MODEL-BASED SHAPE FROM SILHOUETTE
A Solution Involving a Small Number of Views
J. F. Menudet, J. M. Becker, T. Fournel and C. Mennessier
Laboratoire Hubert Curien, UMR CNRS 5516 - Universit
´
e Jean Monnet, 18 rue Pr Beno
ˆ
ıt Lauras, Saint-Etienne, France
Keywords:
Visual Hull, Shape From Silhouette, Contour, Radial Basis Function, Deformable Model, Non-Rigid Regis-
tration.
Abstract:
This article presents a model-based approach to Shape From Silhouette reconstruction. It is formulated as a
problem of 3D-2D non-rigid registration: a surface model is deformed until it correctly matches the detected
silhouettes in the images. An efficient and reliable solution is proposed, based on a Radial Basis Function
deformation driven by control points located on the contour generators of the 3D model. Unlike previous
methods relying on non-linear optimization techniques, the proposed method only requires a linear system
solving. Another advantage of this model-based approach is to produce a surface representation of the visual
hull. Moreover, the introduction of shape priors allows to reduce in a dramatic way the number of views
required to obtain a realistic reconstruction. Application to human body modeling is given.
1 INTRODUCTION
Image-based modeling aims at the acquisition of a 3D
model from multiple images of a real world object. It
is a simple and low cost approach compared to other
methods such as laser scanner or structured-light pro-
jection. Some examples of image-based modeling
techniques are Structure From Motion (SFM) (Hart-
ley and Zisserman, 2004) and Shape From Shading
(Prados and Faugeras, 2005). Another popular tech-
nique is Shape From Silhouette (SFS) which produces
the visual hull of the object, an outer approximation
of the 3D shape, consistent with a set of silhouettes
detected in several images (Laurentini, 1994). Appar-
ent contours give strong clues about the 3D shape of
an object. Furthermore, they are far more easier to
find than point correspondences required for example
in SFM. Nevertheless, the accuracy of the visual hull
is directly related to the number of input images as
well as the different locations of the camera. Another
well-known intrinsic limitation of SFS is the impossi-
bility to reconstruct concavities.
Typically, a good SFS reconstruction requires 15
to 30 views of the object (depending on its complex-
ity), with apparent contours of the object detected in
each view. Automatic detection is possible in some
conditions (almost uniform background) but in most
cases, a human supervision is necessary. Hence, the
number of views is a critical issue, strongly influenc-
ing the overall processing time. An efficient and ac-
curate SFS method with a small number of views is
therefore highly desirable.
When less data is available, the inherent loss of ac-
curacy can be compensated by a priori knowledge as
accounted for in inverse problems literature. If the ob-
ject to reconstruct is not totally unknown, SFS could
also benefit from this approach. The principle of a
deformable model is precisely well suited to incorpo-
rate shape prior. For example, the human body topol-
ogy is always the same and its shape is approximately
known a priori. A generic model is therefore easy to
construct. The main problem is the expression of the
model deformation to match the data, i.e. the apparent
contours in the context of SFS.
The method presented in this article is space de-
formation driven by silhouettes’ fitting: the projec-
tion of the model after this deformation should coin-
cide exactly with the silhouettes. Our main contri-
bution is a linear solution to the underlying problem
of 3D/2D non-rigid registration, unlike prior works
379
F. Menudet J., M. Becker J., Fournel T. and Mennessier C. (2007).
MODEL-BASED SHAPE FROM SILHOUETTE - A Solution Involving a Small Number of Views.
In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IU/MTSV, pages 379-386
Copyright
c
SciTePress
based on non-linear optimization framework. It is
therefore an efficient and reliable (no local minima)
technique. This is achieved by the displacements of
some control points located on the contour generators
of the surface model. Radial Basis Function (RBF) is
then used to construct a space deformation interpolat-
ing these displacements. A special case of RBF de-
formation is the Thin Plate Spline warping presented
in (Bookstein, 1989). In section 2, a review of related
works is presented. Then, section 3 details the differ-
ent steps of the proposed method: computation of the
control points, estimation of their displacements and
construction of the space deformation. Finally, some
results are shown in section 4. We illustrate our ap-
proach throughout the paper with human body mod-
eling, believed to be a relevant and important applica-
tion of our method.
