2007 IEEE International Conference on
Robotics and Automation
Roma, Italy, 10-14 April 2007

ThD8.2

3D reconstruction of a femoral shape using a parametric model andtwo 2D fluoroscopic images
Ryo Kurazume, Kaori Nakamura Toshiyuki Okada, Yoshinobu Sato, Nobuhiko Sugano,Tsuyoshi Koyama, Yumi Iwashita, and Tsutomu Hasegawa
Abstract— In medical diagnostic imaging, an X-ray CT scanner or a MRI system have been widely used to examine 3D
shapes or internal structures of living organisms or bones.
However, these apparatuses are generally very expensive andof large size. A prior arrangement is also required before
an examination, and thus, it is not suitable for an urgent
fracture diagnosis in emergency treatment. This paper proposesa method to estimate a patient-specific 3D shape of a femur fromonly two fluoroscopic images using a parametric femoral model.
Firstly, we develop a parametric femoral model by statisticalanalysis of a number of 3D femoral shapes created from CT
images of 51 patients. Then, the pose and shape parameters of
the parametric model are estimated from two 2D fluoroscopic
images using a distance map constructed by the Level Set
Method. Experiments using synthesized images and fluoroscopic
images of a phantom femur are successfully carried out andthe usefulness of the proposed method is verified.

I. INTRODUCTION
In medical diagnostic imaging, an X-ray CT (Computed
Tomography) scanner or a MRI (Magnetic Resonance Imaging) system have been popular apparatuses to examine 3D
shapes or internal structures of living organisms or bones.
However, these apparatuses are generally very expensive andof large size, and thus, they are usually installed in large
medical institutions rather than small clinics in town. A
prior arrangement is also required before an examination,and thus, it is not suitable for an urgent fracture diagnosis
in emergency treatment.
On the other hand, X-ray/fluoroscopy has been widelyused as traditional medical diagnosis. Recently digital fluoroscopy has been developed and widely used in many
hospitals. The cost of this fluoroscopic inspection system
is much lower than CT or MRI systems and the system
can be dealt with more conveniently. Furthermore, the riskof radiation exposure is also lower than the CT inspection
system.
From the above consideration, if it can realize to reconstruct precise 3D shapes of living organisms or bones
from few conventional 2D fluoroscopic images, it might
be very useful in practice in views of cost, labor, andR. Kurazume and T. Hasegawa are with Graduate Faculty of Information
Science and Electrical Engineering, Kyushu University, 744 Motooka, Nishiku, Fukuoka, Japan kurazume@is.kyushu-u.ac.jp
K. Nakamura and Y. Iwashita is with Graduate School of Information
Science and Electrical Engineering, Kyushu University, 744 Motooka, Nishiku, Fukuoka, 819-0395, JapanT. Okada is with Graduate School of Information Science and Technology,
Osaka University, 1-3, Machikaneyama-cho, Toyonaka-shi, 560-8531, Japan
Y. Sato, N. Sugano, and T. Koyama are with Graduate School of
Medicine, Osaka University, 2-2 Yamadaoka, Suita-shi, 565-0871, Japan

1-4244-0602-1/07/$20.00 ©2007 IEEE.

