Digital Geometry in Image Processing

174 Digital Geometry in Image Processing

digitization of the hyperbola H

o

: x

2

/a

2

− y

2

/b

2

= 1. As in the ca se of an

ellipse, we treat the OBQ image of H

o

as the union of the following two sets

of digital points, namely D

X

and D

Y

.

D

o

X

= {(x

i

, i) : 0 6 i 6 y

n

and x

i

= ⌈a

o

p

(1 + i

2

/b

2

o

)⌉} and

D

o

Y

= {(i, y

i

) : ⌈a

o

⌉ 6 i 6 n and y

i

= ⌊b

o

p

(i

2

/a

2

o

− 1)⌋}.

The iterative refinement equations for computing tight bounds of a

o

and

b

o

are given in the following theorem.

Theorem 5.8. Let the upper and lower bounds of a

o

and b

o

are defined by

the following iterative algorithm where k > 0:

a

0

l

= x

0

− 1; a

k+1

l

= max

i

((x

i

− 1)/

p

(1 + i

2

/(b

k

l

)

2

)),

b

0

l

= y

r

/

p

((r/(r − 1)

2

− 1) where r = x

0

,

b

k+1

l