ABayesianApproachtoSparseModelSelectioninStatisticalShapeModels∗.pdf

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### 关于《基于贝叶斯方法的统计形状模型中稀疏模型选择》的知识点解析 #### 标题：A Bayesian Approach to Sparse Model Selection in Statistical Shape Models 本论文主要探讨了如何利用贝叶斯方法来进行统计形状模型（Statistical Shape Models, SSMs）中的稀疏模型选择问题。在构建统计形状模型时，一个核心步骤是对多个个体进行群体配准（groupwise registration）。当不同个体上的点集数量不同时，通常会将每个点集视为一个空间变换后的高斯混合模型（Gaussian Mixture Model, GMM）样本。因此，模型的选择过程实际上可以转化为从训练样本中估计潜在的高斯混合模型的问题。 #### 描述：A Bayesian Approach to Sparse Model Selection in Statistical Shape Models 描述部分进一步强调了本文的研究内容是采用贝叶斯方法来处理统计形状模型中的稀疏模型选择问题。这里的核心是通过一种稀疏性准则，从一个初始的最大模型出发，在配准过程中逐步剔除不重要的点，以避免过拟合和欠拟合的问题。这种方法不仅能够减少模型的复杂度，还能够在一定程度上提高模型的泛化能力和特异性。 #### 标签：学习算法学习算法标签表明了本文研究的方法属于机器学习领域的一种算法，即通过统计学习的方式来进行模型的选择和优化。 #### 部分内容详解 **背景介绍** - **统计形状模型**：是一种用于表示一组物体（如医学图像中的器官）形状变化的统计模型。这些模型通过分析一组形状实例来捕捉其内在的变化模式。 - **点集配准**：在构建统计形状模型的过程中，需要对来自不同个体的点集进行配准，以确保它们在空间上的一致性。 - **高斯混合模型**：是一种概率模型，常被用来表示具有多个峰的数据分布。在统计形状模型中，每个高斯成分代表一个地标点或模型点。 - **EM算法**：即期望最大化算法，是一种用于求解含有隐变量的概率模型参数估计问题的迭代算法。 **方法论** - **稀疏性准则**：为了减少模型的复杂度并提高模型的性能，作者提出了一种稀疏性准则，该准则允许从一个初始的大模型中逐步剔除那些对模型贡献较小的点。 - **贝叶斯框架下的EM算法**：作者提出了一种改进的EM算法框架，该框架采用了对称狄利克雷分布作为先验分布，以强制执行高斯混合模型中各个成分权重的稀疏性。这种改进的EM算法还能够迭代更新点集到均值模型的刚体配准参数估计。 - **二次规划技术**：在EM迭代过程中，通过二次规划技术来识别并剔除那些可忽略的模型点。 **实验验证** - **数据集**：本文选取了15个手动分割的尾状核数据集、24个海马体数据集以及20个前列腺数据集进行了实验验证。 - **评价指标**：包括模型的泛化能力（generalization）、特异性（specificity）以及紧凑性（compactness），并通过与传统的基于EM算法的模型进行比较，证明了所提方法的有效性。《基于贝叶斯方法的统计形状模型中稀疏模型选择》一文通过对统计形状模型中关键步骤——点集配准的深入研究，提出了一个创新的基于稀疏性的贝叶斯框架下的EM算法，有效地解决了模型选择中的过拟合和欠拟合问题，并通过实验证明了该方法在多个数据集上的优越性能。这对于提高统计形状模型的准确性和鲁棒性具有重要意义。

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SIAM J. IMAGING SCIENCES



2015 Society for Industrial and Applied Mathematics

Vol. 8, No. 2, pp. 858–887

A Bayesian Approach to Sparse Model Selection in Statistical Shape Models

∗

Ali Go oya

†

, Christos Davatzikos

‡

, and Alejandro F. Frangi

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 regis-

tration 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.

Key words. statistical shape models, Gaussian mixture model, sparse inference, model selection, EM algorithm,

caudate, hippocampi, prostate

AMS subject classiﬁcations. 62F15, 62P10

DOI. 10.1137/140982039

Notation.

α Concentration parameter of Dirichlet distribution

π =[π

,...,π

]

Vector of Gaussian priors

δ Thickness of the level set narrow bands

∗

Received by the editors August 12, 2014; accepted for publication (in revised form) January 20, 2015; published

electronically April 21, 2015. The research of the authors was partially funded by the European Commission’s

Framework Program 7th (contracts agreements VPH-DARE@IT FP7-ICT-2011-9-601055 and MD-Paedigree FP7-

ICT-2011-9-600932) and by the Marie Skodowska-Curie Individual Fellowship (contract agreement 625745).

http://www.siam.org/journals/siims/8-2/98203.html

†

Center of Computational Imaging and Simulation Technologies in Biomedicine, Department of Electronic and

Electrical Engineering, University of Sheﬃeld, Sheﬃeld S1 3JD, UK (a.gooya@sheﬃeld.ac.uk).

‡

Center for Biomedical Image Computing and Analytics, Department of Radiology, University of Pennsylvania,

Philadephia, PA 19104 (christos.davatzikos@uphs.upenn.edu).

Center of Computational Imaging and Simulation Technologies in Biomedicine, Department of Electronic and

Electrical Engineering, University of Sheﬃeld, Sheﬃeld S1 3JD, UK (a.frangi@sheﬃeld.ac.uk).

858

BAYESIAN MODEL SELECTION IN SSMs 859

(n)

Responsibility of jth model point

T = {T

} Set of K rigid transformations

X = {X

} Set of K training point sets

Σ =diag(σ

,...,σ

,σ

2

) Covariance matrix of Gaussians

Position vector of jth point on M

= {s

, R

, b

} Rigid transformation mapping M to X

= {x

} kth training point set constructed from kth shape

˜x

=[x

,φ

)]

Hybrid vector of ith point on kth shape

˜x

=[x

,φ

)]

] Hybrid vector of ith point on kth shape

φ(m

) Signed distance function estimated at m

Signed distance function of kth training shape

Θ={T, M, Σ, π} Set of unknowns

D Dimension of x

(n)

kij

Posterior of jth Gaussian, given ˜x

at nth iteration

K Number of training sample shapes

N =



k=1

Total number of points on the shapes

Number of points on X

Number of points on the mean model

Q(Θ | Θ

(n)

) Lower bound on likelihood at nth iteration

z Sparsity parameter (linear in α)

1. Introduction. Statistical shape models (SSMs), originally proposed by

Cootes and Taylor in [13], have been historically applied to the automatic segmentation of

anatomical structures [37, 36, 41, 55, 57, 11, 47, 39, 50, 28]. By exploiting statistical shape

priors, these models increase the robustness of the segmentations against imperfect imaging

features such as weak edges and clutter. Constructing an SSM from a training set often in-

volves estimating both the mean shape and the plausible (principal) modes of its variations.

For a comprehensive review of these methods, see Heimann and Meinzer [26]. The constructed

SSM depends on the shape representation, e.g., a popular approach is binary image repre-

sentation [49, 15, 31, 2]. In these models, the mean shape is registered to the training sets,

usually through diﬀeomorphic deformation ﬁelds as nonfolding and invertible mappings. Next,

to identify the modes of variations, principal component analysis (PCA) is directly applied to

either the deformation [15] or the velocity ﬁelds [31] in the tangent spaces. In a closely related

approach, Leventon, Grimson, and Faugeras [37] and Rousson, Paragios, and Deriche [48]

represented the shapes by signed distance functions (SDFs). The level set values computed

over the entire image domain was concatenated into a vector and subjected to PCA. Although

this method conveniently handles topological variations, nonsparse representation frequently

requires large computational resources.

More compact forms of shape expression are oﬀered by surface representation. For in-

stance, in one category, the surface is expressed as a vector in a coordinate frame spanned by

a set of orthonormal basis functions. Examples include Fourier descriptors [51] and spherical

harmonics [6, 33], obtained by solving the Laplace equation on a sphere. The corresponding

shape vectors are statistically analyzed by PCA. These methods were originally designed for

shapes homeomorphic to a sphere and are not easily extendible to more complicated and

nonconvex structures.

860 A. GOOYA, C. DAVATZIKOS, AND A. F. FRANGI

Shape representation using point sets has been widely reported in the literature [14, 13,

21, 8, 18, 17] because it is simple to implement. Landmarks are sparsely chosen on the surfaces

of the training sets, and concatenated into a vector representing each shape. Evidently, the

PCA outcome depends on the order of the points in the concatenation: inconsistent ordering

can result in artifactual variation modes. In other words, a pointwise “correspondence” should

be speciﬁed for every landmark across the training samples. In addition, the actual location

of the landmarks aﬀects the quality of PCA. Established ways of tacking these problems are

summarized in section 1.1.

Here, we estimate the number of landmarks needed to represent the mean model. This

problem can be important when creating an unbiased SSM from unregistered point sets con-

taining diﬀering number of points. Intuitively, including many landmarks on the mean model

increases the degree of freedom and results in overﬁtting in the from of loose registration (and

incorrect modes of variations). Conversely, few landmarks deteriorate geometrical details. In

fact, the relation between the degree of freedom in a shape model and the under/overﬁtting

problem is also evident in the context of geodesic active contours [10]. In its simplest form, an

evolving contour is attracted toward image edges, while its shape is regularized by penalizing

its length. This leads into a curve evolution scheme that consists in a curvature weighted term

to regularize the shape of the curve [23]. Reducing the weight of curvature often results in

an unregularized curve that does not necessarily match the target borders and could overﬁt

to image background, whereas increasing the curvature weight yields overly simpliﬁed and

underﬁtted curves. Therefore, motivated by these observations, we hypothesize that there

must exist an optimal number of landmarks (or sparsity level as discussed in the rest of the

paper) that minimizes generalization and speciﬁcity errors.

The optimal number of landmarks should ideally minimize both generalization and speci-

ﬁcity errors [18]. In practice, however, the errors may not be minimized by the same number

of model points. Thus, rather than obtaining an SSM with a single optimal number of land-

marks, we use cross-validation and directly measure the generalization/speciﬁcity errors in

the left-out samples. We then identify a range of model point numbers giving reasonable error

values.

More importantly, rather than searching in the discrete space, we propose a Bayesian

framework replacing the discrete number by a continuous hyper-parameter. Starting from a

large number of model points, the hyper-parameter controls the sparsity rate of model point

pruning during the registration. To achieve this, we formulate groupwise registration as the

estimation of a Gaussian mixture model (GMM) from the training points, corresponding each

model point to a Gaussian component. In this setting, the sparsity hyper-parameter deﬁnes

the “concentration” parameter of a symmetric Dirichlet prior imposed on the mixture weights

in the GMM. Thus, the problem of selecting the number of model points becomes that of

determining the correct sparsity level. We show that this process compacts the SSMs and

reduces their generalization and speciﬁcity errors, relative to those of established SSMs [30].

In machine learning studies, estimating the number of Gaussians in a GMM is regarded as

a particular example of the “model selection” problem [16], in which models with intermediate

The ﬁrst error is related to the distance between an actual and the model predicted shapes. The latter

speciﬁes the ability of the model to instantiate correct shapes resembling the training data.

BAYESIAN MODEL SELECTION IN SSMs 861

complexity (ratio of free parameters to training data) are known to be optimal in generalization.

However, to our knowledge, estimating number of model points as a particular model selection

problem has not been fully considered by the medical imaging community. Here, we demon-

strate that an intermediate number of model points (sparsity levels) generally improves model

eﬃciencies. To correctly highlight the contributions of this study, we brieﬂy review some of

the seminal SSM studies in section 1.1

1.1. Review of related state-of-the-art SSM methods. Manual selection of landmarks

and setting their correspondences across training shapes [13] can be a time-consuming and

nonreproducible procedure. In fact, landmarks are matched by machine generated approaches,

traditionally by the iterative closest point (ICP) [42] algorithm. Other automatic methods

improve the matching by including curvatures and local neighborhood features [7, 54, 52]. The

robustness of matchings is enhanced by spatial regularization using Markov random ﬁelds [43],

currents [53], and diﬀeomorphic mappings [22]. In an alternative approach, Frangi et al. [21]

extracted a set of landmarks from a preconstructed atlas and assigned each set an example

shape by nonrigid B-spline mappings. However, these methods do not necessarily optimize

groupwise statistics. Regarding this problem as homologous to the minimum variance princi-

ple, Hill and Taylor [27] and Baumberg and Hogg [3] minimized the trace of the covariance

matrix by displacing the landmarks in the surface normal direction. In an alternative ap-

proach, Kotcheﬀ and Taylor [35] minimized the determinant of the covariance matrix and

introduced a free parameter to avoid zero eigenvalues. Inspired by the Occam’s razor prin-

ciple, which favors simple models over complex ones, Davis et al. [18, 17] minimized the

length of the binary code specifying the Gaussian distribution of the data. These methods

are mathematically elegant but computationally expensive, and because of their parametric

surface representation, they are not readily extendible to nonspherical topologies.

