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Nonparametric Mixture Models for Supervised Image Parcellation

Mert R. Sabuncu1, B.T. Thomas Yeo1, Koen Van Leemput1,2,3, Bruce Fischl1,2, and Polina

Golland1

1 Computer Science and Artificial Intelligence Lab, MIT, USA 2 Department of Radiology, Harvard

Medical School, USA 3 Dep. of Information and Comp. Sci., Helsinki University of Technology,

Finland

Abstract

We present a nonparametric, probabilistic mixture model for the supervised parcellation of images.

The proposed model yields segmentation algorithms conceptually similar to the recently developed

label fusion methods, which register a new image with each training image separately. Segmentation

is achieved via the fusion of transferred manual labels. We show that in our framework various

settings of a model parameter yield algorithms that use image intensity information differently in

determining the weight of a training subject during fusion. One particular setting computes a single,

global weight per training subject, whereas another setting uses locally varying weights when fusing

the training data. The proposed nonparametric parcellation approach capitalizes on recently

developed fast and robust pairwise image alignment tools. The use of multiple registrations allows

the algorithm to be robust to occasional registration failures. We report experiments on 39 volumetric

brain MRI scans with expert manual labels for the white matter, cerebral cortex, ventricles and

subcortical structures. The results demonstrate that the proposed nonparametric segmentation

framework yields significantly better segmentation than state-of-the-art algorithms.

1 Introduction

Supervised image parcellation (segmentation) tools traditionally use atlases, which are

parametric models that summarize the training data in a single coordinate system [1–9]. Yet,

recent work has shown that more accurate segmentation can be achieved by utilizing the entire

training data [10–16], by mapping each training subject into the coordinates of the new image

via a pairwise registration algorithm. The transferred manual labels are then fused to generate

a segmentation of the new subject. There are at least two advantages of this approach: (1)

across-subject anatomical variability is better captured than in a parametric model, and (2)

multiple registrations improve robustness against occasional registration failures. The main

drawback of the label fusion (multi-atlas) approach is the computational burden introduced by

the multiple registrations and the manipulation of the entire training data.

Early label fusion methods proposed to transfer the manual labels to the test image via nearest

neighbor interpolation after pairwise registration [12,14]. Segmentation labels of the test image

were then estimated via majority voting. Empirical results suggested that errors in the manual

labeling and in registration are averaged out during label fusion, resulting in accurate

segmentation. More recent work has shown that a weighted averaging strategy can be used to

improve segmentation quality [11]. The basic idea is that training subjects more similar to the

test subject should carry more weight during label fusion. The practical advantages of various

strategies based on this idea have lately been demonstrated [11,13,16]. Some of these strategies

use the whole image to determine a single, global weight for each training subject [11,15,16],

whereas others use local image intensities for locally adapting the weights [11,13].

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Published in final edited form as:

Med Image Comput Comput Assist Interv. 2009 September 1; 12(WS): 301–313.

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This paper presents a novel unified probabilistic model that enables local and global weighting

strategies within a label fusion-like segmentation framework. We formulate segmentation

using MAP, where a nonparametric approach is used to estimate the joint density on the image

intensities and segmentation labels of the new subject. The proposed framework generalizes a

model we recently presented at MICCAI 2009 [16], which is based on the assumption that the

test subject is generated from a single, unknown training subject. That specific model leads to

a segmentation algorithm that assigns greater importance to the training subjects that are

globally more similar to the test subject and can be viewed as a particular instantiation of the

more general approach presented in this paper. In addition, the proposed approach further

enables two possible variants within the same framework. First, we present a local mixture

model that assumes each voxel in the test image is generated from some training subject with

a uniform prior, independently of other voxels. This local model yields a pixel-wise weighting

strategy in segmentation. Second, we develop a semi-local mixture model that relaxes the

independence assumption of the local model with a Markov Random Field prior. This model

leads to a weighting strategy where intensity information in a local neighborhood is pooled in

a principled manner.

In related literature, soft weighting of training subjects was recently used for shape regression

[17], where the weights depended on the subjects’ age. The proposed nonparametric

parcellation framework is also parallel to STAPLE [18], which fuses multiple segmentations

of a single subject. In contrast, our framework handles multiple subjects and accounts for inter-

subject variability through registration.

The paper is organized as follows. The next section presents the non-parametric generative

model for image segmentation. In section 3, we discuss three instantiations of the framework.

