digital geometry an introduction
DESCRIPTION
A brief introduction to Digital Geometry and Topology for students to get started working in this area and in related algorithms in Image ProcessingTRANSCRIPT
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Digital Geometry – An Introduction
Partha Pratim Das
Indian Institute of Technology, Kharagpur
Research Promotion Workshop on Digital Geometry
Indian Institute of Engineering, Science and Technology (IIEST)
June 23, 2014
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Outline
1 History of Geometry
2 Digital World
3 Fundamentals of Digital GeometryTessellation & DigitizationAdjacency, Connectivity, and NeighbourhoodDigital PicturePaths & Distances
4 Digital Distance GeometryMetric SpacesNeighbourhoods, Paths, and DistancesHypersheresComputations
5 World IS Digital
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Geometry is the study of measurements on Earth.
Transformations Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Rotations Yes Yes Yes YesTranslations Yes Yes Yes YesUniform Scalings No Yes Yes YesNon-Uniform Scalings No No Yes YesShears No No Yes YesCentral Projections No No No Yes
Invariants Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Lengths Yes No No NoAngles Yes Yes No NoRatios of Lengths Yes Yes No NoParallelism Yes Yes Yes NoIncidence Yes Yes Yes YesX-ratios of Lengths Yes Yes Yes Yes
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Geometry is the study of measurements on Earth.
Transformations Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Rotations Yes Yes Yes YesTranslations Yes Yes Yes YesUniform Scalings No Yes Yes YesNon-Uniform Scalings No No Yes YesShears No No Yes YesCentral Projections No No No Yes
Invariants Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Lengths Yes No No NoAngles Yes Yes No NoRatios of Lengths Yes Yes No NoParallelism Yes Yes Yes NoIncidence Yes Yes Yes YesX-ratios of Lengths Yes Yes Yes Yes
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Geometry is the study of measurements on Earth.
Transformations Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Rotations Yes Yes Yes YesTranslations Yes Yes Yes YesUniform Scalings No Yes Yes YesNon-Uniform Scalings No No Yes YesShears No No Yes YesCentral Projections No No No Yes
Invariants Euclidean Similarity Affine ProjectiveGeometry Geometry Geometry Geometry
Lengths Yes No No NoAngles Yes Yes No NoRatios of Lengths Yes Yes No NoParallelism Yes Yes Yes NoIncidence Yes Yes Yes YesX-ratios of Lengths Yes Yes Yes Yes
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Projective Geometry
Fuzzy Geometry
Fractal Geometry
Digital Geometry
Computational Geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Projective Geometry
Fuzzy Geometry
Fractal Geometry
Digital Geometry
Computational Geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Projective Geometry
Fuzzy Geometry
Fractal Geometry
Digital Geometry
Computational Geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Projective Geometry
Fuzzy Geometry
Fractal Geometry
Digital Geometry
Computational Geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Geometry?
Projective Geometry
Fuzzy Geometry
Fractal Geometry
Digital Geometry
Computational Geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Pioneer Geometers
Euclidean Astronomy Euclidean CartesianEuclid Aryabhata Brahmagupta Descartes
325-265 BC 476-550 597-668 1596-1650
Algebraic Digital Computational FractalCoxeter Rosenfeld Edelsbrunner Mandelbrot
1907-2003 1931-2004 1958- 1924-
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
What is Digital Geometry?
Digital geometry is the Geometry of the Computer Screen.The images we see on the TV screen, the raster display ofa computer, or in newspapers are in fact digital images.
Digital geometry deals with discrete sets (usually discretepoint sets) considered to be digitized models or images ofobjects of the 2D or 3D Euclidean space.
Digitizing is replacing an object by a discrete set of itspoints.
Digital Geometry has been defined for nD as well.
Main application areas:
Computer GraphicsImage Analysis
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Why Digital Geometry?
Points, straight lines, planes, circle, ellipses and hyperbolasetc 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 straightline, but rather a finite collection of dots on the screen,which the eye nevertheless perceives as a connected linesegment.
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Why Digital Geometry?
Computers have offered new paradigm of computing byDiscretization 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 SamplingDiscretization of f (x , y): Quantization
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Effects of Digitization on Euclidean Geometry
Euclidean Geometry Digital GeometryPropertiesthat hold
• Euclidean distanceis a metric in nD
• Euclidean distanceis a metric in nD
Propertiesthat holdafter exten-sion
• Jordan’s Curve the-orem holds in 2-D &3-D
• Jordan’s theoremin 2-D & 3-D holds ifmixed connectivity isused
• Every shortest pathwhich connects twopoints has a uniquemid-point
• A shortest path hasa unique mid-point ora mid-point pair
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Effects of Digitization on Euclidean Geometry
Euclidean Geometry Digital GeometryPropertiesthat do nothold
• The shortest pathbetween any pair ofpoints is unique
• The shortest pathbetween pair of pointsmay not be unique
• Only parallel linesdo not intersect
• Lines may not inter-sect but may not beparallel
• Two intersectinglines define an anglebetween them
• Angle is unlikely.Digital trigonometryhas been ruled out
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Focus of Digital Geometry
Task Examples
• Constructing digitized • Bresenham’s algorithmrepresentations of objects • Digitization & processing
• Study of properties of dig-ital sets
• Pick’s theorem, Convex-ity, straightness, or planarity
• Transforming digitized • Skeletons & MATrepresentations 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 derivativeSource: http://en.wikipedia.org/wiki/Digital geometry
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
11 Archimedean Lattices
All polygons are regular and each vertex is surrounded by thesame sequence of polygons. For example, (34, 6) means thatevery vertex is surrounded by 4 triangles and 1 hexagon.
Source: http://en.wikipedia.org/wiki/Percolation threshold
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
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 arraywe assume that the array is unbounded in all directions.Each pixel or voxel is associated with a lattice point (i.e., apoint with integer coordinates) in the plane or in 3D-space.
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Connectivity in 2D
Two lattice points in the plane are said to be:
8-adjacent if they are distinct and and their correspondingcoordinates differ by at most 1.
4-adjacent if they are 8-adjacent and differ in at most oneof 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
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Connectivity in 3D
Two lattice points are said to be:
26-adjacent if they are distinct and their correspondingcoordinates differ by at most 1.
18-adjacent if they are 26-adjacent and differ in at mosttwo of their coordinates.
6-adjacent if they are 26-adjacent and differ in at mostone 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
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Adjacency between a Point and a Set
A point p is said to be adjacent to a set of points S if p isadjacent to some point in S .
Two sets A, B are m-adjacent if there are points: a ∈ A,b ∈ B which are m-adjacent.
Point adjacency to a set Adjacency between Sets
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Simple Closed Curve
A connected curve that does not cross itself and ends at thesame point where it begins.
Simple Closed Curve Non-Simple Closed Curve
DigitalGeometry
Partha PratimDas
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History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Jordan Curve Theorem
• Let C be a Jordan (Simple Closed) Curve in the plane R2.Then its complement, R2 − C , consists of exactly twoconnected components. One of these components is bounded(interior) and the other is unbounded (exterior), and the curveC is the boundary of each component.
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Jordan Curve Theorem
• Let C be a Jordan (Simple Closed) Curve in the plane R2.Then its complement, R2 − C , consists of exactly twoconnected components. One of these components is bounded(interior) and the other is unbounded (exterior), and the curveC is the boundary of each component.
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Jordan Curve Theorem
• Let C be a Jordan (Simple Closed) Curve in the plane R2.Then its complement, R2 − C , consists of exactly twoconnected components. One of these components is bounded(interior) and the other is unbounded (exterior), and the curveC is the boundary of each component.
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Simple Closed Curve - Digital
A subset X of Z 2 is a simple closed curve if each point x of Xhas exactly two neighbours in X .
4 Curve 8 Curve
Not 4 Curve Not 8 Curve
Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Jordan Curve Theorem - Digital
The Jordan property does no hold if X and its complementhave the same adjacency.
(4,4) Adjacency (8,8) Adjacency
Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Jordan Curve Theorem - Digital
The Jordan property does no hold if X and its complementhave the same adjacency.
