2. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Outline
1 History of Geometry
2 Digital World
3 Fundamentals of Digital Geometry
Tessellation & Digitization
Adjacency, Connectivity, and Neighbourhood
Digital Picture
Paths & Distances
4 Digital Distance Geometry
Metric Spaces
Neighbourhoods, Paths, and Distances
Hypersheres
Computations
5 World IS Digital
4. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
What is Geometry?
Geometry is the study of measurements on Earth.
Transformations Euclidean Similarity Affine Projective
Geometry Geometry Geometry Geometry
Rotations Yes Yes Yes Yes
Translations Yes Yes Yes Yes
Uniform Scalings No Yes Yes Yes
Non-Uniform Scalings No No Yes Yes
Shears No No Yes Yes
Central Projections No No No Yes
5. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
What is Geometry?
Geometry is the study of measurements on Earth.
Transformations Euclidean Similarity Affine Projective
Geometry Geometry Geometry Geometry
Rotations Yes Yes Yes Yes
Translations Yes Yes Yes Yes
Uniform Scalings No Yes Yes Yes
Non-Uniform Scalings No No Yes Yes
Shears No No Yes Yes
Central Projections No No No Yes
Invariants Euclidean Similarity Affine Projective
Geometry Geometry Geometry Geometry
Lengths Yes No No No
Angles Yes Yes No No
Ratios of Lengths Yes Yes No No
Parallelism Yes Yes Yes No
Incidence Yes Yes Yes Yes
X-ratios of Lengths Yes Yes Yes Yes
12. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
What is Digital Geometry?
Digital geometry is the Geometry of the Computer Screen.
The images we see on the TV screen, the raster display of
a computer, or in newspapers are in fact digital images.
Digital geometry deals with discrete sets (usually discrete
point sets) considered to be digitized models or images of
objects of the 2D or 3D Euclidean space.
Digitizing is replacing an object by a discrete set of its
points.
Digital Geometry has been defined for nD as well.
Main application areas:
Computer Graphics
Image Analysis
13. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Why Digital Geometry?
Points, straight lines, planes, circle, ellipses and hyperbolas
etc have been studied for ages.
- We can draw them on paper and study.
Computers have offered a new method of drawing pictures
- Raster Scanning
A straight line is not what Euclid understood by a straight
line, but rather a finite collection of dots on the screen,
which the eye nevertheless perceives as a connected line
segment.
14. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Why Digital Geometry?
Computers have offered new paradigm of computing by
Discretization and Approximation
- Sampling - Nyquist Law
- Quantization
- Approximation by Iterative Refinements - Bisection,
Secant, Newton-Raphson, · · ·
An image is a 2D function f (x, y):
- x, y: spatial coordinates
- f : intensity / grey level
- f (x, y): Pixel
If x, y and f are discrete: Digital Image
Digitization of x, y: Spatial Sampling
Discretization of f (x, y): Quantization
15. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Effects of Digitization on Euclidean Geometry
Euclidean Geometry Digital Geometry
Properties
that hold
• Euclidean distance
is a metric in nD
• Euclidean distance
is a metric in nD
Properties
that hold
after exten-
sion
• Jordan’s Curve the-
orem holds in 2-D &
3-D
• Jordan’s theorem
in 2-D & 3-D holds if
mixed connectivity is
used
• Every shortest path
which connects two
points has a unique
mid-point
• A shortest path has
a unique mid-point or
a mid-point pair
16. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Effects of Digitization on Euclidean Geometry
Euclidean Geometry Digital Geometry
Properties
that do not
hold
• The shortest path
between any pair of
points is unique
• The shortest path
between pair of points
may not be unique
• Only parallel lines
do not intersect
• Lines may not inter-
sect but may not be
parallel
• Two intersecting
lines define an angle
between them
• Angle is unlikely.
Digital trigonometry
has been ruled out
17. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Focus of Digital Geometry
Task Examples
• Constructing digitized • Bresenham’s algorithm
representations of objects • Digitization & processing
• Study of properties of dig-
ital sets
• Pick’s theorem, Convex-
ity, straightness, or planarity
• Transforming digitized • Skeletons & MAT
representations of objects • Morphology
• Reconstructing ”real” ob-
jects or their properties
• Area, length, curvature,
volume, surface area, etc.
• Study of digital curves,
surfaces, and manifolds
• Digital straight line, circle,
plane
• Functions on digital space • Digital derivative
Source: http://en.wikipedia.org/wiki/Digital geometry
19. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Pixels and Voxels
The elements of a 2D image array are called pixels.
The elements of a 3D image array are called voxels.
To avoid having to consider the border of the image array
we assume that the array is unbounded in all directions.
Each pixel or voxel is associated with a lattice point (i.e., a
point with integer coordinates) in the plane or in 3D-space.
20. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Connectivity in 2D
Two lattice points in the plane are said to be:
8-adjacent if they are distinct and and their corresponding
coordinates differ by at most 1.
4-adjacent if they are 8-adjacent and differ in at most one
of their coordinates.
An m-neighbour of p is m-adjacent to p. Nm(p), for m = 4, 8,
denotes the set consisting of p and its m-neighbours.
4-Neighbourhood 8-Neighbourhood
21. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Connectivity in 3D
Two lattice points are said to be:
26-adjacent if they are distinct and their corresponding
coordinates differ by at most 1.
18-adjacent if they are 26-adjacent and differ in at most
two of their coordinates.
6-adjacent if they are 26-adjacent and differ in at most
one coordinate.
An m-neighbour of p is m-adjacent to p. Nm(p), for m = 6,
18, 26, denotes the set consisting of p and its m-neighbours.
6-Neighbourhood 18-Neighbourhood 26-Neighbourhood
29. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Jordan Curve Theorem - Digital
The Jordan property does no hold if X and its complement
have the same adjacency.
(4,8) Adjacency (8,4) Adjacency
To avoid topology paradoxes we use different adjacency
relations for black and white points in 2D. In 3D the following
configurations are allowed: (6, 26); (26, 6); (6, 18); (18, 6).
Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
30. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
m-Connected Set and m-Component
A set S is m-connected if S cannot be partitioned into two
subsets that are not m-adjacent to each other.
An m-component of a set of lattice points S is a
non-empty m-connected subset of S that is not
m-adjacent to any other point in S.
An 8-connected Set Its 4-components
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
31. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Digital Picture
A digital picture is a quadruple P = (V , m, p, B), where
V = Z2
or Z3
, and B ⊂ V ,
(m, p) = (4, 8) or (8, 4) if V = Z2
or
= (6, 26), (26, 6), (6, 18), or(18, 6) if V = Z3
The points in B (or V − B) are called the black (or white)
points of the picture.
Usually B is a finite set; so then P is said to be finite.
Two black points in a digital picture (V , m, p, B) are said
to be adjacent if they are m-adjacent
Two white points or a white point and a black point are
said to be adjacent if they are p-adjacent.
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
36. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Black and White Components
A component of the set of all black (white) points of a
digital picture is called a black (white) component.
There is a unique infinite white component called the
background.
(8, 4) digital picture. Pixels from a set S are marked with
a square. {p, q} is 8-component of the set S but it is not
a black component
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
37. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Paths of Points
For any set of points S, a path from p0 to pn in S is a
sequence {pi : pi ∈ S, 0 ≤ i ≤ n} of points such that pi is
adjacent to pi+1 for all 0 ≤ i ≤ n. The path is closed if
pn = p0. A single point {p0} is a degenerate closed path.
In a simple closed curve every point is adjacent to exactly
two other points.
(4,8) Picture (8,4) Picture
Simple closed black curves
39. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Distances in 2D & 3D
Example
Distance Functions in 2D
Distance d(x), x = u − v; u, v ∈ Z2
City Block d4=|x1| + |x2|
Chessboard d8=max(|x1|, |x2|)
d4 > d8
Distance Functions in 3D
Distance d(x), x = u − v; u, v ∈ Z3
Grid d6=|x1| + |x2| + |x3|
d18 d18=max(|x1|, |x2|, |x3|, |x1|+|x2|+|x3|
2 )
Lattice d26=max(|x1|, |x2|, |x3|)
d6 > d18 > d26
42. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Distance is a Fundamental Concept in Geometry
Neighbourhood, Adjacency, and Implicit Graph
Shortest Paths
Straight Lines
Geodesic on Earth
Parallel Lines
Equidistant Ever
Circle
Trajectory of a point equidistant from Center
Least Perimeter with Largest Area
Conics are distance defined
Geometries can be built on Distances
43. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Distance is a Fundamental Concept in Geometry
Divergence from Euclidean Geometry
Preservation of intuitive Properties
Preservation of Metric Properties
Quality of Approximation
How to work in digital domain with Euclidean accuracy?
Circularity of Disks
Computational Efficiency
Distance Transformations
Medial Axis Transform
50. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Metric Space
Common metric spaces are:
Example
< R2, E2 >: Euclidean Plane
< R3, E3 >: Euclidean Space
< R2, L1 >: Real Plane with L1 Metric
< R2, L∞ >: Real Plane with L∞ Metric
< Z2, E2 >: Digital Plane with Euclidean Metric
< Z2, L1 >: Digital Plane with L1 Metric
< Z2, L∞ >: Digital Plane with L∞ Metric
51. Digital
Geometry
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Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Metric Space
Often a metric is defined as Positive Definite, that is, Definite
d(u, v) = 0 ⇐⇒ u = v
as well as Positive:
d(u, v) ≥ 0
However, the property of being Positive actually follows from
properties of being Definite, Symmetric, and Triangular:
d(u, v) =
1
2
(d(u, v) + d(v, u)) ≥
1
2
d(u, u) = 0
52. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Neighbourhood
A neighbourhood of a point is a set containing the point where
one can move that point some amount without leaving the set.
