Image Processing Reference
In-Depth Information
principal coordinate planes as discussed before. The chain codes are then
represented by a fixed number of elements. These representations of chain
code are called tuples. Four different characterization schemes of the chain
code of a 3-D DSLS are discussed in this section. These characterizations are
used in constructing the domain of the DSLS. For details, interested readers
may refer to [39].
3.3.2.1 n-characterization
This is the most rudimentary characterization scheme where a chain code
string C is characterized by the total number of 2-tuples it contains. This is
very similar to the (n)-characterization in the case of a 2-D DSLS.
3.3.2.2 (n,n
o1
,n
o2
)-characterization
This is also an extension of odd-even characterization of a tuple obtained
from a 3-D DSLS. Here, n
0j
, j = 1 or 2, is the total number of 1s in the
j-th component of the tuples in C. Subsequently, (n,n
o1
) and (n,n
o2
) are the
odd-even characterizations of the projections of the 3-D line on XY and XZ
planes, respectively.
3.3.2.3 (n,n
o1
,n
c1
,n
o2
,n
c2
)-characterization
This is yet another improvement over the last characterization scheme.
Here, n
cj
denotes the total number of occurrences of the consecutive pair of
unequal elements in the j-th component of the chain code C. That is, n
cj
is
the cardinality of the set {i| such that C
i,j
= C
i+1
,j}.
3.3.2.4 (n,q
1
,p
1
,s
1
,q
2
,p
2
,s
2
)-characterization
This is similar to and an extension of the faithful characterization of 2-D
DSLSs.
Let C
Z
(C
Y
) be the chain code containing only the first (second) elements
of C
i
for all i in the original string C. q
1
, p
1
, and s
1
(or q
2
, p
2
, s
2
) represent
the period, number of ones in one period, and the phase shift equivalent to
the intercept of the string C
Z
(C
Y
).
Combining Theorem 3.13 and the results in [78] and observing that C
Z
(C
Y
) denote the chain code for projection of the line on the XY (XZ) plane,
we can get a faithful characterization of C in 3-D.
3.3.3 Length Estimators for Different Characterizations
Let L(t) serve as the length attributed to all digitizations with charac-
terization t. For a string of n elements, the total length of the part of the
continuous contour between x = 0 and x = n is given by
f(n,θ,φ) = nsecφsecθ.
