Snakes and Active Contours - Bezier and Splines in Image Processing and Machine Vision - page 184

be satisfied by a straight line with uniformly spaced control points. Note that
if there are no external forces imposed, the sum of the membrane terms in the
snake energy is minimized by contracting all control points into a single point;
just like a soap bubble that becomes a tiny droplet when the air escapes. The
membrane term is often considered to be providing elasticity —it makes the
snake shrink during evolution somewhat like a stretched elastic band.
The membrane term also penalizes curvature indirectly because curvature
increases the snake energy by increasing the distance between the control
points—a straight line is always the shortest distance between two points in
a Euclidean space. Note that in the case of a closed snake, there must always
be some curvature to allow the snake to connect back on to itself.
The second-order or thin-plate term in (9.5) penalizes changes in curvature
and makes the snake behave like a thin metal plate. 3 The thin-plate term only
penalizes changes in the distance between the control points. This becomes
obvious if we rewrite the argument of the modulus in the thin-plate term of
(9.5),

ν i +1 − 2 ν i + ν i− 1 ,

in the form

ν i− 1 ) .
So unlike the membrane term, minimization of the thin-plate term does not
provide the elastic behavior that collapses the snake to a single point under
evolution. Rather it provides stiffness as exhibited by, say, a thin metal plate
that ensures that both the control points and the curvature are uniformly
distributed. This term makes the snake form smooth curves during evolution
just like the traditional wooden spline for lofting in shipbuilding. Thus during
evolution, a closed snake with no external constraints will tend to become
circular due to the stiffness provided by the thin-plate term before it finally
collapses to a single point due to the elasticity provided by the membrane
term.
The image energy is formulated so that its value is minimal at the location
of the desired image features. Kass et al. [88] considered a weighted set of
features based on lines, edges, and terminations ( i.e., the end points of lines)
as follows:

( ν i +1 − ν i ) − ( ν i −

E image = w line E line ( ν s )+ w edge E edge ( ν s )+ w term E term ( ν s ) . (9.6)

In this chapter and henceforth, we will only consider edge energy, so a
suitable image energy term is:

2

E image = E edge ( ν ( s )) = −|∇ I ( x, y ) | (9.7)

where ∇ I ( x, y ) is the gradient of the image intensity.

3 cf a thin-plate spline, is the surface with minimum mean square second derivative
energy that interpolates a given collection of points.