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

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Available from: Heinz HandelsA 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 Ω

k

= {ω

kp

, p = 0, ..., n − 1}.

p(M

k

|Θ) = p(Ω

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

= {Ω

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

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1

This work is supported by a grant from the DFG, HA2355.

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- References (14)
- Cited In (0)

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**ABSTRACT:**Active Shape Models are a popular method for segmenting three-dimensional medical images. To obtain the required landmark correspondences, various automatic approaches have been proposed. In this work, we present an improved version of minimizing the description length (MDL) of the model. To initialize the algorithm, we describe a method to distribute landmarks on the training shapes using a conformal parameterization function. Next, we introduce a novel procedure to modify landmark positions locally without disturbing established correspondences. We employ a gradient descent optimization to minimize the MDL cost function, speeding up automatic model building by several orders of magnitude when compared to the original MDL approach. The necessary gradient information is estimated from a singular value decomposition, a more accurate technique to calculate the PCA than the commonly used eigendecomposition of the covariance matrix. Finally, we present results for several synthetic and real-world datasets demonstrating that our procedure generates models of significantly better quality in a fraction of the time needed by previous approaches. - [Show abstract] [Hide abstract]
**ABSTRACT:**The authors describe a general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces. The method handles the full six degrees of freedom and is based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point. The ICP algorithm always converges monotonically to the nearest local minimum of a mean-square distance metric, and the rate of convergence is rapid during the first few iterations. Therefore, given an adequate set of initial rotations and translations for a particular class of objects with a certain level of `shape complexity', one can globally minimize the mean-square distance metric over all six degrees of freedom by testing each initial registration. One important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model, prior to shape inspection. Experimental results show the capabilities of the registration algorithm on point sets, curves, and surfaces - [Show abstract] [Hide abstract]
**ABSTRACT:**We describe a method for automatically building statistical shape models from a training set of example boundaries/surfaces. These models show considerable promise as a basis for segmenting and interpreting images. One of the drawbacks of the approach is, however, the need to establish a set of dense correspondences between all members of a set of training shapes. Often this is achieved by locating a set of "landmarks" manually on each training image, which is time consuming and subjective in two dimensions and almost impossible in three dimensions. We describe how shape models can be built automatically by posing the correspondence problem as one of finding the parameterization for each shape in the training set. We select the set of parameterizations that build the "best" model. We define "best" as that which minimizes the description length of the training set, arguing that this leads to models with good compactness, specificity and generalization ability. We show how a set of shape parameterizations can be represented and manipulated in order to build a minimum description length model. Results are given for several different training sets of two-dimensional boundaries, showing that the proposed method constructs better models than other approaches including manual landmarking-the current gold standard. We also show that the method can be extended straightforwardly to three dimensions. -
##### Conference Paper: Point-Based Statistical Shape Models with Probabilistic Correspondences and Affine EM-ICP

[Show abstract] [Hide abstract]**ABSTRACT:**A fundamental problem when computing statistical shape models (SSMs) is the determination of correspondences between the in- stances. Often, homologies between points that represent the surfaces are assumed which might lead to imprecise mean shape and variation results. We present a novel algorithm based on the a-ne Expectation Maximization - Iterative Closest Point (EM-ICP) registration method. Exact correspondences are replaced by iteratively evolving correspon- dence probabilities which provide the basis for the computation of mean shape and variability model. We validated our approach by computing SSMs using inexact correspondences for kidney and putamen data. In ongoing work, we want to use our methods for automatic classiflcation applications. -
##### Conference Paper: Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration

[Show abstract] [Hide abstract]**ABSTRACT:**We investigate in this article the rigid registration of large sets of points, generally sampled from surfaces. We formulate this problem as a general Maximum-Likelihood (ML) estimation of the transformation and the matches. We show that, in the specific case of a Gaussian noise, it corresponds to the Iterative Closest Point algorithm(ICP) with the Mahalanobis distance.Then, considering matches as a hidden variable, we obtain a slightly more complex criterion that can be efficiently solved using Expectation-Maximization (EM) principles. In the case of a Gaussian noise, this new methods corresponds to an ICP with multiple matches weighted by normalized Gaussian weights, giving birth to the EM-ICP acronym of the method.The variance of the Gaussian noise is a new parameter that can be viewed as a "scale or blurring factor" on our point clouds. We show that EMICP robustly aligns the barycenters and inertia moments with a high variance, while it tends toward the accurate ICP for a small variance. Thus, the idea is to use a multi-scale approach using an annealing scheme on this parameter to combine robustness and accuracy. Moreover, we show that at each "scale", the criterion can be efficiently approximated using a simple decimation of one point set, which drastically speeds up the algorithm.Experiments on real data demonstrate a spectacular improvement of the performances of EM-ICP w.r.t. the standard ICP algorithm in terms of robustness (a factor of 3 to 4) and speed (a factor 10 to 20), with similar performances in precision. Though the multiscale scheme is only justified with EM, it can also be used to improve ICP, in which case the performances reaches then the one of EM when the data are not too noisy. - [Show abstract] [Hide abstract]
**ABSTRACT:**. The problem of matching shapes parameterized as a set of points is frequently encountered in medical imaging tasks. When the point-sets are derived from landmarks, there is usually no problem of determining the correspondences or homologies between the two sets of landmarks. However, when the point sets are automatically derived from images, the difficult problem of establishing correspondence and rejecting non-homologies as outliers remains. The Procrustes method is a well-known method of shape comparison and can always be pressed into service when homologies between point-sets are known in advance. This paper presents a powerful extension of the Procrustes method to pointsets of differing point counts with correspondences unknown. The result is the softassign Procrustes matching algorithm which iteratively establishes correspondence, rejects non-homologies as outliers, determines the Procrustes rescaling and the spatial mapping between the point-sets. 1 Introduction ... -
##### Conference Paper: Evaluation of 3D Correspondence Methods for Model Building

[Show abstract] [Hide abstract]**ABSTRACT:**The correspondence problem is of high relevance in the con- struction and use of statistical models. Statistical models are used for a variety of medical application, e.g. segmentation, registration and shape analysis. In this paper, we present comparative studies in three anatom- ical structures of four dierent correspondence establishing methods. The goal in all of the presented studies is a model-based application. We have analyzed both the direct correspondence via manually selected landmarks as well as the properties of the model implied by the corre- spondences, in regard to compactness, generalization and specicity. The studied methods include a manually initialized subdivision surface (MSS) method and three automatic methods that optimize the object param- eterization: SPHARM, MDL and the covariance determinant (DetCov) method. In all studies, DetCov and MDL showed very similar results. The model properties of DetCov and MDL were better than SPHARM and MSS. The results suggest that for modeling purposes the best of the studied correspondence method are MDL and DetCov.