Image Processing Reference
In-Depth Information
Lemma 5.5. If r = x q + 1, then y r = q − 1 where p,q,x q ,y p are as defined
in Eq. 5.1 [41].
FIGURE 5.4: Proof of Lemma 5.5. “∗” ∈ E o and “.” ∈ E ul .
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.
Proof: From the definitions of D X and D Y , y r 6 q − 1 (Fig. 5.4). Next,
let us show that x q−1 > x q . In general, x q−1 > x q . If x q−1 = x q , then from
the definition of a u , (x q + 1)/
(1 − (q − 1) 2 /b l )
or q 2 < (q − 1) 2 . This is a contradiction. So, x q−1 > x q + 1 = r. But then
y r > q−1. Hence, y r = q−1.
(1 − q 2 /b l ) < (x q + 1)/
Note that in Example 5.4, q = 3 and x q = 11. So, r = 12 and y r = 2.
Lemma 5.6. D(E ul ) differs from D o only at point(s) like (x q + 1,q) where
a u is (are) achieved [41].
Proof: It is easy to see that D o ⊆ H(E ul ) where H(E) denotes the set
of all grid points lying inside the first quadrant of an ellipse E. Consider
any (x i ,i) ∈ D o X such that (x i ,i) ∈ D X (E ul ). Now, if (x
i ,i) ∈ D X (E ul ),
then x
i > x i + 1. But in that case either i = q and x
i = x
q = x q + 1 or
i = q and x
q > x q + 1. If i = q, then the a u −b l iteration would have stopped
with a higher a u and lower b l at (x i + 1,i). Hence, i = q. Next consider a
(i,y i ) ∈ D Y
and ∈ D Y (E ul ). So, there exists (i,y
i ) ∈ D Y (E ul ) such that
y
i > y i . But this contradicts the construction of b l . Hence, the result.
Now let us introduce a definition for comparing the digitization of an
arbitrary ellipse with the given digitization.
Definition 5.4. We write D X = D X (E) 6 D o X iff ∀i,0 6 i 6
⌊b o
⌋, ⌊a
(1−
i 2 /b 2 )⌋
6 x i where E = E(a,b) is an arbitrary ellipse for some a and b.
D X > D o X ,D Y 6 D Y and D Y > D Y are defined analogously.
For an arbitrary ellipse E, X- and Y-digitizations are said to be consistent
iff D X 6 D o X implies D Y 6 D Y and vice versa. The former case is represented
by D 6 D o and the latter as D > D o .
We prove, in the next theorem, the main result for reconstruction.
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