radiation exposure. Especially, there is a strong demand fromsurgeons that 3D computer aided surgery without laborious
CT imaging should be offered for some simple surgeries
such as artificial joint replacement or fracture treatment.
They have desired a 3D diagnostic system using favorite 2D
fluoroscopic images.
However, 3D shape reconstruction from a 2D image is afundamentally ill-posed problem, and thus a plenty of images
must be given or some constraint conditions for the 3D
shape must be determined. But the shapes of bones havetheir inherent and universal patterns, and thus by modelingsuch inherent patterns, 3D shape reconstruction from few 2D
images becomes possible.
This paper presents a method to estimate a patient-specific
3D shape of a femur from only two fluoroscopic images. This
technique utilizes a parametric femoral model constructed bystatistical analysis of a number of 3D femoral shapes createdfrom by CT images of 51 patients. Then, the pose and shapeparameters of the parametric model are estimated from two
2D fluoroscopic images using a distance map constructed by
the Level Set Method. Experiments using synthesized images
and fluoroscopic images of a phantom femur are successfully
carried out and the usefulness of the proposed method isverified.
II. R ELATED WORKS
2D/3D registration problem is well established in image processing, especially for texture mapping in Computer Graphics or Augmented Reality. For a rigid object,
1) feature-based technique [1],[2],[3], 2) image-based technique using 3D texture, reflectance, brightness, and shading [4],[5],[6], 3) silhouette-based technique [7],[8],[9],[10],have been proposed so far. Especially in surgical navigation system, DRRs (Digitally Reconstructed Radiographs)[22],[23] are widely used in 2D/3D registration for the
fluoroscopy-guided surgery.
On the other hand, in 2D/3D registration of a nonrigid object such as soft tissues in medical imaging,similarity measure [11], affine [12], geometric hashing[13], quadric/superquadric [14], and displacement-fieldbased transformation [15] have been proposed and tested. In
addition, the 3D shape estimation of a parameterized object
has also been studied such as the shape reconstruction ofmathematical plaster models with unknown parameters usinga laser range finder [16], or the comparison of multiple crosssection images of a 3D model and a 3D parametric model[17]. However, these studies assumes the use of a plenty of3002

ThD8.2

VHc
AP
AN
PN c
VGt
VLt
VLp
LR
LV

1

Regioncenter of femoral headprincipal axis of femur
femoral neck axis
cross section of neck center
apex of greater trochanter
lesser trochanter
iliofemoral ligament attachment
ridgeline of greater trochanter
valley of greater trochanter

0.8
Contributions

Features
Point
Line
Line
Surface
Surface
Point
Point
Curve
Curve

AP

AP

0

P Nc
LV

VL

AN

VHc
P Nc

VLt

VGt

(a) Front
Fig. 1.

0

Fig. 2.

LR
VHc

0.4
0.2

TABLE IL IST OF ANATOMICAL FEATURES OF FEMURAN

0.6(b) Back

Anatomical features of femur

images or a precise 3D shape taken by a laser range finder,and only a few studies of 3D non-rigid shape reconstruction
from only two or few 2D images have been proposed so far[29],[27],[28].
III. 3D PARAMETRIC FEMORAL MODEL
A. Construction of parametric femoral model
We utilize a statistical shape model of a femur proposedby Okada [18]. In this technique, a number of 3D femoral
shapes created from CT images is analyzed statistically, andthe parametric femoral model [19] which consists of an
average 3D shape and several shape parameters is created.
With this parametric femoral model, a general 3D shape ofa femur is expressed with the average shape and batch ofshape parameters.
The concrete procedure for creating a parametric 3D
femoral model is as follows:1.
Anatomical features of a femur are determined
automatically or by hand as shown in Table I and
Fig. 1.
2.
According to the extracted anatomical features,the femur is divided by 4 regions (femoral head,
femoral neck, greater trochanter, and femoral shaft).
Spherical and cylindrical coordinate systems aredefined in each region.
3.
In each coordinate system, intersection points oflines which distribute uniformly in the coordinate
system and the bone surface are defined as thesurface points. 3D coordinates of the surface points
are extracted and unique labels are assigned.
4.
Steps 1 to 3 are repeated for a plenty numberof samples and an average shape is calculated.
Next, for each sample, the surface points with

10
20
30
40
Number of parameters

50

Contributions of parametric modelsame labels are compared with the average shapeand displacement vectors are calculated. Then, theprincipal component analysis (PCA) is applied tothe matrix consisting of displacement vectors andseveral shape parameters are extracted as principal
components and vectors.
The parametric femoral model used in the following
experiments was created using CT images of 51 patients.
By applying PCA to 51 samples of 3D femoral shapes,
we extracted the most significant 50 principal components(p1 , p2 , . . . , p50 ), standard deviation (σ1 , σ2 , . . . , σ50 ), andcorresponding principal vectors (v1 , v2 , . . . , v50 ). With theobtained parametric femoral model, the general 3D shape ofa femur is expressed asx= x + (p1 · σ1 · v1 ) + (p2 · σ2 · v2 ) + · · ·(1)where x is the surface point of the average shape and x isthe surface point of the general shape. Therefore, the general
3D shape of a femur is expressed by the parametric femoral
model with• average 3D shape and several principal vectors (predetermined)• several (up to 50) shape parameters (estimated)
Figure 2 shows the contribution ratio of the shape parameters
for the statistical femoral model.
IV. R ECONSTRUCTION OF 3D FEMORAL SHAPE FROMTWO 2D FLUOROSCOPIC IMAGESIn this section, we introduce the 2D/3D registration algorithm and the estimation procedure of the optimum shapeparameters using two fluoroscopic images.
This 2D/3D registration algorithm utilizes the contour
lines of the silhouette of the 2D image and the projected
contour lines of the 3D model. The optimum pose of the 3Dmodel is determined so that both contour lines coincide eachother on the 2D image plane. In popular approaches such as
ICP algorithm, the error metric is usually defined as the sumof the distances between the points on the 2D contour lines
and their nearest points on the projected contour lines of the
3D model. However, the nearest point search is a laborious
task and time consuming even for the kd tree-based algorithm[25].
In our approach, the 2D distance map [7] is utilized.
Firstly, the 2D distance map from the contour lines is createdon the 2D image using the Fast Marching Method [21] or

3003

ThD8.2
raster scan algorithms [24]. Once the 2D distance map iscreated, the error metric is obtained directly from the 2D
distance map as a value at the points on the projected contour
lines of the 3D model. By taking the course to fine strategy
named the “Distance Band” [7], the 2D distance map can beconstructed quite rapidly using the Fast Marching Method.
In case that 2D/3D registration and estimation of the shapeparameters are performed at the same time, the depth fromthe view point and the scale of the 3D model cannot be
distinguished. Therefore, the proposed algorithm utilizes two
fluoroscopic images taken from two viewpoints at different positions. In addition, we assume that the 3D femoral
parametric model is constituted by a large number of small
triangle patches with almost same size.

+20 pixels+40 pixels

Boundary

Contour line in distance map
Fig. 3.

After step 4 of the above procedure, optimum shapeparameters are estimated as follows:

A. Registration of 2D fluoroscopic images and 3D parametric model
The brief registration procedure of the 2D fluoroscopic
images and the 3D parametric model is as follows:1.
Extract contour lines of the femur in the fluoroscopic images using active contour model such as
snakes or Level Set Method [20].
2.
Construct 2D distance map from the extracted
contour lines using the Fast Marching Method [21].
Figure 3 shows an example of the constructed 2D
distance map of a femoral image.
3.
Place the parametric femoral model at an arbitrary
position and calculate the 2D projection image ofthe 3D model.
4.
Extract contour lines of the projected image andcorresponding 3D patches of the 3D model. This
procedure can be executed by the OpenGL hardware accelerator quite rapidly.
5(a). Apply the force which is calculated from the 2D
distance map at the projected contour points directly to the corresponding 3D patch. Details areshown in the following section D.
6(a). Using the robust M-estimator, which is one of therobust estimation techniques, the total force andmoment around the center of gravity is calculated.
7(a). The above procedure from steps 3 to 6(a) is repeated for all the images taken from the different
view points and the total force and moment arecalculated.
8(a). Update the pose of the 3D parametric model according to the total force and moment.
9(a). Repeat from steps 3 to 7(a) until the magnitude of
the total force and moment becomes less than the
pre-defined threshold value.

2D distance map for femoral image

5(b). Calculate the error E which is defined as the sumof the value of the 2D distance map at the projected
contour line of the 3D parametric model.
6(b). Find the optimum shape parameters which minimizes the error E at the current pose using the
conjugate gradient method.
7(b). Reconstruct the 3D shape according the obtained
shape parameters using Eq.(1).
8(b). Repeat from steps 3 to Step 7(b) until the error E
becomes less than the pre-defined threshold value.
C. Fast extraction of the projected contour line of the 3Dparametric femoral model
The contour detection and identifying triangular patcheson the 3D model corresponding to points on the contour
line are computationally expensive and time consuming.
In our implementation, we utilize the high-speed renderingfunction of the OpenGL hardware accelerator and thus theseprocedures are executed quite rapidly.
The detailed algorithm is as follows: Initially, we assign
different colors for all the triangular patches in the 3D modeland draw the projected image of the 3D model on the image
buffer using the OpenGL hardware accelerator. The contour
points of the 3D model are detected by raster scanning of
the image buffer. By reading colors of the detected contour
points, we can identify the corresponding triangular patcheson the 3D geometric model.
D. 2D/3D registration using the robust M-estimator
After obtaining the distance map on the 2D fluoroscopic
image and the list of the triangular patches of the 3D model
corresponding to the contour points, the force fi is applied
to all the triangular patches of the contour points (Figs.4 and5) as explained in Step 5(a).

B. Estimation of the shape parameters
Estimation procedure of the optimum shape parameters of
the 3D parametric femoral model is shown in this section.
This procedure also uses the 2D distance map from thecontour line of the femur in the fluoroscopic image, which
has already been constructed as the above section.

fi = Di∇Di|∇Di |(2)

where Di is the value of the distance map at the contour
point, which corresponds to the triangular patch i, and ∇Diis the gradient of Di .

3004

ThD8.2
BoundaryApfiFig. 6.
Fig. 4.

Directions of fluoroscopic images

Force fi is applied to 3D triangular patch i on contour line

fi
ri
M

fi

ObjectF

Image planefiProjected
contour pointy(s, t)
Steepest descentdirectionx

Fig. 7.

Side view

Reconstructed fluoroscopic imageszvHere, we define the weight function w(z) as the following
equation in order to evaluate the error term.

Focul point(a)Fig. 5.

Front view(b)

Total force and moment around center of gravity

In step 6(a), the total force and moment around the centerof gravity is calculated with the following equations.
ψ(fi )
(3)
F =
i

M=



ψ(ri × fi )(4)iwhere ri is a vector from the COG to the triangular patch
i and ψ(z) is a particular estimate function. In practicalscenario, the contour of the femur is occasionally occluded
or blurred, or the 2D image is corrupted by noise. In thesecases, the obtained boundary does not coincide with the
projected contour of the 3D model and the correct distancevalue cannot be obtained. To deal with this problem, we
introduces the robust M-estimator to ignore contour points
with a large amount of errors.
Let’s consider the force fi and the moment ri × fi as anerror zi and the sum of the error as
E(P ) =
ρ(zi )(5)i

where P is the pose of the 3D parametric model and ρ(z)is a particular estimate function which is defined as∂ρ(z)= ψ(z)∂z(6)

The pose P which minimizes the error E(P ) is obtained as
the following equation.
 ∂ρ(zi ) ∂zi
∂E
=
=0
∂P
∂zi ∂Pi

1 ∂ρ(z)1
ψ(z) =(8)z
z ∂z
From the above equation, we obtain the following weighted
least squares method.

∂E
∂zi
=
=0
(9)
w(zi )zi∂P∂Piw(z) =(7)

In our implementation, the optimum pose which minimizes
the error E(P ) is obtained by the steepest gradient method
as shown in step 9(a).
V. E XPERIMENTS
A. Simulation using DRRs
Firstly we conducted the experiments using DRRs (Digitally Reconstructed Radiographs) to evaluate the fundamental
performance of the proposed method. In the experiment, theestimation accuracy for 10 femoral models is examined using
two reconstructed fluoroscopic images. Among 10 models,
5 models (modeldata1 ∼ 5) are used for the construction of
the 3D parametric model and 5 models (testdata1 ∼ 5) are
not used.
We determine the directions of the fluoroscopic images
as shown in Fig.6 considering the possible direction in
actual radiographic examination. In this condition, two view
directions meet at right angles at the main axis of the femur
AP . Examples of the reconstructed fluoroscopic images areshown in Fig.7.
All of the 3D femoral shape used in the experiments werereconstructed precisely by the CT scanner beforehand, andthe optimized shape parameters which minimizes distanceerrors between surface points were determined by comparing
the 3D actual shape and the 3D parametric model andsearching all possible candidates (ground truth).
Firstly, we chose up to 10 principle components andestimated the pose and the optimum shape parameters of3005

ThD8.2

B. Experiments using phantom femur
We conducted the experiments using a dry bone of a femur
and fluoroscopic images. In the experiments, the special
fluoroscopic imaging apparatus (Siemens, Siremobil ISOC) was used for the fluoroscopic photography from various
directions around the phantom femur.
Firstly, we captured images of calibration markers with
9 glass bubbles (left of Fig.11) at 50 positions around(a) Average shape

Fig. 8.

(b) Actual shape(c) Estimated shape

Femoral model used for shape parameter estimation

2.5
Average error [mm]the femur on the fluoroscopic images. In this experiment,
the pose estimation of the femur and the optimum parameter
estimation were repeated alternatively and independently. An
example of the experimental results for testdata 4 is shownin Fig.8 which illustrates the average shape, the actual shapeand the estimated shape, respectively.
Figure 9 indicates the average error between the estimatedshape and the actual shape. The average error is defined as
the average of the minimum distance from the surface pointof the estimated shape to the triangle patches of the actual
shape. In this figure, “A” on the horizontal axis indicates theaverage error at the initial pose and initial shape parameters(all the parameters are set to “0”) before registration, and“0” indicates the average error when the pose is estimatedbut all the shape parameters are fixed to initial values. This
figure shows that the average error gradually decreases as
the number of the estimated shape parameters increases.
However, the average error converges when the number ofthe shape parameters is around 5 and no significant difference
is observed even if the number of the shape parameters
increases.
In addition, Fig.10 shows kinds of errors defined asfollows in case that the number of the estimated shapeparameters is 5. Figure II also indicates the average of theerror, the standard deviation, maximum value, and minimum
value for 10 models.
Average error 1
The average error between the average model andthe actual shape at the initial pose
Average error 2
The average error between the 3D optimized estimated shape and the 3D actual shape by comparing
the actual shape and the parametric model (groundtruth).
Average error 3
The average error between the estimated shapeand the actual shape by comparing the two 2D
fluoroscopic images and the 3D parametric model(proposed method).
The experimental results show that the average error 3
between the estimated shape and the actual shape is lessthan 1.1 mm at worst and it is verified that the 3D shape
can be estimated using the two 2D fluoroscopic images
with the same accuracy in case that the 3D shapes arecompared directly. Moreover, it is confirmed that there isno significant difference between models which are used forthe construction of the parametric femoral model and not
used.

1.5
1
0.5
0

Fig. 9.

testdata 1~5
modeldata 1~5

2

0

2
4
6
8
Number of estimated parameters

10

Average error for numbers of estimated shape parameters

the markers from 0 to 190 degrees using the fluoroscopic
apparatus. 3D positions of the markers were also measured
by the CT scanner precisely. Next, the intrinsic and extrinsic
parameters of the fluoroscopic apparatus were calibrated by
the Tsai’s method [26].
After the calibration, we replaced the markers with the
dry bone of a femur and captured 50 images at the same
positions. In addition, the precise 3D shape of the dry bone
was measured by the CT scanner. Next, we chose two
fluoroscopic images from 50 images as mentioned below andestimated the pose and the optimum parameters in fluoroscopic images using the propose techniques. The examplesof the fluoroscopic images are shown in Fig.12.
Figure 13 shows one example of the pair (No.4 and 24)of the fluoroscopic images, which were captured from thedirections crossing at right angles. The average errors of
the estimated femoral shape are shown in Fig.14 and Table
III for the various number of shape parameters used forthe estimation. The estimation process and the estimated
3D shape in case that the number of the estimated shapeparameters is 10 are shown in Figs.15 and 16. The calculation
time is about 1 minute by Pentium IV, 3.2GHz, which
includes the contour detection by the Level Set Method andthe shape parameter estimation.
Finally, the average errors for various pairs of the fluoroscopic images are shown in Fig.17 in case that the numberof the estimated shape parameters is 10. As the resultsof a series of experiments using the phantom femur, weconcluded that the 3D shape can be estimated with theaverage error of less than 1.2 mm if we choose the proper
images captured from the directions crossing at about rightangles.

3006

ThD8.2

No.5

No.4

No.1
3

No.1
3

2
1
0

2
1
0

No.2 No.5

No.3

testdata

average error1

No.4

Femoral dry bone
No.2

modeldata

average error2

Caribration markers

No.3

average error3

Fig. 10.
Average error after shape parameter estimation (number ofprinciple component is 5)

Average error 1
Average error 2
Average error 3(testdata)(modeldata)

average
1.69
0.90

STD.
0.54
0.13

maximum
2.52
1.06

minimum
0.90
0.60

0.91
0.81

0.15
0.07

1.08
0.90

0.66
0.71

Fig. 11.

TABLE IIC OMPARISON OF AVERAGE ERRORS ( MM )Fluoroscopic images of calibration markers and phantom femur

No.1
0 deg.

VI. C ONCLUSIONSWe proposed a method to estimate a 3D shape of patient’s
femur from only two fluoroscopic images using a parametricfemoral model. Though a precise 3D shape of a femur isusually measured by a CT scanner or a MRI system, ourmethod enables to estimate a precise 3D shape with only
two fluoroscopic images taken by a low cost fluoroscopic
inspection apparatus. Thus the cost of the inspection system
can be dramatically reduced and the 3D image-based medical
diagnosis becomes available even in small clinics.
The experimental results show that the average error
between the estimated shape and the actual shape is less than
1.1 mm at worst, and it is verified that the 3D shape can be
estimated using the two 2D fluoroscopic images taken fromthe different view points with the same accuracy in case thatthe 3D shapes are compared directly.
Our future works includes study of the optimum conditions such as the optimum number and directions of thefluoroscopic images, and clinical experiments in fluoroscopic
image diagnosis.
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No.4
Fig. 13.
Number ofparameters
0
1
2
3
4
5
6
7
8
9
10

Average error [mm]

2.5

No.24

Fluoroscopic images (No.4 and No.24)

1.5
1

STD.
1.497
1.431
1.153
0.865
0.936
0.886
0.866
0.807
0.784
0.838
0.812

max.
7.269
6.801
5.151
3.973
4.476
4.316
4.130
3.980
3.963
4.240
3.952

min.
0.00250.00010.00110.00100.00070.00130.00100.00100.00100.00030.0013Use of two radiographs(Average error 3, Proposed method)

0.5
0average
1.973
1.943
1.460
1.039
1.085
1.024
1.017
0.989
0.984
1.041
0.992

Estimated by 3D shapes(Average error 2)

2

Fig. 14.

0

2

4
6
8
Number of parameters

Estimation errors for numbers of parameters

Contour line in radiograph

3D paremetric model

Initial shape/pose

TABLE IIIAVERAGE OF ESTIMATION ERRORS OF PHANTOM FEMUR [ MM ]Fig. 15.

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10

After convergence

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3008

Average shape

Estimated shape

Fig. 16.

Actual shape

Estimated shapes of femur
[mm]
4

14

24

34

44

1.246

0.992

1.128

1.377

0.980

1.050

1.223

1.143

1.091

Image No.
4 (∼ 15 deg.)

Fig. 17.

14 (∼ 53 deg.)-

24 (∼ 91 deg.)--

34 (∼ 129 deg.)

-

-

-

44 (∼ 167 deg.)

-

-

-

1.504
-

Estimation errors for various pairs of fluoroscopic images