Maximum-Likelihood Estimation of Biological Growth Variables

  • Anuj Srivastava
  • Sanjay Saini
  • Zhaohua Ding
  • Ulf Grenander
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3757)


Shapes of biological objects, such as anatomical parts, have been studied intensely in recent years. An emerging need is to model and analyze changes in shapes of biological objects during, for example, growths of organisms. A recent paper by Grenander et al. [5] introduced a mathematical model, called GRID, for decomposing growth induced diffeomorphism into smaller, local deformations. The basic idea is to place focal points of local growth, called seeds, according to a spatial process on a time-varying coordinate system, and to deform a small neighborhood around them using radial deformation functions (RDFs). In order to estimate these variables – seed placements and RDFS – we first estimate optimal deformation from magnetic resonance image data, and then utilize an iterative solution to reach maximum-likelihood estimates. We demonstrate this approach using MRI images of human brain growth.


Radial Displacement Grid Model Biological Object Magnetic Resonance Image Data Seed Location 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bookstein, F.L.: Size and shape spaces for landmark data in two dimensions. Statistical Science 1, 181–242 (1986)zbMATHCrossRefGoogle Scholar
  2. 2.
    Christensen, G.E., Rabbitt, R.D., Miller, M.I.: A deformable neuroanatomy textbook based on viscous fluid mechanics. In: Prince, J., Runolfsson, T. (eds.) Proceedings of the Twenty-Seventh Annual Conference on Information Sciences and Systems, Baltimore, Maryland, March 24-26, pp. 211–216. Department of Electrical Engineering, The Johns Hopkins University (1993)Google Scholar
  3. 3.
    Grenander, U., Miller, M.I.: Representations of knowledge in complex systems. Journal of the Royal Statistical Society 56(3) (1994)Google Scholar
  4. 4.
    Grenander, U., Miller, M.I.: Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics LVI(4), 617–694 (1998)Google Scholar
  5. 5.
    Grenander, U., Srivastava, A., Saini, S.: A pattern-theoretic characterization of biological growth. IEEE Transactions on Medical Imaging, in review (2005)Google Scholar
  6. 6.
    Khaneja, N., Miller, M.I., Grenander, U.: Dynamic programming generation of curves on brain surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(11), 1260–1264 (1998)CrossRefGoogle Scholar
  7. 7.
    Kim, B., Boes, J.L., Frey, K.A., Meyer, C.R.: Mutual information for automated unwarping of rat brain autoradiographs. Neuroimage 5(1), 31–40 (1997)CrossRefGoogle Scholar
  8. 8.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision 41(1/2), 61–84 (2002)CrossRefGoogle Scholar
  9. 9.
    Miller, M.I., Christensen, G.E., Amit, Y., Grenander, U.: Mathematical textbook of deformable neuroanatomies. Proceedings of the National Academy of Science 90(24) (December 1993)Google Scholar
  10. 10.
    Sherratt, J.A., Chaplain, M.A.: A new mathematical models for avascular tumour growth. Journal of Mathematical Biology 43(4), 291–312 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Thompson, P.M., Toga, A.W.: A framework for computational anatomy. Computing and Visualization in Science 5, 13–34 (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Trouve, A.: Diffemorphisms groups and pattern matching in image analysis. International Journal of Computer Vision 28(3), 213–221 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anuj Srivastava
    • 1
  • Sanjay Saini
    • 1
  • Zhaohua Ding
    • 2
  • Ulf Grenander
    • 3
  1. 1.Department of StatisticsFlorida State UniversityTallahasseeUSA
  2. 2.Institute of Imaging SciencesVanderbilt UniversityNashvilleUSA
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations