Computation of a Probabilistic Statistical Shape Model in a Maximum-a-posteriori Framework

Summary

Objectives: When analyzing shapes and shape variabilities, the first step is bringing those shapes into correspondence. This is a fundamental problem even when solved by manually determining exact correspondences such as landmarks. We developed a method to represent a mean shape and a variability model for a training data set based on probabilistic correspondence computed between the observations.

Methods: First, the observations are matched on each other with an affine transformation found by the Expectation-Maximization Iterative-Closest-Points (EM-ICP) registration. We then propose a maximum-a-posteriori (MAP) framework in order to compute the statistical shape model (SSM) parameters which result in an optimal adaptation of the model to the observations. The optimization of the MAP explanation is realized with respect to the observation parameters and the generative model parameters in a global criterion and leads to very efficient and closed-form solutions for (almost) all parameters.

Results: We compared our probabilistic SSM to a SSM based on one-to-one correspondences and the PCA (classical SSM). Experiments on synthetic data served to test the performances on non-convex shapes (15 training shapes) which have proved difficult in terms of proper correspondence determination. We then computed the SSMs for real putamen data (21 training shapes). The evaluation was done by measuring the generalization ability as well as the specificity of both SSMs and showed that especially shape detail differences are better modeled by the probabilistic SSM (Hausdorff distance in generalization ability ≈ 25% smaller).

Conclusions: The experimental outcome shows the efficiency and advantages of the new approach as the probabilistic SSM performs better in modeling shape details and differences.

Keywords

Statistical shape model - registration - ICP - EMICP - correspondence problem

• References

• 1 Werner R, Ehrhardt J, Frenzel T, Säring D, Lu W, Low D, Handels H. Motion Artifact Reducing Reconstruction of 4D CT Image Data for the Analysis of Respiratory Dynamics. Methods Inf Med 2007; 46 (Suppl. 03) 254-260.
• 2 Handels H, Werner R, Frenzel T, Säring D, Lu W, Low D, Ehrhardt J. 4D Medical Image Computing and Visualization of Lung Tumor Mobility in Spatio-temporal CT Image Data. International Journal of Medical Informatics 2007; 76S: 433-439.
• 3 Säring D, Ehrhardt J, Stork A, Bansmann MP, Lund GK, Handels H. Computer-Assisted Analysis of 4D Cardiac MR Image Sequences after Myocardial Infarction. Methods Inf Med 2006; 45 (Suppl. 04) 377-383.
• 4 Hacker S, Handels H. Framework for Representation and Visualization of 3D Shape Variations of Organs in an Interactive Anatomical Atlas. Methods Inf Med 2009; 48 (Suppl. 03) 272-281.
• 5 Lorenz C, Krahnstoever N. Generation of Point-Based 3D Statistical Shape Models for Anatomical Objects. Computer Vision and Image Understanding 2000; 77: 175-191.
• 6 Bookstein FL. Landmark methods for forms without landmarks: morphometrics in group differences in outline shapes. Medical Image Analysis 1996; 1: 225-243.
• 7 Styner M, Rajamani KT, Nolte LP, Zsemlye G, Székely G, Taylor CJ, Davies RH. Evaluation of 3D Correspondence Methods for Model Building. In: Proceedings of the International Conference on Information Processing in Medical Imaging 2003 pp 63-75
• 8 Vos FM, de Bruin PW, Streeksa GI, Maas M, van Vliet LJ, Vossepoel AM. A statistical shape model without using landmarks. In: Proceedings of the International Conference of Pattern Recognition 2004 pp 714-717.
• 9 Besl PJ, McKay ND. A method for registration of 3D shapes. IEEE Transactions Pattern Analysis and Machine Intelligence 1992; 14: 239-256.
• 10 Zhao Z, Theo EK. A novel framework for automated 3D PDM construction using deformable models. Medical Imaging 2005; Proc of SPIE: Vol 5747: 303-314.
• 11 Chui H, Win L, Schultz R, Duncan JS, Rangarajan A. A unified non-rigid feature registration method for brain mapping. Medical Image Analysis 2003; 7: 113-130.
• 12 Davies RH, Twining CJ, Cootes TF. A Minimum Description Length Approach to Statistical Shape Modeling. IEEE Medical Imaging 2002; 21 (Suppl. 05) 525-537.
• 13 Heimann T, Wolf I, Williams T, Meinzer HP. 3D Active Shape Models Using Gradient Descent Optimization of Description Length. In: Proceedings of the International Conference of Pattern Recognition 2005 pp 566-577.
• 14 Cates J, Meyer M, Fletcher PT, Whitaker R. Entropy-Based Particle Systems for Shape Correspondences. In: Proceedings of the Medical Image Computing and Computer Assisted Intervention 2006 pp 90-99.
• 15 Tsai A, Wells WM, Warfield SK, Willsky AS. An EM algorithm for shape classification based on level sets. Medical Image Analysis 2005; 9: 491-502.
• 16 Kodipaka S, Vemuri BC, Rangarajan A. et al. Kernel Fisher discriminant for shape-based classification in epilepsy. Medical Image Analysis 2007; 11: 79-90.
• 17 Peter A, Rangarajan A. Shape Analysis Using the Fisher-Rao Riemannian Metric: Unifying Shape Representation and Deformation. IEEE Transactions on the International Symposium on Biomedical Imaging 2006 pp 1164-1167.
• 18 Chui H, Rangarajan A, Zhang J, Leonard C. Un-supervised learning of an atlas from unlabeled point-sets. IEEE Transactions Pattern Analysis and Machine Intelligence 2004; 26: 160-172.
• 19 Granger S, Pennec X. Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration. European Conference on Computer Vision 2002 in LNCS Vol 2353 418-432
• 20 Hufnagel H, Pennec X, Ehrhardt J, Ayache N, Handels H. Generation of Statistical Shape Models with Probabilistic Point Correspondences and Expectation Maximization – Iterative Closest Point Algorithm. International Journal of Computer Assisted Radiology and Surgery 2008; 2 (Suppl. 05) 265-273.
• 21 Hufnagel H, Pennec X, Ehrhardt J, Handels H, Ayache N. Shape Analysis Using a Point-Based Statistical Shape Model Built on Correspondence Probabilities. In: Proceedings of the Medical Image Computing and Computer Assisted Intervention 2007; Part 1:. pp 959-967.
• 22 Styner M, Gerig G, Lieberman J, Jones D, Weinberger D. Statistical shape analysis of neuroanatomical structures based on medial models. Medical Image Analysis 2003; 7: 207-220.