A Bayesian approach to sparse model selection in statistical shape models

Ali Gooya, Christos Davatzikos, Alejandro F. Frangi

Research output: Contribution to journalArticle

Abstract

Groupwise registration of point sets is the fundamental step in creating statistical shape models (SSMs). When the number of points on the sets varies across the population, each point set is often regarded as a spatially transformed Gaussian mixture model (GMM) sample, and the registration problem is formulated as the estimation of the underlying GMM from the training samples. Thus, each Gaussian in the mixture specifies a landmark (or model point), which is probabilistically corresponded to a training point. The Gaussian components, transformations, and probabilistic matches are often computed by an expectation-maximization (EM) algorithm. To avoid over- and under-fitting errors, the SSM should be optimized by tuning the required number of components. In this paper, rather than manually setting the number of components before training, we start from a maximal model and prune out the negligible points during the registration by a sparsity criterion. We show that by searching over the continuous space for optimal sparsity level, we can reduce the fitting errors (generalization and specificities), and thereby help the search process for a discrete number of model points. We propose an EM framework, adopting a symmetric Dirichlet distribution as a prior, to enforce sparsity on the mixture weights of Gaussians. The negligible model points are pruned by a quadratic programming technique during EM iterations. The proposed EM framework also iteratively updates the estimates of the rigid registration parameters of the point sets to the mean model. Next, we apply the principal component analysis to the registered and equal-length training point sets and construct the SSMs. This method is evaluated by learning of sparse SSMs from 15 manually segmented caudate nuclei, 24 hippocampal, and 20 prostate data sets. The generalization, specificity, and compactness of the proposed model favorably compare to a traditional EM based model.

Original languageEnglish (US)
Article numberA003
Pages (from-to)858-887
Number of pages30
JournalSIAM Journal on Imaging Sciences
Volume8
Issue number2
DOIs
StatePublished - Apr 21 2015
Externally publishedYes

Fingerprint

Bayesian Approach
Model Selection
Expectation Maximization
Registration
Sparsity
Point Sets
Model
Gaussian Mixture Model
Number of Components
Specificity
Dirichlet Distribution
Generalization Error
Symmetric Distributions
Overfitting
Expectation-maximization Algorithm
Training Samples
Landmarks
Quadratic Programming
Quadratic programming
Set of points

Keywords

  • Caudate
  • EM algorithm
  • Gaussian mixture model
  • Hippocampi
  • Model selection
  • Prostate
  • Sparse inference
  • Statistical shape models

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

A Bayesian approach to sparse model selection in statistical shape models. / Gooya, Ali; Davatzikos, Christos; Frangi, Alejandro F.

In: SIAM Journal on Imaging Sciences, Vol. 8, No. 2, A003, 21.04.2015, p. 858-887.

Research output: Contribution to journalArticle

Gooya, Ali ; Davatzikos, Christos ; Frangi, Alejandro F. / A Bayesian approach to sparse model selection in statistical shape models. In: SIAM Journal on Imaging Sciences. 2015 ; Vol. 8, No. 2. pp. 858-887.
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