Image Processing Reference
In-Depth Information
2.1.1 Properties of Integer Functions
Next we state a few properties of floor (⌊.⌋) and ceiling (⌈.⌉) integer func-
tions without proof. These are frequently used in this chapter.
Property 2.1. ∀x ∈ R, x ≤⌈x⌉ < x + 1 and x≥⌊x⌋ > x−1
Property 2.2. For p ∈ N, p = ⌈x⌉ iff p ≥x > p−1
Property 2.3. ∀x∈ R and ∀a ∈ Z, ⌈x+a⌉ = ⌈x⌉+a and ⌊x+a⌋ = ⌊x⌋+a
Property 2.4. ∀x,y ∈ R and x≥ y, ⌈x⌉≥⌈y⌉ and ⌊x⌋≥⌊y⌋
Property 2.5. ∀a,b,m∈ N, ⌈a/m⌉+⌈b/m⌉≥⌈(a + b)/m⌉
Property 2.6. ∀a,b,m∈ N and a ≥b, ⌈a/m⌉≥⌈b/m⌉
Property 2.7. ∀a,r,s ∈ N and r ≤ s, ⌈a/r⌉≥⌈a/s⌉
Property 2.8. ∀a,b,m∈ N, b ≥a and m = ⌈b/a⌉, a ≥⌈b/m⌉
Property 2.9. ∀a,b ∈ P and b= 0,
a
b
a + b−1
b
=
.
Property 2.10. ∀r,p,t ∈ P, p = 0 and 0 ≤t ≤ p,
t−1
p−t−1
r−j
p
r + j
p
+
= r.
j=0
j=0
2.2 Neighborhoods, Paths, and Distances
Unlike Euclidean geometry, where neighborhoods are based on the contin-
uum limit, shortest paths are unique and straight, and distances are Euclidean,
digital spaces, due to their very nature of discreteness, offer a wide variety
of neighborhood and path structures and associated shortest paths and dis-
tances. The discrete points are conceived to be vertices of an underlying graph
where neighborhoods define varying forms of adjacency, costs are attached to
the constraints of adjacency, paths are based on the continuity of adjacency,
shortest paths are non-unique, and distances are interesting sets of metrics
that offer a wealth of structural and mathematical properties to explore.
In this section, we introduce the notions of neighborhoods (adjacency),
paths and distances in a generic framework of an underlying graph before
exploring their specific properties in the subsequent sections.
