unified view of the theory of directional statistics, 1975-1988. International Statistical Review 57(3), 261–294 (1989)

    Article  MATH  Google Scholar 

  75. Karcher, H.: Riemannian center of mass and mollifier smoothing. Communications in Pure and Applied Mathematics 30, 509–541 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  76. Kass, R.E., Vos, P.W.: Geometric Foundations of Asymptotic Inference. Wiley series in Probability and Statistics. John Wiley & Sons, Chichester (1997)

    Book  MATH  Google Scholar 

  77. Kaya, C.Y., Noakes, J.L.: Geodesics and an optimal control algorithm. In: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, U.S.A, pp. 4918–4919 (1997)

    Google Scholar 

  78. Kendall, D.G.: A survey of the statistical theory of shape (with discussion). Statistical Science 4, 87–120 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  79. Kendall, M.G., Moran, P.A.P.: Geometrical probability. Griffin’s statistical monographs and courses, vol. 10. Charles Griffin & Co. Ltd. (1963)

    Google Scholar 

  80. Kendall, W.S.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. London Math. Soc. 61(2), 371–406 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  81. Kendall, W.S.: Convexity and the hemisphere. Journal of the London Mathematical Society 43(2), 567–576 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  82. Kent, J.T.: The art of Statistical Science. In: Mardia, K.V. (ed.) New Directions in Shape Analysis, ch. 10, pp. 115–127. John Wiley & Sons, Chichester (1992)

    Google Scholar 

  83. Klassen, E., Srivastava, A., Mio, W., Joshi, S.: Analysis of planar shapes using geodesic path on shape spaces. IEEE Trans. on PAMI 26(3), 372–383 (2004)

    Article  Google Scholar 

  84. Klingenberg, W.: Riemannian Geometry. Walter de Gruyter, Berlin (1982)

    MATH  Google Scholar 

  85. Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Interscience tracts in pure and applied mathematics, vol. 15. John Wiley & Sons, Chichester (1969)

    MATH  Google Scholar 

  86. Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape space: a novel environment for statistics. Annals of Statistics 21, 1225–1271 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  87. Le Bihan, D., Mangin, J.-F., Poupon, C., Clark, C.A., Pappata, S., Molko, N., Chabriat, H.: Diffusion tensor imaging: Concepts and applications. Journal Magnetic Resonance Imaging 13(4), 534–546 (2001)

    Article  Google Scholar 

  88. Le Goualher, G., Procyk, E., Collins, D., Venugopal, R., Barillot, C., Evans, A.: Automated extraction and variability analysis of sulcal neuroanatomy. IEEE Transactions on Medical Imaging 18(3), 206–217 (1999)

    Article  Google Scholar 

  89. Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. Journal of Mathematical Imaging and Vision 25(3), 423–444 (2006)

    Article  MathSciNet  Google Scholar 

  90. Lepore, N., Brun, C., Chou, Y.-Y., Lee, A.D., Barysheva, M., Pennec, X., McMahon, K., Meredith, M., de Zubicaray, G.I., Wright, M.J., Toga, A.W., Thompson, P.M.: Best individual template selection from deformation tensor minimization. In: Proc. of the 2008 IEEE Int. Symp. on Biomedical Imaging: From Nano to Macro (ISBI 2008), Paris, France, May 14-17, pp. 460–463 (2008)

    Google Scholar 

  91. Lovrić, M., Min-Oo, M.: Multivariate normal distributions parametrized as a Riemannian symmetric space. Journal of Multivariate Analysis 74(1), 36–48 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  92. Mahony, R., Manton, R.: The geometry of the Newton method on non-compact Lie groups. Journal of Global Optimization 23, 309–327 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  93. Mangin, J.-F., Riviere, D., Cachia, A., Duchesnay, E., Cointepas, Y., Papadopoulos-Orfanos, D., Collins, D.L., Evans, A.C., Régis, J.: Object-based morphometry of the cerebral cortex. IEEE Transactions on Medical Imaging 23(8), 968–982 (2004)

    Article  Google Scholar 

  94. Mangin, J.-F., Rivière, D., Cachia, A., Duchesnay, E., Cointepas, Y., Papadopoulos-Orfanos, D., Scifo, P., Ochiai, T., Brunelle, F., Régis, J.: A framework to study the cortical folding patterns. NeuroImage 23(suppl. 1), 129 (2004)

    Article  Google Scholar 

  95. Mardia, K.V.: Directional statistics and shape analysis. Journal of applied Statistics 26, 949–957 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  96. Mardia, K.V., Jupp, P.E.: Directional statistics. Wiley, Chichester (2000)

    MATH  Google Scholar 

  97. Mazziotta, J., et al.: A probabilistic atlas and reference system for the human brain: International consortium for brain mapping (ICBM). Philos. Trans. R Soc. Lond B Biol. Sci. 356, 1293–1322 (2001)