(4,8) Adjacency (8,4) Adjacency
To avoid topology paradoxes we use different adjacencyrelations for black and white points in 2D. In 3D the followingconfigurations are allowed: (6, 26); (26, 6); (6, 18); (18, 6).Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
m-Connected Set and m-Component
A set S is m-connected if S cannot be partitioned into twosubsets that are not m-adjacent to each other.
An m-component of a set of lattice points S is anon-empty m-connected subset of S that is notm-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
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Digital Picture
A digital picture is a quadruple P = (V ,m, p,B), where
V = Z 2 or Z 3, and B ⊂ V ,
(m, p) = (4, 8) or (8, 4) if V = Z 2 or
= (6, 26), (26, 6), (6, 18), or(18, 6) if V = Z 3
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 saidto be adjacent if they are m-adjacent
Two white points or a white point and a black point aresaid to be adjacent if they are p-adjacent.
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Digital Picture
Example
A digital picture (V ,m, p,B) will also be shortly called an(m, p) digital picture.
(4,8) Picture (8,4) Picture
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Components in a Digital Picture
Consider the digital picture below:
How many components does it have?
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
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Digital World
Fundamentals
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World ISDigital
Components in a Digital Picture
As an (8, 4) digital picture it has:
3 8-components and 3 4-components.
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Components in a Digital Picture
As a (4, 8) digital picture it has:
5 4-components and 2 8-components.
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Black and White Components
A component of the set of all black (white) points of adigital picture is called a black (white) component.
There is a unique infinite white component called thebackground.
(8, 4) digital picture. Pixels from a set S are marked witha square. {p, q} is 8-component of the set S but it is nota black component
Source: http://www.kis.p.lodz.pl/g/content/file/DENIDIA/intro2DT.pps
DigitalGeometry
Partha PratimDas
Agenda
History
Digital World
Fundamentals
Tessellation
Neighbourhood
Picture
Distances
nD Geometry
Metric Spaces
nD Graph
Hypersheres
Computations
World ISDigital
Paths of Points
For any set of points S , a path from p0 to pn in S is asequence {pi : pi ∈ S , 0 ≤ i ≤ n} of points such that pi isadjacent to pi+1 for all 0 ≤ i ≤ n. The path is closed ifpn = p0. A single point {p0} is a degenerate closed path.In a simple closed curve every point is adjacent to exactlytwo other points.
(4,8) Picture (8,4) PictureSimple closed black curves
DigitalGeometry
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Fundamentals
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Paths
Example
2D 3D
DigitalGeometry
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Distances in 2D & 3D
Example
Distance Functions in 2D
Distance d(x), x = u− v; u, v ∈ Z 2
City Block d4=|x1|+ |x2|Chessboard d8=max(|x1|, |x2|)
d4 > d8
Distance Functions in 3D
Distance d(x), x = u− v; u, v ∈ Z 3
Grid d6=|x1|+ |x2|+ |x3|d18 d18=max(|x1|, |x2|, |x3|,
⌈|x1|+|x2|+|x3|
2
⌉)
Lattice d26=max(|x1|, |x2|, |x3|)d6 > d18 > d26
DigitalGeometry
Partha PratimDas
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Fundamentals
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nD Geometry
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Hypersheres
Computations
World ISDigital
Digital Distance Geometry
Generalize Digital Geometry to n dimensions based onnotions of Distance
Distance Function:
d : Rn × Rn → R
is a function of two points in a space measuring theirseparation or dissimilarity.
Digital Distance Function:
d : Zn × Zn → P
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Examples of Distance Function
Example
For u ≡ (u1, u2, · · · , un), v ≡ (v1, v2, · · · , vn) ∈ Rn
Lp(u, v) = (∑n
i=1 |ui − vi |p)1p
L1(u, v) =∑n
i=1 |ui − vi |L2(u, v) = En(u, v) =
√∑ni=1 |ui − vi |2
L∞(u, v) = maxni=1 |ui − vi |
DigitalGeometry
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Digital World
Fundamentals
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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 CenterLeast Perimeter with Largest Area
Conics are distance defined
Geometries can be built on Distances
DigitalGeometry
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Digital World
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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 TransformationsMedial Axis Transform
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Metric Space
Any distance function d : X × X → R over a set X is called aMetric if it satisfies the following properties:
∀u, v,w ∈ X
Definite: d(u, v) = 0 ⇐⇒ u = v
Symmetric: d(u, v) = d(v,u)
Triangular: d(u, v) + d(v,w) ≥ d(u,w)
< X , d > is called a Metric Space.