2 RELATED WORK
A large literature covers the problem of the represen-
tation and construction of the visual hull. Two cate-
gories of methods can be distinguished: surface-based
and volume-based approaches. The former is based
on the intersection of the visual cones generated by
the apparent contours. Intuitive at first, it is actually
difficult to implement for complex objects but some
efficient algorithms exist for polyhedral reconstruc-
tion (Franco and Boyer, 2003). Because of the com-
plexity of surface-based approach, volumetric recon-
struction is widely used (Szeliski, 1993). The space
of interest is first discretized by a voxel representa-
tion. Then, each voxel is projected on the different
image planes and voxels falling completely outside of
any silhouette are removed. This simple approach has
nevertheless some drawbacks. First, a surface repre-
sentation is often preferred (for rendering purpose for
example). Secondly, voxel representation is only an
approximation of the exact visual hull due to quanti-
zation error. Decreasing the voxel size obviously re-
duces the problem at the expense of rapidly growing
computational and storage costs (octree hierarchies
are therefore often used).
Model-based reconstruction of objects has re-
cently received great attention. It has been used to
incorporate complementary data (mainly silhouette
and texture). In (Isidro and Sclaroff, 2003), a sur-
face mesh is iteratively deformed in order to obtain
a photometrically-consistent solution while enforcing
silhouette constraint. A similar snake-like technique
is described in (Hernandez and Schmitt, 2004): the
texture driven force deforming the model is here com-
puted with a multi-stereo correlation voting approach
and a gradient vector flow diffusion (Xu and Prince,
1998). A silhouette force is added to constraint the
model. A method combining explicit (mesh) and im-
plicit (metaball) surface has been proposed in (Ilic and
Fua, 2006): the model is fitted to silhouette and stereo
data with a Dirichlet Free Form Deformation incor-
porated in an optimization scheme. Another solution
(Sullivan and Ponce, 1998) in the case of pure SFS
(when only silhouette data is available) is based on an
initial spline surface deformed to match the silhou-
ette. Again, it relies on an optimization framework
(gradient-descent) subject to local minima and long
computation time. In the field of human body mod-
eling, the most closely related work to ours is (Hilton
et al., 2000) which deforms a generic human model in
order that its silhouettes approximate the given ones.
Is is a fast technique but the model deformation is
restricted to affine transformation. Another different
approach is statistical shape model as used in (Fleute
and Lavall
´
ee, 1999) but the training phase (inherent
to this kind of model) is by itself a problem.
3 MODEL-BASED SHAPE FROM
SILHOUETTE
3.1 Overview of the Proposed Method
The main idea is to compute some control points on
the contour generators of the 3D model and move
them with the objective that their projection match the
silhouettes. RBF finally provides a smooth deforma-
tion of the 3D space (hence of the embedding model),
based on the displacement of these control points. Re-
call that for a given camera position, a contour gener-
ator (or rim) of a surface
S is the curve on S which
projects to the apparent contour in the image (Rosten
and Drummond, 2003). Moreover, surface normals
along rim curve are orthogonal to the viewing direc-
tion (see figure 1).
Let us now detail the different steps of the pro-
posed method. Apparent contours are assumed de-
tected in calibrated images (Zhang, 2000), i.e. in-
trinsic and extrinsic parameters of the camera are
known. Vectors and matrices are respectively type-
set using bold and sans-serif fonts, e.g. vector P and
matrix K. Vectors are always column vectors such as
P = [X Y Z]
T
. The cross-product of two 3D vectors is
denoted by × and the symbol means equality up to
a non zero scale factor.
Figure 1: Contour generator (or rim) on a surface is the
locus of points where surface normal is orthogonal to the
line of sight.
3.2 Initial Registration
The first step is a registration of the 3D model in the
world coordinate system. It may be based on few
landmarks (at least 3) located on the model and iden-
tified in various images. 3D reconstruction of the
image points is possible thanks to the known cam-
era parameters. Finally, a transformation between the
model landmarks and the reconstructed points can be
computed and applied to the model. With 3 land-
marks, only a similarity (rotation, translation plus
scaling) can be estimated. When more landmarks are
available, affine or projective transformation is a valid
alternative, bringing a first deformation of the model
in addition to the registration of the coordinate sys-
tems.