V ∈ N(p) V /∈ N(p)
In a metric space M =< X, d >, a set V is a neighbourhood of
a point p if there exists an open ball with centre p and radius
r > 0, such that
Br (p) = B(p; r) = {x ∈ X | d(x, p) < r}
is contained in V .
53. Digital
Geometry
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Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Neighbourhood Examples and Properties
L1 Norm L2 Norm L∞ Norm
Source: http://en.wikipedia.org/wiki/File:Vector norms.svg
Well-behaved Neighbourhoods are:
Isotropy: Isotropic in all (most) directions.
Symmetry: Symmetric about (multiple) axes.
Uniformity: Identical at all points of the space.
Convexity: In the sense of Euclidean geometry.
Self-similar: Similar structure at varying resolution.
57. Digital
Geometry
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Das
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History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Digital Neighbourhoods in nD
• The Neighbourhood of a point u ∈ Zn is a set of points
Neb(u) from Zn that are adjacent to u in some sense.
• We associate a non-negative (finite or infinite) cost (called
Neighbourhood or Neighbour Cost)
δ : Zn
× Zn
→ R+
∪ {0}
between u and its neighbour v so that
δ(u, v) = c
where v ∈ Neb(u).
The cost is usually integral though it may be real-valued too.
Example
In 2-D, u = (2, 3) has a neighbourhood Neb(2, 3) =
{(3, 3), (1, 3), (2, 2), (2, 4)} with all 4 costs being 1.
58. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Digital Neighbourhoods in nD
Neighbourhood-induced Graph:
Neb(u), naturally, defines adjacency between points of Zn
.
With the associated with Neighbourhood cost, Neb(u)
therefore induces a weighted graph over Zn
.
We can define shortest paths and distances over this graph.
And once distances are defined, several geometric concepts
can be implied.
Structure in Neighbourhoods:
Impractical to enumerate the neighbourhood of every
vertex (point) in an infinite graph.
A compact repeatable structure for the neighbourhood at
every point is needed to build up a geometry.
Hence the Neighbourhood Sets.
59. Digital
Geometry
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Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Digital Neighbourhood Sets
A Neighbourhood Set N is a (finite) set of (difference)
vectors from Zn such that
∀u ∈ Zn
, Neb(u) = {v : ∃w ∈ N, v = u ± w}
With N, we associate a cost function δ : N → P, where
δ(w) is the incremental distance or arc cost between
neighbours separated by w. Hence, ∀v ∈ Neb(u),
δ(u, v) = δ(u − v).
Neighbourhood Sets are Translation Invariant. The
choice of origin has no effect on the overall geometry.
60. Digital
Geometry
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Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Digital Neighbourhood Sets
We often denote a Neighbourhood Set as N(·) to indicate
the existence of one or more parameters on which the set
may depend.
Various choices of Neighbourhood Sets and associated
Cost Function, therefore, induces different graph
structures with different notions of paths and distances.
61. Digital
Geometry
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Das
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Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
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nD Graph
Hypersheres
Computations
World IS
Digital
Characterizations of Digital Neighbourhood Sets
Neighbourhood Sets are characterized by the following factors
to make the distance geometry interesting and useful.
∀w ∈ N(·) ⊂ Zn:
Proximity: Any two neighbours are proximal and share a
common hyperplane. That is, maxn
i=1 |wi | ≤ 1.
Separating Dimension: The dimension m of the separating
hyperplane is bounded by a constant r such that
0 ≤ r ≤ m < n. That is, n − m = n
i=1 |wi | ≤ n − r.
Separating Cost: The cost between neighbours is integral.
That is, δ(w) ∈ P. Often the cost is taken to be unity.
62. Digital
Geometry
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Digital World
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Tessellation
Neighbourhood
Picture
Distances
nD Geometry
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World IS
Digital
Characterizations of Digital Neighbourhood Sets
Isotropy & Symmetry: The neighbourhood is isotropic in
all (discrete) directions. That is, all permutations and/or
reflections of w, φ(w) ∈ N(·).
Uniformity: The neighbourhood relation is identical at all
points along a path and at all points of the space Zn.
Translation Invariance follows directly from the difference
vector definition of neighbourhood sets.
63. Digital
Geometry
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Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
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nD Graph
Hypersheres
Computations
World IS
Digital
Digital Neighbourhoods in 2D
Example
Cityblock or 4-neighbours have r = 1, m = 1 and
consequently only line separation is allowed.
