An affine shape constraint for geometric active contours - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 420

(1)
Let γ 1 and γ 2 be centered (according to the center of mass) and normalized arc length para-
meterizations of two closed planar curves having shapes F 1 and F 2 . Suppose that γ 1 and γ 2 have
a similar shape under rigid transformation.
In shape space this is equivalent to have:
(2)
where α is the scaling factor, θ the rotation angle, t 0 is the difference between the two starting
points of γ 1 and γ 2 .
Rigid motion estimation is obtained by minimizing the following quantity introduced by
[ 18 ] :
(3)
If α is equals to 1 [ 19 ], showed that Equation (3) is a metric in shape space and corresponds to
the Hausdorff distance between the two shapes. In practice, this result implies the uniqueness
of the motion's parameters so obtained. Using Fourier descriptors C k ( γ 1 ) and C k ( γ 2 ), minimiz-
ing E rr becomes equivalent to minimize f ( θ , t 0 ) in Fourier domain
(4)
[ 18 ] proposed an analytical solution to compute t 0 and θ . t 0 is one of the zeros of the following
function: