Distance Functions in Digital Geometry - Digital Geometry in Image Processing - page 65

polynomials are rational. This is due to the fact that in discrete cases,
all operations are performed with integers and hence in no way can an
irrational coe cient arise.

3. These polynomials are order preserving over m and r as the underlying
distance measure d m maintains an order.
Corollary 2.10. The following hold for surfaces and volumes:

1. H(m,n;r) ⊂ H(m,n;r + 1) and vol(m,n;r) < vol(m,n;r + 1).

2. S(m,n;r)∩S(m,n;r + 1) = φ, but surf(m,n;r) < surf(m,n;r + 1).

3. H(m,n;r) ⊂ H(m + 1,n;r) and vol(m,n;r) < vol(m + 1,n;r).

4. surf(m,n;r) < surf(m + 1,n;r), but S(m,n;r) and S(m + 1,n;r) are
unrelated.

€

vol(m,n;r), being a polynomial of degree n in r (Corollary 2.9), its coef-
ficients are computed from the values of volume for n + 1 distinct values of
r = 0,1,2,...,n and solving for n+1 simultaneous equations in n+1 coe cients
using matrix methods. [70] illustrates this approach in depth.

2.5.3.1 Vertices of Hyperspheres
In Table 2.8, we present the vertices of the hyperspheres of d m in 2-D and
3-D. The following theorem generalizes the result.

Theorem 2.20. The vertices of H(m,n;r) are given by φ(x) where

x = (r,r,··· ,r
m

,0,0,··· ,0
n−m

)

and φ(·) is the 2 n symmetry function (Definition 2.2). €

2.5.3.2 Errors in Surface and Volume Estimations
We present the error between digital and Euclidean hyperspheres from
[70].

Theorem 2.21. ∀n,n ≥ 1 and neighborhood m ≤ n, the volumetric errors
between Euclidean and digital hyperspheres satisfy the following:

• Absolute Error is unbounded and

• Relative Error is bounded