Îling, segmenta- 
influence of in- 
nents have been 
ges The method 
astline retrieval 
detect an edge 
es as the negati- 
ge. 
lulus of a wave- 
ze wavelet tran- 
ale. The desired 
active contour 
1, the obtained 
he snake on a 
> accurate scale. 
of the original 
fferent features 
"s. 
e defined in the 
a set of control 
(1) 
hich is used to 
ntrol point, the 
orhood and the 
d to update the 
ttles, one has 
, which can be 
points. 
| as 
ds (2) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
where E, represent the internal deformation energy defined as 
Eu = lap of epee Js © 
0 
where ¢ and f are weighting parameters that control the 
snake's tension and rigidity, respectively, and y (s) and v (s) 
denote the first and second derivatives of v(s) with respect to 
s. 
The second term in (2) is an external image energy. Typical 
forms of image energy are 
B = -IVrG, y) - (4) 
image 
2 
E 
7 
image 
zj(xy) (5) 
In (4), I(x,y) is a grey-level function (intensity); in (5), the 
intensity is a binary function (black and white, line-art image). 
A snake that minimizes E must satisfy the Euler equation 
ov (s)— Bv (s) - VE, =0> (6) 
which can be viewed as a force balance equation 
F_+F  _=0. (7) 
int image 
The internal force F, prevents stretching and bending, while 
the external force F pull the snake toward the desired 
image 
image edges. 
To find a solution to (6), the snake is made dynamic by treating 
V as function of time f as well as s, v(s,t)- 
A solution is obtained by seeking the snake position for which 
the velocity, defined by 
v (Sf) 2 av (s.t) - Bv (G,t) - VE, ue? (8) 
vanishes. 
3. NUMERICAL IMPLEMENTATION 
In the original model (Kass, 1987) a parametric contour 
representation is used to implement a semi-implicit integration 
scheme for discretizing the law of motion. 
Several authors have proposed different representations (Menet 
at al., 1993) including the use of finite element models (Cohen 
et al, 1992), subdivision curves (Hug et al, 1999) and 
analytical curve models (Metaxas and Terzopoulos, 1991) 
which work better to determine different features on the image. 
Various formulations of the image energy have also been 
447 
proposed to improve the original model, including the 
“Balloon” force field (Cohen, 1991) and the Gradient Vector 
Flow force field (Xu and Prince, 1998). 
In this paper, we use an algorithm based on a hierarchical 
filtering procedure, known as the scale-space continuation 
method (Within, 1983; Leymarie and Levine, 1993), 
subsequently generalized to fit the wavelet-based snake (Liu 
and Hwang, 1992). 
The idea of the scale-space continuation method (Leymarie and 
Levine, 1993) is to calculate the snake in a coarsely smoothed 
image; then the result at the coarse scale is used as an initial 
contour on a finer image and so on, until the native image 
resolution is reached. The original image is filtered through a 
family of Gaussian filters with different resolutions. Then, a 
differentiating filter, such as the Sobel filter, is applied to these 
Gaussian filtered images to produce approximations of the 
gradients of the Gaussian smoothed image. 
The next advance was to implement the gradient-based scale- 
space continuation method by means of a wavelet transform 
(Liu and Hwang, 1992). In this connection it has been shown 
(Mallat and Zhong, 1992) that the first derivatives of a family 
of Gaussian filters are equivalent to the corresponding wavelet 
transform coefficients multiplied by a scaling constant. 
Let the family Gaussian filters be suitably chosen so as to 
satisfy the 2-D dilation equation 
wx 
= ‘ (9) 
> s 
the 2-D wavelet functions are defined, in the x- and y-direction 
as 
1 
0, (x y) = s 4 
Vx, y) a 06(x, y) ; 
ox 
(10) 
me. 
eM 
then the wavelet transform of a (gray-scale) image /(x, y) in 
the x- and y-direction at scale s are 
wli(x,y) = I s V. (x, y) 
(11) 
Winx zi * yr (x, y)- 
It can be shown (Mallat and Zhong, 1992) that 
Wi (x,y) (12) 
; z sV(1*0,)(x. y) 
W,;1(x,y) 
therefore, the above equation implies that applying wavelet 
transform is equivalent to applying both smoothing and gradient 
operations.