Chapter 4 Vortex detection - Computer Graphics and Visualization

Chapter 2. Geometry extraction techniques
where Ú × Ü × Ý × are the positions of the snake elements as a function of the
curve parameter ×, ÒØ is the internal spline energy, Ñ is the image energy which
causes a snake to be attracted to images features, and ÓÒ is the constraint energy
imposed by the user.
The internal energy ÒØ, which controls the continuity of a snake, is defined as a
weighted average: ÒØ « × Ú× × ¬ × Ú×× × . It consists of a first-order
derivative term Ú× × and a second-order derivative term Ú×× × . The first term makes
the snake first-order continuous, the second makes it second-order continuous, with
« × and ¬ × determining the weight of each term. If ¬ × , the snake is secondorder
discontinuous.
The image energy Ñ causes the snake to be attracted to lines, edges, and
(line segment) terminations, and is also defined as a weighted average: Ñ
ÛÐÒÐÒ Û ÛØÖÑØÖÑ.
The last term of the snake energy, the constraint energy ÓÒ, allows the user to
fix certain points of a snake to the background, or to connect two points of adjacent
snakes to each other with a spring. If two points have coordinates Ü and Ü , then
a spring between those two points could add a term Ü Ü to the constraint
energy, where is the stiffness of the spring.
When all the energy terms have been defined and substituted in the total energy
(2.1), the energy is minimized by a variational calculus approach. This basically means
that new snake positions are calculated as small variations of the old ones, such that
the energy gradually decreases at each iteration, and ultimately is minimized. This
will cause an initial snake defined around an object to shrink and lock onto the actual
object contour.
The snakes article has been the basis for numerous other types of deformable
curves. One further development were the fast active contours described in [Williams
& Shah, 1990]. There are several differences with the original snakes. One difference
is that the calculations are not done in continuous Ê space, but on a discrete grid of
pixels, which makes sense for image processing applications. Another difference with
the original snakes is the use of a greedy algorithm to compute new positions of snake
elements. For each of the neighbour pixels of a snake element, the greedy algorithm
calculates the effect of moving that element to that pixel on the new energy of the entire
contour. The algorithm then moves each element to the neighbour pixel which leads
to the lowest energy. One of the advantages of the greedy algorithm is that it works
much faster than variational calculus.
Yet another type of deformable curves is the Discrete Dynamic Contours described in
[Lobregt & Viergever, 1995]. Like previous models, they are based on minimization of
an energy, composed of external energy related to image features, and internal energy
related to local curvature. In this case, the energy is not minimized by variational
calculus or a greedy algorithm, but by means of forces, which are used in Newtonian
motion equations to determine the new positions of the nodes. The force on a node
is given by:
ÛÜ ÜÖ ÛÒ Ò ÑÔ (2.2)
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