Descriptors) (Crimmins, 1982; Kunttu et al., 2004)
and afﬁne case (Arbter et al., 1990). In (Kuthirummal
et al., 2004), the authors have proposed an algebraic
afﬁne recognition constraint.
Although, differential invariants remain constant
in the case of projectivities, they still generally de
pend on the curve parameterization. The parameter
ization is chosen arbitrary and would not be neces
sary the same for different views. Thus, we need to
deal with both parameterization and geometric trans
formation invariance. Some works haveconsider such
problem and have proposed projective invariant de
scriptors which are independent with respect parame
terization (Weiss, 1992; Van Gool et al., 1992).
In this paper, we propose a projective and param
eterization invariant generation framework based on
the harmonic analysis theory and differential geome
try. In fact, we perform a projective curve reparam
eterization with a projective arc length. Thus two
equiprojective reparameterized contours from two
different views are equivalent up to a starting point.
Then, a complete and stable set of projective har
monic invariants is introduced by computing the C
3

Fourier coefﬁcients on projective arc length reparam
eterized contours.
The next section characterises the transformation
in the case of a projection by a pinhole camera. Then,
the equiprojective reparameterization process is intro
duced. In section 4, we construct the complete and
stable set of projective invariants. Next, the NURBS
curve ﬁtting is introduced. Section 6 presents some
experimental results.
2 GEOMETRIC
TRANSFORMATION AND
PERSPECTIVE PROJECTION
To characterize the geometric transformation between
two corresponding shape contours, we review the
concept of planar projective homography. Planar pro
jective homography (also called projectivity) is a lin
ear mapping in the planar projective space P
2
, H :
P
2
→ P
2
deﬁned up to an arbitrary factor λ by a 3×3
matrix H.
The relation between corresponding views of
points on a world plane Π in a 3D space, can be
modeled by a planar homography induced by the
plane. Consider two views p and p
′
of a 3D space
point P ∈ Π, in two camera frames f and f
′
respec
tively. We will denote their corresponding homoge
neous coordinates by ep = (x, y, 1),
e
p
′
= (x
′
, y
′
, 1) and
e
P = (X,Y, 1). Let M = K[I0] and M
′
= K
′
[Rt] be
the ﬁrst and the second camera projection matrices
(respectively), where R and t are the relative rotation
and translation between the cameras and K and K
′
are the respective internal calibration matrices. Thus,
ep = K[I0]
e
P and
e
p
′
= K
′
[Rt]
e
P.
Let n be the unit normal vector to the plane Π and
let d > 0 denote the distance of Π from the optical
center of the ﬁrst camera. The linear transformation
from ep to
e
p
′
can be expressed as:
e
p
′
= K
′
(R+
1
d
tn
T
)K
−1
ep = Hep
3 GINVARIANT
REPARAMETERIZATION
It was proven in differential geometry that a simple
curve is homeomorphic to the unit circle S
1
or the
real line R. Here, we consider only the ﬁrst case
which corresponds to closed contours. Thus, planar
shapes are represented by their smooth boundaries as
a closed 2D continuous parametric curve. In homoge
nous coordinates, a parameterization γ(t) of a planar
curve γ is an 1periodic function of a continuous pa
rameter t deﬁned by:
γ : [0, 1] −→ R
3
t 7−→ γ(t) = [x(t), y(t), 1]
t
.
(1)
and noted by γ(t).
Throughout this section, we indicate with γ : S
1
→
R
2
a closed planar contour and G a group acting on
R
2
.
It’s well known that a same parametric curve may
have different parameterizations. The invariants com
puted from two different parameterizations of the
same geometric curve are generally different. This
is due to parameterization dependence on transforma
tions. One solution to this problem consists in per
forming a Ginvariant reparameterization of the curve
where G is the geometric transformations group.
Deﬁnition 3.1. A reparameterization of a curve γ,
noted (γ(
b
t)), is deﬁned as follows:
γ(
˜
t) = γ(τ(t)) = [x(τ(t)), y(τ(t))]
t
, t ∈ [0, 1]. (2)
where τ is an increasing function deﬁned on [0,1].
Deﬁnition 3.2. A Ginvariant reparameterization is
the process of reparameterizing the curve by a G
invariant arc length.
Let γ
1
(t
1
) and γ
2
(t
2
) two parameterizations of a
geometric curve and its image by a geometric trans
formation g. After Ginvariant reparameterization,
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