Conference Paper

A Global Criterion for the Computation of Statistical Shape Model Parameters Based on Correspondence Probabilities

DOI: 10.1007/978-3-540-78640-5_56 Conference: Bildverarbeitung für die Medizin 2008, Algorithmen, Systeme, Anwendungen, Proceedings des Workshops vom 6. bis 8. April 2008 in Berlin
Source: DBLP
ABSTRACT
A fundamental problem when building a statistical shape model (SSM) is the correspondence problem. We present an approach for unstructured point sets where one-to-one correspondences are replaced by correspondence probabilities between shapes which are determined us- ing the Expectation Maximization - Iterative Closest Points registration. We propose a unified MAP framework to compute the model parameters which leads to an optimal adaption of the model to the observations. The optimization of the MAP explanation with respect to the obser- vation and the generative model parameters leads to very efficient and closed-form solutions for (almost) all parameters. Experimental results on synthetic data and brain structures as well as a performance compar- ison with a statistical shape model built on one-to-one correspondences show the efficiency and advantages of this approach.

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Available from: Heinz Handels
A Global Criterion for the Computation
of Statistical Shape Model Parameters
Based On Correspondence Probabilities
Heike Hufnagel
1,2
, Xavier Pennec
1
, Jan Ehrhardt
2
, Nicholas Ayache
1
, Heinz
Handels
2
1
INRIA Asclepios Project, 06902 Sophia Antipolis, France
2
Medizinische Informatik, Universit
˜
At Hamburg, 20246 Hamburg
h.hufnagel@uke.uni-hamburg.de
Abstract. A fundamental problem when building a statistical shape
model (SSM) is the correspondence problem. We present an approach for
unstructured point sets where one-to-one correspondences are replaced
by correspondence probabilities between shapes which are determined us-
ing the Expectation Maximization - Iterative Closest Points registration.
We propose a unified MAP framework to compute the model parameters
which leads to an optimal adaption of the model to the observations.
The optimization of the MAP explanation with respect to the obser-
vation and the generative model parameters leads to very efficient and
closed-form solutions for (almost) all parameters. Experimental results
on synthetic data and brain structures as well as a performance compar-
ison with a statistical shape model built on one-to-one correspondences
show the efficiency and advantages of this approach.
1 Introduction
In order to analyze shape variability it is necessary to determine correspondences
between the observations of the training set. Several techniques were developed
to automatically find exact correspondences [1], some in combination with the
search for the SSM [2,3] or a shape analysis [4]. An interesting approach pro-
poses an entropy based criterion to find shape correspondences [5]. However,
these methods are not easily adaptable to point sets as exact correspondences
can only be determined between continuous surfaces, not between point cloud
representations of surfaces. The SoftAssign algorithm tries to solve this problem
with an initial probabilistic formulation [6]. In order to build a SSM for unstruc-
tured point sets, we advance the probabilistic concept of [7] using the EM-ICP
registration which proved to be robust, precise, and fast [8]. The approach solves
for the mean shape and the eigenmodes in a unique criterion without the need
of one-to-one correspondences as is usually required by the PCA. This article
focuses on the validation of the work presented in [9] where we showed that our
SSM is robust and leads to plausible results for synthetic data as well as brain
structures. Experiments demonstrate that our approach leads to better SSM
quality measures than a classical approach based on exact correspondences.
Page 1
2 Methods and Experiments
We first realize a MAP estimation of the model and observation parameters
which leads to a unique criterion. We then compute the SSM which best fits the
given data set by optimizing the global criterion iteratively with respect to all
model and observation parameters. A key part of our method is that we can find
a closed-form solution for nearly each of the parameters.
2.1 Statistical Shape Model Built on Correspondence Probabilities
In the process of computing the SSM, we distinguish strictly between model
parameters and observation parameters. The generative SSM is explicitly
defined by 4 model parameters:
The mean shape
¯
M R
3N
m
parameterized by N
m
points m
j
R
3
, the eigen-
modes v
p
consisting of N
m
3D vectors v
pj
, the associated standard deviations λ
p
which describe the impact of the eigenmodes, and the number n of eigenmodes.
Using the generative model Θ = {
¯
M, v
p
, λ
p
, n} of a given structure, the shape
variations of that structure can be generated by M
k
=
¯
M +
P
n
p=1
ω
kp
v
p
with
ω
kp
R being the deformation coefficients. The shape variations along the modes
follow a Gaussian probability with variance λ
p
and
k
= {ω
kp
, p = 0, ..., n 1}.
p(M
k
|Θ) = p(
k
|Θ) =
n
Y
p=1
p(ω
kp
|Θ) =
1
(2π)
n/2
Q
n
p=1
λ
p
exp
n
X
p=1
ω
2
kp
2λ
2
p
!
