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A Biplanar Reconstruction Method Based on 2D and 3D
Contours: Application to the Distal Femur
S. LAPORTE
a,
*, W. SKALLI
a
, J.A. DE GUISE
b
, F. LAVASTE
a
and D. MITTON
a,†
a
Laboratoire de BioMe
´
canique—ENSAM CER de Paris, 151, Boulevard de l’Ho
ˆ
pital, 75013, Paris, France
b
Laboratoire de recherche en Imagerie et Orthope
´
die, ETS/CRCHUM, Montre
´
al, Canada
(Received 18 March 2002; In final form 11 November 2002)
A three-dimensional (3D) reconstruction algorithm based on contours identification from biplanar
radiographs is presented. It requires, as technical prerequisites, a method to calibrate the biplanar
radiographic environment and a surface generic object (anatomic atlas model) representing the
structure to be reconstructed.
The reconstruction steps consist of: the definition of anatomical regions, the identification of 2D
contours associated to these regions, the calculation of 3D contours and projection onto the radiographs,
the associations between points of the X-rays contours and points of the projected 3D contours, the
optimization of the initial solution and the optimized object deformation to minimize the distance
between X-rays contours and projected 3D contours.
The evaluation was performed on 8 distal femurs comparing the 3D models obtained to CT-scan
reconstructions. Mean error for each distal femur was 1 mm.
Keywords: 3D contour reconstruction; Stereoradiography; Personalized modeling; Femur; Anatomic
atlas model; Deformable surfaces
INTRODUCTION
Accurate geometrical three-dimensional (3D) reconstruc-
tion of human bones is currently required for clinical
studies as well as for computer assisted surgery or
personalized biomechanical finite element models
[8,14,20,22]. CT-scan with millimeter slices is commonly
used to obtain these accurate reconstructions [6,19,23].
However, this method is quite irradiant for the patient and
necessitates a large volume of image data sets, typically,
more than 200 transverse images for the distal femur.
Methods based on MRI are also used especially to
calculate cartilage volume or to study articular surfaces
[3,18]. Existing reconstruction methods for stereoradio-
graphy require the identification of specific anatomical
landmarks projection from the radiographs. These
methods are based on several algorithms like the Direct
linear transformation (DLT), dealing with the reconstruc-
tion of points visible on each X-ray image (stereo-
corresponding points) [1,5], or more recently like NSCP,
used for points visible on only one radiograph (non stereo-
corresponding points) [15]. Even though these methods
are now accurate enough to allow reconstruction of
structures like vertebra [17] or pelvis [12,16], they are not
adapted to bones with a continuous shape, like knee joint,
because of the lack of specific anatomical landmark
points.
The method proposed in this paper is an improvement
of the stereoradiographic reconstruction method
especially in such cases. The principle of the method is
to associate identifiable 2D-contours from radiographs to
3D lines defined on the surface of a reference object. This
method will be called the non stereo corresponding
contour (NSCC) method. The NSSC method validation
was assessed in the distal femur specific case.
MATERIAL AND METHODS
The general principle of the method is first globally
presented and then detailed on the distal femur example.
NSCC Method General Principle
NSCC method needs two prerequisites that are only
mentioned as they are detailed elsewhere:
ISSN 1025-5842 print/ISSN 1476-8259 online q 2003 Taylor & Francis Ltd
DOI: 10.1080/1025584031000065956
*Corresponding author. Tel.: þ 33-1-44-24-63-64. Fax: þ 33-1-44-24-63-66. E-mail: sebastien.laporte@paris.ensam.fr
†
E-mail: david.mitton@paris.ensam.fr
Computer Methods in Biomechanics and Biomedical Engineering 2003
Vol. 6 (1), pp. 1–6
. Prerequisite 1: calibration of the stereoradiographic
environment. This leads to X-rays sources and films
localization in a global referential. Any algorithm can
be used, standard DLT calibration [1], self calibration
[4] or Explicit Linear Calibration (ELC) [9].
. Prerequisite 2: a generic geometrical surface object
(anatomical atlas ) of the considered bone structure.
This object can be obtained using either direct
measurement methods or CT-scan reconstruction
from dry bone specimens.
From these prerequisites the NSCC algorithm can be
subdivided into six steps:
. Preliminary step: calculation of an initial solution. This
can be obtained thanks to 2D/3D matching methods
[10,13] or existing 3D reconstructions methods (DLT
and/or NSCP algorithms)—an example will be
presented for the application to distal femur.
. Step 1: definition of anatomical regions from the generic
object. A set of points
G
i
associated to each region
G
si
is
defined.
. Step 2: 2D contours identification onto the radiographs.
Each 2D contour, associated to regions defined in step 1,
yields the creation of a set of points
L
j
(groups of regions
can be made).
. Step 3: 2D contours generation from the 3D initial
solution object. For each incidence, the 3D contours are
lines from the initial object surface where the X-rays are
tangent (Appendix 1). A set of points C
j
is thus defined.
These 3D contours are also sorted by anatomical
regions. These 3D contours C
j
are then back projected
onto their associated radiograph to define a new set of
2D points
V
j
. Each point from
V
j
is associated to a point
of C
j
.
. Step 4: association between points from
L
j
and
V
j
. This
2D association is based on point-to-point distances and
contours derivations. This enables to obtain a
correspondence between the 2D set of points
L
j
and
the 3D contours C
j
.
. Step 5: optimization of the initial solution. A rigid
deformation associated to a global homothetic coeffi-
cient is applied to the initial solution in order to reduce
the distance between
L
j
and
V
j
. The complete
optimization is obtained using step 3 to step 6 until
there is no more transformation (no translation, no
rotation and no homothetic transformation).
. Step 6: deformation of the optimized solution. A kriging
algorithm [7,21] is then applied to the optimized
solution considering control points defined as in step 3
and 4. Finally, the reconstructed object is obtained
iterating step 6 as long as the distance between
L
j
and
V
j
is superior to a given precision value.
Distal Femur Application
The NSCC technique was applied to 8 dry femurs
(5 left, 3 right) from the “Laboratoire d’anatomie des
Saints-Pe
`
res” (Paris, France). Two calibrated X-rays
films of each femur (postero-anterior (PA) and
lateral (LAT)) were acquired at the Saint-Vincent
de Paul Hospital (Paris, France) using the ELC
protocol [9].
The generic model used for reconstruction was obtained
by projection of a mesh (556 points, 1100 triangles) on the
surface of a 3D computerized CT-scan tomographic
reconstruction of a ninth dry femur (using a slices
thickness of 1 mm and a distance of 1 mm between two
slices, a pixel resolution of 0.25 mm and SliceOmatic
q
software) [2,6,17]. The initial solution object was obtained
as follows:
. A 3D diaphysal axis was created from the 2D diaphysal
axis in each X-rays film (defined by inertial properties).
. Superior points of the medial and lateral condyles were
reconstructed with the DLT and NSCP methods.
. These geometrical constructions were used in order to
get an approximation of the localization and the shape
of the reconstructed femur.
The following four anatomical regions were defined
from the generic surface (Fig. 1): the diaphysis (
G
1
), the
medial and lateral condyles (
G
2
and
G
3
) and the trochlea
and intercondylar notch region (
G
4
). The 2D contours
were extracted semi-automatically by means of active
contours [11] (Fig. 2). Three contours were obtained
from the frontal radiograph: the medial and lateral
diaphysis (
L
PA1
and
L
PA2
) associated to
G
1
and the
epiphysis (
L
PA3
) associated to
G
2
,
G
3
and
G
4
. Five
FIGURE 1 Anatomical regions for the reference femur.
S. LAPORTE et al.2
contours came from the lateral radiograph: the anterior
and posterior diaphysis (
L
LAT1
and
L
LAT2
) associated to
G
1
, the internal and external condyles (
L
LAT3
and
L
LAT4
) associated, respectively, to
G
2
and
G
3
, and the
trochlea and intercondylar notch (
L
LAT5
) associated to
G
4
. Then the reconstructed object was finally obtained
with NSCC.
Validation
The femurs models obtained with the NSSC method
were compared with 3D computerized tomographic
models of each femur (using a slices thickness of 1 mm
and a distances of 1 mm between two slices, a pixel
resolution of 0.25 mm and SliceOmatic
q
software) with
an overall accuracy of 1 mm [2,6,17]. The comparison
consisted in superimposing the two models by using
geometrical transformations (rotations and translations)
and a least square matching method. The comparison of
the shapes between the NSCC and CT-scan reconstructed
femurs was made using the point to surface distance
[15,17] between each point obtained from the NSCC
reconstruction and the surface of the CT-scan model.
The mean errors, RMS (Root mean square) and
maximum errors were calculated for the whole set of
556 points.
Initial Solution Sensitivity
So as to evaluate the importance of the initial solution on
the results of the NSCC method, ^ 1 mm translations
along the diaphysal axis and ^ 58 rotations around the
same axis were applied to one femur. Shape errors were
calculated for each reconstruction with regards to the
CT-scan reconstruction.
RESULTS
Validation Results on the Femurs
Qualitative evaluation was obtained by superimposing
the NSCC and CT-scan models for each femur (Fig. 3).
Quantitative comparisons between the CT-scan and the
NSCC reconstructions using the point to surface distance
for the 8 femurs are given in Table I. A 3D map of these
distances were also created (Fig. 3). These comparisons
yielded, respectively, mean/RMS errors of 1.0/1.4 mm
for all the 4448 points (8 femurs). Local maximum
errors for 95% of the points are smaller than 2RMS
(2.8 mm) while the 5% remaining were found lower than
5 mm. In all cases, errors were mainly located in regions
incompletely defined by 2D contours, such as the
proximal diaphysis. A faithful articular surfaces recon-
struction was then obtained (mean, RMS and maximum
errors for the articular surfaces were, respectively, 0.9,
1.1 and 4.2 mm).
Initial Solution Sensitivity
The sensitivity study of the initial solution on the NSCC
algorithm leads to the following results. Variations on
mean and RMS errors were lower than 0.05 mm;
maximal errors were always between 4.5 and 5 mm.
Shape errors are given in Table II. This indicates the
slight dependence of the NSCC method with regards to
the initial solution.
DISCUSSION
A 3D reconstruction method (NSCC), based on 2D-
contours extracted from biplanar radiographs, has been
FIGURE 2 X-rays information: anatomical contours and points, calibration steel balls.
BRM FOR DISTAL FEMUR 3
presented. The requirements for this method are the
following: (1) a precise calibration of the X-rays
environment with at least two images of different
incidences, (2) a location and shape initial solution of
the object to be reconstructed. Whereas, in the DLT and/or
NSCP methods, the reconstruction is based on the precise
identification of anatomical landmarks, the NSCC
method enables the reconstruction from contour data.
This simplifies the information extraction from the
radiographs because contours can be more easily extracted
from images than specific landmark points using semi-
automated segmentation methods. The NSCC method
yields realistic and accurate geometrical models of the
distal femur using only two radiographs. The results are as
accurate as the one obtained from computerized tomo-
graphic reconstructions based on semi-manual treatment
from more than 200 image slices (global mean error,
2RMS and maxima: 1.0, 2.8 and 5 mm). The given results
indicate that the proposed method is close to CT-scan
reconstruction with less irradiation (i.e. only two radio-
graphs). Moreover, the presented method allows a better
description of condyles curvature in regions where the CT-
scan slices are tangent to the condyle surfaces. The
reconstruction of regions without any 2D contours
associated is generally less accurate, but these regions
are not functional areas (i.e. proximal diaphysis). This is
due to the fact that these regions do not undergo
constrained deformations.
The sensitivity study, for the influence of the pre-
reconstructed models on the final reconstruction, shows
the slight importance of 2D–3D matching methods.
TABLE I Shape errors (point to surface distances, mm)
for the 8 reconstructed femurs by NSCC method vs. CT-scan
reconstructions
Global surfaces (mm) Articular surfaces (mm)
Femurs Mean RMS Max Mean RMS Max
1 1.1 1.3 4.6 0.8 1.0 3.4
2 1.1 1.4 4.5 1.0 1.2 3.2
3 1.1 1.5 5.0 1.0 1.3 4.1
4 1.0 1.3 4.9 0.9 1.1 3.0
5 0.9 1.2 3.8 0.8 1.0 3.2
6 1.1 1.4 4.7 0.9 1.1 3.6
7 1.0 1.3 4.4 0.8 1.1 3.7
8 1.1 1.4 4.9 1.0 1.3 4.2
Global 1.0 1.4 5.0 0.9 1.1 4.2
TABLE II Variation of the shape errors (CT-scan model vs. NSCC
method for the sensitivity study of the initial solution object relative to
rotations and translations along the diaphysal axis—evaluation for one
femur
Diaphysal axis modifications
Global point to surface errors
(mm) CT-scan model vs.
NSCC model
Rotation Translation (mm) Mean RMS Max
08 0 1.0 1.3 4.9
58 1 1.0 1.3 5.0
08 1 1.0 1.3 4.5
2 58 1 1.0 1.3 4.6
2 5821 1.0 1.3 4.8
0821 1.0 1.3 4.6
5821 1.0 1.3 4.5
58 0 1.0 1.3 5.0
2 58 0 1.0 1.3 4.6
Variations
Min: 1.0 1.3 4.5
Max: 1.0 1.3 5.0
FIGURE 3 (a) Qualitative evaluation: superimposition of the NSCC and CT-scan models. (b) Quantitative evaluation: colored bars represent the point
to surface distances on the CT-scan models (gray surface).
S. LAPORTE et al.4
CONCLUSION
This paper presents a method of accurate 3D reconstruc-
tion from biplanar X-rays using 2D contours extracted
from the radiographs.
The NSCC method improves existing stereoradio-
graphic methods based on the localization of specific
anatomical landmarks. The method principle is elastic 3D
model deformation with regards to 2D contours available
onto the different X-rays films.
The mean point to surface distance between the femur
models reconstructed using the NSCC algorithm and the
reference femur models obtained thanks to CT-scan
reconstruction was around 1 mm. Therefore, the results are
close to those obtained generally using CT-scan.
The originality of the method consists in using 2D
visible contours from radiographs. This method can be
adapted to any structure as long as the requirements are
fulfilled. This technique seems to be an interesting
alternative to CT-scan 3D reconstruction (based on semi-
manual treatment on millimetric slices), with the
advantage of low radiation. Once fully validated, it will
be of great usefulness for clinics and 3D applications such
as finite elements studies.
Acknowledgements
The authors wish to acknowledge X. Maillotte, E. Brugere,
J.B. Roy and A. Gabrion for their participation in the this
study, A. Magnin and E. Viguier for the CT-scan
reconstruction, The “Departement de radiologie de
l’ho
ˆ
pital Saint-Vincent de Paul” for the radiographs, The
“Laboratoire d’anatomie des Saint-Pe
`
res” for the dry
femurs, as well as the donors whose femurs were used in
this study. The authors would also like to thank Biospace
company for their financial support.
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APPENDIX 1
Determination of the 3D Contours for a 3D Object
Defined by Nodes and Triangles
If
a is a 3D surfacic object, defined by N nodes a
i
(
!
Oa
i
)
and P triangles
t
ijk
ða
i
; a
j
; a
k
Þ; and Sð
!
OSÞ a defined point of
view, the 3D contour on
a defined by this point of view is
the set of points
M where the normal vector to the surface
is orthogonal to the vector
!
SM
k
(M
k
[ M if
!
n
M
k
:
!
SM
k
¼ 0
and M
k
a point of the surface). This set of points is
calculated as follows.
1. For each triangle, the normal vector is calculated. The
normal vector to a triangle
t
ijk
is defined by
!
n
ijk
¼
!
Oa
i
^
!
Oa
j
þ
!
Oa
j
^
!
Oa
k
þ
!
Oa
k
^
!
Oa
i
k
!
Oa
i
^
!
Oa
j
þ
!
Oa
j
^
!
Oa
k
þ
!
Oa
k
^
!
Oa
i
k
:
2. Then for each node, the normal vector is calculated.
The normal vector for a given node
a
i
is the mean
vector of the normal vectors obtained for the triangles
BRM FOR DISTAL FEMUR 5
t
ijk
where a
i
is a vertex. If L is the list of these N
i
triangles we get
!
n
a
i
¼
1
N
i
L
X
!
n
t
ijk
:
3. The normal vector field is defined for each triangle
t
ijk
(variation of the orientation along a given triangle).
D is a point of a triangle
t
ijk
if
!
OD ¼ x:
!
Oa
i
þ y:
!
Oa
j
þ
z:
!
Oa
k
with x þ y þ z ¼ 1andðx; y; zÞ [ R
3þ
.The
normal vector to D [ t
ijk
can then be defined by
!
n
D
¼ x:
!
n
a
i
þ y:
!
n
a
j
þ z:
!
n
a
k
with x, y and z defined
above.
4. For each triangle, we search the solution of the
equation
!
n
D
:
!
SD ¼ 0 : bx:
!
Sa
i
þ y:
!
Sa
j
þ z:
!
Sa
k
c
£bx:
!
n
a
i
þ y:
!
n
a
j
þ z:
!
n
a
k
c ¼ 0
with x þ y þ z ¼ 1 and ðx; y; zÞ [ R
3þ
:
5. The whole three dimensional contours are then created
by the association of the set of points defined for all
triangles verifying the equation above.
S. LAPORTE et al.6
