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Generation of a statistical shape model with probabilistic point correspondences and the expectation maximization- iterative closest point algorithm

  • Heike HufnagelEmail author
  • Xavier Pennec
  • Jan Ehrhardt
  • Nicholas Ayache
  • Heinz Handels
Original article

Abstract

Objective:

Identification of point correspondences between shapes is required for statistical analysis of organ shapes differences. Since manual identification of landmarks is not a feasible option in 3D, several methods were developed to automatically find one-to-one correspondences on shape surfaces. For unstructured point sets, however, one-to-one correspondences do not exist but correspondence probabilities can be determined.

Materials and methods:

A method was developed to compute a statistical shape model based on shapes which are represented by unstructured point sets with arbitrary point numbers. A fundamental problem when computing statistical shape models is the determination of correspondences between the points of the shape observations of the training data set. In the absence of landmarks, exact correspondences can only be determined between continuous surfaces, not between unstructured point sets. To overcome this problem, we introduce correspondence probabilities instead of exact correspondences. The correspondence probabilities are found by aligning the observation shapes with the affine expectation maximization-iterative closest points (EM-ICP) registration algorithm. In a second step, the correspondence probabilities are used as input to compute a mean shape (represented once again by an unstructured point set). Both steps are unified in a single optimization criterion which depe nds on the two parameters ‘registration transformation’ and ‘mean shape’. In a last step, a variability model which best represents the variability in the training data set is computed. Experiments on synthetic data sets and in vivo brain structure data sets (MRI) are then designed to evaluate the performance of our algorithm.

Results:

The new method was applied to brain MRI data sets, and the estimated point correspondences were compared to a statistical shape model built on exact correspondences. Based on established measures of “generalization ability” and “specificity”, the estimates were very satisfactory.

Conclusion:

The novel algorithm for building a generative statistical shape model (gSSM) does not need one-to-one point correspondences but relies solely on point correspondence probabilities for the computation of mean shape and eigenmodes. It is well-suited for shape analysis on unstructured point sets.

Keywords

Iterative Close Point Minimum Description Length Point Correspondence Variability Model Iterative Close Point 
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.

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References

  1. Lorenz C and Krahnstoever N (2000). Generation of point-based 3D statistical shape models for anatomical objects. Comput Vis Image Understanding 77: 175–191 CrossRefGoogle Scholar
  2. Bookstein FL (1996). Landmark methods for forms without landmarks: morphometrics in group differences in outline shapes. Medi Image Anal 1: 225–243 CrossRefGoogle Scholar
  3. Styner M, Gerig G, Lieberman J, Jones D and Weinberger D (2003). Statistical shape analysis of neuroanatomical structures based on medial models. Medi Image Anal 7: 207–220 CrossRefGoogle Scholar
  4. Vos F, de Bruin P, Streeksa G, Maas M, van Vliet L, Vossepoel A (2004) A statistical shape model without using landmarks. In: Proceedings of the ICPR’04, vol 3, pp 714–717Google Scholar
  5. Besl PJ and McKay ND (1992). A method for registration of 3D shapes. IEEE Trans PAMI 14: 239–256 Google Scholar
  6. Zhao Z, Theo EK (2005) A novel framework for automated 3D PDM construction using deformable models. In: Medical imaging 2005, SPIE proc, vol 5747, pp 303–314Google Scholar
  7. Chui H, Win L, Schultz R, Duncan J and Rangarajan A (2003). A unified non-rigid feature registration method for brain mapping. Medi Image Anal 7: 113–130 CrossRefGoogle Scholar
  8. Davies R, Twining C and Cootes T (2002). A minimum description length approach to statistical shape modeling. IEEE Trans Medi Imaging 21(5): 525–537 CrossRefGoogle Scholar
  9. Heimann T, Wolf I, Williams T, Meinzer H (2005) 3D active shape models using gradient descent optimization of description length. In: Proceedings of the IPMI’05, vol 3565, pp 566–577Google Scholar
  10. Cates J, Meyer M, Fletcher P, Whitaker R (2006) Entropy-based particle systems for shape correspondences. In: Proceedings of the MICCAI’06, vol 1, pp 90–99Google Scholar
  11. Tsai A, Wells WM, Warfield SK and Willsky AS (2005). An EM algorithm for shape classification based on level sets. Medi Image Anal 9: 491–502 CrossRefGoogle Scholar
  12. Kodipaka S, Vemuri B, Rangarajan A, Leonard C, Schmallfuss I and Eisenschenk S (2007). Kernel fisher discriminant for shape-based classification in epilepsy. Medi Image Anal 11: 79–90 CrossRefGoogle Scholar
  13. Peter A, Rangarajan A (2006) Shape analysis using the Fisher-Rao Riemannian metric: unifying shape representation and deformation. In: IEEE transactions ISBI’06, pp 1164–1167Google Scholar
  14. Chui H, Rangarajan A, Zhang J and Leonard C (2004). Unsupervised learning of an atlas from unlabeled point-sets. IEEE Trans PAMI’ 04(26): 160–172 Google Scholar
  15. Granger S, Pennec X (2002) Multi-scale EM-ICP: a fast and robust approach for surface registration. In: Proceedings of the ECCV’02, LNCS, vol 2525, pp 418–432Google Scholar
  16. Rangarajan A, Chui H, Bookstein FL (1997) The softassign procrustes matching algorithm. In: Proceedings of the IPMI’97, vol 1230, pp 29–42Google Scholar
  17. Dempster A, Laird N and Rubin D (1977). Maximum likelihood from incomplete data via the EM algorithm. Royal Stat B 39: 1–38 Google Scholar
  18. Granger S (2003) Une approche statistique multi-échelle au recalage rigide de surfaces: application à l’implantologie dentaire. PhD thesis, Ecole des Mines de ParisGoogle Scholar
  19. Cootes T, Taylor C, Cooper D and Graham J (1995). Active shape models—their training and application. Comput Vis Image Understanding 61: 38–59 CrossRefGoogle Scholar
  20. Schroeder WJ, Zarge JA and Lorensen WE (1992). Decimation of triangle meshes. Comput Graph 26: 65–70 CrossRefGoogle Scholar
  21. Styner M, Rajamani K, Nolte L, Zsemlye G, Székely G, Taylor C, Davies R (2003) Evaluation of 3D correspondence methods for model building. In: Proceedings for the IPMI’03, vol 2732, pp 63–75Google Scholar

Copyright information

© CARS 2007

Authors and Affiliations

  • Heike Hufnagel
    • 1
    • 2
    Email author
  • Xavier Pennec
    • 1
  • Jan Ehrhardt
    • 2
  • Nicholas Ayache
    • 1
  • Heinz Handels
    • 2
  1. 1.Institut National de Recherche en Informatique et en Automatique (INRIA), Asclepios ProjectSophia AntipolisFrance
  2. 2.Department of Medical InformaticsUniversity Medical Center Hamburg-EppendorfHamburgGermany

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