Fast and robust parameter estimation for statistical partial volume models in brain MRI

Article (PDF Available)inNeuroImage 23(1):84-97 · October 2004with38 Reads
DOI: 10.1016/j.neuroimage.2004.05.007 · Source: PubMed
Abstract
Due to the finite spatial resolution of imaging devices, a single voxel in a medical image may be composed of mixture of tissue types, an effect known as partial volume effect (PVE). Partial volume estimation, that is, the estimation of the amount of each tissue type within each voxel, has received considerable interest in recent years. Much of this work has been focused on the mixel model, a statistical model of PVE. We propose a novel trimmed minimum covariance determinant (TMCD) method for the estimation of the parameters of the mixel PVE model. In this method, each voxel is first labeled according to the most dominant tissue type. Voxels that are prone to PVE are removed from this labeled set, following which robust location estimators with high breakdown points are used to estimate the mean and the covariance of each tissue class. Comparisons between different methods for parameter estimation based on classified images as well as expectation--maximization-like (EM-like) procedure for simultaneous parameter and partial volume estimation are reported. The robust estimators based on a pruned classification as presented here are shown to perform well even if the initial classification is of poor quality. The results obtained are comparable to those obtained using the EM-like procedure, but require considerably less computation time. Segmentation results of real data based on partial volume estimation are also reported. In addition to considering the parameter estimation problem, we discuss differences between different approximations to the complete mixel model. In summary, the proposed TMCD method allows for the accurate, robust, and efficient estimation of partial volume model parameters, which is crucial to a variety of brain MRI data analysis procedures such as the accurate estimation of tissue volumes and the accurate delineation of the cortical surface.

Figures

Fast and robust parameter estimation for statistical partial volume
models in brain MRI
Jussi Tohka,
a,
*
Alex Zijdenbos,
b
and Alan Evans
b
a
Digital Media Institute / Signal Processing, Tampere University of Technology, FIN-33101, Finland
b
McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University, Montreal, Canada
Received 5 December 2003; revised 24 April 2004; accepted 11 May 2004
Due to the finite spatial resolution of imaging devices, a single voxel in a
medical image may be composed of mixture of tissue types, an effect
known as partial volume effect (PVE). Partial volume estimation, that
is, the estimation of the amount of each tissue type within each voxel,
has received considerable interest in recent years. Much of this work
has been focused on the mixel model, a statistical model of PVE. We
propose a novel trimmed minimum covariance determinant (TMCD)
method for the estimation of the parameters of the mixel PVE model.
In this method, each voxel is first labeled according to the most
dominant tissue type. Voxels that are prone to PVE are removed from
this labeled set, following which robust location estimators with high
breakdown points are used to estimate the mean and the covariance of
each tis sue class. Compariso ns b etween different methods fo r
parameter estimation based on classified images as well as expecta-
tion maximization-like (EM-like) procedure for simultaneous param-
eter and partial volume estimation are reported. The robust estimators
based on a pruned classification as presented here are shown to
perform well even if the initial classification is of poor quality. The
results obtained are comparable to those obtained using the EM-like
procedure, but require considerably less computation time. Segmenta-
tion results of real data based on partial volume estimation are also
reported. In addition to considering the parameter estimation problem,
we discuss differences between different approximations to the
complete mixel model. In summary, the proposed TMCD method
allows for the accurate, robust, and efficient estimation of partial
volume model parameters, which is crucial to a variety of brain MRI
data analysis procedures such as the accurate estimation of tissue
volumes and the accurate delineation of the cortical surface.
D 2004 Elsevier Inc. All rights reserved.
Keywords: Partial volume effect; Segmentation; Robust estimation; Mixel
model
Introduction
The quantitative analysis of magnetic resonance (MR) images in
the study of human brain anatomy is becoming more and more
important. For example, a range of brain disorders as well as brain
development and healthy aging can cause structural changes in the
brain. These changes can be quantified by measuring volumes or
other properties of anatomical structures of interest providing
information, for example, on disease severity. Before measure-
ments can be performed, the structures of interest must be extracted
from the image data. This often includes the labeling of voxels
according to their tissue type. This labeling, or classification, can be
performed based on a single MR image or based on a multispectral
image constructed by combining series images of the same subject
acquired with different pulse sequence parameters. Typically, the
tissue types of interest are white matter (WM), gray matter (GM),
and cerebrospinal fluid (CSF), but also more anatomical labels can
be assigned to image voxels (Collins et al., 1999; Fischl et al.,
2002). However, voxel classification with three basic tissue types
has itself rather direct applications such as quantifying disease
burden in multiple sclerosis by estimation of the amount of brain
atrophy (Collins et al., 2001). Moreover, many procedures aiming
at the extraction of particular brain structures, such as cerebral
cortex, can gain from the initial tissue classification (MacDonald et
al., 2000; Xu et al., 1999; Zeng et al., 1999) as can partial volume
correction in positron mission tomography (Rousset et al., 1998).
Because of the finite resolution of the imaging devices, a single
voxel may contain several tissue types. This is known as partial
volume effect (PVE). Due to PVE, the classification of a voxel
reflecting the dominant tissue type (WM, GM, or CSF), does not
reveal all possible information about the tissue content of that
voxel. This can be problematic in small structures or highly
convoluted areas of the brain. For example, algorithms aimed at
extraction of the cortical surface often omit entire sulci due to the
PVE between the thin ribbon of sulcal CSF and the surrounding
gray matter. These problems are especially serious when pediatric
subjects are considered as illustrated in Fig. 1. Estimation of the
amount of each tissue type present in each voxel, that is, partial
volume estimation, provides an interesting possibility to improve
the accu racy of cortical surface extraction (cf. Fig. 1). Other
applications that gain from modeling of PVE have been as well
considered within brain MRI. Santago and Gage (1993) apply
information about partial volume voxels to improve tissue quan-
tification. Gonza
´
lez Ballester et al. (2002) study the asymmetry of
temporal horns taking PVE into account, and in an earlier work
(Gonza
´
lez Ballester et al., 2000 ), they suggest that PVE and
1053-8119/$ - see front matter D 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2004.05.007
* Corresponding author. Digital Media Institute/Signal Processing,
Tampere University of Technology, P.O. Box 553, FIN-33101, Finland.
Fax: +358-3-31153087.
E-mail address: jussi.tohka@tut.fi (J. Tohka).
Available online on ScienceDirect (www.sciencedirect.com.)
www.elsevier.com/locate/ynimg
NeuroImage 23 (2004) 84 97
discrete sampling at boundary locations can l ead to v olume
measurement errors in the range 20 60%.
Partial volume effect and PV estimation have been addressed in
various ways in MR imaging literature. For example, Pham and
Prince (1999) have proposed a fuzzy C-means algorithm. They
have also studied the relationship between the fuzzy C-means
objective function and statistical models of PVE in the simplified
case (only two type is of tissues and single-spectral data), showing
that in this case, these two approaches can be considered equivalent
(Pham and Prince, 1998). However, these considerations do not
extend to more realistic situations where one would have more than
two tissue types and where the data would be multispectral.
Wang et al. (2001) propose to use a Bayesian classifier with a
variable number of tissue classes, including classes of mixed tissue
types. However, as the authors model the image histogram with a
finite mixture of normal distributions and parameters (means and
variances) for mixed tissue classes, which are independent of the
parameters of the related pure tissue classes, there is no explicit
model of the PVE. While the procedure may be reasonable for
detection of voxels containing PVE, estimation of the mixing
proportions can be challenging.
