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POINT-SET REGISTRATION OF TAGGED HE-3 IMAGES USING A

STRUCTURALLY-BASED JENSEN-SHANNON DIVERGENCE MEASURE WITHIN A

DETERMINISTIC ANNEALING FRAMEWORK

N.J. Tustison, S.P. Awate, T.A. Altes, J.C. Gee

University of Pennsylvania

Department of Radiology

Philadelphia, PA USA

J. Cai, G.W. Miller, E. E. de Lange, J.P. Mugler III

University of Virginia

Department of Radiology

Charlottesville, VA USA

ABSTRACT

Helium-3 tagged magnetic resonance imaging has demon-

strated potential for calculating pulmonary deformation from

medical imagery. Such measurements are useful for deter-

mining the biomechanical properties of the lung. Unfortu-

nately, the relative facility of visually tracking deformation

via the high contrast tag lines has not transferred readily to

the algorithmic domain of automatically establishing tag-line

correspondences. We proffer a solution to this dilemma by

translating the problem into a unique point-set registration

scenario. Not only does this permit capitalizing on cer-

tain spectral aspects of tagged MRI but registration can be

performed within a deterministic annealing framework for

decreased susceptibility to local minima.

Index Terms— B-splines, free-form deformation, Gabor

filter bank, Jensen-Shannon entropy, point-set registration

1. INTRODUCTION

Introducing our previous research [6], we outlined the com-

parative maturity of myocardial tagged MRI analysis and

the reliance on that research field for forwarding analysis of

tagged helium-3 MRI. The current research we present con-

stitutes a step towards more automated analysis of generic

tagged MR imagery although, in this paper, we specifically

apply our methodology to tagged helium-3 MRI of the lung.

2. METHODOLOGY

Given two configurations of a deformable body (such as the

lung), extraction of salient biomechanical quantities requires

establishing a continuous mapping between the configura-

tions. Assuming the reliable permanency of the tagging fidu-

cials, we can establish that continuous mapping by finding

the correspondence between the sets of tag planes in the two

configurations via an underlying continuous mapping defined

over the entire transformation domain. The general algorith-

mic workflow for calculating this mapping is summarized as

follows:

1. Images are preprocessed which includes an automated

coarse segmentation of the lungs and inhomogeneity

correction of the image.

2. A separate Gabor filter bank is applied to the prepro-

cessed images for each set of tag planes. Those vox-

els within the coarse segmentation with a high response

are thresholded to produce candidate points providing

a sampling of the tag planes.

3. Convert each of the three point sets from each configu-

ration into separate probability density functions.

4. For each of the three sets of tag planes, minimize the

Jensen-Shannon entropy using a specified transforma-

tion model and minimization technique. Each of the

three sets of corresponding tag planes provides the de-

formation field in one of the three canonical Cartesian

directions.

5. Conjoin the results to provide a full 3-D continuous

mapping which establishes correspondence between

the two configurations. From this mapping, kinematic

measures can be calculated.

Elaboration of the algorithmic details is provided in subse-

quent sections.

2.1. Gabor Filter Banks for Tag Points Extraction

After preprocessing the images, we extract the sampled tag

plane points using a Gabor filter bank [3]. The Gabor func-

tion is complex sinusoid modulated by a Gaussian envelope

formulated (in 1-D) as

?

?

g(x) =

1

√2πσexp

−x2

2σ2

?

? ??

Gaussian

·exp(−j2πωx + θ)

?

???

complex sinusoid

.

(1)

772 978-1-4244-2003-2/08/$25.00 ©2008 IEEEISBI 2008

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(a)(b)

Fig. 1. (a) Coronal view of a resampled helium-3 MRI lung

image and (b) the superimposed results for the set of sagit-

tal (red points) and axial (green points) tag planes. The tag

plane spacing sampling was {14,28,32} mm and the angular

sampling was {−20◦,0◦,20◦}. The threshold level was 90%.

2-D and 3-D extensions of the Gabor filter have found use

in different areas of computer vision research (e.g. [5]). Al-

though the tag planes have such structure, due to the defor-

mation of the lung during the respiratory cycle, the tag plane

surfaces will also deform which necessitates the use of multi-

ple Gabor filters for enhancing the local planar structure. For

eachimage, theGaborfilterbankisappliedthreetimes—once

for each of the three sets of mutually orthogonal tag planes.

The user selects the number of Gabor filters in the filter bank

by sampling both in terms of the orientation of the filters as

well as the tag spacing. A user-selected threshold percentage

determines the number of voxels included as candidate tag

plane points. Note that it is expected that some true tag plane

sample points will be excluded from the results and that out-

liers will inevitably be included in the result. Such misclassi-

fication relies on the robustness of the subsequent registration

step for accurate results. Using typical values given in [3],

sample results are provided in Figure 1.