In section 4, we present inference algorithms for these instantiations. We conclude with

experiments in section 5. We report experiments on 39 brain MRI scans that have

corresponding manual labels, including the cerebral cortex, white matter, and sub-cortical

structures. Experimental results suggest that the proposed nonparametric parcellation

framework achieves better segmentation than the existing state-of-the-art algorithms.

2 Theory

Let {Ii} be N training images with corresponding label maps {Li}, i = 1, . . . ,N. We assume

the label maps take discrete values from 1 to that indicate the label identity at each spatial

location. We treat these training images as spatially continuous functions on

a suitable interpolator. Let denote a new, previously unseen test image defined on

a discrete grid . Let denote the spatial mapping (warp) from the test image

coordinates to the coordinates of the training image i. We assume that {Φi} have been

precomputed using a pairwise registration procedure, such as the one described in Section 4.1.

by assuming

Our objective is to estimate the label map L̂ associated with the test image I. One common

formulation to compute L̂ is via MAP:

(1)

where p(L, I|{Li, Ii, Φi}) denotes the joint probability of the label map L and image I given the

training data.

Rather than using a parametric model for p(L, I|{Li, Ii, Φi}), we employ a non-parametric

estimator, which is an explicit function of the entire training data, not a summary of it. Let

denote an unknown (hidden) random field that, for each voxel in test image

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I, specifies the training image Ii that generated that voxel. Given M, the training data, and warps,

and assuming the factorization depicted in the graphical model of Fig. 1, we can construct the

conditional probability of generating the test image and label map:

(2)

(3)

(4)

where pM(x)(L(x), I(x)|LM(x), IM(x), ΦM(x)(x)) is the conditional probability of (L(x), I(x)) given

that voxel x ∈ Ω of the test subject was generated from training subject M(x). Note that Eq. (4)

assumes that (L(x), I(x)) are conditionally independent given the membership M(x),

corresponding warp ΦM(x) and training data. Given a prior on M, we can view p(L, I|{Li, Ii,

Φi}) as a mixture:

(5)

where

(4) into Eq. (5) yields:

denotes the marginalization over the unknown random field M . Substituting Eq.

(6)

In the next section, we present instantiations of the individual terms in Eq. (6).

3 Model Instantiation

3.1 Image Likelihood

We adopt a Gaussian distribution with a stationary variance σ2 as the image likelihood:

(7)

3.2 Label Likelihood

We use the distance transform representation to encode the label prior information, cf. [19].

Let denote the signed distance transform of label l in training subject i, assumed to be positive

inside the structure of interest. We define the label likelihood as:

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(8)

where ρ > 0 is the slope constant and

of labels including a background label. pi(L(x) = l|Li,Φi(x)) encodes the conditional probability

of observing label l at voxel x ∈ Ω of the test image, given that it was generated from training

image i.

, where is the total number

3.3 Membership Prior

The latent random field

image and training data. We place a Markov Random Field (MRF) prior on M:

encodes the local association between the test

(9)

where β ≥ 0 is a scalar parameter,

function that only depends on β, and δ(M(x), M(y)) = 1, if M(x) = M(y) and zero otherwise. In

our implementation, includes the immediate 8 neighbors of each voxel. Similar models

have been used in the segmentation literature, e.g. [5,9], mainly as priors on label maps to

encourage spatially contiguous segmentations. In contrast, we use the MRF prior to pool local

intensity information in determining the association between the test subject and training data.

is a spatial neighborhood of voxel x, Zβ is the partition

The parameter β influences the average size of the local patches of the test subject that are

generated from a particular training subject. In this work, we consider three settings of the

parameter β. For β = 0, the model effectively assumes that each test image voxel is

independently generated from a training subject, drawn with a uniform prior. β → +∞ forces

the membership of all voxels to be the same and corresponds to assuming that the whole test

subject is generated from a single unknown training subject, drawn from a uniform prior. A

positive, finite β favors local patches of voxels to have the same membership.

The β → +∞ case reduces to a model similar to the one we presented in [16], except now we

make the simplifying assumption that the training data is apriori mapped to the test subject's

coordinate frame as a preprocessing step. Due to this simplification, the warp cost in registration

plays no role in the segmentation algorithms we present in this paper. Without this

simplification, however, inference for finite values of β becomes intractable. As demonstrated

in the next section, the resulting inference algorithms allow us to determine the association

between the training data and test subject using local intensity information.