Identifying correspondences between landmarks becomes even more challenging when the

training sets cannot be exactly matched [44], either because of intrinsic anatomical diﬀerences

or because the number of points diﬀers among the point clouds. In such cases, hard-binds

established between the landmarks could introduce dummy modes in the PCA variations.

To alleviate this problem, Rangarajan, Chui, and Bookstein [45] proposed the soft-assign

procrustes algorithm for matchings two sets of landmarks with diﬀerent counts. Granger

and Pennec [25] proposed EM-ICP, a combination of expectation-maximization (EM) [19]

and conventional ICP that enables partial or probabilistic matching between points. Since

EM based methods can deal with complex variations, they have been popularly applied in

groupwise registrations and SSM constructions.

Chui et al. [12] proposed a forward-backward consistent deformable registration of point

sets to the mean model in an EM-like framework. Their algorithm iteratively updates the

point correspondences, mean model, and deformations, but it does not consolidate the results

by PCA. In a similar but computationally less expensive framework, Hufnagel et al. [30]

appliedanaﬃnetransformationtoregisterthe point sets. Based on the EM outcome (i.e.,

the registered point clouds with partial matches between points), they then performed a

heuristic PCA on the “virtually correspondent” points. Recently, Rasoulian Rohling, and

Abolmaesumi [46], replaced the aﬃne with a B-spline based registration model, and they

applied PCA to the computed deformation ﬁelds.

862 A. GOOYA, C. DAVATZIKOS, AND A. F. FRANGI

In the above studies, groupwise registration of point sets was reformulated as estimating

the underlying probability density function (pdf) from spatially transformed noisy observa-

tions of points. The pdf is often represented as a Gaussian mixture model, whose means

(estimated by EM) determine the landmark locations. Determining an appropriate number

of Gaussian components for a speciﬁed number of data points, the so-called model selection

problem in machine learning, is ignored in these approaches. However, model selection can

avoid under- and overﬁtting errors and thereby improve the generalization eﬃciency of the

trained pdf. In an SSM context, model complexity can be inﬂuenced by the number of points

representing the mean shape, which is usually selected before the training phase [25, 12, 30, 46].

The contributions of this study are twofold. First, we propose a Bayesian framework for

sparse SSM construction. Within this framework, the number of model points is indirectly

determined during the training phase. Starting from a maximal number of model points,

sparsity is enforced by employing a symmetric Dirichlet prior on the mixture weights. Neg-

ligible model points are pruned during each EM iteration by a quadratic programming (QP)

technique. The sparsity rate is controlled by a hyper-parameter, namely, the concentration

parameter of the Dirichlet prior. As is well known, the weights of the Gaussians in the GMM

become sparse when this parameter is less than unity. We optimize the value of this param-

eter in cross-validation experiments and identify an approximate range that generalizes well

globally over various data sets. Second, we simplify the reconstruction of surfaces from points

by representing each landmark as a hybrid vector containing the level set and coordinates.

Note that our representation is denser than that of Leventon, Grimson, and Faugeras [37],

who disregarded the coordinates.

This paper extends our preliminary work in [24]. Here, we provide the mathematical

details and apply our method to diﬀerent and a larger number of data sets, two of which

are nonhomeomorphic to a sphere. Topologically nonspherical data pose special challenges to

training SSMs by Davis et al. [17] and Keleman, Szekely, and Gerig [33]. Thus, we compare our

method to a closely related approach proposed by Hufnagel et al. [30], which is more ﬂexible

compared to [17, 33] and has been used as a reference in an independent recent study [46].

Sparsity has been applied to SSMs by several research groups. For instance, in the sparse

shape representation of Zhang et al. [56], any unseen shape is represented by a sparse linear

combination of the training samples. Alternatively, Durrleman et al. [20] enforced sparsity

on the deformation ﬁelds using an L1-type prior, which encodes the shape variations. By

contrast, our approach focuses on the number of model points.

The remainder of this paper is organized as follows. Our model is formulated in section 2.

Sections 3 and 4 present the results and conclusions, respectively. Detailed mathematical

derivations are given in the appendix (section 5).

2. Methods. This section introduces our EM algorithm. This general framework allows

us to (1) align given training point clouds and (2) prune out negligible points during the

registration. The result is a sparse mean model. We emphasize that in addition to the

number of model points, the properties of the principal modes of variations aﬀect the eﬃcacy

of the SSM. However, because of its complexity, we do not explicitly optimize the variation

modes but rather obtain them by independent PCA after registration. In fact, this problem

has been extensively studied in other works, such as [17, 27, 3, 35]. Having achieved these

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