The most commonly used, statistically based model of PVE is
the mixel model proposed by Choi et al. (1991). (A similar model
was proposed earlier by Kent and Mardia, 1988, but without
consideration of medical imaging applications.) This approach
assumes that each intensity value in the image is a realization of
a weighted sum of random variables (RVs), each of which
characterizes a pure tissue type. We call these weighting factors
partial volume coefficients (PVCs). The method involves maxi-
mum-likelihood estimation of the PVCs for each voxel that model
PV fractions of pure tissue types. Some authors have studied the
identification of voxels containing PVE based on the mixel or a
closely related model without trying to estimate the PVCs for each
voxel (Laidlaw et al., 1998; Ruan et al., 2000; Santago and Gage,
1993). Our interest in this study is in estimating PVCs and not in
merely identifying voxels containing PVE.
Before statistical PV estimation can be performed, the proba-
bility density functions (pdfs) of the RVs describing pure tissue
types must be specified. In practice, this typically means that the
parameters of the pdfs—usually normal distributions—need to be
estimated. Unfortunately, errors in the parameter estimation often
have a major impact on the quality of PV estimation. In the case
of statistical ‘hard’ classification, where each voxel is classified
according to the most dominant tissue typ e, the parameter
estimation problem differs from the one studied here as each
intensity is a realization of a single RV. On the other hand, the
mixel model assumes that each voxel’s intensity represents a
weighted sum of several RVs and aims to estimate these unknown
weight parameters.
In general, there are three approaches to the parameter estima-
tion problem: histogram analysis (Santago and Gage, 1993),
simultaneous parameter, and partial volume estimation by expec-
tation maximization (EM)-like algorithms (Noe and Gee, 2001),
and estimation based on a hard segmentat ion of the image
(Shattuck et al., 2001). These three approaches, however, each
have their drawbacks. Histogram analysis requires a mixture
probability density to be fit to an image histogram by parameter
optimization. This involves finding the minimizer of a multimodal
objective function and therefore reliability of histogram analysis
for parameter estimation depends heavily on the optimi zation
algorithm used for the fitting task. If a standard nonlinear optimi-
zation algorithm aimed at local minimization (e.g., Levenberg
Marquee algorithm used commonly for curve fitting) is used, the
initialization for the algorithm has to be chosen carefully to avoid
convergence to a poor local minimum. These considerations call
for the use of advanced global optimization algorithms, for
example, Santago and Gage (1993) propose to use the tree-
annealing method (Bilbro and Snyder, 1991). The problem with
global optimization methods is that they are usually far more time
consuming than local optimization methods.
Like histogram analysis, expectation maximization schemes
for parameter estimation are time consuming. Besides, the use of
spatial information in the form of Markov Random Fields (Besag,
1974, 1986) causes practical problems with the E-step of the EM-
algorithm (Van Leemput et al., 2003). To solve this, statistical
dependency between voxel labels can be ignored during the E-step
Fig. 1. PVE in the cortical region, in the top row, a pediatric subject, and in the bottom row, a middle aged subject. Pediatric brain sulci are more compacted
with less (or no) intrasulcal CSF in the segmented image (see left column). However, the partial volume skeleton, estimated automatically by using the method
presented (TMCD) in this study, can penetrate down into that intrasulcal CSF (see middle column) for the pediatric case (top row). Different colors correspond
to different amounts of CSF within the voxels. Consequently, partial volume estimation allows, for example, for more accurate cortical surface extraction than
that could be achieved by methods relying on discrete classification.
J. Tohka et al. / NeuroImage 23 (2004) 84–97 85
as in Noe and Gee (2001). However, this leads to an algorithm that
is merely a heuristic and does not necessarily share the conver-
gence properties of the original EM-algorithm (Dempster et al.,
1977). An elegant solution to this problem based on a Monte Carlo
EM-algorithm (Wei and Tanner, 1990) was recently proposed by
Van Leemput et al. (2003). The algorithm was implemented only
for 2D image slices and the authors reported that partial volume
estimation typically requires about 20 min for a single slice of an
MR image with a very fast 1.7 GHz processor. Hence, the total
time consumption for an (moderately sized) image of 100 slices
would be over 30 h. The authors claimed that the computation time
can be significantly reduced but did not offer any figures to support
this claim.
Parameter estimation based on a hard labeling can be compu-
tationally efficient due to the prior knowledge of labels of voxels
that can be utilized in parameter estimation. However, due to the
PVE and classification errors in the hard labeling, each class in the
hard segmented image contains a large number of outliers. This
fact is taken into the account in Shattuck et al. (2001), but their
approach for the parameter estimation applies only to single-
spectral images and assumes that each tissue type has a Gaussian
noise distribution with the same variance. Also, since this method
involves detecting modes of histograms, it is sensitive to noise and
the estimates may not be unique.
In this study, using minimum volume ellipsoid and minimum
covariance determinant estimators (Rousseeuw, 1984; Rousseeuw
and Leroy, 1987), we propose routines for parameter estimation
based on segmented images that are well-defined and can be used
in single-spectral as well as multispectral cases. The studied
estimators are robust; in other words, they tolerate deviations from
the parametric form of the distribution assumed for the data. This is
important because in our case, part of the data is good (correctly
classified voxels of pure tissue), but the data contain also outliers
(e.g. PV voxels). There are fast algorithms for computing the
estimators and hence increasing the robustness of the parameter
estimation does not lead to significantly increased running time for
the PV estimation. We compare different techniques of parameter
estimation using simulated MR data (Kwan et al., 1999) and
demonstrate how the errors in the parameter estimation affect the
final PV estimation results. The results are also compared against
those obtained from an EM-like method, similar to Noe and Gee
(2001), but with an advanced initialization technique. Furthermore,
we compare our results with a fast PV estimation technique
proposed by Shattuck et al. (2001) and demonstrate that our
technique can yield clear improvements in the accuracy of PV
estimates without considerable loss in time efficiency. As a
secondary contribution, we consider differences between material
and sampling noise models for PVE also on a more theoretical
level (cf. Santago and Gage, 1995).
Methods
Statistical model for the partial volume effect
In this section, we state the PVE parameter estimation
problem and describe the mixel model more formally. In the
following, random variables (RVs) are denoted by boldface
letters, while both scalars and vectors are shown in italics. Let
us denote the observed image by X ={x
i
: i =1,..., N}, with
x
i
aR
K
, and K the number of data channels. Let the set of
possible tissue types present in the image be L = {1,. . ., M}.
Moreover, l
j
is the RV describing the tissue type j and the pdf of
l
j
is Gaussian g(Al
j
, R
j
) with mean l
j
and covariance R
j
. Each x
i
is now a realization of the RV
x
i
¼
X
M
j¼1
w
i
j
l
j
þ a ð1Þ
where a denotes white Gaussian noise and partial volume coef-
ficients (PVCs) w
i
j
a [0,1] for all i, j, and
P
M
j¼1
w
i
j
¼ 1 for all i.A
partial volume context image is denoted by W ={w
i
j
: i =1,...N, j =
1,..., M}. The problem is now to find an estimate W* for the true
partial volume context image, given the observed image X. This
is similar to the fuzzy classification problem, but in this case, the
coefficients w
i
j
specifically model the fraction of tissue type j
present in the voxel i, as opposed to the likelihood that voxel i is
of type j for example. For notational convenience, we set we w
i
=
[w
i
1
,..., w
i
M
]
T
.
The model in Eq. (1) for the formation of image intensities
within MRI is a simplification. Perhaps most importantly, MR
images are known to contain low frequen cy spatial intensity
variations often called RF inhomogeneity or shading artifact. The
correction for this artifact can be assumed to be performed before
PV estimation and there exist several methods for the task (e.g.,
Sled et al., 1998; Wells et al., 1996). Another somewhat contro-
versial issue of the model is the assumed Gaussian distributions for
tissue classes and for the noise component. The assumption is
common within classifiers and PV estimation routines applied to
brain MR images.
Since there are no methods that would allow us to estimate
parameters for l
j
, j =1,..., M, and the parameters for the noise
variable a at the same time, Eq. (1) needs to be simplified. Using
the terminology from Santago and Gage (1995), the material-
dependent noise model is obtained by embedding the sampling
noise component a into RVs l
j
describing the tiss ue types.
Accordingly, the sampling noise model is obtained by ignoring
variations within tissue types, that is, each tissue type is described
by a deterministic but initially unknown intensity value as opposed
to describing tissue types by RVs. Our attention is centered on the
material-dependent noise model, because, for our topic, it can be
considered to be a generalization of the sampling noise model.
Indeed, the methods for the parameter estimation to be proposed
are as well applicable for the sampling noise case, where only a
single covariance matrix describing the sampling noise is required.
This issue can be solved by considering the pooled covariance of
the estimated covariance matrices for the tissue classes (Dougherty,
1990, p. 467).
Algorithm for statistical partial volume estimation
Problem statement
The maximum a posteriori (MAP) criterion is used to find W*
given the observed image, that is,
W * ¼ arg max
W
pðW AX Þ: ð2Þ
We make the assumption that the observed image intensity
value x
i
of voxel i depends only on the RV x
i
related to the same
voxel. In other words, we assume that the intensity at voxel i does
J. Tohka et al. / NeuroImage 23 (2004) 84–9786
not depend on the tissue content of the other voxels. Hence, for the
material-dependent noise model,
pðW AX Þ~pðW ÞpðX AW Þ¼pðW Þ
Y
N
i¼1
pðx
i
Aw
i
Þ; ð3Þ
pðx
i
Aw
i
Þ¼gðx
i
A
X
M
j¼1
w
i
j
l
j
;
X
M
j¼1
w
2
i
j
R
j
Þ; ð4Þ
where
gðxAl; RÞ¼
1
ð2pÞ
K=2
detðRÞ
1=2
exp
1
2
ðx lÞ
T
R
1
ðx lÞ

denotes a multivariate Gaussian pdf with the mean l and the
covariance R. The expression for the likelihood term (4) is derived
using the fact that a weighted sum of Gaussian RVs is a Gaussian
RV with the mean and the covariance as in Eq. (4), (cf., e.g.,
Dougherty, 1990, Theorem 5.9). The prior probability p(W) can be
ignored or modeled by a Markov Random Field (MRF) (Besag,
1974, 1986; Geman and Geman, 1984). However, in MR images,
labels w
i
of nearby voxels are typically correlated. Ignoring these
correlations would lead to loss of relevant information, and hence
MRF modeling of the prior probability is preferable. This has been
also confirmed empirically in some studies (Shattuck et al., 2001;
Van Leemput et al., 2003).
The estimation problem (2) is challenging because both the
PVCs w
i
and the model parameters l
j
, R
j
are initially unknown.
Indeed, even if the model parameters would have been identified,
the estimation problem (2) would be under constrained and could
not be solved directly if the number of data channels K < M 1
(Choi et al., 1991; Kent and Mardia, 1988). In anatomical MRI of
the brain of normal subjects, three labels [white matter (WM), gray
matter (GM) and cerebrospinal fluid (CSF)] are of primary interest;
therefore, at least two data channels are required to solve the
estimation problem. However, often only an image from a single
channel is available, and therefore the requirement for the data to
be at least from two channels is quite restrictive.
An interesting way for solving the PV estimation problem even
with single-spectral data has been presented in Noe and Gee (2001)
and Shattuck et al. (2001) by extending ideas proposed by Santago
and Gage (1993, 1995). For this method, which will be referred as
indirect, we need only to assume that each voxel contains at most
two types of tissue. The indirect method involves the identification
of the tissue types present in each voxel, followed by a simplified
MAP estimation of the proportion of the each tissue type per voxel.
Here, the first step is called partial volume classification and the
second step is called PVC estimation. An overview of the entire
process, including a separate parameter estimation step, is presented
in Fig. 2. Parameter estimation is discussed in the next subsection.
PV classification
In the PV classification step, a label from the set L
ˆ
= L [
{{j, k}: j, k a L} is assigned to each voxel. Labels of type {j,
k} indicate mixed voxels of the tissue types j and k. Let us
Fig. 2. Depiction of the overall process for the PV estimation.
J. Tohka et al. / NeuroImage 23 (2004) 84–97 87
denote a context image, consisting of the labels of each voxel by
C ={c
i
: i =1,..., N, c
i
a L
ˆ
.
}
For the PV classification, the probability densities for mixed
tissue classes consisting of two types of tissue need to be obtained.
This is achieved by marginalizing the densities over all the possible
values of the PVC w (Santago and Gage, 1993, 1995). Since pure
tissue RVs are Gaussian, also their weighted sums are Gaussian
RVs and the marginal density is
pðx
i
Ac
i
¼fj; k ¼
Z
1
0
g½x
i
; lðwÞ; RðwÞdw; ð5Þ
where
lðwÞ¼wl
j
þð1 wÞl
k
; RðwÞ¼w
2
R
j
þð1 wÞ
2
R
k
; ð6Þ
for the PVC w a [0, 1]. The integral in Eq. (5) has no closed form
solution and hence numerical integration must be used to evaluate
it. Now the context image can be estimated as
C* ¼ arg max
C
PðCA X Þ¼arg max
C
PðCÞ
Y
N
i¼1
pðx
i
Ac
i
Þ; ð7Þ
where the prior term P(C) is modeled by an Markov Random Field
(MRF). Here we use a simple Potts model suggested in Shattuck et
al. (2001):
PðCÞ~exp b
X
N
i¼1
X
kaN
i
a
ik
dði; kÞ
!
; ð8Þ
where b is a user tunable parameter, N
i
is the 26-neighborhood
around voxel i, d(i, k) is the distance between centers of voxels i
and k, and
a
ik
¼
2:c
i
¼ c
k
1:c
i
and c
k
share a component
1 : otherwise
8
>
>
>
>
<
>
>
>
>
:
ð9Þ
The iterated conditional modes (ICM) algorithm (Besag, 1986)
is used to solve Eq. (7) locally. A well-known approach by Geman
and Geman (1984) to solve the optimization problem (7) globally
could also be employed, but since this method is much more time
consuming than ICM, we prefer to use the latter.
PVC estimation
In the second step, PVC estimation, we estimate the proportions
of each tissue type within each voxel. If c
i
*=j, that is labeling of
voxel i indicates that it is a pure voxel, w
i
j
*
= 1 and for all k p jw
i
k
*
=
0. If i is a mixed voxel, c
i
*={j, k}, then we employ the maximum-
likelihood principle:
w
i
j
*
¼ arg max
wa½0;1
gðx
i
AlðwÞ; RðwÞÞ
¼ arg max
wa½0;1
lngðx
i
AlðwÞ; RðwÞÞ
¼ arg min
wa½0;1
½ðx
i
lðwÞÞ
T
RðwÞ
1
ðx
i
lðwÞÞ
þ lndetðRðwÞÞ; ð10Þ
where l(w), R(w) are as in Eq. (6). Furthermore, w
i
k
*
=1 w
i
j
*
and all
the other PVCs are zero. The maximum-likelihood PVC estimate
(10) can be solved by a simple grid search, which we have found to
be surprisingly efficient in our experiments.
Since Shattuck et al. (2001) and Noe and Gee (2001) adopted
the sampling noise model for PVC estimation, we briefly compare
Eq. (10) to the solution derivable from the sampling noise model.
In that case, we would have only a single covariance matrix, which
we denote by R
a
that describes the sampling noise a. Otherwise,
the notation is as above. The sampling noise model leads to the
objective function
½ðx
i
lðwÞÞ
T
R
1
a
ðx
i
lðwÞÞ ð11Þ
to be minimized with respect to w. This minimization problem has
the closed form solution
w
i
j
*
¼ r
ðl
k
x
i
Þ
T
R
1
a
ðl
k
l
j
Þ
ðl
k
l
j
Þ
T
R
1
a
ðl
k
l
j
Þ
!