2.2. Minimizing the Jensen-Shannon Entropy for Point-

Set Registration

Wang et al. [9] present recent research for point-set registra-

tion which employs the Jensen-Shannon (JS) entropy diver-

gence measure and a thin-plate spline transformation model.

Each point-set is represented as a probability density function

through the use of a Gaussian mixture model (GMM) where

each point, xi, specifies a Gaussian center with a constant

isotropic covariance.

Given K probability densitys {P1,...,PK}, the JS di-

vergence is given by

JS(P1,...,PK) = H

?K

k=1

?

πkPk

?

−

K

?

k=1

πkH (Pk) (2)

where H(·) is the Shannon entropy and the set of weights

{π1,...,πK|πk> 0,?K

k=1πk= 1} determines the relative

contribution of the corresponding probability density func-

tion to the divergence measure. Associating each Pkwith a

mapping function, Tk, allows for the determination of the pa-

rameters of the respective mapping functions which map each

point set to an unbiased average atlas. Please note that in our

case, we are not interested in calculating an average atlas but

wish to map the deformed configuration (end-expiration) to

the undeformed configuration (end-inspiration). This is done

by maintaining Tundeformed = I throughout the minimiza-

tion process where I is the identity transformation.

Using the core algorithm presented above, we made sev-

eral improvements to enhance the performance of our algo-

rithm when applied to tagged MR images. Two major dif-

ferences are the construction of a structurally-based probabil-

ity density function from each point-set and the employment

of an enhanced B-spline deformation model within a deter-

ministic annealing framework. These two improvements are

described below.

2.2.1. Manifold Parzen Windowing for Structurally-Based

Density Functions

Given that point-sets often represent a sampling of an under-

lying structure, we modify the conversion process transform-

ing a point-set to its corresponding probability density func-

tion to capture that local structure. Whereas previous work

used isotropic Gaussians, we use the local point-set neighbor-

hood to estimate an appropriate covariance matrix where the

local structure is reflected in the anisotropy of that covariance

[8]. For each point, xi, the associated weighted covariance

matrix, CKi, is given by

?

CKi=

xj∈Ni,xj?=xiK(xi;xj)(xi− xj)T(xi− xj)

?

where Niis the local neighborhood of the point xiand K is

a user-selected neighborhood weighting kernel. We use an

isotropic Gaussian for K as well as a k-d tree structure for

efficient determination of Ni.

For the optimization routine the inverse of each covari-

ance matrix is required. Determination of CKifrom Equation

(3) could potentially result in an ill-conditioned matrix. For

this reason, we use the modified covariance, Ci= CKi+σ2I

where I is the identity matrix and σ is a user-provided param-

eter denoting added isotropic Gaussian noise (this parameter

is used in our annealing scheme discussed in a later section).

Thus, the kthprobability density function calculated from the

kthpoint-set is given by the GMM

xj∈Ni,xj?=xiK(xi;xj)

(3)

Pk(s) =

1

Nk

Nk

?

i=1

G(s;xk

i,Ck

i)

(4)

where G(s;xk

and covariance Ck

i,Ck

i) is a normalized Gaussian with mean xk

ievaluated at s.

i

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2.2.2. B-spline Transformation Model

Thus far we have described how distinct point-sets extracted

from tagged helium-3 MR images are converted into proba-

bility density functions which allows their dissimilarity to be

quantified by the JS divergence measure. Associating each

point-set (i.e. probability density function) with a continu-

ous mapping function, Tk, and minimizing the JS divergence

with respect to the parameters of all K transformation models

brings the point-sets into correspondence.

Optimization requires evaluation of Equation (2) in addi-

tion to the corresponding gradient with respect to the transfor-

mation model parameters. Both evaluations require the gener-

ation of a set of sample points from each of the K probability

density functions (further details are given in [9]). We desig-

nate the number of sample points generated for each of the K

probability density functions as {M1,...,MK} and the kth

set of samples as {sk

Calculation of the gradient of the JS divergence with re-

spect to the transformation parameters is slightly different

than that given in [9] and so we reproduce our gradient for-

mulation for the benefit of the reader. The partial derivative

of Equation (2) with respect to the lthparameter, βl

given by

1,...,sk

Mk}.

k, of Tkis

∂JS

∂βl

k

=∂Tk

∂βl

⎡

k

·

Nk

?

i=1

⎣

−

1

MN

K

?

k?=1

Mk?

