Image Processing Reference
In-Depth Information
(a) Initial contour
(b) After iteration 1
(c) After iteration 2
(d) After iteration 3
Figure 6.5
Effect of removing control by spacing
localised curvature. Setting γ to zero would force the snake to ignore image data and
evolve under its own forces. This would be rather farcical. The influence of γ is reduced
in applications where the image data used is known to be noisy. Note that one fundamental
problem with a discrete version is that the final solution can oscillate when it swaps
between two sets of points which are both with equally low energy. This can be prevented
by detecting the occurrence of oscillation. A further difficulty is that as the contour becomes
smaller, the number of contour points actually constrains the result as they cannot be
compressed into too small a space. The only solution to this is to resample the contour.
(a) Initial contour
(b) After iteration 1
(c) After iteration 2
(d) After iteration 3
Figure 6.6
Effect of removing low curvature control
6.3.3
Complete (Kass) snake implementation
The Greedy method iterates around the snake to find local minimum energy at snake
points. This is an
approximation
, since it does not necessarily determine the 'best' local
minimum in the region of the snake points, by virtue of iteration. A
complete snake
implementation
, or
Kass snake
, solves for all snake points in one step to ensure that the
snake moves to the best local energy minimum. We seek to choose snake points (
v
(
s
) =
(
x
(
s
),
y
(
s
))) in such a manner that the energy is minimised, Equation 6.8. Calculus of
variations shows how the solution to Equation 6.7 reduces to a pair of differential equations
that can be solved by finite difference analysis (Waite, 1990). This results in a set of
equations that iteratively provide new sets of contour points. By calculus of variations, we
shall consider an admissible solution
∧
(
s
perturbed by a small amount,
v
(
s
), which
achieves minimum energy, as:
dE
snake
(
v
( ) +
s
ε δ
v
( ))
= 0
s
(6.15)
d
