Digital Geometry in Image Processing

230 Digital Geometry in Image Processing

FIGURE 7.5 : An object A in Z

3

(left) and its outer is othetic cover for g = 2

(right). (See color insert.)

7.1.1 3-D Isothetic Covers

Given a grid G and a 3-D digital object A, we can approximate A by its 3-

D isothetic cover, which is defined as the minimum-volume isothetic polytope

P

G

(A) that contains A (Fig. 7.5). Mathematically, the following conditions

are satisfied:

• A ⊆ P

G

(A)

• for each p ∈

P

G

(A), 0 6 d

⊤

(p, A) < g

Here, g denotes the grid size, and P

G

(A) denotes the entire cover including

its surface P

G

(A) and interior region. The first condition implies that each

point of A lies on or inside

P

G

(A), and the