TWO-LEVEL METHOD FOR 3D NON-RIGID REGISTRATION

With an Application to Statistical Atlases Construction

C. Wu*, P. E. Murtha**, A. B. Mor** and B. Jaramaz* **

* Carnegie Mellon University, School of Computer Science

Robotics Institute, Pittsburgh, PA, 15213

** Institute for Computer Assisted Orthopaedic Surgery, The Western Pennsylvania Hospital Mellon Pavilion

Suite 242, 4815 Liberty Avenue, Pittsburgh, PA 15224

Keywords:

3D deformable registration, Two-level method, Statistical atlas.

Abstract:

We propose a two-level method for 3D non-rigid registration and apply the method to the problem of building

statistical atlases of 3D anatomical structures. 3D registration is an important problem in computer vision and

a challenge topic in medical image ﬁeld due to the geometrical complexity of anatomical shapes and size of

medical image data. In this work we adopt a two-level strategy to deal with these problems. Compared with

a general multi-resolution framework, we use an interpolation to propagate the matching instead of repeating

registration scheme in each resolution. Our algorithm is divided into two main parts: a low-resolution solution

to the correspondences and mapping of surface models using Chui and Rangarajan’s robust point matching

algorithm, followed by an interpolation to achieve high-resolution correspondences. Experimental results

demonstrate our approach for solving the non-rigid registration and correspondences within complicated 3D

data sets. In this paper we present an example of this method in the construction of a statistical atlas of the

femur.

1 INTRODUCTION

Registration has been studied for years in computer

vision, which is still a critical problem in medi-

cal image ﬁeld due to the geometrical complexity

of anatomical shapes, and computational complexity

caused by the enormous size of volume data. It has

numerous clinical applications such as statistical atlas

construction for group study and statistical parame-

ters analysis (Hill et al., 2001), mapping anatomical

atlases to individual patient images for disease anal-

ysis (Fleute et al., 2002) and segmentation (Rohlﬁng

et al., 2004).

According to the type of the transformation be-

ing applied, registration can be rigid or non-rigid.

In other words, as long as the shape has no change

between two images, the registration should be

rigid, such as the intra-subject(same patient)-inter-

modality(different imaging system) registration by

capturing images at the same time. However, when

we take into account the time, i.e., when two im-

ages are captured at different time, most of intra-

subject registration will be non-rigid due to the

shape variance of the anatomical structures for ex-

ample swelling, prostate poking, bone fractures, tu-

mor growth changes, intestinal movements etc. Be-

sides, inter-subject(different patient) registrations are

usually non-rigid because of the local anatomical dif-

ferences between patients. Therefore, non-rigid (also

known as deformable) registration has been an active

topic in recent years. In general, a non-rigid transfor-

mation is represented by a global rigid or afﬁne trans-

formation plus a local non-linear deformation, which

can be described by radial basis functions (RBF) (Yu,

2005), octree-spline (Szeliski and Lavallee, 1996),

thin-plate spline (TPS) (Chui and Rangarajan, 2003),

geometric splines (Farin, 1993), ﬁnite elements (Park

et al., 1996), or free form B-spline (Rueckert et al.,

2003) etc. In order to evaluate the registration, dif-

ferent similarity measurements have been utilized ac-

cording to different features and imaging modalities.

For example, sum of squared distances (SSD) is used

for geometric features (Besl and McKay, 1992). For

the intensity features, correlation coefﬁcients (CC)

(Kim and Fessler, 2004), Ratio Image Uniformity

(RIU) (Woods et al., 1994), or mutual information

356

Wu C., E. Murtha P., B. Mor A. and Jaramaz B. (2007).

TWO-LEVEL METHOD FOR 3D NON-RIGID REGISTRATION - With an Application to Statistical Atlases Construction.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 356-362

 SciTePress

(MI) (Wells et al., 1996) are usually considered. Reg-

istration problem can be simpliﬁed given some known

correspondences, for example using markers (Maurer

et al., 1997). Nevertheless, markers are not allowed

to use or available in many scenarios. Alternate es-

timation of correspondences and transformations are

therefore widely used for both rigid case (Besl and

McKay, 1992) and non-rigid case (Chui and Rangara-

jan, 2003; Chui et al., 2004; Glaunes et al., 2004).