4 Algorithms

4.1 Efficient Pairwise Registration

To perform pairwise registration, we employ an efficient algorithm [20,21] that uses a one-

parameter subgroup of diffeomorphisms, where a warp Φ is parameterized with a smooth,

stationary velocity field

condition Φ(x, 0) = x. The warp Φ(x) = exp(v)(x) can be computed efficiently using scaling

and squaring and inverted by using the negative of the velocity field: Φ−1 = exp(−v) [22].

via an ODE [22]: and initial

We impose an elastic-like regularization on the stationary velocity field:

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(10)

where λ > 0 is the warp stiffness parameter, Zλ is a partition function that depends only on λ,

and xj and vk denote the j'th and k'th component (dimension) of position x and velocity v,

respectively. A higher warp stiffness parameter λ yields more rigid warps.

To derive the registration objective function, we assume a simple additive Gaussian noise

model, consistent with the image likelihood term described in Section 3.1. This model leads

to the following optimization problem for registering the i-th training image to the test subject:

(11)

where σ2 is the stationary image noise variance, and

the bidirectional log-domain Demons framework [20], which decouples the optimization of

the first and second terms by introducing an auxiliary transformation. The update warp is first

computed using the Gauss-Newton method. The regularization is achieved by smoothing the

updated warp field. It can be shown that the smoothing kernel corresponding to Eq. (10) can

. To solve Eq. (11), we use

be approximated with a Gaussian;

controls the size of the Gauss-Newton step.

, where and γ > 0

4.2 Segmentation Algorithms

Here, we present algorithms to solve the optimization problem of Eq. (6) for the three cases of

β in the model presented in Section 3.

4.3 Global Mixture

First, we consider β → +∞, which is equivalent to a global mixture model, where the test subject

is assumed to be generated from a single, unknown training subject. In this case, the

segmentation problem in Eq. (6) reduces to

(12)

Eq. (12) cannot be solved in closed form. However, an efficient solution to this MAP

formulation can be obtained via Expectation Maximization (EM). Here, we present a summary.

The E-step updates the posterior of the membership associated with each training image:

(13)

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where L̂(n−1)(x) is the segmentation estimate of the test image from the previous iteration and

. The M-step updates the segmentation estimate:

(14)

The E-step in Eq. (13) determines a single membership index for the entire training image,

based on all the voxels. The M-step in Eq. (14) performs an independent optimization at each

voxel x ∈ Ω; it determines the mode of a vector, where is the number of labels.

The EM algorithm is initialized with

Equations (14) and (13), until convergence.

and iterates between

4.4 Local Mixture: Independent Prior

The second case we consider is β = 0, which corresponds to assuming a voxel-wise independent

mixture model with a uniform prior on M:

(15)

where |Ω| is the cardinality of the image domain, i.e., the number of voxels. It is easy to show

that the segmentation problem reduces to

(16)

where the image and label likelihood terms in the summation can be computed using Eqs. (7)

and (8). The optimization problem can be solved by simply comparing numbers at each

voxel.

4.5 Semi-local mixture: MRF Prior

Finally, we consider a finite, positive β. This leads to an MRF prior, which couples neighboring

voxels and thus the exact marginalization of Eq. (6) becomes computationally intractable. An

efficient approximate solution can be obtained using variational mean field [23]. The main idea

of variational mean field is to approximate the posterior distribution of the membership p(M |

I, L, {Ii, Li, Φi}), with a simple distribution q that is fully factorized:

(17)

The objective function of Eq. (6) can then be approximated by an easier-to-optimize lower

bound, which is a function of q. This approximate problem can be solved via coordinate-ascent,

where the segmentation L and approximate posterior q are updated sequentially, by solving the

optimization for each variable while fixing the other. One particular formulation leads to a

straightforward update rule for L:

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(18)

where q(n−1) is an estimate of the posterior at the (n − 1)'th iteration. Eq. (18) is independent

for each voxel and entails determining the mode of a

estimate L̂(n)(x) the optimal q is the solution of the following fixed-point equation:

vector. For a fixed segmentation

(19)

and

alternates between Eqs. (19) and (18), until convergence.

. We solve Eq. (19) iteratively. The variational mean field algorithm

5 Experiments

We validate the proposed framework on 39 T1-weighted brain MRI scans of dimensions 256

× 256 × 256, 1mm isotropic. Each MRI volume is an average of 3-4 scans and was gain-field

corrected and skull-stripped. These volumes were then manually delineated by an expert

anatomist into left and rightWhite Matter (WM), Cerebral Cortex (CT), Lateral Ventricle (LV),

Hippocampus (HP), Thalamus (TH), Caudate (CA), Putamen (PU), Pallidum (PA) and

Amygdala (AM). We use volume overlap with manual labels, as measured by the Dice score

[24], to quantify segmentation quality. The Dice score ranges from 0to 1, with higher values

indicating improved segmentation.

5.1 Setting Parameters Through Training

The proposed nonparametric parcellation framework has two stages with several input

parameters. The registration stage has two independent parameters: γ that controls the step size

in the Gauss-Newton optimization and α that determines the smoothness of the final warp. The

segmentation stage has two additional input parameters: σ2, which is the intensity variance of

the image likelihood in Eq. (7) and the slope ρ of the distance transform uses to compute the

label prior in Eq. (8). Furthermore, the semi-local model of Section 4.5 has a non-zero, finite

β parameter.

Nine subjects were used to determine the optimal values of these parameters. First, 20 random

pairs of these nine subjects were registered for a range of values of γ and α. Registration quality

was assessed by the amount of pairwise label overlap and used to select the optimal (γ*, α*)

pair.

We used the optimal (γ*, α*) pair to register all 72 ordered pairs of the nine training subjects.

We performed nine leave-one-out segmentations using both the global and local models to

determine the corresponding optimal pairs of σ2 and ρ. The optimal pair for the local model

was then used to determine the optimal value for β in the semi-local model. The optimal

parameter values were finally used to segment the remaining 30 subjects.

5.2 Benchmarks

The first benchmark we consider is the whole-brain parcellation tool available in the Freesurfer

software package [25]. The Freesurfer parcellation tool uses a unified registration-

segmentation procedure that models across-scanner intensity variation [2,3]. We consider this

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as a state-of-the-art benchmark, since numerous imaging studies across multiple centers have

shown Freesurfer's a robustness and accuracy as a segmentation tool.

As a second benchmark, we use our implementation of the Label Fusion algorithm [12,14].

We employ the pairwise registrations obtained with (γ*, α*) to transfer the labels of the training

subjects via the trilinear interpolation of the probability maps, obtained by assigning 1 to entries

corresponding to the manual labels and zero elsewhere. Segmentation is then computed through

majority voting at each voxel. We use trilinear interpolation instead of nearest neighbor

interpolation because we find that trilinear interpolation yields better results.

5.3 Results

We report test results for the 30 subjects not included in the group used for setting the algorithm

parameters γ, α, σ2, ρ, and β. For each test subject, we treated the remaining subjects as training

data in a cross-validation evaluation.

Fig. 2 illustrates a typical automatic segmentation result obtained with the local mixture model

and overlaid on the MRI volume. Fig. 3 shows box-plots of Dice scores for the two benchmarks

and the proposed non-parametric parcellation algorithms. Table 1 provides the mean Dice

scores averaged over all subjects and both hemispheres. Fig. 4 provides an overall comparison

between the average dice scores achieved by the algorithms.

On average, the local and semi-local mixture models yield better segmentations than the global

mixture model, mainly due to the large improvement in the white matter, cerebral cortex and

lateral ventricles, the segmentation of which clearly benefits from the additional use of local

intensity information. A paired t-test between the local and semi-local models reveals that a

statistically significant improvement is achieved with the MRF model that pools local intensity

information. Yet, this improvement is overall quite modest: about 1% per ROI.

As discussed earlier, the global mixture model is similar to that of [16], except that [16]

incorporates registration into the model. Despite this, we find that both algorithms achieve

similar segmentation accuracy (results not shown).

A paired sample t-test implies that the difference in accuracy between the proposed semi-local

mixture model and Freesurfer is statistically significant (p < 0.05, Bonferroni corrected) for

all ROIs, except the cerebral cortex and right Caudate, where the two methods yield comparable

results. The same results are obtained when comparing the local mixture model and Freesurfer.

Compared to the Label Fusion benchmark, the nonparametric parcellation algorithms (global,

local and semi-local) yield significantly better segmentation (paired sample t-test, p < 0.05,

Bonferroni corrected) in all regions, except Pallidum and Putamen, where the improvement

over Label Fusion does not reach statistical significance. We note, however, that the results

we report for our Label Fusion implementation are lower than the ones reported in [12]. This

might be due to differences in the data and/or registration algorithm. Specifically, normalized

mutual information (NMI) was used as the registration cost function in [12]. Entropy-based

measures such as NMI are known to yield more robust alignment results. We leave a careful

analysis of this issue to future work.

Table 2 lists the average run-times for all five algorithms. The parametric atlas-based Freesurfer

algorithm is the fastest, mainly because it needs to compute only a single registration. The

remaining algorithms take up more than 20 hours of CPU time on a modern machine, most of

which is dedicated to the many registrations performed with the training data. The two iterative

algorithms that solve the global mixture and semi-local mixture models (EM and variational

mean field, respectively) require significantly longer run-times. The local-mixture model, on

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the other hand, requires minimal computation time once the registrations are complete, since

it simply performs a voxelwise weighted averaging. Its run-timeis similar to that required by

Label Fusion.

6 Conclusion

This paper presents a novel, nonparametric mixture model of images and label maps, that yields

accurate image segmentation algorithms. The resulting algorithms are conceptually similar to

recent label fusion (or multi-atlas) methods that utilize the entire training data, rather than a

summary of it, and register the test subject to each training subject separately. Segmentation

is then achieved by fusing the transferred manual labels. In the proposed framework,

similarities between the test image and training data determine how the transferred labels are

weighed during fusion. As we discuss in this paper, different settings of a model parameter

yields various weighting strategies. Our experiments suggests that a semi-local strategy that is

derived from an MRF model that encourages local image patches to be associated with the

same training data provides the best segmentation results. We also show that a computationally

less expensive local strategy that treats each voxel independently leads to accurate

segmentations that are better than the current state-of-the-art.

We leave an investigation of various registration algorithms within the proposed framework

to future work. It is clear that alternative strategies can be used to improve the alignment

between the training data and test subject. For example, one could use a richer representation

of diffeomorphic warps, cf. [4], or a more sophisticated registration cost function, cf. [12].

Since any multi-atlas segmentation algorithm will be robust against occasional registration

failures, whether a better alignment algorithm will lead to more accurate segmentation remains

an open question.

Acknowledgments

Support for this research is provided in part by: NAMIC (NIH NIBIB NAMIC U54-EB005149), the NAC (NIH NCRR

NAC P41-RR13218), the mBIRN (NIH NCRR mBIRN U24-RR021382), the NIH NINDS R01-NS051826 grant, the

NSF CAREER 0642971 grant, NCRR (P41-RR14075, R01 RR16594-01A1), the NIBIB (R01 EB001550,

R01EB006758), the NINDS (R01 NS052585-01), the MIND Institute, and the Autism & Dyslexia Project funded by

the Ellison Medical Foundation. B.T. Thomas Yeo is funded by the A*STAR, Singapore.

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Fig. 1.

Generative model for (L(x),I(x)) given M(x) = i and (Li,Ii,Φi). Φi is the mapping from the image

coordinates to the template coordinates. Squares indicate non-random parameters, while circles

indicate random variables. Shaded variables are assumed to be observed.

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Fig. 2.

A typical segmentation obtained with the local mixture model.

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Fig. 3.

Boxplots of Dice scores for Freesurfer (red), Label Fusion (yellow), the global mixture model

(green), the local mixture model (blue) and the semi-local mixture model (purple). Top row is

left hemisphere. Bottom row is right hemisphere. Medians are indicated by horizontal bars.

Boxes indicate the lower and upper quartiles and vertical lines extend to 1.5 inter-quartile

spacing. ‘+’s indicate outliers.

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Fig. 4.

Average Dice scores for each algorithm (FS: Freesurfer, LF: Label Fusion, Global: Global

Mixture, Local: Local Mixture, and Semi-Local: MRF-based model). Error bars show standard

error. Each subject and ROI was treated as an independent sample with an equal weight.

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Table 1

Comparison of average dice scores. Boldface font indicates best scores for each structure. As a reference, the last row lists approximate average volumes.

WM

CT

LV

HP

TH

CA

PU

PA

AM

Freesurfer

0.92

0.85

0.87

0.84

0.88

0.85

0.85

0.80

0.75

Label Fusion

0.85

0.66

0.84

0.77

0.86

0.80

0.86

0.81

0.75

Global Mixture

0.88

0.77

0.87

0.83

0.90

0.84

0.89

0.83

0.81

Local Mixture

0.93

0.84

0.90

0.86

0.90

0.86

0.88

0.82

0.82

Semi-local Mixture

0.93

0.86

0.91

0.87

0.91

0.87

0.89

0.83

0.82

Volumes (×l03mm3)

450

448

25

7

14

7

10

3

3

Med Image Comput Comput Assist Interv. Author manuscript; available in PMC 2010 September 1.