was proposed by Osher and Sethian [Osh88] in which the original curve y(O) is
first embedded in the surface of an arbitrary 2D Lipschitz continuous function
qbo(x,y) as its level set (contour line) at zero level. For example, we can select
450 (x, y) to be equal to the signed distance function from the boundary of y(0),
positive (negative) in the interior (exterior) of y (0). Then, the evolving planar curve
is embedded as the zero-level set of an evolving space-time function qb (x, y, t):
u = { (x, y) : r y, t) = 0},
y(0) = {(x, y) : ~,(x, y, 0) - ~o(X, y) - 0}.
Geometrical properties of the evolving curve can be obtained from spatial deriva-
tives of the level function. Thus, at any point on the front the curvature and
outward normal of the level sets can be found from 4>:
llV~ll' llV ll
The curve evolution PDE of Eq. (10.54) induces a PDE generating its level function:
qbt = VIIV~II,
4~(x, y, O) = 4>o(X, y).
If V = 1, the above function evolution PDE is identical to the fiat circular dilation
PDE of Eq. (10.3 5) by equating scale with time. Thus, we can view this specific
dilation PDE as a special case of the general function evolution PDE of Eq. (10.60)
in which all level sets expand in a homogeneous medium with V = 1. Propagation
in a heterogeneous medium with V = V (x, y) > 0 leads later to the eikonal PDE.
10.5 Distance Transforms
10.5.1 Distance Transforms and Wave Propagation
For binary images, the distance transform is a compact way to represent their
multiscale dilations and erosions by convex polygonal structuring elements whose
shape depends upon the norm used to measure distances. Specifically, a binary
image can be divided into the foreground set S _c R 2 and the background set
S c
= { (x, y) : (x, y) ~ S}. For shape analysis of an image object S, it is often
more useful to consider its inner distance transform by using S as the domain to
measure distances from its background. However, for the applications discussed
herein, we need to view S as a region marker or a source emanating a wave that will
propagate away from it into the domain of
S c.
Thus, we define the
outer distance
of a set S with respect to the metric induced by some norm II 9 lip,
p = 1, 2, .... oo, as the distance function:
=~ A II(x -
v,y -
w)llp. (10.61)

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