N4((x, y)) = {(x, y)} ∪ {(x − 1, y), (x + 1, y), (x, y − 1), (x, y + 1)}
{(±1, 0), (0, ±1)}, k = 4
Chessboard or 8-neighbours have r = 0, m = 0, 1 and
both point- and line-separations are allowed. N8((x, y)) =
N4((x, y))∪{(x −1, y −1), (x +1, y −1), (x +1, y +1), (x −1, y +1)}
{(±1, 0), (0, ±1), (±1, ±1)}, k = 8
64. Digital
Geometry
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Das
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Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
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Exceptional Neighbourhood Sets
At times the characteristic properties are violated:
1 Knight’s distance: NKnight((x, y)) = {(x, y)} ∪
{(x − 1, y − 2), (x − 1, y + 2), (x + 1, y − 2), (x + 1, y + 2),
(x − 2, y − 1), (x − 2, y + 1), (x + 2, y − 1), (x + 2, y + 1)}
{(±1, ±2), (±2, ±1)}, k = 8
does not obey Proximity.
2 t-Cost distances use non-Unity Costs. ∀w ∈ N(·) ⊂ Zn:
• n
i=1 |wi | = r ≤ n: Separating plane of any dimension
• δ(w) = min(t, n − r), where t, 1 ≤ t ≤ n
3 Hyperoctagonal distances use path-dependent
neighbourhoods, albeit cyclically, and thus violates
Uniformity For example, octagonal distance use an
alternating sequence of 4- and 8- neighbourhoods.
65. Digital
Geometry
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Digital World
Fundamentals
Tessellation
Neighbourhood
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World IS
Digital
Digital Paths
Given a Neighbourhood Set N(·), a Digital Path Π(u, v; N(·))
between u, v ∈ Zn, is defined as a sequence of points in Zn
where all pairs of consecutive points are neighbours. That is,
Π(u, v; N(·)) : {u = x0, x1, x2, ..., xi , xi+1, ..., xM−1, xM = v}
such that ∀i, 0 ≤ i < M, xi , xi+1 ∈ Zn and xi+1 ∈ N(xi ).
66. Digital
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Digital World
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Distances
nD Geometry
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Digital Paths
The Length of a Digital Path denoted by |Π(u, v; N(·))|, is
defined as
|Π(u, v; N(·))| =
M−1
i=0
δ(xi+1 − xi)
Usually there are many paths from u to v and the path with
the smallest length is denoted as Π∗(u, v; N(·)). It is called the
Minimal Path or Shortest Path.
If the neighbourhood costs are all unity, then the length of the
minimal path is given by |Π∗(u, v; N(·))| = M. It is the number
of points we need to touch after starting from u to reach v.
69. Digital
Geometry
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Digital World
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Neighbourhood
Picture
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Digital
m-Neighbour Distance
∀m, n ∈ N and ∀u, v ∈ Zn, we define m-neighbor distance
dn
m(u, v) between u and v as
dn
m(u, v) = max(
n
max
k=1
|uk − vk|,
n
k=1 |uk − vk|
m
)
Example
Distance d(u, v) = d(x), x = u − v; u, v ∈ Z2
City Block d1
2 = d4=|x1| + |x2|
Chessboard d2
2 = d8=max(|x1|, |x2|)
Distance d(u, v) = d(x), x = u − v; u, v ∈ Z3
Grid d1
3 = d6=|x1| + |x2| + |x3|
d18 d2
3 = d18=max(|x1|, |x2|, |x3|, |x1|+|x2|+|x3|
2 )
Lattice d3
3 = d26=max(|x1|, |x2|, |x3|)
74. Digital
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Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Hypersurface
S(N(·); r) is the Hypersurface of radius r in n-D for
Neighborhood Set N(·). It is the set of n-D grid points
that lie exactly at a distance r, r ≥ 0, from the origin
when d(N(·)) is used as the distance.
S(N(·); r) = {x : x ∈ Zn
, d(x; N(·)) = r}
The Surface Area surf (N(·); r) = ||S(N(·); r)|| of a
hypersurface S(N(·); r) is defined as the number of points
in S(N(·); r).
In the digital space surf (N(·); r) often is a polynomial in r
of degree n − 1 with rational coefficients.
75. Digital
Geometry
Partha Pratim
Das
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World IS
Digital
Hypersheres
H(N(·); r) is the Hypersphere of radius r in n-D for
Neighborhood Set N(·). It is the set of n-D grid points
that lie within at a distance r, r ≥ 0, from the origin when
d(N(·)) is used as the distance.
H(N(·); r) = {x : x ∈ Zn
, 0 ≤ d(x; N(·)) ≤ r}
The Volume vol(N(·); r) = ||H(N(·); r)|| of a hypersphere
H(N(·); r) is defined as the number of points in
H(N(·); r).
In the digital space vol(N(·); r) often is a polynomial in r
of degree n with rational coefficients.