Image Processing Reference
In-Depth Information
FIGURE 5.1: Digitization of an ellipse with a
o
= 12.5 and b
o
= 10.5. Grid
points inside the ellipse have been marked with '*'. The boundary (that is
the OBQ image) has been highlighted by underlining. The chaincode of the
boundary (from left to right, i.e., clockwise) is '0007007077676766'.
Reprinted from
CVGIP: Graphical Models and Image Processing
, 54(5)(1992), S. Chattopadhyay et al., Parameter Es-
timation and Reconstruction of Digital Conics in Normal Positions, 385-395, Copyright(1992), with permission
from Elsevier.
The above result is significant because D can be directly constructed from
the image, whereas the formal characterization can be carried out in D =
D
X
∪D
Y
. In the rest of the section we view the OBQ simply as a pair of sets
D
X
and D
Y
. We have another interesting property of D or D
∗
.
Lemma 5.1. For any a,b denoting semi-major and minor axes of an ellipse
in canonical form, D is 8-connected.
€
Hence, D can be represented by a chain code such as 0007007077676766.
The above discussion, though reasoned for an ellipse, holds equally well for
any convex closed curve, like a circle. In case of open, unbounded convex curves
like parabolas or hyperbolas we assume it to be truncated by a horizontal or a
vertical line, say y = n or x = n, where n is the maximum range up to which
the digitization is performed. Then the above analysis can again be applied.
We consistently use the sub-(super-)script 'o' to represent the parame-
ters and quantities related to the original conic and use symbols without
postscripts for estimated ones. For example, a
o
and b
o
are the semi-axes of
the original ellipse E
o
with digitization D
o
(= D(E(a
o
,b
o
))), whereas a and b
stand to mean the estimates of a
o
and b
o
, respectively. We define the domain
Domain(D
o
) of D
o
as follows.
Definition 5.3. Domain(D
o
) of an ellipse given a set of digital points, D
o
,
is the set of all possible values of a and b of an ellipse that allow for recon-
struction, that is,