3.3 Computation of Control Points on
the 3d Model
We want to place some control points on the rims of
the 3D model. The first step is the computation of
the rims, a classical problem occurring for example in
non-photorealistic rendering (Hertzmann and Zorin,
2000). Different techniques have been proposed, de-
pending on the way the surface is represented. Im-
plicit surfaces are probably the most convenient rep-
resentation for rim computation: an efficient algo-
rithm based on the solution of an ordinary differential
equation is available (Rosten and Drummond, 2003).
Nevertheless, implicit surfaces are difficult to render
and their deformation is not intuitive. Surfaces rep-
resented by triangular meshes are therefore preferred
for their versatility and fast rendering ability.
We have adopted the solution presented in (Hertz-
mann, 1999), well suited for triangular meshes rep-
resenting smooth surfaces (via subdivision technique
for example). Let N
v
be the surface normal at a ver-
tex V of the mesh. Let C be the position of the camera
center. As mentioned above (see figure 1), rim curve
is the locus of surface points P whose normal is per-
pendicular to the line of sight:
N
p
.(PC) = 0. (1)
For each vertex of the mesh, it is possible to compute
the normalized dot product between surface normal
and viewing direction:
d
v
=
N
v
.(VC)
k
N
v
kk
VC
k
. (2)
The surface is assumed continuous, hence if two ver-
tices V
1
and V
2
forming an edge of the mesh have
different signs for values d
v
, then there must be a rim
point on this edge - that is, a zero crossing of the dot
product function (see figure 2). The location of this
zero crossing along the edge can be estimated with a
linear interpolation between the 2 vertices using val-
ues d
V
1
and d
V
2
. The rim curve is then constructed by
connecting the rim points located on edges sharing a
common vertex.
Figure 2: The sign of the dot product between surface nor-
mal and viewing direction is computed at each vertex of the
mesh. A change of signs between two adjacent points al-
lows to identify an edge with a rim point (blue circle) which
can be connected to create the rim curve (red line).
This algorithm is applied to the different locations
of the camera. The control points are taken as the vis-
ible rim points previously obtained. Note that only
the visibility of these points has to be checked. It is
significantly simpler than the removal of hidden por-
tions of curves, as it would be required in a com-
puter graphics application. The result of this algo-
rithm clearly depends on the resolution of the mesh:
the smaller the triangles, the better the rim estimation
and the higher the number of control points. But in re-
turn, the number of triangles also influences the com-
putation time. A multi-resolution approach would be
clearly an efficient solution. Subdivision surface pre-
cisely allows to refine locally the mesh around the ini-
tial rim curve. In this way, only edges of the newly
created triangles are tested, speeding up the computa-
tion . This process is repeated until the desired reso-
lution is reached. As will be seen in the sequel (see
section 3.5), a large number of control points (more
than a few thousand) is actually undesirable.
3.4 Estimation of Control Points
Displacements
The present issue is to find for every previously com-
puted control point P, the displacement toward the
corresponding point Q. The projection of Q must be
located on the apparent contours, but an infinite num-
ber of points satisfy this constraint. Two observations
will guide us to compute Q:
Obviously, Q is located on the optical rays passing
through silhouette pixels.
To ensure a coherent deformation of the model,
control points P should be moved along the sur-
face normal N
P
. This is a common assumption
(Hernandez and Schmitt, 2004) which has the ad-
vantage to preserve the initial quality of the mesh
(especially an even repartition of its vertices).
Therefore, we propose the following idea to estimate
Q given a rim point P on the 3D model: compute λ
such that the point Q = P+ λN
p
projects to the appar-
ent contour (see figure 3).
Figure 3: Estimation of Q, the rim point on the real object
corresponding with the rim point P of the 3D model.
Because the camera is calibrated, coordinates
[X Y Z]
T
of P are known in the 3D camera frame. For
the same reason, homogeneous coordinates of its pro-
jection p can also be determined from the intrinsic
parameters matrix K (Zhang, 2000):
p = [uv1]
T
KP. (3)
λ can be computed in two steps:
1. Starting from the point p in image, find the point
q = [u
v
1]
T
on apparent contour situated in the
direction of n
p
(the projection of N
p
).
