Image Processing Reference
In-Depth Information
TABLE 5.1: Listing of distinct PDSVs for one-parameter planar curves in
the first quadrant.
PDSV
1
= ( -, +, - )
PDSV
2
= ( +, +, - )
PDSV
3
= ( +, +, + )
Reprinted from
Sadhana
18(2)(1993), S. Chattopadhyay et al., A Generalized Approach to the Reconstruction of
a Restricted Class of Digitized Planar Curves, 349-364, Copyright (1993), with permission from Indian Academy
of Sciences.
(or '-') in π is changed to a '-' (or '+') in π
c
, denotes the same monotonicity
matrix as π.
Property 2(a): (Permutation of spatial variables) Since x
i
and x
j
can
be interchanged (with proper adjustment of the coordinate system)
π = (π
1
,.,π
i−1
,π
i
,π
i+1
,..,π
j−1
,π
j
,π
j+1
,..,π
d
,..,π
n
), and
′
= (π
1
,.,π
i−1
,π
j
,π
i+1
,..,π
j−1
,π
i
,π
j+1
,..,π
d
,..,π
n
)
define equivalent monotone classes of curves.
π
Property 2(b): (Permutation of non-spatial variables) As a
i
and a
j
can be interchanged by renaming, their order also becomes immaterial in a
PDSV.
In this chapter we have treated only one (k = 1) or two (k = 2) parameter
planar (d = 2 and n = 3 or 4) curves. Thus, at most eight PDSVs may arise in
the first case while the number of PDSVs may go up to sixteen in the latter.
But there are three distinct PDSVs to consider for a planar curve with one
parameter. Similarly, the number of distinct PDSVs reduce to five for curves
with two parameters. The different PDSVs are listed in Tables 5.1 and 5.2.
The MM for PDSV
1
in Table 5.1 and PDSV
3
in Table 5.2 are shown in Tables
5.3 and 5.4, respectively.
We have selected the OBQ scheme for our approach. Now let us revisit the
formal definition of the OBQ scheme.
Definition 5.10. I(f), the OBQ image of a curve given by f(x,y,a,b) = 0,
is the set of digital points obtained as follows: While traversing f clockwise
(1) whenever f passes through a grid point P, then P belongs to I(f), (2)
whenever f crosses a grid line L but not a grid point, the nearest grid point
to the right of the curve and on L is a point of I(f), (3) no other point is
included in I(f).
€
Thus, for a closed contour f = 0, I(f) is the set of grid points inside f and
nearest to its boundary. Note also that we should collect the points on the
left of f if we traverse it counter-clockwise. This definition, though precises is