; ð12Þ
where r(x)=x when x a [0,1], r(x) = 0 when x < 0, and r(x)=
1 when x > 1. The two objective functions (Eqs. (10) and (11))
differ principally by the additional term ln det(R(w)) of Eq. (10).
Multiplying R
a
by a positive constant s does not affect the
solution (12) of the sampling noise problem. On the other hand,
multiplying R(w)bys changes the solution of the problem (10).
More precisely, if s > 1, more weight is put to the term ln
det(R(w)), which does not depend on the data x
i
. Therefore, the
material-depe ndent noise problem (10) can be considered a
regularized version of the sampling noise problem. The regular-
ization term ln det(R(w)) is a logarithm of high degree polyno-
mial in the multispectral case making the precise analysis of its
effect challenging. However, in the single-spectral case, the
minimum of ln det(R(w)) can be solved explicitly as R
k
/(R
j
+
R
k
) if it is in [0, 1]. These considerations are illustrated in Fig.
3, where on the left panel, the minimum of ln det(R(w)) is
shown when R
j
is varied. It can be seen that when the variances
have equal values, then the minimum of the regularization term
is at w
i
j
= 0.5 and hence the regularization term favors the
solution w
i
j
= 0.5. When R
j
is incremented, the value of the
favored solution decreases meaning that the voxel is a priori
assumed to contain less of the tissue type j than the tissue type
k, which has a smaller variance. On the right panel of Fig. 3, all
variances, R
j
, R
k
, R
a
, are assumed to have an equal value. When
this value is incremented, the sampling noise solution does not
change but the material noise solution tends slowly towards the
value 0.5.
Parameter estimation
As already noted, the model parameters l
j
, R
j
, j =1,.. ., M must
be estimated before (or during) the PV estimation. An efficient way
to do this is to first label each voxel in the image X by a hard label
belonging to L, and then to estimate the parameters based on the
intensity distribution of the labeled voxels for each tissue class. In
the following, we denote the set of the intensity values of voxels
labeled as belonging to the class j by D
j
. However, now each D
j
contains values that do not repr esent t he pure tissue type in
question due to the PVE as well as to misclassified voxels. These
values are called outliers. From this, it follows that the set of
samples D
j
contains samples that are not drawn from the distribu-
J. Tohka et al. / NeuroImage 23 (2004) 84–9788
tion describing the tissue type j that leads to problems when
applying conventional techniques, such as the maximum-likeli-
hood principle, for parameter estimation. Particularly, in the
Gaussian case, the sample mean and covariance are poor estimators
of l
j
and R
j
when the data are contaminated by PVE.
However, we can tackle the problem by using point estimation
techniques from robust statistics (Hampel et al., 1985). These
robust estimators are designed to tolerate significant deviations
from the assumed model while still being able to use information
about the model. Especially, we apply the minimum variance
ellipsoid (MVE) and the minimum covariance determinant
(MCD) estimators (Rousseeuw, 1984; Rousseeuw and Leroy,
1987) for the parameter estimation task and also combine a simple
outlier detection procedure with robust estimators. We will show
experimental ly that these techniques combined produce better
parameter estimates than either of the techniques alone. We start
by introducing the method for outlier detection. It is reasonable to
assume that voxels lying on the boundaries between tissue types in
hard-segmented image are likely to contain PVE, and that their
intensity values are therefore outliers. For this reason, voxels with
at least one of their 6 neighbors belonging to a different class than
the voxel itself are discarded from parameter estimation. We could
as well consider 26-neighborhoods instead of 6-neighborhoods, but
6-neighborhoods lead to faster computations. We will refer to the
estimates computed based on this kind of reduced data set as
‘trimmed estimates’’.
Since the above procedure does not guarantee outlier-free
sample sets, robust estimation techniques, namely the minimum
variance ellipsoid (MVE) and the minimum covariance determi-
nant (MCD) estimators are used to estimate parameters l
j
, R
j
.
MVE estimates the mean l of the set of data samples DoR
K
by
setting (Rousseeuw and Leroy, 1987):
l is the center of the ellipsoid with the minimal volume
covering at least h points in D.
Here, h is t
ADA
2
b þ 1, that is, we employ approximately 50% of the
data samples to estimate the mean. The covariance is then the matrix
defining the minimal ellipsoid around l multiplied by (v
2
K, 0.5
)
1
,
where (v
2
K, 0.5
) is the median of the chi-squared distribution with K
degrees of freedom. The MCD estimate is obtained by setting
(Rousseeuw and Leroy, 1987):
l is the mean of h points of D for which the determinant of the
covariance matrix is minimal.
Again, h is t
ADA
2
b þ 1 . The covariance matrix is then the
covariance matrix of those minimizing h points inflated or defeated
to contain exactly h points multiplied by (v
2
K, 0.5
)
1
.
In the single-spectral case, both estimators can be computed
exactly in O(jDj log jDj) time (Rousseeuw and Leroy, 1987). For
the multispectral case, there are computationally efficient algo-
rithms for approximating the estimators (Rousseeuw and Leroy,
1987, MVE, Rousseeuw and Van Driessen, 1999, MCD).
Besides that they can be computed efficiently, the MVE and
the MCD estimators for location and scatter have several favor-
able properties due to which we adopted them for parameter
estimation. The global reliability of a robust estimator can be
measured using the concept of a breakdown point. The (finite-
sample) breakdown point of an estimator is the smallest percent-
age of contaminated data that can cause the estimator to take on
arbitrarily large aberrant values. The breakdown points of MVE
and MCD estimators converge to 50% as the number of samples
approaches infinity, whereas the breakdown point of the sample
mean tends toward 0%. Note that 50% may be considered as the
best possible breakdown point. In addition, the MVE and the
MCD estimators are affine equivariant. This means that these
estimators commute with the affine transformations, which implies
that reparametrization of the sample space does not affect the
estimate. This is clearly an important property in multispectral
MRI. The combination of the affine equivariance and a high
breakdown point is not a common property for robust estimators
of multivariate location. For example, affine equivariant M
estimators have rather low breakdown points (cf. Rousseeuw
and Leroy, 1987 ). The MCD estimator is asymptotically normal
and has a better convergence rate than the MVE-estimator (see
Rousseeuw and Van Driessen, 1999), but the algorithm for the
multivariate MVE is easier to implement and therefore both
estimators are implemented.
Simultaneous parameter estimation and PV classification
To compare our results with other techniques, we also imple-
mented an EM-like algorithm for simultaneous parameter estima-
tion and PV classification. The outline of the algorithm is
(1) Initialize parameters l
j
, R
j
for every pure voxel class;
(2) Initialize ICM by maximum-likelihood estimation;
(3) Carry out one round of ICM to maximize Eq. (7);
(4) Reestimate the parameters l
j
, R
j
for j =1,..., M based on
posterior probabilities resulting from step 3;
(5) Finish if the termination condition is satisfied; else go to step 3.
Fig. 3. Differences between the material-dependent and sampling noise models. On left, the minimum of the regularization term ln det R(w) is shown when
R
k
= 1 and the value of R
j
is varied. On right, the PVC estimates obtained by sampling noise Eq. (12) (dotted line) and by material noise Eq. (10) (solid
line) are shown when the variances (R
a
, R
j
, R
jk
) increase, x
i
=7,l
j
=0,l
k
= 10.
J. Tohka et al. / NeuroImage 23 (2004) 84–97 89
This algorithm, here referred to as EM-ICM, was suggested by
Besag (1986). A similar algorithm was used for PV estimation by
Noe and Gee (2001), from which our EM-ICM algorithm differs
only with respect to its initialization. For initialization, we incor-
porated an advanced routine based on statistical probability of
anatomy maps (SPAMs) (Kamber et al., 1995; Kollokian, 1996).
