Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps

  • Chapter
Geometry, Mechanics, and Dynamics

Part of the book series: Fields Institute Communications ((FIC,volume 73))

  • 2268 Accesses

Abstract

These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Alexander, D.C., Pierpaoli, C., Basser, P.J., Gee, J.C.: Spatial transformations of diffusion tensor magnetic resonance images. IEEE Trans. Med. Imaging 20(11), 1131–1139 (2001)

    Article  Google Scholar 

  3. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  4. Artaechevarria, X., Munoz-Barrutia, A., Ortiz-de Solorzano, C.: Combination strategies in multi-atlas image segmentation: application to brain MR data. IEEE Trans. Med. Imaging 28(8), 1266–1277 (2009)

    Article  Google Scholar 

  5. Ashburner, J.: A fast diffeomorphic image registration algorithm. NeuroImage 38(1), 95–113 (2007)

    Article  Google Scholar 

  6. Ashburner, J., Friston, K.J.: Diffeomorphic registration using geodesic shooting and Gauss-Newton optimisation. NeuroImage 55(3), 954–967 (2011)

    Article  Google Scholar 

  7. Babalola, K.O., Patenaude, B., Aljabar, P., Schnabel, J., Kennedy, D., Crum, W., Smith, S., Cootes, T., Jenkinson, M., Rueckert, D.: An evaluation of four automatic methods of segmenting the subcortical structures in the brain. NeuroImage 47(4), 1435–1447 (2009)

    Article  Google Scholar 

  8. Bai, J., Trinh, T.L.H., Chuang, K.H., Qiu, A.: Atlas-based automatic mouse brain image segmentation revisited: model complexity vs. image registration. Magn. Reson. Imaging 30(6), 789–798 (2012)

    Google Scholar 

  9. Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50, 60–97 (2014). doi:10.1007/s10851-013-0490-z

    Article  MATH  MathSciNet  Google Scholar 

  10. Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)

    Article  Google Scholar 

  11. Bruveris, M.: Geometry of diffeomorphism groups and shape matching. Ph.D. thesis, Imperial College, London (2012)

    Google Scholar 

  12. Bruveris, M., Gay-Balmaz, F., Holm, D.D., Ratiu, T.: The momentum map representation of images. J. Nonlinear Sci. 21, 115–150 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993). doi:10.1103/PhysRevLett.71.1661. http://dx.doi.org/10.1103/PhysRevLett.71.1661

  14. Cao, Y., Miller, M.I., Winslow, R.L., Younes, L.: Large deformation diffeomorphic metric mapping of vector fields. IEEE Trans. Med. Imaging 24(9), 1216–1230 (2005)

    Article  Google Scholar 

  15. Collins, D.L., Peters, T.M., Dai, W., Evans, A.C.: Model-based segmentation of individual brain structures from MRI data. In: Robb, R.A. (ed.) Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 1808, pp. 10–23 (1992)

    Google Scholar 

  16. Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. Int. J. Comput. Vis. 90, 255–266 (2010)

    Article  Google Scholar 

  17. Durrleman, S., Pennec, X., Trouvé, A., Ayache, N.: Statistical models on sets of curves and surfaces based on currents. Med. Image Anal. 13(5), 793–808 (2009)

    Article  Google Scholar 

  18. Faria, A.V., Hoon, A., Stashinko, E., Li, X., Jiang, H., Mashayekh, A., Akhter, K., Hsu, J., Oishi, K., Zhang, J., Miller, M.I., van Zijl, P.C., Mori, S.: Quantitative analysis of brain pathology based on MRI and brain atlases—applications for cerebral palsy. NeuroImage 54(3), 1854–1861 (2011)

    Article  Google Scholar 

  19. Ferri, C.P., Prince, M., Brayne, C., Brodaty, H., Fratiglioni, L., Ganguli, M., Hall, K., Hasegawa, K., Hendrie, H., Huang, Y., Jorm, A., Mathers, C., Menezes, P.R., Rimmer, E., Scazufca, M.: Global prevalence of dementia: a Delphi consensus study. Lancet 366(9503), 2112–2117 (2006)