. (1)
In order to account for the unknown position and orientation of the model in
space, we introduce the linear transformation T
k
. A model point m
j
can then
be deformed and placed in space by the linear transformation T
k
with T
k
?
m
kj
= T
k
? ( ¯m
j
+
P
p
ω
kp
v
p
). Finally, we model each observation point s
ki
as a
Gaussian measurement of a (transformed) model point m
j
. As we do not know
the originating model point for each observation point s
ki
, the probability of
a given s
ki
is described by a Mixture of Gaussians and the probability for the
whole scene S
k
becomes:
p(S
k
|M, T
k
) =
N
k
Y
i=1
1
N
m
N
m
X
j=1
p(s
ki
|m
j
, T
k
) (2)
with p(s
ki
|m
j
, T
k
) = (2π)
3/2
σ
1
exp(
1
2σ
2
(s
ki
T
k
? m
j
)
T
.(s
ki
T
k
? m
j
)). We
summarize the observation parameters as Q
k
= {
k
, T
k
}.
2.2 Derivation of the Global Criterion Using a MAP Approach
When building the SSM, we use N observations S
k
R
3N
k
, and we are interested
in the parameters linked to the observations Q = {Q
k
} as well as the unknown
model parameters Θ. In order to determine all parameters of interest, we optimize
a MAP on Q and Θ.
MAP =
N
X
k=1
log(p(Q
k
, Θ|S
k
)) =
N
X
k=1
log
p(S
k
|Q
k
, Θ)p(Q
k
|Θ)p(Θ)
p(S
k
)
. (3)
Page 2
As p(S
k
) does not depend on Θ and p(Θ) is assumed to be uniform, the global
criterion integrating our unified framework is the following:
C(Q, Θ) =
N
X
k=1
(log(p(S
k
|Q
k
, Θ)) + log(p(Q
k
|Θ))) (4)
The first term describes the ML criterion (eq. (2)) whereas the second term is
the prior on the deformation coefficients ω
kp
as described in eq.(1). Dropping
the constants, our criterion simplifies to C(Q, Θ)
P
N
k=1
C
k
(Q
k
, Θ) with
C
k
(Q
k
, Θ) =
n
X
p=1
log(λ
p
) +
ω
2
kp
2λ
2
p
!
N
k
X
i=1
log
N
m
X
j=1
exp
ks
ki
T
k
? m
kj
k
2
2σ
2
.(5)
This equation is the heart of the unified framework for the model computation
and its fitting to observations. By optimizing it alternately with respect to the
operands in {Q, Θ}, we are able to determine all parameters we are interested
in. In a first step, all observations are aligned with the initial mean shape by es-
timating the T
k
using the EM-ICP. In order to robustify, we used a multi-scaling
scheme concerning the variance σ
2
, for details please refer to [9]. Starting from
the initial model parameters Θ, we then fit the model to each of the observations.
Next, we fix the observation parameters Q
k
and update the model parameters.
This is iterated until convergence.
2.3 Experiments
We present two experiments we ran in order to evaluate our approach.
A typical correspondence problem: One of our arguments against the con-
cept of enforcing homologies between points representing surfaces is the fact that
certain shape structures of one observation might not appear on all observations
of the training set. We constructed a synthetic data set containing 20 observa-
tions in order to illustrate this problem, see figure 1 a). Half of the ellipsoids
are equipped with a bump, and the other half are not. A reliable SSM should
be able to represent both classes by including this distinction in its variability
model. In order to compare the results, we generated two SSMs, one by using
our new method, the other one by using the traditional ICP and PCA.
Generalization ability and specificity: Our data set consists of N = 21 left
segmented putamen observations (approximately 20mm×20mm×40mm) which
are represented by min 994 and max 1673 points (Fig. 1c)). In order to assess the
quality of a SSM, we measure two performance measures as proposed in [10]. A
good generalization ability is important for recognition purposes as a SSM must
be able to adopt the shape of an unseen observation which comes from the same
structure type. We test it in a series of leave-one-out experiments. The SSM
is first aligned with the unseen observation, then the matching is optimized by
finding the best deformation coefficients ω. Finally, the distance of the deformed
SSM to the left-out observation is measured. The specificity of a SSM must be
Page 3
Table 1. Performance Measures: Shape distances found in generalization experiments
(7 leave-one-out tests) and in specificity experiments (500 random shapes) with our
SSM approach and with an ICP+PCA approach using 18 eigenmodes.