Given that images to be segmented are registered to a common
stereotaxic space, SPAMs provide information on the probability of
voxels to be dominantly of certain tissue type. Based on SPAMs, it
is possible to automatically select a training sample for each tissue
class. The training sample then consists of the intensity values of
those voxels, which are very likely to be of the tissue type in
question. This sample can then be used for training a classifier or
for initializing clustering algorithms. Obviously, supervised clas-
sifiers trained this way are sensitive to misregistrations and unusual
anatomy (Cocosco et al., 2002).
The EM-ICM algorithm is considerably more computationally
expensive compared to the one suggested in the previous section
for two reasons: (1) it usually takes more iterations to converge and
(2) each iteration is more time consuming than a simple iteration of
ICM, which has fixed model parameters. This is because each
iteration involves computing new likelihoods based on updated
parameter estimates and new parameter estimates in addition to the
standard ICM cycle.
For example, whereas the procedure for PV estimation imple-
mented in C for a single-spectral 181
217
181 image takes
about 20 min using the ideas proposed in Parameter estimation
section, the EM-ICM algorithm can be expected to take slightly
over 2 h, both on a 400 MHz MIPS R12000 processor
(SGIROriginR 3800 compute server).
Experiments
Simulated data
Different methods for parameter estimation and the influence of
the quality of parameter estimates to PVC estimates were studied
using the BrainWeb Simulated Brain Database of Montreal Neu-
rological Institute (Cocosco et al., 1997) that is available at http://
www.bic.mni.mcgill.ca/brainweb. The images in the database are
generated by an MRI simulator (Kwan et al., 1999), which models
the MRI data acquisition process starting from the Bloch equation.
The input for simulations is a fuzzy realistic brain phantom
(Collins et al., 1998) as opposed to a discrete phantom, which
makes the evaluation of PV estimation algorithms possible. Exam-
ples of transverse slices of simulated T1-, T2-, and proton density
(PD)-weighted images are shown in Fig. 2.
All experiments were performed using single-spectral T1-
weighted as well as multispectral (T1-, T2-, and PD-weighted)
images. Four image sets for both single-spectral and multispectral
cases were used. Each set contained three images with varying
noise levels. The voxel size of the images of the first and the
second sets was 1
1
1 mm. The intensity across the images of
the first set was uniform, but the images in the second set were
simulated with 40% intensity nonuniformity (see Cocosco et al.,
1997). The third set contained images with no intensity nonuni-
formity but the voxel size was 1
1
3 mm. The images in the
fourth set were as those in the first set except that as opposed to the
other sets, they were not properly registered to the stereotaxic,
Talairach-based brain space used by BrainWeb. The initial classi-
fication algorithm used here (see Implementation section) is
sensitive to misregistrations and hence the fourth image set allows
us to study the dependence of the parameter estimates on the
quality of the initial hard classification. The image sets are
summarized in Table 1.
Implementation
Preprocessing steps included the correction of the intensity
inhomogeneity where necessary (Sled et al., 1998) and the classifi-
cation into hard labels (WM, GM, and CSF) by a neural network
classifier that was trained automatically using SPAMs (Kollokian,
1996). (The classifier used was a standard back propagation artificial
neural network with one hidden layer with 10 nodes.) As described
in Simultaneous parameter estimation and PV classification section,
the use of SPAMs for automatic training of unsupervised classifiers
is sensitive to errors in registering images to the stereotaxic space.
Voxels containing primarily nonbrain tissues were labeled as back-
ground before parameter estimation. Six methods for the parameter
estimation based on a hard labeling were examined. In the maxi-
mum-likelihood (ML) method, parameters for each class were the
sample-mean and the sample-covariance of that class in the hard
labeled image. Trimmed maximum-likelihood (TML) estimates
were the sample-mean and the sample-covariance of that class in
the trimmed hard labeled image. By trimming, it is meant that
outliers were detected and removed for each class as described in
Parameter estimation section. Minimum volume ellipsoid (MVE)
and minimum covariance determinant (MCD), respectively, esti-
mates were obtained by applying MVE and MCD estimators to the
classes in the hard labeled image. Trimmed minimum volume
ellipsoid (TMVE) and trimmed minimum covariance determinant
(TMCD), respectively, estimates were obtained by applying MVE
and MCD estimators to the trimmed hard labeled image. Moreover,
we listed parameter estimates resulting from the EM-ICM algorithm
and parameter estimates used to initialize it. For the purpose of the
PV classification, intensities of background voxels were assumed to
have a normal distribution with the zero mean and the covariance
matrix 0.1R
CSF
where R
CSF
is the covariance matrix of the class
CSF. In reality, the image intensities in magnitude MR images are all
greater than zero by definition and the intensity distribution of the
background voxels is Rician. Hence, this assumption is a rough
approximation of the reality and it is made purely for convenience.
During the PV classification, pure tissue classes were WM,
GM, and CSF. Mixed tissue classes were CSF/backgro und,
CSF/GM, and GM/WM. Because nonbrain voxels were masked
off, the background class was not required. The parameter b
(see Eq. (8)) controlling the relative strength of the MRF was
set to 0.1 unless otherwise mentioned. The grid search for
solving the PVC estimation step (10) used an equally spaced
Table 1
The image sets
Image set INU (%) Voxel size Registration
101
1
1 mm succeeded
2401
1
1 mm succeeded
301
1
3 mm succeeded
401
1
1 mm failed
The abbreviation INU stands for the strength of intensity nonuniformity
field in a simulated image.
J. Tohka et al. / NeuroImage 23 (2004) 84–9790
grid G = {0, 0.01, 0.02 ..., 1} of 101 points at which the
objective function in Eq. (10) was evaluated. The value of w
i
j
a
G yielding optimal value of the objective function was then
selected as the estimate w
i
j
*
.
Error criteria
To quantitatively evaluate location estimates l
j
, the mean of the
Mahalanobis distances between the true sample mean and the
estimated mean of each pure tissue class was computed. (We name
this criterion ‘Mahalanobis error ’’.) The true mean and the
covariance matrix were computed from voxels that contained over
99% of a certain tissue type in the phantom image. This error
criterion was selected due to its simplicity and its applicability to
both the single and multispectral case. Direct comparison of
Mahalanobis errors obtained at different noise levels would require
normalization of the covariances used in calculation of errors.
However, this kind of normalization would make the error criterion
and its analysis more complicated.
For covariance estimates, the definition of an error criterion that
would be meaningful for our problem is more difficult. In experi-
ments, errors in location estimates usually had a greater effect to
the PV estimation accuracy than errors in estimates of covariances
R
j
. For thes e two reasons, we only describe the quality of
covariance estimates verbally and we do not attempt to evaluate
it quantitatively.
For estimates of PVCs, the following error criterion is used
E
PVE
¼
1
N
X
N
i¼1
X
M
j¼1
Aw
i
j
*
w
i
j
A; ð13Þ
where w
i
j
*
is the estimated PVC, w
i
j
is the true PVC, M is the number
of pure tissue types, and N is the number of voxels in the brain
volume. Here, M = 3. Note that the value of this criterion does not
depend on the number of the pure tissue types in the image, because
we assume that each voxel contains at most two types of tissue. The
range of this error criterion is from 0 to 2. We have selected this
criterion closely related to mean absolute error (MAE) because we
are interested in the average accuracy that can be obtained in PV
estimation. Error criteria that are based on mean squared error (MSE)
could be as well employed, but they put more weight to gross errors,
hence evaluating more robustness than accuracy.
To establish a link between Mahalanobis errors in the model
parameter estimates and the error criterion for the PVC estimates,
a simulation was performed. For the simulation, we generated
intensities of WM/GM mixed voxels with varying mixing pro-
portions. Statistical models for the required tissue classes were
obtained from the simulated images in the same manner as when
computing the error criterion for parameter estimates. That is,
parameters for the densities were computed based on voxels that
contained over 99% of a certain tissue type (WM or GM) in the
phantom image. We also randomly generated a set of erroneous
model parameter estimates yielding a fixed Mahalanobis error.
