### 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 language | English (US) |
---|---|

Article number | A003 |

Pages (from-to) | 858-887 |

Number of pages | 30 |

Journal | SIAM Journal on Imaging Sciences |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - Apr 21 2015 |

Externally published | Yes |

### Fingerprint

### 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

*SIAM Journal on Imaging Sciences*,

*8*(2), 858-887. [A003]. https://doi.org/10.1137/140982039

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

Research output: Contribution to journal › Article

*SIAM Journal on Imaging Sciences*, vol. 8, no. 2, A003, pp. 858-887. https://doi.org/10.1137/140982039

}

TY - JOUR

T1 - A Bayesian approach to sparse model selection in statistical shape models

AU - Gooya, Ali

AU - Davatzikos, Christos

AU - Frangi, Alejandro F.

PY - 2015/4/21

Y1 - 2015/4/21

N2 - 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.

AB - 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.

KW - Caudate

KW - EM algorithm

KW - Gaussian mixture model

KW - Hippocampi

KW - Model selection

KW - Prostate

KW - Sparse inference

KW - Statistical shape models

UR - http://www.scopus.com/inward/record.url?scp=84936762826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84936762826&partnerID=8YFLogxK

U2 - 10.1137/140982039

DO - 10.1137/140982039

M3 - Article

VL - 8

SP - 858

EP - 887

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 2

M1 - A003

ER -