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

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

 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) =

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

−1

(t)dt. (2)

Then, the total deformation of the bar is

δL = P

−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

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

−1

(t)dt

−1

(t)dt

. (4)

Let w(t) ∈ [0, 1] be the deformation distribution

function (DDF) along the bar such that,

w(t) =

−1

(t)dt

−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) =

= 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)

A(t)

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

(t)

is constant along C(t). Therefore, for regions with

constant cross section the vulnerability to stretching

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

(t) along C(t),

δL

(t) ∼ k∇χ

(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),

δΘ

(t) ∼

χ(t)

k∇r

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

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,

δL

(t) =

δL

(t)dt

δL

(t)dt

, (12)

δΘ

(t) =

δΘ

(t)dt

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)

, 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

δL

(t).

2.4 Weighted Rotation Minimizing

Frame

Establishing an appropriate frame ﬁeld F(t) =

(t), e

(t)) is a basic task in curve design and

analysis (Farin, 1997). While e

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

) , (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

, (18)

with initial condition e

(0) = e

. 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

along the curve (Klok,

1986).

With additional boundary constraint e

(1) = e

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). M’s shape is assumed to be

completely deﬁned by position of ﬁnite set of con-

trol points P = {p

(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

7→ R

(Lazarus et al., 1994):

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

(a) (b)

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

∈ M is equipped with triple

, θ(t

), r(t

)), where t

is parameter of the han-

dle axis C(t), θ(t

) is the rotation angle relative to

the moving frame deﬁned in Section 2.4, and r(t

) is

the distance between correspondent point C(t

) and

(see Figure 5).

Once M is embedded in the parametric space, de-

formation transfer from C(t) to M is straight forward,

= C(t

) + R(θ

)r(t

), (21)

where R(θ

) 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)

theta

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

. (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

(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.

, 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 M’s bounding box, as shown in

Figure 6(a). Thus, parameter t

of the control point

can be directly computed from z

max

− z

min

. (23)

These values are associated with each control point

and stored. In addition, initial curve length L = z

max

−

min

is stored.

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

(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

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

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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