Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms

Part of the Advances in Computer Vision and Pattern Recognition book series (ACVPR)


In this chapter, we present Bayesian models for diffeomorphic shape variability in populations of images. The first model is a probabilistic formulation of the image atlas construction problem, which seeks to compute an atlas image most representative of a set of input images. The second model adds diffeomorphic modes of shape variation, or principal geodesics. Both of these models represent shape variability as random variables on the manifold of diffeomorphic transformations. We define a Gaussian prior distribution for diffeomorphic transformations using the inner product in the tangent space to the diffeomorphism group. We develop a Monte Carlo Expectation Maximization (MCEM) algorithm for the Bayesian inference, due to the lack of closed-form solutions, where the expectation step is approximated via Hamiltonian Monte Carlo (HMC) sampling of diffeomorphisms. The resulting inference produces estimates of the image atlas, principal geodesic modes of variation, and model parameters. We show that the advantage of the Bayesian formulation is that it provides a principled way to estimate both the regularization parameter of the diffeomorphic transformations and the intrinsic dimensionality of the input data.


Diffeomorphism Group Image Atlas Hamiltonian Monte Carlo Image Registration Problem Latent Random Variable 
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.



This work was supported by NIH Grant R01 MH084795, NIH Grant 5R01EB007688, and NSF CAREER Grant 1054057.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of UtahSalt Lake CityUSA

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