    Article  Google Scholar 

  20. Fiot, J.B., Risser, L., Cohen, L.D., Fripp, J., Vialard, F.X.: Local vs global descriptors of hippocampus shape evolution for Alzheimer’s longitudinal population analysis. In: Durrleman, S., Fletcher, T., Gerig, G., Niethammer, M. (eds.) Spatio-Temporal Image Analysis for Longitudinal and Time-Series Image Data. Lecture Notes in Computer Science, vol. 7570, pp. 13–24. Springer, Berlin/Heidelberg (2012)

    Chapter  Google Scholar 

  21. Fischl, B., Salat, D.H., Busa, E., Albert, M., Dieterich, M., Haselgrove, C., van der Kouwe, A., Killiany, R., Kennedy, D., Klaveness, S., Montillo, A., Makris, N., Rosen, B., Dale, A.M.: Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain. Neuron 33(3), 341–355 (2002)

    Article  Google Scholar 

  22. Gee, J.C., Reivich, M., Bajcsy, R.: Elastically deforming a three-dimensional atlas to match anatomical brain images. J. Comput. Assist. Tomogr. 17(2), 225–236 (1993)

    Article  Google Scholar 

  23. Glaunès, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. J. Math. Imaging Vis. 20, 179–200 (2004)

    Article  Google Scholar 

  24. Glaunès, J., Qiu, A., Miller, M.I., Younes, L.: Large deformation diffeomorphic metric curve mapping. Int. J. Comput. Vis. 80, 317–336 (2008)

    Article  Google Scholar 

  25. Grenander, U.: Regular Structure. Lectures in Pattern Theory III. Applied Mathematical Science, vol. 33. Springer, Berlin/Heidelberg (1981)

    Google Scholar 

  26. Grenander, U.: General Pattern Theory. Oxford University Press, Oxford (1993)

    Google Scholar 

  27. Grenander, U., Miller, M.I.: Computational anatomy: an emerging discipline. Q. Appl. Math. 56, 617–694 (1998)

    MATH  MathSciNet  Google Scholar 

  28. Grenander, U., Miller, M.I.: Pattern Theory: From Representation to Inference. Oxford University Press, Oxford (2007)

    Google Scholar 

  29. Holden, M.: A review of geometric transformations for nonrigid body registration. IEEE Trans. Med. Imaging 27(1), 111–128 (2008)

    Article  Google Scholar 

  30. Holm, D.D., Marsden, J.E.: Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation. In: Marsden, J.E., Ratiu, T. (eds.) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 203–235. Birkhäuser, Basel (2005)

    Chapter  Google Scholar 

  31. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Holm, D.D., Rathanather, J.T., Trouvé, A., Younes, L.: Soliton dynamics in computational anatomy. NeuroImage 23, 170–178 (2004)

    Article  Google Scholar 

  33. Holm, D.D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimension. Clarendon Press, Oxford (2009)

    Google Scholar 

  34. Jenkinson, M., Smith, S.: A global optimisation method for robust affine registration of brain images. Med. Image Anal. 5(2), 143–156 (2001)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Klein, A., Andersson, J., Ardekani, B.A., Ashburner, J., Avants, B., Chiang, M.C., Christensen, G.E., Collins, D.L., Gee, J., Hellier, P., Song, J.H., Jenkinson, M., Lepage, C., Rueckert, D., Thompson, P., Vercauteren, T., Woods, R.P., Mann, J.J., Parsey, R.V.: Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. NeuroImage 46(3), 786–802 (2009)

    Article  Google Scholar 

  37. Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)

    Google Scholar 

  38. Lorenzi, M., Ayache, N., Pennec, X.: Schild’s ladder for the parallel transport of deformations in time series of images. In: Székely, G., Hahn, H. (eds.) Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 6801, pp. 463–474. Springer, New York (2011)

    Chapter  Google Scholar 

  39. Maes, F., Collignon, A., Vandermeulen, D., Marchal, G., Suetens, P.: Multimodality image registration by maximization of mutual information. IEEE Trans. Med. Imaging 16(2), 187–198 (1997)

    Article  Google Scholar 

  40. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol. 17, 2nd edn. Springer, New York (1999)

    Google Scholar 

  41. McDowell, K.S., Vadakkumpadan, F., Blake, R., Blauer, J., Plank, G., MacLeod, R.S., Trayanova, N.A.: Methodology for patient-specific modeling of atrial fibrosis as a substrate for atrial fibrillation. J. Electrocardiol. 45(6), 640–645 (2012)

    Article  Google Scholar 

  42. Micheli, M., Michor, P.W., Mumford, D.: Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds. Izv. Math. 77(3), 541–570 (2013). doi:10.1070/IM2013v077n03ABEH002648

    Article  MATH  MathSciNet  Google Scholar 

  43. Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. American Mathematical Society, Providence (2008)

    Google Scholar 

  44. Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23(1), 74–113 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Michor, P.W., Mumford, D.: A zoo of diffeomorphism groups on \(\mathbb{R}^{n}\). Ann. Glob. Anal. Geom. 44(4), 529–540 (2013). doi:10.1007/s10455-013-9380-2

    Article  MATH  MathSciNet  Google Scholar 

  46. Miller, M.I., Qiu, A.: The emerging discipline of computational functional anatomy. NeuroImage 45(1, Suppl. 1), S16–S39 (2009)

    Google Scholar 

  47. Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: a general framework. Int. J. Comput. Vis. 41, 61–84 (2001)

    Article  MATH  Google Scholar 

  48. Miller, M.I., Christensen, G.E., Amit, Y., Grenander, U.: Mathematical textbook of deformable neuroanatomies. Proc. Natl. Acad. Sci, 90(24), 11944–11948 (1993)

    Article  MATH  Google Scholar 

  49. Miller, M.I., Trouvé, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375–405 (2002)

    Article  Google Scholar 

  50. Miller, M., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24, 209–228 (2006)

    Article  Google Scholar 

  51. Mumford, D., Desolneux, A.: Pattern Theory: The Stochastic Analysis of Real-World Signals. A K Peters, Natick (2010)

    Google Scholar 

  52. Omori, H.: On Banach-Lie groups acting on finite dimensional manifolds. Tôhoku Math. J. 30(2), 223–250 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  53. Peng, T., Wang, W., Rohde, G., Murphy, R.: Instance-based generative biological shape modeling. In: IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2009. ISBI ’09, pp. 690–693 (2009)

    Google Scholar 

  54. Qiu, A., Bitouk, D., Miller, M.: Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator. IEEE Trans. Med. Imaging 25(10), 1296–1306 (2006)

    Article  Google Scholar 

  55. Qiu, A., Younes, L., Wang, L., Ratnanather, J.T., Gillepsie, S.K., Kaplan, G., Csernansky, J., Miller, M.I.: Combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia. NeuroImage 37(3), 821–833 (2007)

    Article  Google Scholar 

  56. Qiu, A., Younes, L., Miller, M.I., Csernansky, J.G.: Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the alzheimer’s type. NeuroImage 40(1), 68–76 (2008)

    Article  Google Scholar 

  57. Qiu, A., Albert, M., Younes, L., Miller, M.I.: Time sequence diffeomorphic metric mapping and parallel transport track time-dependent shape changes. NeuroImage 45(1, Suppl. 1), S51–S60 (2009)

    Google Scholar 

  58. Qiu, A., Wang, L., Younes, L., Harms, M.P., Ratnanather, J.T., Miller, M.I., Csernansky, J.G.: Neuroanatomical asymmetry patterns in individuals with schizophrenia and their non-psychotic siblings. NeuroImage 47(4), 1221–1229 (2009)

    Article  Google Scholar 

  59. Qiu, A., Fennema-Notestine, C., Dale, A.M., Miller, M.I.: Regional shape abnormalities in mild cognitive impairment and Alzheimer’s disease. NeuroImage 45(3), 656–661 (2009)

    Article  Google Scholar 

  60. Qiu, A., Younes, L., Miller, M.: Principal component based diffeomorphic surface mapping. IEEE Trans. Med. Imaging 31(2), 302–311 (2012)

    Article  Google Scholar 

  61. Rohde, G.K., Ribeiro, A.J.S., Dahl, K.N., Murphy, R.F.: Deformation-based nuclear morphometry: capturing nuclear shape variation in HeLa cells. Cytometry A 73A(4), 341–350 (2008)

    Article  Google Scholar 

  62. Rueckert, D., Sonoda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Nonrigid registration using free-form deformations: application to breast MR images. IEEE Trans. Med. Imaging 18(8), 712–721 (1999)