2. Solve for λ the equation
K(P+ λN
p
) q (4)
which can be rewritten
K(P+ λN
p
) ×q p×q+ λn
p
×q = 0, (5)
giving 3 scalar equations for finding λ. One of
them may be preferred to avoid the smallest com-
ponents of n
p
×q.
3.5 Space Deformation
3.5.1 Problem Statement
The previous steps define only the displacements
P
i
Q
i
of N control points whereas a deformation of
space is required: an underlying R
3
R
3
mapping
S(P) = [s
x
(P) s
y
(P) s
z
(P)]
T
(6)
has to be estimated. It would allow to interpolate the
displacement of any 3D point, especially those of the
3D model. The problem is to find the three R
3
R
interpolant functions (s
x
, s
y
, s
z
) such that
s
x
(P
i
) = Q
x
i
s
y
(P
i
) = Q
y
i
for i = 1, ..., N (7)
s
z
(P
i
) = Q
z
i
with Q
i
=
Q
x
i
Q
y
i
Q
z
i
T
. There are obviously an infi-
nite number of functions verifying the interpolation
conditions (7). The problem is to find the “best one”.
3.5.2 Radial Basis Function Interpolation
Radial Basis Function (RBF) is a popular technique
to solve this problem of scattered data interpolation
(Carr et al., 2003). In general, suppose that the val-
ues f
i
of a R
d
R function f are known at N arbi-
trary points c
i
R
d
: the problem is to find an inter-
polant function s(x) such that s(c
i
) = f
i
= f(c
i
) for
i = 1, ..., N. A RBF is a function s of the form
s(x) = p(x) +
N
i=0
w
i
φ(
k
xc
i
k
) (8)
where p(x) is a polynomial with a low degree m and
φ(r) is the basis function defined for r [0, +[. The
w
i
s and c
i
s are respectively the weights and the cen-
ters of the RBF. Some common RBFs are:
gaussian: φ(r) = e
cr
2
(and optional polynomial)
multiquadric: φ(r) =
r
2
+ c
2
with p(x) = a
1
thin-plate spline (d=2): φ(r) = r
2
ln(r) with
p(x) = a
1
+ a
2
x+ a
3
y
biharmonic spline (d=3): φ(r) = r with p(x) =
a
1
+ a
2
x+ a
3
y+ a
4
z
triharmonic spline (d=3): φ(r) = r
3
with p(x) a
trivariate quadratic polynomial
Depending on the choice of φ and p, the resulting
function s will have different properties. For exam-
ple, splines give smooth interpolant functions because
they minimize energy in second (thin-plate, bihar-
monic) or third (triharmonic) derivative of s (Book-
stein, 1989; Carr et al., 2003). Note also that a closely
related technique to RBF is kriging with spline in-
terpolation as a special case (Trochu, 1993). Mainly
used in geostatistics, its goal is to find the best linear
unbiased estimator of a random function (best means
with minimal variance). Interestingly, in this frame-
work, the basis functions φ are interpreted as a spatial
correlation between the data which yields some in-
sights on their choice.
The interpolation problem is reduced now to the
computation of the weights w = [w
1
... w
N
]
T
and coef-
ficients a = [a
1
... a
k
]
T
of
p(x) = a
1
p
1
(x) + ... + a
k
p
k
(x) (9)
where {p
1
, ..., p
k
}is a basis for polynomials of degree
at most m (for example {1, x, y} in the case of the 2D
thin-plate spline). Let f be the vector [ f
1
... f
N
]
T
, Φ the
N ×N matrix with Φ
ij
= φ(
c
i
c
j
) and P the N ×k
matrix with P
ij
= p
j
(c
i
). The interpolation conditions
can then be rewritten in matrix form:
Φ P
w
a
= f. (10)
This system is underdetermined (N+k unknowns and
N equations). Therefore, the so-called “side condi-
tions” are imposed on w to ensure that the transfor-
mation has square integrable second derivatives:
P
T
w = 0. (11)
(10) and (11) can be gathered into the following
symmetric linear system:
Φ P
P
T
0
w
a
=
f
0
. (12)
The weights w and polynomial coefficients a are
the solution of (12) and completely define the inter-
polant function s(x). The different choices of φ(r)
and p(x) presented above guarantee an invertible sys-
tem under very mild conditions on the centers’ lo-
cations (not aligned for thin-plate spline while gaus-
sian and multiquadric place no restrictions). Never-
theless, using a large number of data points can even-
tually give rise to a ill-conditioned matrix in equation
(12). Moreover, basis functions with non-compact
support (e.g. polyharmonic splines) yield a dense sys-
tem whose solution requires
O (N
3
) operations.
3.5.3 Radial Basis Function Deformation
As said before, we are looking for 3 functions (s
x
, s
y
,
s
z
) describing a space deformation. These 3 RBFs
share the same centers (namely the N control points
P
i
), hence the matrix of system (12) is identical for the
3 functions which saves computation time. Because a
smooth transformation is desired, biharmonic or tri-
harmonic splines seem appropriate. The latter have
been used in free-form modeling because of their pre-
dictable and intuitive behavior (Botsch and Kobbelt,
2005). We adopt it for the same reason, using gener-
ally less than 1000 control points.
An interesting variant of the interpolation prob-
lem comes from the introduction of a regularization
term (i.e. smoothing), yielding approximation rather
than interpolation. It may be a better choice when the
reliability of the control points is low. The solution
to this approximation scheme is very similar to the
interpolation one: the only modification is the addi-
tion of a parameter λ in the diagonal of the matrix Φ
which is therefore replaced by Φ + λI. The parame-
ter λ controls the smoothness of the transformation:
with λ = 0, we obtain the interpolant transformation
whereas higher values of λ bring smoother transfor-
mations, until the weights w vanish when λ (sim-
ple polynomial transform, e.g. affine).
3.5.4 Progressive Deformation
The proposed deformation scheme relies on the fun-
damental assumption that the deformed rims coincide
with the rims of the transformed model. This assump-
tion looks very reasonable because the deformation is
smooth and normal to the surface. However, it is not
strictly true, especially when a large deformation is
applied. In other words, the rims computed after the
model deformation may not be projected exactly on
the apparent contours in images. This drawback oc-
curs especially when the initial model is far from the
real object. Moreover, in this case the computed rim
points are usually misplaced on the initial surface, i.e.
they do not correspond with the apparent contours of
the real object. It can generate some artifacts in the
space deformation.
We overcome both problems with a progressive
and regularized space deformation. It is achieved with
successive RBF deformations steered by a decreasing
smoothing parameter λ. After each deformation, con-
trol points and their displacements are recomputed.
Therefore, their reliability increases as the surface is
getting closer to the target shape, justifying the pro-
gressive decrease of λ. At the last step, λ is set to
0 to obtain exact interpolation. In our current imple-
mentation, the number of steps (namely 6) and the
successive values of λ are fixed a priori; affine trans-
formation (λ ) is forced during the first 2 iter-
ations, then λ = (1, 0.1, 0.01, 0) for the last 4 steps.
Note that these values are related to the overall size of
the model. Here, the control points are scaled to fit in
a unit cube before forming the RBF system.
4 APPLICATION TO HUMAN
BODY MODELING
4.1 Experimental Setup
We demonstrate an application of our method with the
modeling of a mannequin female torso based on only
4 images (see figure 4). These images (1600×1200
pixels) were acquired with a digital still camera. The
photographs are only taken from the right side and
front side of the dummy. The apparent contours in the
different views are found in a semi-automatic way:
an initial contour is hand drawn, then deformed using
a snake method (Xu and Prince, 1998). All images
are calibrated using a plane-based calibration method
(Zhang, 2000) (calibration grid not shown in images).
4.2 Initialization
We use a generic model of female torso, represented
by a mesh of 9000 triangles. The initial affine reg-
istration of this model (see section 3.2) is computed
by pointing at 4 anatomical landmarks in the images
(seen as green dots in figure 4). Several views of the
3D model after this first registration step are presented
in the figure 5. The projection in the images of the
model vertices is shown in figure 4. At this step, vol-
ume, position and orientation of the model are rather
good but the matching of silhouettes is not respected.
Especially, breast, waist and spinal shapes are not co-
herent with the images.
4.3 Results
The rim curves are computed on the initial surface
model (black curves in figure 5), according to the
calibration parameters. The same figure shows the
displacements of 344 control points estimated from
the apparent contours (green segments). The control
points are well distributed all around the shape though
the photographs are taken from restricted viewpoints.
The displacement of the control points can be signifi-
cant in some areas (up to 4 cm).
The progressive deformation described previously
is finally applied. The resulting surface model after
space deformation is presented in figure 6 and its pro-
jection in the 4 images is shown in figure 7. Note
how the model perfectly matches the silhouettes (es-
pecially the shape of the waist). Nevertheless, the pro-
jection of the final model in two images not used in
Figure 4: The 4 images of a dummy torso with the anatomi-
cal landmarks used to compute the initial registration (green
dots). Projection of the initial model is superimposed.
Figure 5: Three different views of the initial 3D model. The
computed rim curves are shown in black. The green seg-
ments represent the displacements of control points used to
computed the RBF deformation.
the deformation reveals some differences (especially
around the right shoulder blade) between the recon-
structed and the actual shape (see figure 8). Indeed,
deformable model method is obviously not a com-
plete substitute for image data that could give, for ex-
ample, the shape of the shoulder blade. Accuracy has
been manually assessed by measuring some circum-
ferences on both real object and 3D model. A 3%
relative mean error of has been found, probably due
to camera parameters and manual measurements un-
certainties.
The computation time is only 3 seconds on a com-
puter equipped with a P4 3GHz processor and 1024
MB of RAM. This time is based on a basic C++ im-
plementation, but there is a lot of place for improve-
ment of speed. The most time consuming tasks are
rims computation and RBF system solving. In both
cases, more efficient algorithms are available (Hertz-
mann and Zorin, 2000; Botsch and Kobbelt, 2005).
Figure 6: Different views of the final 3D model, i.e. after
the space deformation.
4.4 Influence of the Initial Model
In another experiment, the generic female torso model
has been replaced by a cylinder. The goal is to evalu-
ate the influence of the initial surface. The 3D shape
obtained after deformation is shown in figure 9. The
final result is globally similar to the previous surface
and its projection in images still matches the appar-
ent contours. The main problem is the shape of the
chest which is not satisfying, especially the loss of
the concave part between the breasts. However, this
is not surprising since it is a well-known limitation
of SFS. The resulting shape is however rather accept-
able, given the small number of images. It confirms
that our model-based SFS is able to bring a signifi-
cant improvement of quality when the model is well
chosen.
5 CONCLUSION
A model-based Shape From Silhouette technique has
been presented. The result is a surface representa-
tion of the visual hull, more convenient than volu-
metric representation. Moreover, the introduction of
a shape prior allows to reduce the number of input
images while keeping a good reconstruction quality.
Figure 7: Projection in the images of the final 3D model.
Figure 8: Projection of the final 3D model in two additional
images, not used during the deformation.
Our main contribution is a reliable and efficient so-
lution to the underlying problem of 3D/2D non-rigid
registration between the 3D model and the 2D silhou-
ettes. The core idea is that the apparent contours give
sufficient strong clues to construct a progressive de-
formation of space based on Radial Basis Function.
Figure 9: The 3D shape obtained from the deformation of a
cylinder.
Unlike previous works, the method does not use
non-linear optimization strategies: a simple linear
system has to be solved. Efficiency is closely con-
nected with the size of this system, i.e. the number
of control points used to drive the deformation. Our
method is clearly intended for smooth objects, hence
a few hundred of control points are sufficient. An-
other factor influencing the computation time is the
number of images but the method is designed to keep
it small. The main limitations are the availability of
a generic 3D model of the object and the requirement
of some landmarks for the initial registration. A point
to improve is the decreasing scheme of the smoothing
parameter used in the deformation. We plan to find
a dynamic adjustment of this sequence, based on an
analysis of the displacements of control points.
Human body is a good subject for our method.
We have demonstrated the modeling of a human torso
with as few as 4 images. A classical SFS technique
would need at least 20 images to produce such a re-
alistic 3D model. The next step is the modeling of
the whole body. The work presented in (Hilton et al.,
2000) may, for example, be adapted to fit our ap-
proach. A lot of applications in medical, garment or
virtual reality fields would be possible.
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