?

j=1

G(sk?

j;xk

i,Ck

i)(Ck

P∗(sk?

i)−1(xk

j)

i− sk?

j)

1

MkN

Mk

?

j=1

G(sk

j;xk

i,Ck

i)(Ck

Pk(sk

i)−1(xk

j)

i− sk

j)

⎤

⎦

(5)

where

P∗(s) =

1

N

K

?

k=1

Nk

?

i=1

G(s;xk

i,Ck

i)

(6)

N =

K

?

k=1

Nk,M =

K

?

k=1

Mk.

(7)

Traditional gradient approaches, which are intrinsically

susceptible to hemstitching during the gradient descent, cal-

culate∂Tk

∂βl

kas

∂Tk

∂βl

k

=

D

?

j=1

Blj,dj(uj)

(8)

whereas we calculate a preconditioned version

?D

?

∀uj

where D is the parametric dimensionality of the B-spline ob-

ject, d is the degree of B-spline used, and the set of B(·) are

the B-spline basis functions governing Tk. Additional details

discussing the deficiency of traditional gradient approaches as

well as the derivation of Equation (9) can be found in [7].

∂Tk

∂βl

k

=

j=1Blj,dj(uj) ·?D

j=1B2

?D

lj,dj(uj)

lj,dj(uj)

kj,dj(uj)

?

?d1+1

·

k1=1...?dD+1

?

kD=1

1

j=1B2

?D

j=1B2

(9)

2.2.3. Deterministic Annealing

The core of the optimization by conjugate gradient descent is

relatively standard [2]. However, we have placed this mini-

mization routine within a deterministic annealing framework

[4], both in terms of the transformation model as well as the

JS divergence, which decreases the susceptibility to local

minima. This is extremely important since, at the local level,

determination of correct tag correspondence is difficult.

At the initial stage of the optimization, the B-spline trans-

formation model is defined by a low-resolution mesh to deter-

mine more global correspondence. At each subsequent level,

the mesh-resolution is doubled [7] for increased local, refined

registration. In coordination with this hierarchical registra-

tion, we specify an annealing schedule for the isotropic gaus-

sian noise discussed in Section 2.2.1. At the pthlevel the

covariance is calculated as Ci = CKi+ λpσ2I where λ is

the annealing rate. This also has the effect of increasing the

localization during the course of the optimization.

3. EXPERIMENTATION

To demonstrate the methodology described above, sample

registration results are acquired using the tagged helium-3

images shown in Figure 2. MR imaging was performed using

a 1.5 T whole-body scanner (Siemens Sonata), and helium-3

gas was polarized to approximately 40% using a commercial

system (Model 9600, MITI). Following inhalation of a 1 L

mixture of ∼400 ml hyperpolarized helium-3 and ∼600 ml

N2, 3-D tagging grids were created at breath hold by applying

sinc-modulated RF-pulse trains consecutively along all three

principal axes. Tag spacing was 22 mm. The grid-tagging

preparation was applied first, followed by a FLASH-based

3D image acquisition at full inspiration with the following

parameters: TR/TE = 2.0/ 0.7 ms; FOV= 340x280x198 mm;

matrix = 64x64x22; flip angle = 1. The subject was then

instructed to exhale completely, and the FLASH-based acqui-

sition was repeated following a pause of 2-3 s. Total scan time

is 7.4s. The short TE of the FLASH sequence was achieved

by using the asymmetric readout, yielding a partial k-space

acquisition.

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(a)(b)

Fig. 2. Superimposed sample points on the (a) end-inspiration

and (b) end-expiration images after manual refinement.

The acquired images were then processed to extract point

sets (one set from each set of tag planes from each image).

Extracted points were then manually refined to remove spu-

rious sample points. The results are provided in Figure 2. A

cubic B-spline transformation domain was then defined over

the image domain. Starting from an initial mesh resolution of

one B-spline element, four levels were used during the course

of the optimization where the B-spline mesh resolution was

doubled at each level. The resulting displacement field and

corresponding principal strain field are shown in Figure 3(a)

and Figure 3(b), respectively.

4. DISCUSSION

We have presented a potentially useful framework for analyz-

ing pulmonary biomechanics from tagged helium-3 MRI with

possible application to other tagged MR imagery. Equally

as important, we implemented this framework using the In-

sight Toolkit [1] which allows for open-source dissemination

and efficient portability of useful components of our research.

This allows others in relevant areas of research to apply and

build upon our methods.

5. ACKNOWLEDGMENTS

This work was supported by NIH grant R01 HL079077,

Siemens Medical Solutions and the Flight Attendant Medical

Research Institute.

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