    Article  Google Scholar 

  63. Saitoh, S.: Theory of Reproducing Kernels and Its Applications. Pitman Research Notes in Mathematics. Longman, England (1988)

    MATH  Google Scholar 

  64. Steinert-Threlkeld, S., Ardekani, S., Mejino, J.L., Detwiler, L.T., Brinkley, J.F., Halle, M., Kikinis, R., Winslow, R.L., Miller, M.I., Ratnanather, J.T.: Ontological labels for automated location of anatomical shape differences. J. Biomed. Inform. 45(3), 522–527 (2012)

    Article  Google Scholar 

  65. Thirion, J.P.: Image matching as a diffusion process: an analogy with Maxwell’s demons. Med. Image Anal. 2(3), 243–260 (1998)

    Article  Google Scholar 

  66. Thompson, D.W.: On Growth and Form. Dover, New York (1992). Reprint of 1942 2nd edn. (1st edn. 1917)

    Google Scholar 

  67. Trouvé, A.: Diffeomorphic groups and pattern matching in image analysis. Int. J. Comput. Vis. 28, 213–221 (1998)

    Article  Google Scholar 

  68. Vaillant, M., Glaunes, J.: Surface matching via currents. In: Christensen, G., Sonka, M. (eds.) IPMI. Lecture Notes in Computer Science, vol. 3565, pp. 381–392. Springer, Berlin (2005)

    Google Scholar 

  69. Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Symmetric log-domain diffeomorphic registration: a demons-based approach. In: Metaxas, D., Axel, L., Fichtinger, G., Székely, G. (eds.) Medical Image Computing and Computer-Assisted Intervention MICCAI 2008. Lecture Notes in Computer Science, vol. 5241, pp. 754–761. Springer, New York (2008)

    Google Scholar 

  70. Vialard, F.X.: Hamiltonian approach to shape spaces in a diffeomorphic framework: from the discontinuous image matching problem to a stochastic growth model. Ph.D. thesis, École Normale Supérieure de Cachan (2009)

    Google Scholar 

  71. Vialard, F.X., Risser, L., Rueckert, D., Cotter, C.J.: Diffeomorphic 3D image registration via geodesic shooting using an efficient adjoint calculation. Int. J. Comput. Vis. 97, 229–241 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  72. Warfield, S., Zou, K., Wells, W.: Simultaneous truth and performance level estimation (STAPLE): an algorithm for the validation of image segmentation. IEEE Trans. Med. Imaging 23(7), 903–921 (2004)

    Article  Google Scholar 

  73. Weiner, M.W., Veitch, D.P., Aisen, P.S., Beckett, L.A., Cairns, N.J., Green, R.C., Harvey, D., Jack, C.R., Jagust, W., Liu, E., Morris, J.C., Petersen, R.C., Saykin, A.J., Schmidt, M.E., Shaw, L., Siuciak, J.A., Soares, H., Toga, A.W., Trojanowski, J.Q.: The Alzheimer’s disease neuroimaging initiative: a review of papers published since its inception. Alzheimers Dement. 8(1 Suppl.), S1–S68 (2012)

    Article  Google Scholar 

  74. Younes, L.: Jacobi fields in groups of diffeomorphisms and applications. Q. Appl. Math. 65(1), 113–134 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  75. Younes, L.: Shapes and Diffeomorphisms. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  76. Younes, L., Qiu, A., Winslow, R., Miller, M.I.: Transport of relational structures in groups of diffeomorphisms. J. Math. Imaging Vis. 32, 41–56 (2008)

    Article  MathSciNet  Google Scholar 

  77. Younes, L., Arrate, F., Miller, M.I.: Evolution equations in computational anatomy. NeuroImage 45, 40–50 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

Both authors gratefully acknowledge partial support by Advanced Grant 267382 from the European Research Council.

Author information

Authors and Affiliations

  1. Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, UK

    Martins Bruveris

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Darryl D. Holm

Authors
  1. Martins Bruveris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Darryl D. Holm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martins Bruveris .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Dong Eui Chang

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl D. Holm

  3. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    George Patrick

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Tudor Ratiu

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this chapter

Cite this chapter

Bruveris, M., Holm, D.D. (2015). Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation