GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 105

Fig. 4.15. Examples of equidistance surfaces (surfaces for the distance values d s =
r, r = integer are shown for distance functions d s ,s = 1 , 2 ,..., 7 , in Table 4.6). In
d 3 of (3), slash line = double structure. In d 4 of (4), slash line = double structure
for r =even.In d 5 of (5), slash line = triple structure for r = even, double structure
for r = odd. In d 6 of (6), slash line = double structure for r = even, dark = triple
structure for r =even.In d 7 of (7), slash line = double except for r = 3 n − 2 ,dark
= double structure for r = n .Here n -fold structure means

that n equidistance surfaces exist on the plane perpendicular to the coordinate axis
such as k

n − 1 ,triplefor r =

3 3

− j plane.

Then, for an arbitrary β M and arbitrary pairs of pixels, if and only if the
length of the minimal path for the variable neighborhood path between two
pixels with the neighborhood sequence β M is never longer than that with β M ,
the length of the variable neighborhood minimal path with β M

becomes a

distance measure.
Furthermore, for an arbitrarily given constant c , the relative error of the
distance according to the above minimal path length to the Euclidean distance
can be kept smaller than c by finding a suitable neighborhood sequence. How-
ever, the absolute difference between such distance and the Euclidean distance
cannot necessarily become smaller than c . In other words, there exists a pair
of pixels such that the distance between them measured by the minimal path
length with the neighborhood sequence differs from the Euclidean distance
by the amount larger than c . For proof of these theorems, see [Yamashita84].
The similar results may be expected to be correct for a 3D image, but have
not been reported yet.