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Metric Space
Any distance function d : X × X → R over a set X is called aMetric if it satisfies the following properties:
∀u, v,w ∈ X
Definite: d(u, v) = 0 ⇐⇒ u = v
Symmetric: d(u, v) = d(v,u)
Triangular: d(u, v) + d(v,w) ≥ d(u,w)
< X , d > is called a Metric Space.
DigitalGeometry
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Metric Space
Any distance function d : X × X → R over a set X is called aMetric if it satisfies the following properties:
∀u, v,w ∈ X
Definite: d(u, v) = 0 ⇐⇒ u = v
Symmetric: d(u, v) = d(v,u)
Triangular: d(u, v) + d(v,w) ≥ d(u,w)
< X , d > is called a Metric Space.
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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
< Z 2,E2 >: Digital Plane with Euclidean Metric
< Z 2, L1 >: Digital Plane with L1 Metric
< Z 2, L∞ >: Digital Plane with L∞ Metric
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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
< Z 2,E2 >: Digital Plane with Euclidean Metric
< Z 2, L1 >: Digital Plane with L1 Metric
< Z 2, L∞ >: Digital Plane with L∞ Metric
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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
< Z 2,E2 >: Digital Plane with Euclidean Metric
< Z 2, L1 >: Digital Plane with L1 Metric
< Z 2, L∞ >: Digital Plane with L∞ Metric
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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
< Z 2,E2 >: Digital Plane with Euclidean Metric
< Z 2, L1 >: Digital Plane with L1 Metric
< Z 2, L∞ >: Digital Plane with L∞ Metric
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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 fromproperties of being Definite, Symmetric, and Triangular:
d(u, v) =1
2(d(u, v) + d(v,u)) ≥ 1
2d(u,u) = 0
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Neighbourhood
A neighbourhood of a point is a set containing the point whereone 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 ofa point p if there exists an open ball with centre p and radiusr > 0, such that
Br (p) = B(p; r) = {x ∈ X | d(x , p) < r}
is contained in V .
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Neighbourhood Examples and Properties
L1 Norm L2 Norm L∞ NormSource: 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.
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Digital Neighbourhoods in 2D
Example
City-block Chessboard
Cityblock or 4-neighbours:N4((x , y)) = {(x , y)} ∪ {(x − 1, y), (x + 1, y), (x , y − 1), (x , y + 1)}
Chessboard or 8-neighbours: N8((x , y)) =
N4((x , y))∪{(x−1, y−1), (x+1, y−1), (x+1, y+1), (x−1, y+1)}
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Digital Neighbourhoods in 2D
Example
Knight
Knight’s neighbours: 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)}
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Digital Neighbourhoods in 3D
Example
Face (6) Edge (18) Corner (26)
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Digital Neighbourhoods in nD
• The Neighbourhood of a point u ∈ Zn is a set of pointsNeb(u) from Zn that are adjacent to u in some sense.• We associate a non-negative (finite or infinite) cost (calledNeighbourhood 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.
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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 conceptscan be implied.
Structure in Neighbourhoods:
Impractical to enumerate the neighbourhood of everyvertex (point) in an infinite graph.A compact repeatable structure for the neighbourhood atevery point is needed to build up a geometry.Hence the Neighbourhood Sets.
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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, whereδ(w) is the incremental distance or arc cost betweenneighbours separated by w. Hence, ∀v ∈ Neb(u),δ(u, v) = δ(u− v).
Neighbourhood Sets are Translation Invariant. Thechoice of origin has no effect on the overall geometry.
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Digital Neighbourhood Sets
We often denote a Neighbourhood Set as N(·) to indicatethe existence of one or more parameters on which the setmay depend.
Various choices of Neighbourhood Sets and associatedCost Function, therefore, induces different graphstructures with different notions of paths and distances.
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Characterizations of Digital Neighbourhood Sets
Neighbourhood Sets are characterized by the following factorsto make the distance geometry interesting and useful.∀w ∈ N(·) ⊂ Zn:
Proximity: Any two neighbours are proximal and share acommon hyperplane. That is, maxni=1 |wi | ≤ 1.
