Snakes and Active Contours - Bezier and Splines in Image Processing and Machine Vision - page 181

For example, when presented with a photograph of an unobstructed person,
almost everyone would agree on the same partitioning of such an image into
person and background. Yet, this important and seemingly trivial task of im-
age labeling is extraordinarily dicult to achieve automatically. It turns out
that snakes and general energy minimization techniques are some of the most
promising methods for automated analysis—though all methods have their
weaknesses.

9.2 Classical Snakes

An active contour or snake as proposed by Kass et al. [88] is a closed or open
curve defined within a 2D image domain that is able to evolve or deform to
conform to features, such as edges and lines, in the image under analysis.
The evolution of the snake is formulated as an iterative energy minimization
process in which the snake is deformed to reach a locally minimum energy
configuration.
The total energy associated with the snake is defined as the sum of an internal
energy term ,an external energy term ,andan external force term . The internal
energy influences the shape and smoothness of the snake and depends only on
the properties of the snake itself, independent of the underlying image ( cf the
bending strain in a wooded spline). The external energy is what causes the
snake to align itself with image features and is derived from the underlying
image. The force term allows the user to manually force the snake to move in
particular directions to aid in finding the best solution.
In general, curves cannot be described by one-dimensional functions as
they may double back on themselves, so we parameterize the snakes along
their length as follows:

ν ( s )=( x ( s ) ,y ( s )) ,s ∈ [0 , 1] . (9.1)

Thus as s varies from 0 to 1 inclusive, we traverse the entirety of the snake.
In practice, we discretize this parameterization and evaluate the energy of the
snake at, say, N sample points, often called control points, along the contour.
These points actually define the snake so they must be spaced somewhat
closer than would be the case for the control points of a Bernstein-Bezier
spline—generally they are spaced just a few pixels apart so that small image
features are not missed.
Thus we have initially a set of N points such that

ν n = ν ( s ) | s = k/N ,k ∈ [0 ..N − 1] . (9.2)

In other words, we place the N control points successively along the length
of the snake at locations ( x n ,y n )= ν n = ν ( s ) evaluated at monotonically
increasing values of s by assumption. Normally, we try to space the points
evenly along the snake initially. However, even if we don't, the membrane