    Article  Google Scholar 

  98. Medioni, G., Lee, M.-S., Tang, C.-K.: A Computational Framework for Segmentation and Grouping. Elsevier, Amsterdam (2000)

    MATH  Google Scholar 

  99. Meijering, E.: A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE 90(3), 319–342 (2002)

    Article  Google Scholar 

  100. Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc (JEMS) 8, 1–48 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  101. Miller, M.I., Trouvé, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annual Re-view of Biomedical Engineering, 375–405 (2003)

    Google Scholar 

  102. Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision 41(1/2), 61–84 (2001)

    Article  MATH  Google Scholar 

  103. Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision (2006)

    Google Scholar 

  104. Moakher, M.: Means and averaging in the group of rotations. SIAM Journal of Matrix Analysis and Applications 24(1), 1–16 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  105. Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM Journal of Matrix Analysis and Applications 26(3), 735–747 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  106. Mohammadi, B., Borouchaki, H., George, P.L.: Delaunay mesh generation governed by metric specifications. Part II: applications. Finite Elements in Analysis and Design, 85–109 (1997)

    Google Scholar 

  107. Moran, B., Suvorova, S., Howard, S.: Sensor management for radar: a tutorial. In: Advances in Sensing with Security Applications, Il Ciocco, Italy, July 17–30. NATO Advanced Study Institute (2005)

    Google Scholar 

  108. Nomizu, K.: Invariant affine connections on homogeneous spaces. American J. of Math. 76, 33–65 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  109. Oller, J.M., Corcuera, J.M.: Intrinsic analysis of statistical estimation. Annals of Statistics 23(5), 1562–1581 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  110. Owren, B., Welfert, B.: The Newton iteration on Lie groups. BIT Numerical Mathematics 40(1), 121–145 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  111. Pennec, X.: L’incertitude dans les problèmes de reconnaissance et de recalage – Applications en imagerie médicale et biologie moléculaire. Thèse de sciences (Ph.D. thesis), Ecole Polytechnique, Palaiseau (France) (December 1996)

    Google Scholar 

  112. Pennec, X.: Computing the mean of geometric features - application to the mean rotation. Research Report RR-3371, INRIA (March 1998)

    Google Scholar 

  113. Pennec, X.: Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements. In: Cetin, A.E., Akarun, L., Ertuzun, A., Gurcan, M.N., Yardimci, Y. (eds.) Proc. of Nonlinear Signal and Image Processing (NSIP 1999), Antalya, Turkey, June 20-23, vol. 1, pp. 194–198. IEEE-EURASIP (1999)

    Google Scholar 

  114. Pennec, X.: Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision 25(1), 127–154 (2006); a preliminary appeared as INRIA RR-5093 (January 2004)

    Article  MathSciNet  Google Scholar 

  115. Pennec, X.: Left-invariant Riemannian elasticity: a distance on shape diffeomorphisms? In: Pennec, X., Joshi, S. (eds.) Proc. of the International Workshop on the Mathematical Foundations of Computational Anatomy (MFCA 2006), pp. 1–13 (2006)

    Google Scholar 

  116. Pennec, X.: Statistical Computing on Manifolds for Computational Anatomy. Habilitation à diriger des recherches, Université Nice Sophia-Antipolis (December 2006)

    Google Scholar 

  117. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. International Journal of Computer Vision 66(1), 41–66 (2006); a preliminary version appeared as INRIA Research Report 5255 (July 2004)

    Article  MATH  Google Scholar 

  118. Pennec, X., Guttmann, C.R.G., Thirion, J.-P.: Feature-based registration of medical images: Estimation and validation of the pose accuracy. In: Wells, W.M., Colchester, A.C.F., Delp, S.L. (eds.) MICCAI 1998. LNCS, vol. 1496, pp. 1107–1114. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  119. Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., Ayache, N.: Riemannian elasticity: A statistical regularization framework for non-linear registration. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 943–950. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  120. Pennec, X., Thirion, J.-P.: A framework for uncertainty and validation of 3D registration methods based on points and frames. Int. Journal of Computer Vision 25(3), 203–229 (1997)

    Article  Google Scholar 

  121. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI) 12(7), 629–639 (1990)

    Article  Google Scholar 

  122. Picard, J.: Barycentres et martingales sur une variété. Annales de l’institut Poincaré - Probabilités et Statistiques 30(4), 647–702 (1994)

    MathSciNet  Google Scholar 

  123. Poincaré, H.: Calcul des probabilités, 2nd edn., Paris (1912)

    Google Scholar 

  124. Provenzale, J.M., Mukundan, S., Barboriak, D.P.: Diffusion-weighted and perfusion MR imaging for brain tumor characterization and assessment of treatment response. Radiology 239(3), 632–649 (2006)

    Article  Google Scholar 

  125. Rajamani, K.T., Joshi, S.C., Styner, M.A.: Bone model morphing for enhanced surgical visualization. In: IEEE (ed.) Proc. of IEEE Symp. on Biomedical Imaging: Nano to Macro (ISBI) 2004, vol. 2, pp. 1255–1258 (2004)

