Hybrid Geometric Active Models for Shape
Recovery in Medical Images *
B. C. Vemuri1 and Y. Guo2
1 Department of Computer & Information Science & Engg.
2 Department of .I. I i r .il & Computer Engg.
University of Florida, Gainesville, FL. 32611
Abstract
Abstract. In this paper, we propose extensions to a powerful geomet-
ric shape modeling scheme introduced in [23]. This modeling allows for
the representation of global shapes with local
shaping via physics-based control. \\. extend the model to automati-
cally cope with topological changes by introducing the concept of hybrid
geometric snake (active contour/surface) models wherein, object shapes
are represented using a parameterized function called the generator -
which accounts for the objects global shape and the pedal curve/surface
of the global shape with respect to a geometric snake to account for
any local I i.n!! Traditionally, pedal curves/surfaces are defined as the
loci of the feet of perpendiculars to the tangents of the generator from
a fixed point called the pedal point. \\. introduce physics-based control
for shaping these geometric models by using distinct pedal points lying
on a snake for each point on the generator. The model dubbed as a
"snake pedal" allows for interactive manipulation via forces applied to
the snake. Automatic topological changes of the model may be achieved
by replacing the traditional snake in the snake pedal with a geomet-
ric active contour which is realized via a level-set implementation. \\.
demonstrate the applicability of this modeling scheme via examples of
shape estimation from medical image data.
1 Introduction
I!, 'I". modeling is a fundamental task in computer vision as well as computer
graphics. In the context of shape modeling, there are two primary tasks of im-
portance in computer vision namely, shape recovery from image data and shape
recognition. The shape recovery task requires that the model have large number
of degrees of freedom, while shape recognition requires that the model represen-
tation be compact in terms of the number of parameters. These two requirements
are potentially conflicting and designing shape models that can possibly "- i1-
-!; these requirements has been the focus of !1 ii~ researchers in the recent
past in the computer vision .!,i!iin!- [1, 4, 5, 7, 11, 13, 15, 22].
*Submitted to the International Conference on Medical Image Computing and
Computer-Assisted Intervention 1998.
Geometric models a.k.a. lumped parameter models require very few pa-
rameters for their descriptions, are non-reactive and good for object recogni-
tion applications for example, cylinders, cones, superqudrics etc. On the other
hand, ,1i -i' --based model are distributed parameter models and are charac-
terized by a large number of parameters. They are well suited for shape recov-
ery/reconstruction. In [22], Terzopoulos and Metaxas developed a dynamically
deformable superquadric model which combines the descriptive power of geomet-
ric and ,1! -i, --based models via superposition and can characterize a plethora
of shapes (o 1'I' i, both ends of the spectrum one end of which is occupied by
lumped parameter models and the other with distributed parameter models. An
alternative modeling scheme which also involves the superposition principle was
presented in [1]. This scheme involved superposing a core superquadric with a
global volumetric deformation called the free form deformation (FFD) [18]. This
modeling scheme requires relatively fewer parameters to describe global defor-
mations. In Vemuri and ,1- i- li, [24], a multiresolution 1i- 1., 1 modeling
scheme was developed which involved representing the membrane displacement
function of a deformable superquadric model in a wavelet basis. In [15], Pentland
et al. introduced a novel shape modeling technique based on modal representa-
tions. This scheme permits the representation of global shapes with local detail.
The number of parameters required for the representation depends on the level
of detail needed in the representation.
In [4], Cohen describes a i! 1'i 1 shape modeling scheme wherein the prim-
itive used for modeling is a 1i i L i ., in i ..1 i.it. r1. with an exponential
term containing a i! I '1, !,li [i in the exponent. This modeling scheme 'h!I i -
from the above discussed schemes primarily because the model is geometric and
allows for global as well as local deformations. The number of parameters re-
quired to describe the model is much smaller than in the models involving the
superposition principle described earlier. The model fitting algorithm however
performs at non-interactive rates.
More recently, O'Donnell et al., [13] developed a novel solid shape modeling
scheme wherein the model included built-in offsets from a core base model. The
off-sets are functions of the parameters of the core base model. Local deformation
over the scaled-offset model is allowed via displacement vectors attached to the
default model. Model fitting to data is achieved in a dynamic framework as in
[22]. This modeling scheme is well suited for cardiac motion .... i fi .i!i tagged
lII: images.
In [5], DeCarlo et. al., developed an interesting topologically adaptable de-
formable model via the use of parameterized blending. These models have the
advantage of adapting to the l' ,I" *1' *- of an object via the use of an appropriate
error of fit criteria. In the context of I, .1. ,,_ adaptation, numerous techniques
based on the level-set approach to shape recovery introduced in Malladi et al.,
[10] and Caselles et al., [2] have been developed and the reader is referred to
[3, 21, 25]. In [12], M, I. i. i!i. et. al., introduced a novel version of snakes called
the topologically adaptable snakes which allow for automatic changes in 1 .1. ,1_ -*
during shape recovery. These models [3, 21, 25, 12] do not posses a mechanism
to characterize the global/core shape of an object.
1.1 Overview of Our Modeling Scheme
In [23] a powerful geometric shape modeling scheme was proposed which per-
mitted the representation of global shapes with local detail using only a "-- ,,i-
global" characterization in ,iiiLnii, rii with relatively simple and It. !,I nu-
merical methods. This geometric model allowed for -1 'il /sculpting via I 1! -
based control. The salient features of this the model were, it was inherently ge-
ometric but allowed for local and global deformations as would a i'1! -1. --based
model. Also, the model was compact, capable of global deformations e.g., bend-
ing, twisting, tapering etc. without the introduction of additional parameters.
In this paper, we extend the work in [23] by developing a method for estimat-
ing shapes of unknown I 1. .1 _-**- without resorting to blending operations -
through embedding of the modeling scheme into a level-set framework [10, 19].
In doing so, we introduce the novel concept of a global/core model into the now
popular PDE-based curve/-. ..-.. evolution framework. This global/core model
allows us to express ;,!! shape as a combination of a global shape (e.g., ellipse,
circle etc.) and a variable offset defined with respect to this global shape and
an evolving curve. We demonstrate the l.1I 1, -- change I 1.1 ,;l; of the model
via a synthesized example and illustrate several examples of shape recovery from
medical image data using this new modeling scheme.
The rest of the paper is organized as follows. In section 2 we 'i i discuss
the the "-! I:, i. 1 I a modeling scheme introduced in [23] and show examples
of the model fitting in 3D. In section 3 we introduce the 1i- '' 1 1 geometric active
contour/surface model (snake pedals in a level-set formulation) and demonstrate
the performance of the modeling scheme via examples of model fitting to 11-
thetic and real image data. We conclude in section 4.
2 The Snake Pedal Model
In this section we will first present definitions for the pedal curves and surfaces
and illustrate the ideas via examples. We will then define the -11 I: ,. '1 I and
illustrate the power of the modeling scheme via some model fitting examples in
3D.
Let a be a planar curve, the pedal curve [6] of a is defined as the locus of
points on the foot f of the perpendicular from a fixed point p called the pedal
point to a variable tangent of a. Let 3 be the pedal curve of a with respect
to the pedal point p, and let a(t) = g,/ (t) = f, as shown in the Fig. 1. The
projection of a (t) p in the direction Ja (t) must be 3 (t) p, where a' (t)
is the tangent line of plane curve a (t), J : R2 _" > R2 is a linear map given by
J(!',p2) = (-p2,P1). J can be geometrically interpreted as a rotation by 7r/2
in a counterclockwise direction. We can thus define a pedal curve as follows [6]:
Definition : I ., pedal curve of a regular curve a : (c, d) -> R2 with
respect to a fixed (pedal) point p E R2 is given by
(a)
Fig. 1. f is on the pedal curve of a with respect to the pedal point p.
pedal[p, a] (t)
In figure 2, we present examples depicting the pedal curves of an ellipse for
,I!!. I i- positions of the pedal point as shown. Note that the pedal curve is
capable of exhibiting local as well as a global deformations and the location of
the local deformation is in the 1... l i of the pedal point. By moving the position
of the pedal point, it is possible to -- i all! -i .. a i- ii. of local deformations as
depicted in the figure 2. The curve a(t) will be referred to as the generator for
the pedal curve 3 (t) and process of generating a pedal curve will be referred to
as the pedaling operation.
Fig. 2. Examples of pedal curves of an ellipse for I t. i. iw pedal point positions. Pedal
points are shown by a dot in each case.
More general shapes !i be !ll. -i .1 by letting the pedal point be dif-
ferent for each point of the generator. We can let the pedal points be specified
by another curve p(t), which can be represented by a standard snake/B-snake
[8, 11] and then apply the pedaling operation to each point on the generator
ai = a (ti) with respect to corresponding pedal point pi = p(ti). The generator
can be either a parameterized or an implicit function representing a curve. The
pedaling operation generates a new curve that we dub a snake pedal x(t) as
shown in Fig. 3. If the generator is an ellipse as shown in Fig. 3 (a), we can
P (t) p) Ja'(t) Ja'(t)
snake edal
Snake edal
snake
Generator snake
(a) (b)
Fig. 3. (a) The process of generating a snake pedal with an ellipse as a generator. (b)
"snake pedal" controlled by the snake using an ellipse generator.
represent it in a parametric form by
cos9 sinr6~ a cost \mi (2)
I-sinO cosO b sint] J m21
where a, b are aspect ratio parameters, 0 is the rotation angle between the in-
trinsic (material) coordinates and inertial coordinates, m = (ml, m2)T is the
centroid of the generator in the world coordinates. We collect the generator
parameters into the global parameter vector q = (a, b, 0, mi, m2)T.
For the ellipse, the term Ja'(t) in the equation of the pedal curve is given by
J'(t) [J Ja f -1 a sint sinr b cost cos 1 (3)
Ja 2 j =[ -a sint cosO + b cost sin (
Note that |Ja'' = Ja + J. + = a2sin2t + b2cos2t.
In figure 2(a)-(b) we depict some examples of snake pedals, curves generated
using snakes and an ellipse as the generator. Note the !- i. of local defor-
mations that can be generated using this modeling technique. We remind the
reader that the snake pedal itself is a geometric model and that it is not i/'.'.. ,
responsive to the application of external forces unlike the standard snake models
[8].
The pedal curve definition can be modified slightly by subtracting the second
term from the first term in equation 1. The key feature of using this modifica-
tion is that the snake pedal curve allows for more local deformations including
shrinkage and expansion of the snake pedal. The shrinkage and expansion be-
havior is controlled by the location of the snake which should be on the inside
of the generator for the former and outside of the generator for the latter case.
I i,!i.- 2(c)-(d) depicts examples of shapes generated using this modification.
Note the amount of local deformations achieved using this modification.
A pedal surface is the surface analog of the pedal curve. It is the locus of the
points on the foot of the perpendicular from a fixed pedal point to a variable
tangent plane of the surface. As in the 2D case, we can let the pedal point vary
(a) (b) (c) (d)
(e) (f)
Fig. 4. Examples of "snake pedals" (a)-(b) and modified "snake pedals" (c)-(d) us-
ing an ellipse generator and a snake. (e)-(f) are comparisons of original (red) and
modified (blue) "snake pedals" using the same generators and same snakes in both
case.
for each point on the generator surface. Thus we have the snake pedal surfaces in
3D whose shape can be controlled by snakes which are either curves or surfaces
in 3D [23]. For the 3D snake pedal examples we refer the reader to the next
section.
2.1 Model Fitting to Data
1 11!!, the "-i i!:.- pedal" to data is posed as a nonlinear minimization and we
use the Levenberg-Marquardt nonlinear optimization algorithm [16] to achieve
the fitting.
To fit the snake pedal to a given data set of n points Di in 3D, we minimize
the sum of the squared distances from the data points Di to the corresponding
closest points xi on the pedal curve/surface. Let h be the function that returns
the closest point on the snake pedal from a given data point Di i.e., xi =
hi(q,p, t). Let {pi,P2, ...,PN} be a set of N discrete points along the snake
p(t), {x, X2, ..., XN} be the corresponding points on the snake pedal curve. Let
D = {D } and p = {p }7, then the i. !. -_ to be minimized for the model
fitting operation can be written as:
ED(D,q,p)= (Di -x)2 (4)
i=1
where xi = hi(q, pi, ti) and the magnitude of parameter P is related to the
uncertainty of data measurements.
There are two I- i" of parameters which need to be determined in order to
find the minimum of above i!. i function. The first one is the generator param-
eters q = (a, b, 0, mi, m2)T and the second one is the snake position p = {p }i=.
We apply the Levenberg-Marquardt (LM) method to update the generator
parameter vector q. For each iteration of the LM method, we solve a sparse lin-
ear - -1. i of equations for determining the position of the parameterized snake
which responds to the applied forces synthesized from the data.
In order to fit the snake pedal curve to salient features in an image, we use the
following equation as the i i l o be minimized: E = IG,*VI((xx(t),y(t))' :
Where, x(x(t), y(t)) describes the snake pedal curve, G, is a Gaussian with stan-
dard derivation a, and denotes the convolution operator. By minimizing the
above ii. i the model is made to fit to points of high gradient in the image.
More sophisticated potential energies using region-based information can easily
be synthesized and incorporated into this framework.
2.2 Example of Model Fitting
We now present a set of three experiments, one of these demonstrate model
fitting to sparse 3D data points placed by an expert Neuro-scientist along the
boundaries of a _- in selected slice of an -: i:I brain scan. Such a scenario arises
in the semi-automatic construction of anatomical models for possible use as prior
information in shape recovery from unknown data. The last two experiments
demonstrate the power of the model in fitting to 3D synthetic data from a (hair
pin) bent shape and a twisted shape respectively.
The medical data example is organized as follows: left to right, the images
depict a slice of an -I1i: brain scan in which the shape of interest a i ,-
- has been identified by a Neuro-scientist via sparsely placed points (only in
selected slices of the 1 lI: scan) on the shape boundary. The next image shows
the collection of these 3D points in red and the initialized snake pedal model
followed by an images depicting the intermediate stage of fitting and the final
fitted model respectively. As evident, the model achieves a visually accurate fit
to the data. In addition, we have verified that these fits are in agreement with
what the expert Neuro-scientist would achieve via manual segmentation.
The two ii !. -I, data examples are organized as follows: (e) & (g) are the
images depict a 3D i 1 1. i data set (in red) along with the model initialization
(in green) superimposed where (f) & (h) are images of the final model fit. What
is to be noted in these synthetic data examples is that in one case, the data
exhibit large bending and in another a large amount of twist. What we would
like to stress here, is the fact that the model did not require ;,i! explicit bending
or twisting transformations (unlike the deformable superquadric) to achieve the
fits. As mentioned earlier, modeling schemes that require explicit incorporation
of bending, twisting transforms are highly nonlinear and are known to introduce
numerical instabilities into the fitting algorithms.
(a) (b)
(e) (f)
(g) (h)
Fig. 5. Top row left-right: I. i. brain scan depicting region of interest (a gyrus), initial-
ized model, intermediate fitting stage and final model fit. Bottom row: (e) & (g) and (f)
& (h) show initialized models superposed on data and final fitted models respectively.
3 Hybrid Geometric Active Contours/Surfaces
Topological changes can be achieved in the snake pedal model by considering
discontinuous snakes and generating the pedal curves/surfaces for these snake
segments. The most natural way to automatically allow the snake curve/surface
to split/merge is via a level-set implementation. Numerous researchers have used
the PDE-based curve/surface evolution in a level-set implementation framework
for shape segmentation [2, 3, 9, 10, 14, 17, 20, 21].
For the snake pedal to have some amount of regularization built into it, we
can impose this regularization constraint via Euclidean are-length/area mini-
mization. This would lead to the standard geometric active contour/surface.
However, we would like to know how the "-- I!:.." would evolve if the snake-
pedal were evolving as a function of its local curvature. This will give us the
snake position and the pedal curves/surfaces can be determined by the pedal-
ing operation defined earlier given the position of the generator. Note that the
li, 1 ii1 geometric active contours/surfaces we propose allows incorporation of a
global parameterized shape into the curve/surface evolution framework. The role
of this global parameterized shape is served by the generator of the snake pedal
model discussed in earlier sections. We formalize the above mentioned concepts
in the following.
If we denote the pedal and the snake positions respectively by pe = ,pe2}
(c) (d)
and p = {pl,p2}. Denote the generator by a {a=1, a 2} with normal J, =
{Jel, Ja2}, then from the definition of the pedal curve, we have
(a p) J (
Pe = P- .j" -j (5)
Eqn. (5) can be rewritten in component form as
| |Pl + Jll J 2 J lat 2 a21 ( "J l + (2 "J a2 [J ql1
2 Where
jal la2 1J 2 J+ +Ja2
[1 +;2 1 2(8)
(6)
Or simply,
pe = JA P b (7)
b 1 + J2 2 ja2 + Ja2
JA = al J 2 a J a2 (8)
S2 1 22
al a2 al a2
and
b- a, -Ja+ a Jl 2 + 022 (9)
r2 + T2 Ja2
jal+2ja2 I I
Jl + 2a2 Jal Ja2
J = J +a a22 (10)
"al "a2 2j,l+ J,i"2
2 22 12 2
Since we assume that the generator is stationary in time, JA and b are constant
with respect to time, and eqn. (7) can be explicitly written as a function of time:
Pe(t) = JA p(t) b (11)
Taking partial derivatives in both sides of eqn. (11) with respect to time gives
ape(t) ap(t)
)=_ JA (12)
at at
Or
p(t) J pe (t)
-= JB (13)
at at
As mentioned earlier, the smoothness constraint is imposed on the pedal
curve and one form of this constraint can be expressed as a variational princi-
ple involving minimization of the Euclidean are-length of the snake pedal. The
solution of the length minimization can be obtained by solving the following
gradient descent equation, 0p(t) = kN. Where, k, and N, are the curva-
at
ture and the normal of the snake pedal respectively, and k, is a speed function
controlling the evolution of the snake pedal. Using eqn. (13), upon substitu-
tion we have Ja) = JAlkeNe = kJBN. Which gives the evolution equation
for the "---i I!:" in terms of a curvature-based speed function defined on the
"-,i I!:,- pedal". A level-set implementation of this equation will naturally facil-
itate changes in 1, ,I. .1. _-- of the snake and hence the snake pedal. In order to
apply the above model to extract shapes of interest from image data, we can
use either of the techniques suggested in Caselles et al., [3] or Tek et al., [21].
In either case, the minimization of the Euclidean are-length of the snake pedal
has to be modified to a weighted length i.e., a Riemanian metric has to be used
instead of the Euclidean metric.
In the top row of figure 6, we demonstrate a level-set implementation of
the snake pedal model described above with an ellipse generator. The example
contains a 2D slice from an abdominal CT scan. Left to right, the images show
model initialization depicting the snake pedal in yellow and the generator in
red, intermediate stages of evolution and final model fit respectively. Several
more examples of the 1i 1'il contour model fitting are shown in (e),(f) and
(g) of the same figure. These images contain the 2D slices from CT and II;
images. In the middle row of figure 6, from left to right, the images depict model
initialization for liver metastasis in a CT scan of the human liver, an endocardium
and a i11- ., 1 1,i,11ii in the IlI: scan of the human heart respectively. The snake
pedal is shown in green and the generator in red. I ,'iL,- 6 (h)-(j) depict the
final model fits. As is evident, the snake pedal was successful in extracting the
shape of interest in these images via the model fitting process wherein an image-
based speed function was used to deter the model evolution. Note that the
".,. .1, !" shapes depicted by the generators in these examples are scaled versions
of "o, I.. i!" shape descriptors of the objects. The "-.1. I," shape provides us with
a "lI, I orientation of the shape of interest.
I ,oii.- 7 depicts four topological change examples. In the top row, from left
to right, images depict model initialization, intermediate stages of evolution and
final fit respectively. In this example, the concept of a -"ol. .1, shape is not
very meaningful however, the concept of orientation 1!! i be associated with a
"oI.Il. .1" orientation of a cluster of shapes (in this case, two). More examples
depicting topological changes are shown in the same figure. Fig. 7 (e)-(g) depict
the model initialization and (h)-(j) are final model fits. In figure 7 (e) and (g),
the snake pedals are initialized to include all the objects in the images, then the
models shrink and split to fit to all the object boundaries. While in figure 7 (f),
the snake pedal is initialized as a single small ellipse, as the fitting proceeds, the
model expands and splits, and finally fits to all the object contours in the whole
image. The global shapes of the generator are not shown here since the meaning
of the ",-l. 1,I" shape in these examples is not very meaningful.
In all these examples, we implemented the level-set form of the equation:
OP(t= J ( J (V )keNe (Vg Ne)Ne )= ( g(VI)k Vg N, )JBNe.
Where, g(VI) = 1/(1 + I'(G I, "- K) with G I being a Gaussian con-
volved with the image and K being a scaling constant. More sophisticated speed
functions that use region-based information can be synthesized from the image
data and incorporated in this framework to achieve better and more accurate
segmentation.
(b) (c) (d)
(e) (f)
I'r
(h) (i)
Fig. 6. Hybrid geometric active contour model fit examples: (a)-(d) Model initialization
(snake pedal in yellow and generator in red), intermediate stages of evolution and final
fit. (e)-(h) topology change example.
(a) (b) (c) (d)
(e)
DdD
(h)
(g)
(4
(j)
Fig. 7. Hybrid geometric active contour model fit examples: (a)-(d) Model initialization
(snake pedal in yellow and generator in red), intermediate stages of evolution and final
fit. (e)-(h) topology change example.
4 Conclusion
In this paper we have presented a new 1- 1. 1i geometric active contour model
which allows for representing global shapes with local detail. For representing the
global shape, we use a parameterized function and the local detail is represented
by the snake pedal of a geometric active contour. A level-set implementation of
the geometric active contour facilitates topological changes in the snake pedal
model. The 1i 11 i 1 geometric active contour model in essence introduces a vehicle
for incorporating a global parameterized shape descriptor into the now popular
framework of geometric active contours/surfaces. We demonstrate the power of
the modeling scheme via model fitting to -- !i!1l 1, and medical image data.
Acknowledgments: This research was in part funded by the NIH grant
RO1-LM05944. Authors would like to thank Dr. C. M. Leonard and the Dept.
of ,I11r .1 -_- at UFL for the medical image data.
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