Moreover, with the increase of data size and geomet-

rical complexity, multi-resolution strategy has been

adopted into the registration framework (Ellingsen

and Prince, 2006; Jaume et al., 2002; Shen, 2002).

Sparse matrices are also used to handle the computa-

tional complexity (Papademetris et al., 2003).

In this paper we propose a two-level non-rigid

registration approach for 3D surface mesh to deal

with the computational and geometrical complexity,

inspired by Chui and Rangarajan’s non-rigid regis-

tration algorithm and the previous multi-resolution

works. Since Chui and Rangarajan’s algorithm is

not able to handle more than 2000 3D points (Pa-

pademetris et al., 2003), in order to deal with more

points, we break down the registration into a two-

level process. We ﬁrst apply their algorithm to the

simpliﬁed low-resolution meshes (We use Garland’s

mesh simpliﬁcation technique (Garland and Heck-

bert, 1997) to compute low-resolution meshes). And

then, instead of successively matching in each reso-

lution from coarse to ﬁne, we directly propagate the

correspondences from low resolution to the high res-

olution by an interpolation.

2 TWO-LEVEL REGISTRATION

2.1 Mesh Simpliﬁcation

We use Garland’s quadric error metrics (QEM) based

mesh simpliﬁcation (Garland and Heckbert, 1997)

technique to obtain low-resolution meshes. QEM is

based on the iterative contraction of vertex pairs. The

cost of contraction is noted by a quadric error and the

whole process is an iteratively minimizing the quadric

error.

A critical parameter in the simpliﬁcation is the

number of vertices in the low-resolution meshes. The

less vertices, the faster low-resolution registration but

less accurate the high-resolution registration. We

make a trade off between accuracy and speed by do-

ing a series of experiments (See Sec. 4).

2.2 Low-Res Non-Rigid Registration

Point-to-Point Registration: We apply Chui and

Rangarajan’s non-rigid registration method on simpli-

ﬁed meshes. Fuzzy correspondences and a determin-

istic annealing technique are adopted for a smoother

optimization process and efﬁciency. A dual update

strategy is utilized to estimate the correspondences

and transformation iteratively. The non-rigid transfor-

mation is parameterized using thin-plate splines for a

smooth spatial mapping.

Initial Alignment: Before applying Chui and Ran-

garajan’s method, we need an extra alignment due to

the particularity of our data. Our data comes from

CT scanned surfaces of human femur. Since we are

more interested in condyles, As Fig. 1 shows, only

distal femur is scanned to build meshes. According

to different patients, some meshes include more fe-

mur shaft such as mesh Y, others include less shaft

for example mesh X. Experiments shows that the reg-

istration might be very slow and may not converge

for some cases, if we only move mesh X to the center

of mesh Y at the very beginning. The reason is that

some part of mesh Y (as showed in a blue rectangle)

has no counterpart in mesh X, and we should not take

into account this part in the registration. To tackle

this problem, we estimate the pseudo center of mesh

Y and a rigid transformation between two meshes.

As Fig. 1 illustrates, we use the height of mesh

X to estimate the pseudo center of mesh Y. Assume

axis z is the scan direction from the knee to hip, we

estimate the pseudo center c

′

as:

′

∑

−minz

)<(maxz

−minz

)

(1)

where N

′

covers points p

′

in mesh Y that satisfy

− minz

) < (maxz

− minz

) (black points in 1.

We apply principal component analysis (PCA) to es-

timate the pose of each mesh. Assume

∑

(2)

is the center of mesh X. Then we can compute the

covariance matrix for {p

} and {p

′

−1

−c

,·· · ,p

−c

] · [p

−c

,·· · ,p

−c

]

′

−1

′

−c

′

,·· · ,p

′

−c

′

] · [p

′

−c

′

,·· · ,p

′

−c

′

]

′

(3)

We compute the principle axes by decomposing the

covariance matrix using moment analysis:

= U

′

, Ψ

′

= U

′

(4)

Each column of U

represents an principle axis of

points set {p

}, and U

′

for {p

′

}. We use three

axes to describe the pose of points set (Fig. 1): red

Mesh X Mesh Y

Move Mesh X to the pseudo center of Mesh Y

Rotate Mesh X to be the same pose as Mesh Y

Mesh X & Y

Figure 1: Illustration of the rigid transformation from mesh

X to mesh Y. The ﬁrst row compares the translated mesh

X with Y. Black points in mesh Y are used to compute the

pseudo center. The second compare the translated and ro-

tated mesh X with Y. Red axes represent the principle com-

ponents of point set in mesh X, blue axes for Y.

for {p

} and green for {p

′

}. We can estimate a

rotation from axes of X to Y

′

, which is given by

′

·U

′

. Therefore, we can apply a rigid transfor-

mation [U

′

·U

′

|(c

′

−c

)] to points set {p

}. Fig.

1 shows the transformed mesh X. Experiment shows

the rate of convergence has been improved from 78%

to 95.2%.

2.3 Local Non-Rigid Registration

Point-to-Surface Registration: In the previous sec-

tion the non-rigid registration is applied to deform

only points and SSD is used as a criteria. Here we

will step further to minimize the SSD by using some

points on the surface instead of original vertices. This

idea is straightforward. As Fig. 2 shows, x

is a point

in the deformed mesh X, whose correspondence in

mesh Y is y

. SSD (

∑

− y

) has been minimized

in the previous section. However, it is possible to de-

crease SSD more if we use some points on the surface

instead of y

. Let’s check the neighboring triangles of

, which are triangles sharing the same vertex y

, for

example S

and S

in Fig. 2. We examine the dis-

tance from x

to each neighboring triangle, such as

, d

and d

in Fig. 2. If any of them is smaller than

= |x

−y

|, we can use the corresponding projected

point to replace y

such that we can have achieve a

smaller SSD.

Since the point-surface registration is a local pro-

cess, we have to take into account the case that differ-

ent points in mesh X which come up with the same

corresponding surface point in mesh Y. We use a sim-

ple rule to handle such interference: the point in mesh

X with smaller SSD will be updated with the surface

point in mesh Y.

Deformed Mesh X

Mesh Y

Figure 2: Illustration of local non-rigid registration between

point and surface. x

is a point in the deformed mesh X,

whose correspondence in mesh Y is y

. The projection of

to each triangle S

sharing y

is denoted by y

. d

is the

distance between x

and y

. (t = 1, 2,3, ·· ·).

2.4 Low-Resolution to High-Resolution

Interpolation

After the registration in low-resolution meshes, we

directly apply a surface interpolation to those coarse

meshes, in order to migrate the registration to high-

resolution meshes. The problem is to build the cor-

respondences between mesh X and Y, given corre-

spondence between a subset of X and a subset of

Y. Radial basis functions(RBF), ﬁnite element, mul-

tivariate spline such as thin-plate spline(2D bivariate

spline) and triharmonic thin-plate spline, are popular

techniques used in surface interpolation. Carr et al

(Carr et al., 1997) include multivariate splines method

into radial basis functions by using splines as kernel

functions. In this work we use Gaussian kernel based

RBF as an illustration due to it’s simple mathematical

representation and less restrictions on nodes. Specif-

ically, we use a linear afﬁne function plus a series of

radial basis functions (RBFs) to construct the interpo-

lation function:

·[ϕ(kx

k),· ·· ,ϕ(kx

k)]

′

·x

{z }

g(x

)

(5)

where x

is a vertex in the low-resolution mesh X

whose correspondence in the low-resolution mesh Y

is y

, i = 1, 2,··· , N (N is the number of vertices in

mesh X

). x

and y

are both 3×1 vectors with three

coordinates. c

is a 3×N coefﬁcient matrix of ra-

dial basis functions. ϕ(kx

k) are radially sym-

metric basis functions. We have chosen a Gaussian

kernel ϕ(u

) = exp(−ku

− u

k/0.5), as suggested

by (Pighin et al., 1998). c

and c

are coefﬁcients for

the afﬁne component. c

is a 3×1 vector and c

is a

3×3 matrix. Given N correspondences, we have N

equations for each axis (

X , Y and Z ):







ϕ(x

) ··· ϕ(x

) 1 x

ϕ(x

) · · · ϕ(x

) 1 x







{z }













{z }













{z }

(6)

where c

, c

and c

denote the k

row of c

, c

and c

, respectively. y

denotes the k

row of y

k can be 1,2 or 3, corresponding to the

X , Y , and Z

axes. Therefore we have 3N equations in all:

P=[P

]

′

,c=[c

]

′

,y=[y

]

′

(7)

P is a 3N×(N+4) matrix. In order to ensure a smooth

interpolation function, we add the additional orthogo-

nality constraints

∑

1,i

= 0 (Carr et al., 2001) to

Eq. 6, where c

1,i

denotes the i

column of c



··· x

4×4



{z }

· c =



4×1



{z }

(8)

The least-squares solution for this linear system,

Qc = w, is given by c = (Q

−1

Finally, the correspondence of a vertex x

in the

high-resolution mesh X

can be computed by using

Eq. 5: y

= g(x

), for j = 1, · ·· , M (M is the number

of vertices in mesh X

). We also apply the local non-

rigid registration for y

as described in Sec. 2.3.

3 APPLICATION: FEMUR ATLAS

CONSTRUCTION

Statistical anatomical atlases are one of powerful

analysis tool for 2D and 3D medical images (Cootes

et al., 1995; Cootes and Taylor, 2001). Due to the

anatomical variance between subjects, construction

of statistical anatomical atlases usually requires non-

rigid registrations between individual models. As an

application, we apply our registration results to build

a statistical atlas for an anatomical structure. A rigid

pose alignment has been applied to eliminate the ef-

fect of imaging pose (Goryn and Hein, 1995) before

atlas construction.

Suppose we have K aligned meshes and each

mesh can be represented by a 3M×1 vector v

(i =

1,·· · ,K), where M is the number of points in each

mesh and 3 denotes three coordinates

X , Y , and Z .

We compute the mean vector c =

∑

v and covari-

ance matrix Ψ=

M−1

−c,· ··,v

−c]·[v

−c,· ··,v

−c]

′

and then apply PCA: Ψ = UΛU

′

Therefore, any mesh vector in the data set can be

represented by a mean vector plus a linear combina-

tion of each principal components (each column of

U):

= c+ Uη

(9)

where η

is a K×1 coefﬁcient vector obtained by

project v

onto each principal axis. New models, not

included in the data set, can be generated by manipu-

lating the elements of η

4 EXPERIMENTAL RESULTS

87 CT-scanned 87 patients with healthy femur: 53

males and 34 females; 43 left femurs and 44 right.

The patients age from 39 to 78 and their femur height

ranges from 400mm to 540mm. The CT volumes are

segmented to provide triangulated surface models us-

ing Marching Cube algorithm. All surface models are

smoothed by the method in (Desbrun et al., 1999).

Each femur data includes two mesh surfaces: femoral

head and distal femur.

Fig. 3 shows two high-resolution mesh X

(21130

vertices, 42256 triangles, 65.84mm in z-axis) and Y

(26652 vertices, 53300 triangles, 105.89mm in z-axis)

for distal femur (Patient X is a 79 years old female,

with 472.55mm height femur; Patient Y is a 53 years

old female, with 477.59mm height femur). We com-

pute point-to-surface distance from X

to Y

(Aspert

et al., 2002):

d(p,Y

) = min

′

∈Y

kp− p

′

, p ∈ X

(10)

where k · k

is Euclidean norm. The HSV

color of each vertex in mesh X

denotes the

distance d(p,Y

). We also compute the mean

error d

) and root mean square error

RMS

) between mesh X

and Y

) =

p∈X

d(p,Y

)dX

RMS

) =

p∈X

d(p,Y

)

(11)

With respect to the bounding box diagonal of mesh

(158.48mm), the mean error is 6.49% and root

mean square error is 7.70%. Fig. 4 shows the low-

resolution mesh X

(169 vertices, 334 triangles) and

(213 vertices, 422 triangles) after simpliﬁcation.

With respect to the bounding box diagonal of mesh

(158.28mm), the mean error is 6.53% and root

mean square error is 7.74%. Fig. 5 shows the de-

formed low-resolution mesh X

L(1)

and Y

after ap-

plying Chui and Rangarajan’s non-rigid registration.

With respect to the bounding box diagonal of mesh

(158.28mm), the mean error is 1.68% and root

mean square error is 2.13%. Surface distance has

been signiﬁcantly decreased after Chui and Rangara-

jan’s non-rigid registration. Fig. 6 shows the de-

formed low-resolution mesh X

L(2)

and Y

after ap-

plying a local deformation discussed in Sec. 2.3.

Mesh X

Mesh Y

High-Resolution

Surface Distance from X

to Y

Figure 3: Input high-resolution meshes.

Low-Resolution

Mesh X

Mesh Y

Surface Distance from X

to Y

Figure 4: After simpliﬁcation.

Low-Resolution

Deformed Mesh X

L(1)

Mesh Y

Surface Distance from X

L(1)

to Y

Figure 5: After low-resolution registra-

tion.

Low-Resolution

Deformed Mesh X

L(2)

Mesh Y

Surface Distance from X

L(2)

to Y

Figure 6: After local deformation.

Mesh Y

High-Resolution

Deformed Mesh X

H(1)

Surface Distance from X

H(1)

to Y

Figure 7: After interpolation.

Mesh Y

High-Resolution

Deformed Mesh X

H(2)

Surface Distance from X

H(2)

to Y

Figure 8: After local deformation.

With respect to the bounding box diagonal of mesh

(158.28mm), the mean error is 0.68% and root

mean square error is 1.42%, which shows local point-

to-surface registration can decrease the surface dis-

tance further. Fig. 7 shows the interpolated high-

resolution mesh X

H(1)

and Y

after applying an inter-

polation. With respect to the bounding box diagonal

of meshY

(158.48mm), the mean error is 1.65% and

root mean square error is 2.10%. The reason why the

surface distance slightly increases after interpolation

is: only 0.80% of vertices in mesh X

H(1)

have cor-

respondences obtained by non-rigid registration, oth-

ers obtain correspondences by interpolation. Fig. 8

shows the deformed high-resolution mesh X

H(2)

and

after applying a local deformation discussed in

Sec. 2.3. With respect to the bounding box diagonal

of meshY

(158.48mm), the mean error is 0.28% and

root mean square error is 1.26%, which once again

demonstrates that local point-to-surface registration is

helpful for decreasing the surface distance.

The critical parameter in our algorithm is the num-

ber of vertices N used in the low-resolution meshes,

which affects the computational complexity and ac-

curacy. Due to the different length of femur shaft

within different meshes, we choose a mesh as a ref-

erence such that we could compare any other mesh

with this reference, following the procedure showed

in Fig. 3-8. By tuning the number N

ref

, we can

make a trade off between accuracy and efﬁciency

(in order to maintain the same points density, we set

other

= N

ref

×height

other

/height

ref

Moreover, in the application of atlas construction,

we can choose N

ref

by comparing the reconstructed

mesh from atlas with the original mesh, such that we

can also ensure the generality and accuracy of the

atlas. We conduct a series of leave-one-out experi-

ments. We ﬁrst select a reference mesh(which has

all correspondences in any other meshes in the data

set) and then change N

ref

. For each N

ref

, we com-

pute N

other

(other = 1,··· , K, other 6= ref) for other

meshes and apply the two-level non-rigid registration.

After that, with K aligned meshes we apply K leave-

one-out experiments: for any v

(i = 1,·· · ,K), we use

other K − 1 to construct an atlas using PCA (Sec. 3).

Let U

denote the ﬁrst S columns of the principal

component matrix U

, which consists of 95% energy.