Subsequently, PVCs were estimated using Eq. (10) based on the
simulated intensity values and the erroneous location estimates.
Errors in covariance estimates were not simulated, because this
simplifies the interpretation of the simulation results. A more
detailed description of the simulation can be found in the
Appendix A.
As can be seen in Fig. 4, the relationship between the PVC
and Mahalanobis errors is approximately linear in most cases.
Only the curve corresponding to the highest level of noise in
the single-spectral case seems to be nonlinear. Based on this
curve, Mahalanobis error values above 1.35 are worse than the
values below 1.35. However, experiments with the simulated
MR images rarely gave as large as 1.35 Mahalanobis error
values with the highest noise level. Also, it can be noticed in
Fig. 4 that the PVC error increases more rapidly relative to
Mahalanobis error in the single-spec tral ca se than in the
multispectral case.
IBSR data set
Simulated images allow us to study in detail each step of the
proposed procedure. To examine the practicality of the proposed
method, it is tested on real data sets. With real data, quantitative
valuation of the PV estimation is restricted to evaluation of the
quality of the subsequent ha rd segmentations. For this, the
segmentation from the PV estimation with TMCD parameter
estimates was evaluated with the internet brain segmentation
repository (IBSR) data set. The IBSR data set comprises of the
20 normal MR brain data sets and their manual segmentations
Fig. 4. The influence on the errors in parameter estimates to the errors in
PVC estimates. The top and bottom panels show results for, respectively,
the single-spectral (T1-weighted only) and multispectral (T1-, T2-, and
proton density-weighted) cases.
J. Tohka et al. / NeuroImage 23 (2004) 84–97 91
that were provided by the Center for Morphometric Analysis at
Massachusetts General Hospital and are available at http://neuro-
www.mgh.harvard.edu/cma/ibsr. These images have been used to
evaluate the quality of several (automatic) tissue classification
methods. The quality measure used is the Tanimoto coefficient
between the tissue class in the expert segmentation and in the
automatic segmentation (cf., e.g., Shattuck et al., 2001).
The preprocessing of the images consisted of correction of
interslice intensity nonuniformity that was done (see Zijdenbos et
al. (1995)) before registering them with stereotaxic (Talairach)
space. After stereotaxic registration, the N3 algorithm (Sled et al.,
1998) was used to correct for the remaining intensity nonunifor-
mity. Images were classified for parameter estimation in the same
way as simulated images. A hard segmentation was generated
from PVC estimates by selecting the class with the largest PVC
for each voxel as its label. The dimensions for the images were
256
256
64.
Results
Simulated data
Parameter estimation
Results for the single-spectral case are presented in Table 2.
Robust estimators combined with trimming were the most reliable
among the segmentation-based parameter estimation techniques.
They were also better than parameter estimates that were used to
initialize EM-ICM and usually better than the final results of EM-
ICM. For example, with image set 1 and 5% noise, TMCD
produced a Mahalanobis error of 0.06, whereas the error resulting
from MCD was 0.19 and the error resulting from EM-ICM was
0.11. Only with the image set 4, where the initial classifications
were not accurate, the EM-ICM algorithm yielded better results
than image-based parameter estimation techniques. Without trim-
ming of the initial classification, robust estimators did well only
with the lowest level of noise. The most extreme case occurred
with image set 3 and the highest noise level, where the errors
produced by robust estimators were 10 times higher than those
produced by trimmed robust estimators. T he performance of
TMVE and TMCD did not differ much in the single-spectral
case. The best variance estimates were achieved using the
trimmed robust estimators, while the trimmed ML estimator
typically underestimated the variances and nontrimmed estimators
overestimated them, although over/underestimation was always
consistent.
In Table 2, error values typically decrease when the noise level
increases. This is because, as was explained in Error criteria
section, the values of the Mahalanobis error at different noise
levels are not necessarily comparable.
Results for the multispectral case are presented in Table 3.
These results are similar to those with single-spectral data, except
that TML estimator was almost as good as TMCD estimator and
even better than TMVE estimator. With the image set 1 and the
noise level of 5%, for example, the error of TML was 0.25, the
error of the TMVE was 0.36, and the error of TMCD was 0.23. It
remains to be studied whether poor results of the TMVE
estimator as compared to the single-spectral case were due to
the estimator itself or to the approximative algorithm used to
compute it. The failure of all image-based estimators with the
image set 4 and a 5% noise level was due to very poor initial
classifications (about two-thirds of GM voxels were labeled as
WM), causing also robust estimators to break down. The quality
of initial classifications was better with the other noise levels.
With all the image sets, the covariance estimates obtained by
trimmed robust estimators were better than the ones of the TML
estimator, and the effect of this can be seen in the quality of PVC
estimates (see Table 5).
The EM-ICM algorithm typically produced the best estimates
when the voxel size was 1 mm
3
(i.e., image sets 1, 2, and 4).
However, with the image set 4 and the lowest noise level, EM-ICM
failed in the parameter estimation (error 5.67) and this can be seen
also in the quality of the corresponding PVC estimates (error
0.783). This failure was probably because the EM-ICM algorithm
failed to converge to a strong maximum due to a poor initialization
based on the ima ge tha t was not pr oper ly reg ist ere d to the
stereotaxic space. Especially, the initial covariance estimates great-
ly overestimated the (scaling of) covariance matrices, which could
explain why EM-ICM failed particularly with the lowest noise
level. The application of the EM-ICM algorithm did not always
improve the parameter estimates when compared the parameter
estimates used for its initialization. Indeed, there are no results
Table 2
Means of Mahalanobis distances between true and estimated mean
intensities of tissue types of T1-weighted data
Image
set
Noise
level
(%)
ML MVE MCD TML TMVE TMCD INIT EM-
ICM
1 1 1.16 0.11 0.13 0.20 0.21 0.19 0.27 0.11
5 0.46 0.18 0.19 0.06 0.07 0.06 0.13 0.09
9 0.28 0.55 0.39 0.17 0.04 0.05 0.06 0.14
2 1 0.99 0.07 0.07 0.11 0.14 0.16 0.26 0.09
5 0.48 0.16 0.17 0.09 0.07 0.07 0.14 0.10
9 0.28 0.48 0.39 0.15 0.06 0.05 0.23 0.19
3 1 2.32 0.30 0.13 0.50 0.14 0.13 0.70 0.20
5 0.66 0.44 0.38 0.13 0.10 0.05 0.22 0.22
9 0.31 0.64 0.51 0.19 0.05 0.04 0.13 0.25
4 1 3.03 0.65 0.54 1.51 0.58 0.61 7.58 0.11
5 1.38 0.39 0.45 0.84 0.39 0.46 2.84 0.10
9 0.92 1.37 1.26 0.58 0.27 0.37 1.62 0.15
Names for different methods for the parameter estimation are given in
Implementation section. INIT is the initialization for the EM-ICM
algorithm.
