Digital Geometry in Image Processing

82 Digital Geometry in Image Processing

upper bound. Hence, only the relative error is used as a measure of goodness

of digital approximation to Euclidean dista nc e.

We quote the theorems below. The proofs are from [69].

Theorem 2.34. The absolute error a(u) is unbounded, that is, for all M ∈

R

+

, ∃u ∈ Σ

n

such that a(u) = |E

n

(u) − d

n

m

(u)| > M. 

Theorem 2.35. ∀m, n, 1 ≤ m ≤ n,

REL

0

(m, n) = max

u∈Σ

n

r(u) <

n

m

.



Corollary 2.16. ∀n, ∀m, 1 ≤ m ≤ n, REL

1−4

(m, n) are all bounded. The

bounds are as follows:

REL

1

(m, n) < max

n

√

m − 1, 1 −

n

n+m

o

REL

2

(m, n) < max

n

1 −

1

√

m

, 1 −

n

n+m

o

< 1

REL

3

(m, n) < max

n

√

m−1

√

m+1

,

n

n+2m

o

< 1

REL

4

(m, n) < max

n

√

m−1

√

m+1

,

n

√

n

2

+2nm+2m

2

o

< 1



The upper ...