Then v

(i = 1, · ·· , K) can be reconstructed by this at-

las:

= c+ U

− c) (12)

We compare the surface distance between each pair of

mesh V

and

and obtain the average mean error and

(mm)

Figure 9: Reconstruction error d

and

RMS

in terms of N

ref

(minutes)

Figure 10: Registration processing time

T in terms of N

ref

Number of Principal Components

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85

Femoral head atlas Distal femur atlas Combined femur atlas

Energy

Figure 11: Atlases of distal femur(pink),

femoral head(blue) and entire fe-

mur(green).

root mean square error for each N

ref

RMS

) (13)

By changing N

ref

we can compute different

and

, also processing time T

of two-level registration.

Fig. 9 shows when N

ref

≥ 290,

will be less than

1mm, which is a practical number in clinical applica-

tion. Fig. 10 shows when N

ref

≥ 330, average pro-

cessing time of registration of two meshes will ex-

ceed 5 mins (2.4GHz Pentium PC with 1G RAM).

Compared with original Chui and Rangranjan’s algo-

rithm, ours only need 5 mins for registering any size

of meshes (interpolation costs less than 1sec when the

number of vertices is less than 200,000), however,

their method costs 5 mins for 350 vertices, 10 mins

for 460 vertices, 20 mins for 610 vertices, etc. Our al-

gorithm signiﬁcantly improve the efﬁciency without

losing accuracy.

Finally we select N

ref

= 290 and construct atlases

for distal femur, femoral head, and entire femur, re-

spectively. Fig. 11 shows the energy in terms of prin-

cipal components in each atlas.

5 CONCLUSION

In this paper we have developed a two-level algorithm

to tackle the registration problem due to the geo-

metrical and computational complexity. Experiments

demonstrate our algorithm signiﬁcantly improve the

efﬁciency without decreasing accuracy, by comparing

with the original Chui and Rangranjan’s algorithm.

Some interesting issues such as gender effect for the

atlases, interpolation with other kernel functions, etc

will be addressed in the future work.

REFERENCES

Aspert, N., Santa-Cruz, D., and Ebrahimi, T. (2002). Mesh:

measuring errors between surfaces using the hausdorff

distance. In Proc. ICME’02, volume 1, pages 705–

708.

Besl, P. and McKay, N. (1992). A method for registration of

3-d shapes. In IEEE Trans. on PAMI’92, volume 14,

pages 239–256.

Carr, J. C., Beatson, R. K., Cherrie, J. B., Mitchell, T. J.,

Fright, W. R., Mccallum, B. C., and Evans, T. R.

(2001). Reconstruction and representation of 3d

objects with radial basis functions. In Proc. SIG-

GRAPH’01, pages 67–76.

Carr, J. C., Fright, W. R., and Beatson, R. K. (1997). Sur-

face interpolation with radial basis functions for med-

ical imaging. In IEEE Trans. on Medical Imaging,

volume 16, pages 96–107.

Chui, H. and Rangarajan, A. (2003). A new point match-

ing algorithm for non-rigid registration. In Computer

Vision and Image Understanding, volume 89, pages

114–141.

Chui, H., Rangarajan, A., Zhang, J., and Leonard, C. M.

(2004). Unsupervised learning of an atlas from un-

labeled point-sets. In IEEE Trans. on PAMI’04, vol-

ume 2, pages 160–172.

Cootes, T. and Taylor, C. (2001). Statistical models of ap-

pearance for medical image analysis and computer vi-

sion. In Proc. SPIE Image Processing, volume 4322,

pages 238–248.

Cootes, T., Taylor, C., Cooper, D., and Graham, J. (1995).

Active shape models: their training and application.

In Computer Vision and Image Understanding, vol-

ume 61, pages 38–59.

Desbrun, M., Meyer, M., Schroder, P., and Barr, A. (1999).

Implicit fairing of irregular meshes using diffusion

and curvature ﬂow. In Proc. SIGGRAPH’99, pages

317–324.

Ellingsen, L. M. and Prince, J. L. (2006). Deformable regis-

tration of ct pelvis images using mjolnir. InProc. SPIE

Medical Imaging Conference, volume 6144, pages

329–337.