Table 3
Means of Ma halanobis distances between true and estimated mean
intensities of tissue types of multispectral data (cf. Table 2)
Image
set
Noise
(%)
ML MVE MCD TML TMVE TMCD INIT EM-
ICM
1 1 1.57 0.52 0.33 0.32 0.40 0.37 0.52 0.30
5 0.54 0.36 0.41 0.25 0.36 0.23 0.26 0.11
9 0.49 0.37 0.44 0.33 0.24 0.31 0.14 0.13
2 1 1.36 0.44 0.32 0.27 0.49 0.30 0.60 0.19
5 0.54 0.58 0.44 0.23 0.39 0.22 0.30 0.17
9 0.51 0.41 0.48 0.33 0.25 0.33 0.23 0.20
3 1 3.26 2.49 2.30 0.61 0.43 0.25 1.05 1.25
5 1.09 1.05 0.94 0.22 0.24 0.18 0.36 0.29
9 0.63 0.67 0.61 0.19 0.21 0.18 0.19 0.26
4 1 2.26 0.52 0.30 0.76 0.43 0.37 9.93 5.67
5 2.71 2.09 1.99 2.16 1.98 1.81 3.93 0.11
9 0.91 0.92 0.87 0.37 0.36 0.34 2.38 0.13
J. Tohka et al. / NeuroImage 23 (2004) 84–9792
concerning the convergence properties of the EM-ICM algorithm
(cf. Besag, 1986). Furthermore, the convergence to a local mini-
mum (or maximum) may actually draw estimates further from the
global minimum (or maximum).
Partial volume coefficients
Errors in the PV estimation are listed in Tables 4 and 5.InFig.
5, PV estimation errors obtained by applying different parameter
estimation methods are compared with some of the image sets.
Improvements in the quality of the model parameter estimates
improved PVC estimates in most cases. In that respect, these
results were consistent with the simulation described in Error
criteria section. However, generally the errors in PVC estimates
obtained with simulated images were lower than those obtained in
the simulation of Error criteria section assuming the same value of
the Mahalanobis error. This is not surprising since in the simulated
images some voxels are classified as pure voxels that yield a PVC
error value of zero in the case of a correct classification. Similar
reasoning could be used to explain why the improvements to an
already small Mahalanobis error led ra rely to a considerably
improved PVC estimation. An example of this phenomenon can
be seen in Table 4 by comparing the results by TMCD (Mahala-
nobis error 0.05, PVC error 0.195) and EM-ICM (Mahalanobis
error 0.22, PVC error 0.196) with the image set 3 and the noise
level of 5%.
PVC estimates obtained with multispectral images were
considerably better than those obtained with single-spectral
images (see Fig. 6). Moreover, it can be seen from PV
estimation results that gain from multispectral images increased
with the noise level. The differences in the errors in PVC
estimates between single and multispectral cases increased with
the noise level. However, it should be noted that in this case,
multispectral images were perfectly registered with each other,
which is rarely the case with real data. In the multispectral
image set 4, better PVC estimates with the noise level of 9%
Table 5
Errors (13) in PV estimation with multispectral data (cf. Table 2)
Image
set
Noise
(%)
ML TML TMVE TMCD EM-
ICM
1 1 0.104 0.086 0.082 0.083 0.088
5 0.140 0.132 0.135 0.131 0.128
9 0.237 0.213 0.210 0.213 0.198
2 1 0.111 0.087 0.086 0.084 0.081
5 0.147 0.137 0.138 0.137 0.134
9 0.243 0.217 0.223 0.216 0.201
3 1 0.153 0.104 0.084 0.085 0.099
5 0.202 0.169 0.165 0.164 0.163
9 0.271 0.257 0.264 0.256 0.252
4 1 0.130 0.095 0.083 0.082 0.783
5 0.642 0.538 0.485 0.443 0.128
9 0.285 0.245 0.235 0.241 0.199
Fig. 5. Errors in PV estimation with different parameter estimation methods
and noise levels. Form top, the sets 1 and 3 of T1-weighted images and the
set 1 of multispectral images.
Table 4
Errors (13) in PV estimation with T1-weighted data (cf. Table 2)
Image
set
Noise
(%)
ML TML TMVE TMCD EM-
ICM
1 1 0.108 0.083 0.084 0.084 0.078
5 0.157 0.156 0.156 0.155 0.156
9 0.249 0.259 0.247 0.248 0.252
2 1 0.112 0.089 0.089 0.089 0.086
5 0.160 0.158 0.157 0.157 0.157
9 0.252 0.262 0.252 0.253 0.256
3 1 0.156 0.106 0.089 0.089 0.088
5 0.206 0.195 0.195 0.195 0.196
9 0.316 0.317 0.306 0.308 0.318
4 1 0.160 0.107 0.082 0.083 0.079
5 0.257 0.207 0.162 0.168 0.156
9 0.364 0.324 0.276 0.286 0.255
J. Tohka et al. / NeuroImage 23 (2004) 84–97 93
compared to those with the noise level of 5% were due to
particularly poor initial classification when the noise level was
5%.
Shattuck et al. (2001) have evaluated their very fast PV
estimation routine using BrainWeb images but using an error
criterion different from ours. Therefore, to compare our results
with theirs, we computed errors of TMCD-based PVC estimates
also using the criterion they applied. The criterion is defined as (cf.
Eq. (13) for notation)
E
MSE
j
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N
X
N
i¼1
Aw
i
j
*
w
i
j
A
2
v
u
u
t
; ð14Þ
for each tissue class j. Particularly, we compared results with
single- spectral images from sets 1 and 2. (In Shattuck et al.,
2001, no results with multispectral data or images corresponding
our sets 3 and 4 were available.) Results are shown in Fig. 7.As
can be seen invariably, the TMCD-based method was better of the
two. Particularly, the PVC estimates for the gray matter were
considerably better with the TMCD estimated parameters. For the
white matter, differences were smaller. Although the methods differ
in all their components, they are both based on a division of the
problem in smaller subproblems. Hence, these results are a good
indication that our paramete r est imation techniques are a real
improvement as compared to previous similar approaches. The
model by which the data were simulated is the material-dependent
noise model, but with very similar covariance for each class.
Hence, the results of the PVC estimation step are similar with
both models (cf. PVC estimation section).
Comparison with the initial hard segmentation
With image set 4, we also list the misclassification rates of the
initial hard labeling and the hard labeling derived from PV
estimation in Tables 6 and 7. For this, the PV estimations were
hardened by labeling each voxel by the most dominant tissue type.
Almost all PV estimation techniques improved the results of the
original labeling (see Fig. 8 for an example). The only exception
was EM-ICM in the multispectral case with the lowest noise level.
In this case, the poor classification result was due to poor
parameter estimates (cf. Parameter estimation section).
Computation times
The entire procedure to estimate PVCs with a single-spectral
181
217
181 image took on average about 20 min using
parameter estimation based on hard classification. Most of the time
(over 10 min) was spent to the PV classification step. Image
Fig. 6. Voxel-wise absolute errors in PVC estimates obtained with single-
spectral (top) and multispectral (center) data. In the bottom, histograms of
voxel-wise absolute errors are shown. The noise level is 5% and the voxel
size is 1
1
3 mm. The parameter estimates were obtained with TMCD
estimator.
Fig. 7. MSE error values as in Eq. (14) with BrainWeb images. PVCs
estimated with the TMCD method are compared to the ones published in
Shattuck et al. (2001).
Table 6
Misclassification rates in the brain area in percent
Noise
(%)
ORIG ML TML TMVE TMCD EM-
ICM
Image set 4 1 18.8 4.5 2.1 1.9 1.9 1.9
5 24.3 8.6 6.8 6.0 6.1 5.9
9 31.6 17.0 15.1 13.6 14.0 13.4
ORIG denotes the initial hard classification. Otherwise, abbreviations are as
in Table 2.
Table 7
Misclassification rates in the brain area in percent with multispectral data
Noise
(%)
ORIG ML TML TMVE TMCD EM-
ICM
Image set 4 1 8.2 3.2 2.9 2.6 2.6 37.8
5 38.2 32.4 25.6 19.7 18.9 4.5
9 21.7 11.4 9.9 9.8 9.6 9.0
ORIG denotes the original hard classification. Otherwise, abbreviations are
as in Table 2.