    Google Scholar 

  126. Rousseeuw, P.J., Leroy, A.M.: Robust Regression and Outliers Detection. Wiley series in prob. and math. stat. J. Wiley and Sons, Chichester (1987)

    Book  MATH  Google Scholar 

  127. Rovaris, M., Gass, A., Bammer, R., Hickman, S.J., Ciccarelli, O., Miller, D.H., Filippi, M.: Diffusion MRI in multiple sclerosis. Neurology 65, 1526–1532 (2005)

    Article  Google Scholar 

  128. Sijbers, J., den Dekker, A.J., Scheunders, P., Van Dyck, D.: Maximum likelihood estimation of Rician distribution parameters. TMI 17(3) (June 1998)

    Google Scholar 

  129. Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Statistics 11, 211–223 (1984)

    MathSciNet  MATH  Google Scholar 

  130. Small, C.G.: The Statistical Theory of Shapes. Springer series in statistics. Springer, Heidelberg (1996)

    Book  Google Scholar 

  131. Spivak, M.: Differential Geometry, 2nd edn., vol. 1. Publish or Perish, Inc. (1979)

    Google Scholar 

  132. Subsol, G., Thirion, J.-P., Ayache, N.: A scheme for automatically building 3D morphometric anatomical atlases: application to a skull atlas. Medical Image Analysis 2(1), 37–60 (1998)

    Article  Google Scholar 

  133. Talairach, J., Tournoux, P.: Co-Planar Stereotaxic Atlas of the Human Brain: 3-dimensional Proportional System: an Approach to Cerebral Imaging. Thieme Medical Publishers, New York (1988)

    Google Scholar 

  134. Thévenaz, P., Blu, T., Unser, M.: Interpolation revisited. IEEE Transactions on Medical Imaging 19(7), 739–758 (2000)

    Article  Google Scholar 

  135. Thompson, P.M., MacDonald, D., Mega, M.S., Holmes, C.J., Evans, A.C., Toga, A.W.: Detection and mapping of abnormal brain structure with a probabilistic atlas of cortical surfaces. Journal of Computer Assisted Tomography 21(4), 567–581 (1977)

    Article  Google Scholar 

  136. Thompson, P.M., Mega, M.S., Woods, R.P., Zoumalan, C.I., Lindshield, C.J., Blanton, R.E., Moussai, J., Holmes, C.J., Cummings, J.L., Toga, A.W.: Cortical change in alzheimer’s disease detected with a disease-specific population-based brain atlas. Cerebral Cortex 11(1), 1–16 (2001)

    Article  Google Scholar 

  137. Trouvé, A.: Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision 28(3), 213–221 (1998)

    Article  Google Scholar 

  138. Tschumperlé, D.: PDE-Based Regularization of Multivalued Images and Applications. PhD thesis, University of Nice-Sophia Antipolis (December 2002)

    Google Scholar 

  139. Vaillant, M., Glaunès, J.: Surface matching via currents. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 381–392. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  140. Vaillant, M., Miller, M.I., Younes, L., Trouvé, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23(supp. 1), S161–S169 (2004)

    Article  Google Scholar 

  141. Vaillant, M., Qiu, A., Glaunès, J., Miller, M.: Diffeomorphic metric surface mapping in subregion of the superior temporal gyrus. NeuroImage 34(3), 1149–1159 (2007)

    Article  Google Scholar 

  142. Wang, Z., Vemuri, B.C.: DTI segmentation using an information theoretic tensor dissimilarity measure. IEEE Trans. on Medical Imaging 24(10), 1267–1277 (2005)

    Article  Google Scholar 

  143. Wang, Z., Vemuri, B.C., Chen, Y., Mareci, T.: A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. on Medical Imaging (2004)

    Google Scholar 

  144. Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging, Providence. Contemporary Mathematics, vol. 313, pp. 251–268. AMS (2002)

    Google Scholar 

  145. Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  146. Westin, C.F., Maier, S.E., Mamata, H., Nabavi, A., Jolesz, F.A., Kikinis, R.: Processing and visualization for diffusion tensor MRI. Medical Image Analysis 6(2), 93–108 (2002)

    Article  Google Scholar 

  147. Younes, L.: Computable elastic distances between shapes. SIAM Journal on Applied Mathematics 58(2), 565–586 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Sophia Antipolis - Asclepios Team, 2004 Rte des Lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    Xavier Pennec

Authors
  1. Xavier Pennec
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIX, Ecole Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    Frank Nielsen

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pennec, X. (2009). Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy. In: Nielsen, F. (eds) Emerging Trends in Visual Computing. ETVC 2008. Lecture Notes in Computer Science, vol 5416. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00826-9_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00826-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00825-2

  • Online ISBN: 978-3-642-00826-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation