Fua - 3
images and corresponding camera models, we write
£i = £ si , (5)
1 <k<N
Si = S(Pr k (x„y i ,z i ),(Pr k (z i+ i, !«+„*.•+,)))/ £ L ?, j+1 , («)
1 <»< n 1 <»< n
where /: denotes the image number, Pr k (x,y, z) the pair of coordinates of the projection of point ( x,y,z )
into image k and L k ■ the length of the projection into image k of the segment i,j.
2.2 Smooth Snakes and Ribbons
These snakes are used to model smoothly curving features such as roads or ridge-lines.
2-D curves. Following Ivass el al. [ 1988], we choose the vertices of such curves to be roughly equidistant
and add to the image energy £j a regularization term £jj of the form
£d{C) = ^(x,- - x,_i) 2 -I- (y, - y,-i) 2 + /i2 ]T^(2x,- - x,_i - x i+ i) 2 + (2y, - y,_i - y l + i) J (7)
1 »
and define the “total energy” £t as
£t(C) = £d(C) + £i(C) (8)
The first term of £d approximates the curve’s tension and the second term approximates the sum of the
square of the curvatures, assuming that the vertices are roughly equidistant. In addition, when starting, as
we do, with regularly spaced vertices, this second term tends to maintain that regularity. To perform the
optimization we could use the steepest or conjugate gradient, but it would be slow for curves with large
numbers of vertices. Instead, it has proven much more effective to embed the curve in a viscous medium
and solve the equation of the dynamics
d£ dS
dS +a dt
with
d£_
dS
0,
d£o d£i
~дS' + ~дS ,
( 9 )
where £ is the energy of Equation 8, a the viscosity of the medium, and S the state vector that defines the
current position of the curve. Since the deformation energy in Equation 7 is quadratic, its derivative
with respect to S is linear and therefore Equation 9 can be rewritten as
Ps$t + ot(St — St- 1)
=> ( I\s + aI)S t
aSt-i -
( 10 )
where
d£ D
dS
I<sS ,
and A's is a sparse matrix. Note that the derivatives of £& with respect to x and y are decoupled so that
we can rewrite Equation 10 as a set of two differential equations in the two spatial coordinates
{I< + aI)X t
(K + aI)Y t
(*Xt-i -
ocY t -i —
d£i_
dX
d £/
dY
X,-,
y ,-1
