INHOMOGENEOUS AXIAL DEFORMATION
FOR ORTHOPEDIC SURGERY PLANNING
Sergei Azernikov
Siemens Corporate Research, 755 College Road East, Princeton NJ 08540, U.S.A.
Keywords:
Feature preserving spatial deformation, Weighted arc-length curve parametrization, Orthopedic surgery plan-
ning.
Abstract:
Intuitive global deformation of complex geometries is very important for many applications. In particular,
in the biomedical domain, where interactive manipulation of 3D organic shapes is becoming an increasingly
common task. Axial deformation is natural and powerful approach for modeling of tubular structures, like
bones. With this approach, the embedding space is associated with deformable curve, the handle axis, which
guides deformation of the embedded model. As a result, the produced deformation is homogeneous and
independent of the model representation and shape. However, in many situations it is beneficial to incorporate
geometric and physical properties of the model into the deformation formulation. This leads to inhomogeneous
axial deformation which allows to achieve more intuitive results with less user interaction. In this work, the
inhomogeneous axial deformation is achieved through deformation distribution function (DDF) induced on
the guiding axis by the embedded model. Since with the proposed formulation the DDF can be pre-computed,
run-time computational complexity of the method is similar to the original axial deformation approach.
1 INTRODUCTION
Growing amount of available high definition 3D
data opens broad opportunities in many different ar-
eas. In particular in biomedical domain, increasing
number of procedures are planned and executed in
computer-aided environment using 3D models of pa-
tient’s anatomy. Manipulation of these anatomical
models, however, often requires new tools and meth-
ods, which are not provided in the traditional geomet-
ric modeling systems (see for example (Azernikov,
2008)).
One of the earliest yet powerful and intuitive
methods for shape manipulation is spatial deforma-
tion pioneered by Barr (Barr, 1984). Later, Sederberg
and Parry (Sederberg and Parry, 1986) extended the
spectrum of possible deformations by introducing the
free-form deformation (FFD) approach. Due to its at-
tractive properties and conceptual simplicity this ap-
proach was extensively used in geometric modeling
and computer animation (Gain and Bechmann, 2008).
The basic idea of the FFD approach is to parameter-
ize the embedding space as a tri-variate tensor product
volume. The shape of this volume can be manipu-
lated by grid of control points. Once control points
are repositioned, the volume is warped and the embe-
dded objects are deformed accordingly. The draw-
back of the grid-based FFD is that because of the high
number of control points that should be moved, it may
require significant user interaction even for simple de-
formations. In order to reduce the amount of required
user interaction, lower dimensional variations of the
original trivariate FFD method were introduced: sur-
face deformation (Feng et al., 1996) and axial defor-
mation (AxDf) (Lazarus et al., 1994). With the AxDf
approach, the ambient space deformation is guided by
axial curve, which can be represented as a straight
segment chain (Lazarus et al., 1994) or parametric
curve (Peng et al., 1997). The major drawback of spa-
tial deformation is that it is global, and as such, does
not preserve shape of the local features.
Recently, mesh-based deformation techniques
gained popularity in the computer graphics commu-
nity (Sorkine, 2005; Lipman et al., 2005). These
techniques are formulated directly on the polygonal
mesh and explicitly designed to preserve local sur-
face features. From the user perspective, with these
techniques the deformable object behaves like elastic
homogeneous material.
Popa et al. (Popa et al., 2006) applied material-
aware scheme for mesh deformation assigning stiff-
ness property to various regions of the mesh. As a
59
Azernikov S..
INHOMOGENEOUS AXIAL DEFORMATION FOR ORTHOPEDIC SURGERY PLANNING.
DOI: 10.5220/0003370300590066
In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2011), pages 59-66
ISBN: 978-989-8425-45-4
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
(a) (b) (c) (d) (e)
Figure 1: Inhomogeneous axial deformation example: (a) original model, (b) applied deformation and constraints cause
stretching, (c) homogeneous axial deformation produces distortion of the spheres, (d) scaling can prevent distortion but
modifies size of the spheres, and(e) proposed inhomogeneous axial deformation preserves shape and size of the spheres,
while transferring the deformation to cylindrical regions invulnerable to stretching.
result, the deformation is non-uniformly distributed.
Kraevoy et al. (Kraevoy et al., 2008) proposed a
method for feature-sensitive scaling of 3D shapes.
With this approach, the embedding volume is vox-
elized and scaling function is evaluated in each voxel
based on the embedded shape variation. When scal-
ing is applied, the deformation is distributed in a non-
uniform manner according to the assigned scaling
function values. These methods attempt to mimic be-
havior of elastic inhomogeneous material. The work
described in this paper is inspired by this idea and
introduces inhomogeneous axial deformation frame-
work.
2 INHOMOGENEOUS AXIAL
DEFORMATION
When the handle axis length is modified during the
modeling session, the aspect ratio of the deformed
object is naturally modified as well. Since the defor-
mation is global, the stretching is distributed equally
along the axis. Although this behavior is often per-
fectly desired, it may cause distortion of local fea-
tures as shown in Figure 1(c). One way to deal with
this problem is to scale the model to compensate for
the length change. This, however, may lead to non-
intuitive results, as can be seen in Figure 1(d). In
order to overcome this issue, the inhomogeneous ax-
ial deformation formulation is proposed in this paper.
The basic idea is to redistribute the deformation along
the axis such that most of the distortion is applied
on invulnerable portions of the model. This behavior
attempts to mimic the physical behavior of an inho-
mogeneous bar under axial load (Timoshenko, 1970),
which is briefly described in Section 2.1. Based on
this analogy, deformation distribution function (DDF)
is introduced in Section 2.3. In order to prevent dis-
tortion of the model features, the axial curve is re-
parameterized according to the DDF, as described in
Section 2.2. Section 2.5 describes embedding of the
model into the axial parametric space. When the ax-
ial curve is deformed, its deformation is transferred
to the embedded model following the established em-
bedding. The flow chart of the proposed deformation
method is shown in Figure 2.
2.1 Inhomogeneous Bar Deformation
under Axial Load
Let B be a bar with variable cross section A(t), , t
[0, 1] and length L, as shown in Figure 3. The local
deformation δs of B under axial load P can be com-
puted as (Timoshenko, 1970):
δs(t) =
Z
t
0
P
E(t)A(t)
dt, (1)
where E(t) is the Young modulus of the bar (Tim-
oshenko, 1970). Introducing stiffness notion k(t) =
A(t)E(t) and assuming constant force P = const,
δs(t) = P
Z
t
0
k
1
(t)dt. (2)
Then, the total deformation of the bar is
δL = P
Z
1
0
k
1
(t)dt. (3)
In the inverse problem, total deformation δL of the
bar is given, when local deformation and stress should
GRAPP 2011 - International Conference on Computer Graphics Theory and Applications
60
Define axial curve
Establish axial
shape parametrization
Define deformation
distribution function
Re-parametrize axial curve
Transfer axial curve
deformation to model
Deform axial curve
Figure 2: Flow chart of the proposed deformation method.
be found. Then, the local deformation δs(t) can be
rewritten in terms of the given total deformation and
bar’s properties,
δs(t) = δL
R
t
0
k
1
(t)dt
R
1
0
k
1
(t)dt
. (4)
Let w(t) [0, 1] be the deformation distribution
function (DDF) along the bar such that,
w(t) =
R
t
0
k
1
(t)dt
R
1
0
k
1
(t)dt
. (5)
Then Eq. (4) can be rewritten in a more compact form,
δs(t) = δLw(t). (6)
For the uniform case when k(t) = const,
w(t) =
R
t
0
dt
R
1
0
dt
= t. (7)
The local deformation δs(t) is then reduces to the uni-
form scaling,
δs(t) = δLt, t [0, 1]. (8)
Similar results can be obtained for inhomoge-
neous bar under torsion (Timoshenko, 1970). By re-
placing the local stretching δs(t) with local rotation
angle δθ(t) and total length change with total rotation
angle δΘ, Eq. (6) can be rewritten as,
δθ(t) = δΘw(t). (9)
P
A(t)
B
t
L
Figure 3: Bar with variable cross section under axial load.
2.2 Deformation Distribution Function
Function w(t) controls deformation distribution along
the handle axis. With the proposed approach, w(t)
is designed to avoid distortion of the important fea-
tures of the model while obeying the boundary con-
ditions imposed by the user. Kraevoy et al. (Kraevoy
et al., 2008) estimate local vulnerability of the model
to non-uniform scaling as a combination of slippage
measure (Gelfand and Guibas, 2004) and normal cur-
vature in the direction of deformation. Since in our
case the deformation of the model M is guided by the
axial curve C(t), the proposed vulnerability measure
is based on shape variation of M along C(t). Assum-
ing that length preserving bending does not introduce
distortion as long as C(t) remains smooth, axial de-
formation can produce two types of distortion:
1. Stretching: by modifying the total length of C(t).
2. Torsion: by imposing twist around the C(t).
Stretching will not introduce any distortion in re-
gions where the cross section of the model χ
M
(t)
is constant along C(t). Therefore, for regions with
constant cross section the vulnerability to stretching
v
δL
(t) should be close to zero, and where cross sec-
tion is rapidly changing, v
δL
(t) should be high. In
other words, v
δL
(t) is proportional to the gradient of
χ
M
(t) along C(t),
v
δL
(t) k∇χ
M
(t)k (10)
On the other hand, torsion should be allowed
where the shape contour is close to circular and pre-
vented elsewhere to avoid distortion. So, the vulner-
ability to torsion v
δΘ
(t) can be estimated integrating
the radius gradient across the shape profile χ(t),
v
δΘ
(t)
Z
χ(t)
kr
χ(t)
k, (11)
where r
χ(t)
is the local radius of the shape contour
χ(t).
INHOMOGENEOUS AXIAL DEFORMATION FOR ORTHOPEDIC SURGERY PLANNING
61
Replacing stiffness k(t) in Eq. (5) with the
above vulnerability measures, deformation distribu-
tion functions (DDFs) w
δL
(t) and w
δΘ
(t) can be can
be formulated as,
w
δL
(t) =
R
t
0
v
δL
(t)dt
R
1
0
v
δL
(t)dt
, (12)
w
δΘ
(t) =
R
t
0
v
δΘ
(t)dt
R
1
0
v
δΘ
(t)dt
. (13)
These functions are used to define weighted arc-
length parametrization of the axial curve C(t) and to
establish weighted rotation minimizing frame along
C(t).
2.3 Weighted Arc-length
Parametrization
Arc-length preserving axial deformation produces
natural results and therefore very useful in anima-
tion (Peng et al., 1997). However, when the length of
the model is modified in purpose, arc-length of the ax-
ial curve C(t) cannot be preserved everywhere. How-
ever, it can be preserved locally by weighting the arc-
length change with deformation distribution function
(DDF) w
δL
(t).
The curve C(t) is initially parameterized w.r.t. its
arc-length s(t) such that (do Carmo, 1976),
t =
s(t)
L
, t [0, 1], (14)
where L is the total length of the curve. When L is
modified by δL, the new parametrization
˜
t is com-
puted as follows:
˜
t =
s(t) + δs(t)
L + δL
, (15)
where δs(t) is the weighted local deformation,
δs(t) = w
δL
(t)δL. (16)
This new parametrization is used to transfer the
deformation of the axial curve C(t) to the model M,
as will be explained in Section 2.5. As a result, the
deformation δL is redistributed according to the DDF
w
δL
(t).
2.4 Weighted Rotation Minimizing
Frame
Establishing an appropriate frame field F(t) =
(e
1
(t), e
2
(t), e
3
(t)) is a basic task in curve design and
analysis (Farin, 1997). While e
3
is usually associated
with the tangent vector of the curve, another axis has
to be defined in order to set up the frame field com-
pletely. In differential geometry, the classical Frenet
frame is commonly used,
(e
1
, e
2
, e
3
) , (n, b, t), (17)
where n, b and t are the normal, binormal and tan-
gent respectively (Koenderink, 1990). The advantage
of Frenet frame is that it can be computed analyti-
cally for any point on a twice differentiable curve.
Unfortunately, for general curves with vanishing sec-
ond derivative or high torsion, Frenet frame fails to
generate stable well-behaving frame field. As a better
alternative, Klok (Klok, 1986) introduced the rotation
minimizing frame (RMF), which is formulated in dif-
ferential form,
e
0
1
=
(C
00
· e
1
)C
0
kC
0
k
2
, (18)
with initial condition e
1
(0) = e
0
1
. Although no ana-
lytic solution is available in that case, there is a sim-
ple and effective method to approximate RMF by dis-
cretizing the curve and propagating the first frame de-
fined by the initial condition e
0
1
along the curve (Klok,
1986).
With additional boundary constraint e
1
(1) = e
1
1
,
twist has to be introduced into the RMF (see Fig-
ure 4). The twisting angle δΘ is then propagated
along the axis according to the parameter t [0, 1],
δθ(t) = δΘt. (19)
The resulted frame field is shown in Figure 4(b).
As was already mentioned, imposing twist may
distort certain features of the model. This effect is
demonstrated in Figure 4(a). Using the proposed in-
homogeneous approach, the twist δθ(t) distribution
will be adapted to the DDF w
δΘ
(t),
δθ(t) = δΘw
δΘ
(t). (20)
With this weighted torsion distribution, shown in Fig-
ure 4(d), the distortion of the features is avoided and
applied in the invulnerable regions, as can be seen in
Figure 4(c).
2.5 Model Embedding and Deformation
Transfer
One of the important advantages of spatial deforma-
tion in general, and axial deformation in particular,
is its independence of the deformed model M repre-
sentation (Barr, 1984). Ms shape is assumed to be
completely defined by position of finite set of con-
trol points P = {p
i
(x, y, z), i = 0..n)}. For polygonal
meshes, P is the list of the vertices, while for para-
metric surfaces, P will represent the control points.
The axial deformation consists of two mapping
R
3
7→ R
3
(Lazarus et al., 1994):
GRAPP 2011 - International Conference on Computer Graphics Theory and Applications
62
(a) (b)
(c) (d)
Figure 4: Comparison of Klok’s RMF with the proposed
weighted RMF: (a),(b) RMF creates distortion of the cubes,
(c),(d) the proposed weighted RMF preserves the cubes
while obeying the twist angle.
1. Embedding of the model M in parametric space of
the axial curve C(t).
2. Transfer of the deformation from C(t) to M.
To embed the model M in the parametric space,
each control point p
i
M is equipped with triple
(t
p
i
, θ(t
p
i
), r(t
p
i
)), where t
p
i
is parameter of the han-
dle axis C(t), θ(t
p
i
) is the rotation angle relative to
the moving frame defined in Section 2.4, and r(t
p
i
) is
the distance between correspondent point C(t
p
i
) and
p
i
(see Figure 5).
Once M is embedded in the parametric space, de-
formation transfer from C(t) to M is straight forward,
˜
p
i
= C(t
p
i
) + R(θ
t
p
i
)e
1
(t
p
i
)r(t
p
i
), (21)
where R(θ
t
p
i
) is the rotation matrix around axial
curve. In some applications, it may be useful to in-
troduce scaling in radial direction (Yoon and Kim,
2006). For example in Figure 1(d), uniform radial
scaling is applied to compensate for stretching δL.
p(x,y,z)
C(t)
e1
e3
theta
r
e2
Figure 5: Axial parametrization of the model.
(a) (b) (c)
Figure 6: Femur model embedding and deformation: (a)
handle axis initialized to the center line, (b) mesh vertices
are parameterized along the center line and model profiles
are extracted, (c) deformation transfer from the axis to the
model; notice the shape preservation of the femural head
and the knee joint.
Then, r(t) is multiplied by a constant scaling factor,
˜r(t) = r(t)
L + δL
L
. (22)
3 IMPLEMENTATION
The axial deformation session starts from setting the
axial curve C(t) and associating control points P =
INHOMOGENEOUS AXIAL DEFORMATION FOR ORTHOPEDIC SURGERY PLANNING
63
(a) (b) (c)
Figure 7: Tibia deformity modeling: (a) X-ray image of the tibia pathology, (b) template tibia model, and (c) overlay of the
deformed 3D tibia on the 2D image.
(a) (b) (c)
Figure 8: Plate deformation (original model is shown as solid and deformed as wire frame): (a) plate with shown boundary
conditions, (b) after simple scaling holes are distorted, (c) with inhomogeneous deformation shape of the holes is preserved;
notice the precise matching of the hole after deformation.
{p
i
, i = 1..n} of the model M with this curve. First, M
is oriented such that z axis is aligned with its maximal
principal component. Afterwards, C(t) is initialized
to the center line of Ms bounding box, as shown in
Figure 6(a). Thus, parameter t
p
i
of the control point
p
i
can be directly computed from z
p
i
,
t
p
i
=
z
p
i
z
max
z
min
. (23)
These values are associated with each control point
and stored. In addition, initial curve length L = z
max
z
min
is stored.
GRAPP 2011 - International Conference on Computer Graphics Theory and Applications
64
(a) (b)
Figure 9: Semi-automatic implant placement on femur: (a) guiding points placed by the user, (b) appropriate plate is auto-
matically picked from the implant database and deformed to fit the bone.
Afterwards, the model M is sliced with m planes
perpendicular to the center line and the shape profiles
are stored (see Figure 6(b). Based on these profiles,
vulnerability measures v
δL
and v
δΘ
are estimated for
each profile using Eqs. (10) (11). And DDFs w
δL
and
w
δΘ
are computed using Eq. (12) and stored.
When the axial curve C(t) is modified to
˜
C(t), it is
re-parameterized according to the new arc-length and
the stored w
δL
DDF using Eq. (15). If twist defor-
mation is introduced, it is distributed according to the
pre-computed w
δΘ
function.
For initially curved models it may be more in-
tuitive to initialize C(t) to follow the shape of the
model (Lazarus et al., 1994). One simple approach is
to extract shape profiles parallel to some pre-defined
plane and to fit C(t) to the centers of these profiles.
In that case, several practical questions arise. First,
for each control point p closest point on C(t) has to
be found, which may take significant computational
effort for big models (Yoon and Kim, 2006). Sec-
ond, models with complex topology (see for example
model shown in Figure 8) may require special treat-
ment in order to compute stable axial curve. More-
over, this simple approach may fail completely for
certain models. In that case, a complete skeletoniza-
tion should be applied (Au et al., 2008) and the re-
sulting skeleton curve may be used as the deformation
axis.
4 APPLICATIONS
The major motivation for this work came from bone
modeling for orthopedic surgery planning. Currently,
most interventions are still planned based on the pre-
operative 2D X-ray images. Although significant
amount of research work was dedicated to 3D ortho-
pedic surgery planning (Hazan and Joskowicz, 2003),
it is assumed that pre-operative 3D image of the pa-
tient’s anatomy is available. Unfortunately, this is not
always the case. One way to overcome this issue is to
recover 3D geometry from the available 2D data. This
can be done by overlaying a template bone model on
the image and deforming the model to approximate
patient’s pathology (Messmer et al., 2001). Figure 7
shows tibia deformity that is going to be treated with
Ilizarov spatial frame (Rozbruch et al., 2006). In or-
der to simulate this procedure, 3D tibia model is de-
formed to match the available 2D image using the pro-
posed inhomogeneous AxDf approach. With this ap-
proach, tibia features can be automatically preserved
during the deformation, without explicit user speci-
fied constraints, as can be seen in Figure 7.
Due to its feature preserving property, the pro-
posed approach is also used for implant modeling for
pre-operative planning of fracture reduction and fixa-
tion. Figure 8 shows how shape of the implant holes is
preserved under axial load with the proposed inhomo-
geneous axial deformation method. This is important
since distortion of the holes will make it difficult or
impossible insertion of the fixation screws. Fracture
care systems that are currently in clinical use perform
planning in 2D and require manual reduction of the
fracture and fitting of the implant to the bone. With
the proposed semi-automatic implant placement, the
surgeon just needs to mark region on the bone where
the implant should be placed. Then, the system will
automatically adapt the implant’s shape and size to
the bone, as shown in Figure 9. This is achieved
by computing implant path on the bone from the
provided guiding points using the shortest-path algo-
rithm (Surazhsky et al., 2005). From the computed
path, appropriate plate length is estimated and the ap-
propriate plate is chosen automatically from the im-
plant database. Finally, the chosen plate is deformed
preserving shape of the holes, while the path is used
INHOMOGENEOUS AXIAL DEFORMATION FOR ORTHOPEDIC SURGERY PLANNING
65
as the deformation axis.
5 CONCLUSIONS AND FUTURE
WORK
In this paper, inhomogeneous axial deformation was
introduced and demonstrated on a number of impor-
tant applications in orthopedic surgery planning. The
proposed method inherits all the attractive proper-
ties of the classical axial deformation and introduces
shape sensitivity to the original formulation. This al-
lows to preserve shape and size of the features while
reducing required user interaction.
Currently, only shape of the model is used for the
DDF formulation. However, physical material prop-
erties can be easily incorporated into the proposed
framework.
Another promising research direction is learning
of deformation modes from provided examples (Popa
et al., 2006). This would allow intelligent adaptation
of the deformation distribution functions to the spe-
cific domain.
By definition, AxDf supports a limited class of
deformations. The proposed approach can be ex-
tended to handle more general deformation schemes.
In particular, it would be interesting to combine
the proposed inhomogeneous formulation with the
sweep-based freeform deformation method (Yoon and
Kim, 2006). It is also possible to formulate two-
dimensional DDFs to consider deformations guided
by parametric surfaces (Feng et al., 1996).
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