Farin, G. (1993). Curves and surfaces for cagd. In Academic

Press. New York.

Fleute, M., Lavall

ee, S., and Desbat, L. (2002). Integrated

approach for matching statistical shape models with

intra-operative 2d and 3d data. In Proc. MICCAI’02,

volume 2, pages 364–372.

Garland, M. and Heckbert, P. (1997). Surface simpliﬁcation

using quadric error metrics. In Proc. SIGGRAPH’97,

pages 209–216.

Glaunes, J., Trouv

e, A., and Younes, L. (2004). Diffeo-

morphic matching of distributions: a new approach

for unlabelled point-sets and sub-manifolds matching.

In Proc. CVPR’04, volume 2, pages 712–718.

Goryn, D. and Hein, S. (1995). On the estimation of rigid

body rotation from noisy data. In IEEE Trans. on

PAMI’95, volume 17, pages 1219–1220.

Hill, D., Batchelor, P., Holden, M., and Hawkes, D. (2001).

Medical image registration. In Physics in Medicine

and Biology, volume 46, pages 1–45.

Jaume, S., Macq, B., and Warﬁeld, S. K. (2002). Labeling

the brain surface using a deformable multiresolution

mesh. In Proc. MICCAI’02, pages 451–458.

Kim, J. and Fessler, J. A. (2004). Intensity-based image reg-

istration using robust correlation coefﬁcients. In IEEE

Trans. on PAMI’04, volume 23, pages 1430–1444.

Maurer, C. R., Fitzpatrick, J. M., Wang, M. Y., Galloway,

R. L., Maciunas, R. J., and Allen, G. G. (1997). Regis-

tration of head volume images using implatable ﬁdu-

cial markers. In IEEE Trans. on PAMI’97, volume 16,

pages 447–462.

Papademetris, X., Jackowski, A., Schultz, R., Staib, L., and

Duncan, J. (2003). Non-rigid brain registration using

extended robust point matching for composite multi-

subject fmri analysis. In Proc. MICCAI’03, volume

2879, pages 788–795.

Park, J., Mataxas, D., Young, A., and Axel, L. (1996). De-

formable models with parameter functions for cardiac

motion analysis from tagged mri data. In IEEE Trans.

on Medical Imaging, volume 15, pages 278–289.

Pighin, F., Hecker, J., Lischnski, D., Szeliski, R., and

Salesin, D. (1998). Synthesizing realistic facial ex-

pressions from photographs. In Proc. SIGGRAPH’98,

pages 75–84.

Rohlﬁng, T., Russakoff, D., and Maurer, C. (2004).

Performance-based classiﬁer combination in atlas-

based image segmentation using expectation-

maximization parameter estimation. In IEEE Trans.

on Medical Imaging, volume 23, pages 983–994.

Rueckert, D., Frangi, A., and Schnabel, J. (2003). Auto-

matic construction of 3-d statistical deformation mod-

els of the brain using nonrigid registration. In IEEE

Trans. on Medical Imaging, volume 22, pages 1014–

1025.

Shen, D. Davatzikos, C. (2002). Hammer: hierarchical at-

tribute matching mechanism for elastic registration. In

IEEE Trans. on Medical Imaging, volume 21, pages

1421–1439.

Szeliski, R. and Lavallee, S. (1996). Matching 3-d anatomi-

cal surfaces with non-rigid deformations using octree-

splines. In IJCV’96, volume 18, pages 171–186.

Wells, W. M., Viola, P., Atsumi, H., Nakajima, S., and Kiki-

nis, R. (1996). Multi-modal volume registration by

maximization of mutual information. In Medical Im-

age Analysis, volume 1, pages 35–51.

Woods, R. P., Mazziotta, J. C., and Cherry, S. R.

(1994). Mri-pet registration with automated algo-

rithm. In Journal of Computer Assisted Tomography,

volume 17, pages 536–546.

Yu, H. (2005). Automatic rigid and deformable medical

image registration. In Ph.D. Dissertation, Mechan-

ical Engineering Department, Worcester Polytechnic

Institute, Worcester, MA.