Generalization ability ICP+PCA model our SSM
average mean distance + standard deviation in mm 0.610 ± 0.089 0.447 ± 0.101
average maximal distance + standard deviation in mm 4.388 ± 0.930 2.426 ± 0.712
Specificity ICP+PCA model our SSM
average mean distance + standard deviation in mm 0.515 ± 0.117 0.452 ± 0.020
a)
b)
c)
d)
e)
1 2 1 2
ICP SSM
Fig. 1. SSM on synthetic data set. a): Ellipsoid observation examples. b) Results
SSM/ICP. c): Results SSM/EM-ICP. Final mean shape (middle), and the mean shapes
deformed with respect to the first eigenmode (
¯
M 3λ
1
v
1
and
¯
M + 3λ
1
v
1
). d): Puta-
men observation examples. e): Exact correspondence versus correspondence probabili-
ties. Left: ICP, right: EM-ICP.
high for shape prediction purposes as the SSM should only adopt shapes similar
to the ones in the underlying training set. We generated random shapes x which
are uniformly distributed with σ being equal to the standard deviation of the
SSM. Next, we computed the distance of the random shapes to the closest obser-
vations in the training data set. As we do not have surface representations, we
chose a distance measure which computes the average minimum point distance
between the deformed model points and the observation points.
3 Results
For our SSM, the following parameters were chosen: σ
start
= 4mm, reduction
factor = 0.85, 10 iterations (EM-ICP) with 5 SSM iterations. For the ICP+PCA
SSM, we iterated the ICP 50 times. The results for the experiments concerning
the typical correspondence problem are shown in figure 1 a),b). For both methods,
we chose the initial model from the ellipsoid class without bump. The first row
shows the mean shape and eigenmodes for the SSM built by the ICP+PCA SSM
whereas the second row shows those for our SSM.
The results of the testing series for the generalization ability and the specificity
for both our SSM and the ICP+PCA SSM on putamen data are depicted in table
1. We performed the leave-one out test for 7 different unknown observations. The
specificity was computed for 500 random shapes. We computed the average and
maximal distances as well as the variances.
Page 4
4 Discussion
We proposed a mathematically sound and unified framework for the computation
of model parameters and observation parameters and succeeded in determining
a closed form solution for optimizing the associated criterion alternately for all
parameters. Experiments showed that our algorithm works well and leads to
plausible results. It seems to be robust to different initial mean shape choices
and is stable even for small numbers of observations. We showed the efficiency of
our approach compared with a SSM built by the traditional ICP and PCA for a
typical correspondence problem on synthetic data: The SSM based on the EM-
ICP models the whole data set, it is able to represent the ellipsoids featuring
a bump and those without as that deformation information is included in its
variability model. On the other hand, the results show that the SSM based on
the ICP is not able to model the bump. This is due to the fact that the ICP
only takes into account the closest point when searching for correspondence,
thus, the point on top of the bump is not involved in the registration process.
The EM-ICP, however, evaluates the correspondence probability of all points,
therefore, also the point on top of the bump is matched. We illustrated these
two concepts in figure 1d). Furthermore, in the test series on putamen data,
our SSM achieved superior results in both performance measures. Especially the
values of the maximal distance illustrate the benefit of the new approach. From
a theoretical point of view, a very powerful feature of our method is that we are
optimizing a unique criterion. Thus, the convergence is ensured.
1
References
1. Besl PJ, McKay ND. A method for registration of 3D shapes. IEEE Transactions
PAMI. 1992;14:239–256.
2. Davies RH, Twining CJ, Cootes TF. A Minimum Description Length Approach
to Statistical Shape Modeling. IEEE Medical Imaging. 2002 May;21(5):525–537.
3. Heimann T, Wolf I, Williams T, Meinzer HP. 3D Active Shape Models Using
Gradient Descent Optimization of Description Length. In: IPMI’05. vol. 3565;
2005. p. 566–577.
4. Tsai A, Wells WM, Warfield SK, Willsky AS. An EM algorithm for shape classi-
fication based on level sets. Medical Image Analysis. 2005;9:491–502.
5. Cates J, Meyer M, Fletcher PT, Whitaker R. Entropy-Based Particle Systems for
Shape Correspondences. In: MICCAI’06. vol. 1; 2006. p. 90–99.
6. Rangarajan A, Chui H, Bookstein FL. The Softassign Procrustes Matching Algo-
rithm. In: IPMI’97. vol. 1230; 1997. p. 29–42.
7. Hufnagel H, Pennec X, Ehrhardt J, et al. Point-Based Statistical Shape Models
with Probabilistic Correspondences and Affine EM-ICP. In: BVM’07; 2007. p.
434–438.
8. Granger S, Pennec X. Multi-scale EM-ICP: A Fast and Robust Approach for
Surface Registration. In: ECCV’02. vol. 2525 of LNCS; 2002. p. 418–432.
9. Hufnagel H, Pennec X, Ehrhardt J, Handels H, Ayache N. Shape Analysis Using
a Point-Based Statistical Shape Model Built on Correspondence Probabilities. In:
MICCAI’07. vol. 1; 2007. p. 959–967.
10. Styner M, Rajamani KT, Nolte LP, et al. Evaluation of 3D Correspondence Meth-
ods for Model Building. In: IPMI’03. vol. 2732; 2003. p. 63–75.
1
This work is supported by a grant from the DFG, HA2355.
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