Snakes and Active Contours - Bezier and Splines in Image Processing and Machine Vision - page 179

9
Snakes and Active Contours
9.1 Introduction
9.1.1 Splines and Energy Minimization Techniques
If we examine the origin and historical usage of splines, there is a clear link
between smooth curves and energy minimization. Splines were originally thin,
wooden strips used in both traditional ship and aircraft building techniques
to help create the curved hulls needed to allow the ship or aircraft to travel
speedily through the water or air. The splines were used for the process of
lofting, which is required to expand the small scale plan from a boat designer
into the full-size plans required for construction. The rescaling was done by
transferring a series of measurements called offsets on to the large lofting floor
of the lofthouse and then interpolating these offset points by bending splines
into smooth interpolative curves. The wooden splines were bent into shape
with the aid of a small number of weights, called ducks, and clamps that kept
the wood in position as shown in Figure 9.1. The ducks provided positional
constraints and the clamps could provide both positional and tangential (i.e.,
derivative) constraints. Wooden splines have a natural tendency to assume a
smooth shape to minimize overall bending energy while satisfying the imposed
constraints.
This energy minimization approach to producing smooth curves is fun-
damentally different from the approach taken with Bernstein-Bezier and B-
splines. In the former approach, the smoothness derives naturally from the
minimization of energy—as in the case of the wooden spline. By contrast,
the latter approach ensures smoothness by representing a shape as a sum
of smooth mathematical functions. So it could be persuasively argued that
energy minimization methods are more faithful to the historical concept of
splines than the modern concept of mathematical splines developed by Bezier
et al.
Indeed Kass, Witken, and Terzopoulos [88, 89] referred to snakes as a
form of spline in their groundbreaking paper presented at the very first ICCV