Digital Geometry in Image Processing

104 Digital Geometry in Image Processing

For a given c, [m

∗

l

(c), m

∗

u

(c)) of a DSLS defines a line interval in the (c, m)-

plane. Taking union of these inter vals for c

l

< c < c

u

, we get a region in the

(c, m)- plane. Thus, Domain(D

o

) is a region in the (c, m)-plane.

Proof: From Theorem 3.7, Domain(D

o

) ⊆ R

ul

. Again from Lemma 3.4, for

given c, c

l

< c < c

u

, and for all m, m ∈ [m

∗

l

(c), m

∗

u

(c)), (c, m) ∈ Dom(D

o

), if

m < m

∗

l

(c), then m < (y

p

− c)/p for some p. In other words, y

p

> mp + c or

y

p

> ⌊mp + c⌋. Thus, m < m

∗

l

(c) implies that (c, m) /∈ Dom(D

o

). Similarly,

m ≥ m

∗

u

(c) implies that (c, m) /∈ Dom(D

o

). Hence, for any c, c

l

< c < c

u

,

(c, m) ∈ Dom(D

o

) if and only if m ∈ [m

∗

l

(c), m

∗

u

(c)). 

Corollary 3.2. R

ul

is the smallest rectangle in the (c, m)-space so that ...