J. Tohka et al. / NeuroImage 23 (2004) 84–9794
preprocessing (hard classification and nonuniformity correction)
typically took 3 min. Parameter estimation and PV estimation were
fast, requiring about 1 min each. In the multispectral case, the total
computation time was approximately 40 min. PV estimation with
EM-ICM algorithm took 2 h in the single-spectral case and 6 h in
the multispectral case. All computation times are on a 400 MHz
MIPS R12000 processor (SGIROriginR 3800 compute server). The
algorithms were implemented in C. These running times, particu-
larly for the PV classification step, could be probably improved by
paying more attention on the efficiency of the implementation.
Results with the IBSR data set
The Tanimoto coefficients (TCs) of segmentations obtained
from PV estimations using the TMCD parameter estimates can
be found in Table 8. The mean value of the Tanimoto coefficient
was 0.6686 for the WM and 0.6760 for the GM. We experimented
the procedure also with a higher value (0.5) for the parameter b
controlling the strength of the MRF prior. The average TC values
in this case were 0.6637 (WM) and 0.6807 (GM).
We selected to evaluate our algorithm with this data set due to its
frequent use in the evaluation of MR segmentation algorithms.
Although our interest in PVE is not in improving hard segmenta-
tions by modeling of it, it is interesting to compare the results of our
method to those obtained with algorithms for hard segmentation.
Rajapakse and Krugge (1998) compared several algorithms and the
best TC values were obtained by Adaptive MAP method, and they
were 0.567 for the WM and 0.564 for the GM. A more recent
method by Marroquin et al. (2002) achieved TCs 0.683 for the WM
and 0.662 for the GM. The average running time for this algorithm
was 19.2 min with the IBSR data set, which is similar to our
methods, but no PV estimation can be obtained with this method.
Discussion
In this paper, various methods for the parameter estimation for a
statistical model of the partial volume effect have been studied. It
has been shown that it is possible to estimate parameters in a
reliable and fast way based on the initial hard labeling of the image
before the actual partial volume estimation procedure. For this,
outliers of each class in the classified image were eliminated using
a simple morphological rule, and thereafter, parameters for model
were computed by us ing robust estimators: MVE and MCD.
Parameters estimated by MCD were of better quality than those
relying on sample means and covariances as estimates. It was also
observed that the trimming of the initial classification is necessary
even when using robust estimators. The parameter estimates based
on the hard classification were almost as good as the ones obtained
by the EM-ICM algorithm for simultaneous PV classification and
parameter estimation. The EM-ICM algorithm is, however, more
time consuming. Another downside of the EM-ICM algorithm for
parameter estimation is that it is essentially a local optimization
algorithm and hence its results depend on its initialization. In our
experiments, EM-ICM failed completely in parameter estimation
and subsequently in PV estimation with one of the multispectral
images. In that case, the parameter estimates that were used to
initialize the EM-ICM algorithm were of particularly poor quality
due to the failed stereotaxic registration.
The influence of improved parameter estimates on the PV
estimation was studied. As expected, better parameter estimates
produced better estimates of partial volume coefficients. Fur-
thermore, we examined the effect of increased error in param-
eter estimation to the error in PV estimation through a
simplified simulation. The simulation showed that the relation-
ship between Mahalanobis errors in parameter estimation and
errors in PV estimation is approximately linear for a fixed level
of noise.
We have studied our parameter estimation procedures in con-
junction with the material-dependent noise model, an indirect PV
estimation algorithm and a particular method for preliminary hard
classification. But in fact, the proposed parameter estimation
methods (TMVE and TM CD) can be applied more generally,
and hence other choices could have been made as well. First,
location parameter estimates for the sampling noise model can be
obtained in the exactly same manner as for the material-dependent
noise model. However, only a single covariance matrix describing
the sampling noise is required. This issue can be solved by
considering the pooled covarianc e of the estimated covariance
for the tissue classes (Dougherty, 1990, p. 467). Secondly, as
parameter estimation is a separate step of the whole procedure, it
could be applied also with direct PV estimation algorithms.
Furthermore, the initial hard classification and correction of inten-
sity nonuniformity can be performed with other methods than those
applied in this study. The state of the art algorithms for these tasks
(such as Marroquin et al., 2002; Van Leemput et al., 1999; Zhang
et al., 2001) are however more time consuming than the methods
applied in this study. Moreover, the robustness in parameter
estimation compensates also for classification errors as was shown
with misregistered Brainweb images.
Fig. 8. The improvement of the hard classification results by using PV
model. Left: the ground truth classification. Center: the initial classification
of the misregistered image with 5% of noise. Right: the hardened partial
volume estimation with TMCD parameters of the same image.
Table 8
IBSR data set results
Image 100
_
23 110
_
3112
_
211
_
312
_
313
_
3111
_
215
_
316
_
317
_
3
GM 0.6783 0.7363 0.7063 0.7381 0.8044 0.7716 0.7765 0.6076 0.6261 0.6026
WM 0.6844 0.6841 0.6632 0.7300 0.7393 0.7550 0.7409 0.6009 0.6275 0.6609
Image 191
_
31
_
24 202
_
3 205
_
32
_
44
_
85
_
86
_
10 7
_
88
_
4
GM 0.6896 0.7106 0.7968 0.7426 0.4535 0.5876 0.6581 0.5632 0.6447 0.6261
WM 0.7399 0.7328 0.7516 0.7519 0.5399 0.5625 0.6006 0.4851 0.6632 0.6591
J. Tohka et al. / NeuroImage 23 (2004) 84–97 95
To conclude, we have presented fast and robust methods to
estimate parameters for the mixel model. These robustified param-
eter estimates were shown to improve the performance of PV
estimation as compared standard estimates and heuristics used in
Shattuck et al. (2001). Moreover, parameter estimation based on
segmented image was demonstrated to be much faster than
competing EM style algorithms (Noe and Gee, 2001; Van Leemput
et al., 2003) . Also, as demonstrated with the IBSR data set, our fast
PV estimation routine produced hard segmentations of a similar
quality than a state of the art algorithm for this task (Marroquin et
al., 2002). In summary, we recommend the use of the TMCD
estimator based on hard labeling for fast and reliable parameter
estimation for a statistical PVE model.
Acknowledgments
Fig. 1 was generously provided by Jason Lerch, McConnell
Brain Imaging Centre, Montreal Neurological Institute. Thanks to
Chris Cocosco, Jason Lerch, and Steve Robbins for help in
implementing the algorithms for this paper. J. Tohka acknowledges
financial support from the Tampere Graduate School in Informa-
tion Science and Engineering, the Academy of Finland, the
KAUTE foundation, and the Jenny and Antti Wihuri fund.
Appendix A
This appendix describes the simulation used to link the Maha-
lanobis errors in parameter estimation and the absolute errors in
PVC estimation. Only mixtures of the white matter and the gray
matter tissue types were considered. The intensities were generated
by using the material-dependent noise model and assuming param-
eters as in GM and WM classes of the simulated images. The
mixing proportions were 0.1, 0.3, 0.5, 0.7, and 0.9. The number of
samples generated per each mixing proportion were 1000.
Different Mahalanobis errors evaluated were 0, 0.05, 0.1,...,
2.5. One hundred different locatio n parameter estimates were
generated for each fixed value of Mahalanobis error. The location
parameter estimates were generated by first randomly selecting the
direction of the error for both WM and GM tissue classes. In the
single-spectral case, this is easy, since only the choice between
positive or negative direction has to be made. This is a bit more
complicated in the multispectral case, where random points lying
on the unit sphere have to be drawn (see Marsaglia, 1972). After
the selection of directions of the error, the proportion of the total
Mahalanobis error rising from the estimate of the mean of WM
class was randomly drawn from the interval [0.2, 0.8]. Thereafter,
the exact covariance and each erroneous location estimate was used
to estimate the true PVC-coefficient from each 5000 samples of
intensity values. The results of the simulation are shown in Fig. 4.
We also performed a simulation where Mahalanobis errors were
assumed to be equally distributed between tissue types, but results
were similar to the more general experiment described above.
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