Separating Dimension: The dimension m of the separatinghyperplane is bounded by a constant r such that0 ≤ r ≤ m < n. That is, n −m =
∑ni=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.
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Characterizations of Digital Neighbourhood Sets
Isotropy & Symmetry: The neighbourhood is isotropic inall (discrete) directions. That is, all permutations and/orreflections of w, φ(w) ∈ N(·).
Uniformity: The neighbourhood relation is identical at allpoints along a path and at all points of the space Zn.
Translation Invariance follows directly from the differencevector definition of neighbourhood sets.
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Digital Neighbourhoods in 2D
Example
Cityblock or 4-neighbours have r = 1, m = 1 andconsequently 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 andboth 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
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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-dependentneighbourhoods, albeit cyclically, and thus violatesUniformity For example, octagonal distance use analternating sequence of 4- and 8- neighbourhoods.
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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 ).
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Digital Paths
The Length of a Digital Path denoted by |Π(u, v;N(·))|, isdefined 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 withthe smallest length is denoted as Π∗(u, v;N(·)). It is called theMinimal Path or Shortest Path.If the neighbourhood costs are all unity, then the length of theminimal path is given by |Π∗(u, v;N(·))| = M. It is the numberof points we need to touch after starting from u to reach v.
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Example of Digital Paths in 2D
Example
O(2) or 8-paths between two points u = 0 and v = (9,5) in2-D. The paths Π1 (marked by ’*’) and Π2 (marked by ’#’) areboth minimal while the path Π (marked by ’$’) is not minimal.Note that |Π∗1|=|Π∗2|=9 and |Π|=14.
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Example of Digital Paths in 3D
Example
A minimal O(2) or 18-path between two points (2,-7,5) and(-8,-4,13) in 3-D.
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m-Neighbour Distance
∀m, n ∈ N and ∀u, v ∈ Zn, we define m-neighbor distancednm(u, v) between u and v as
dnm(u, v) = max(
nmaxk=1|uk − vk |,
⌈∑nk=1 |uk − vk |
m
⌉)
Example
Distance d(u, v) = d(x), x = u− v; u, v ∈ Z 2
City Block d12 = d4=|x1|+ |x2|
Chessboard d22 = d8=max(|x1|, |x2|)
Distance d(u, v) = d(x), x = u− v; u, v ∈ Z 3
Grid d13 = d6=|x1|+ |x2|+ |x3|
d18 d23 = d18=max(|x1|, |x2|, |x3|,
⌈|x1|+|x2|+|x3|
2
⌉)
Lattice d33 = d26=max(|x1|, |x2|, |x3|)
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m-Neighbour Distance
Theorem
∀m, n ∈ N, dnm is a metric over Zn.
Lemma
∀m, n ∈ N, m > n and ∀x ∈ Zn, dnm(x) = dn
n (x)
Corollary
There exists exactly n number of m-neighbor distance functions
in n-D space Zn given by dnm(u, v) = max(dn
n (u, v),⌈dn
1 (u,v)m
⌉)
for 1 ≤ m ≤ n.
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m-Neighbour Distance
Lemma
∀u ∈ Zn, dnr (u) ≥ dn
s (u), ⇐⇒ r ≤ s
Lemma
∀x, y ∈ Zn, x and y are r -neighbors iff dnr (x, y) = 1 and
dns (x, y) > 1, ∀s, s < r
Corollary
∀x, y ∈ Zn are O(r)-adjacent neighbors iff dnr (x, y) = 1
Theorem
∀u, v ∈ Zn, dnm(u, v) = |Π∗(u, v;m : n)|
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t-Cost Distance
∀w ∈ Zn,∑n
i=1 |wi | = r ≤ n; δ(w) = min(t, n − r); 1 ≤ t ≤ n
Example
Cost of a minimal 2-cost path Π∗(2 : 3)from (2,-7,5)to (-8,-4,13) is |Π∗|= 8×2+2×1 = 18.Also D3
2 ((2,−7, 5),(−8,−4, 13)) =D3
2 ((10, 3, 8)) =max(10, 3, 8) +
max(min(10, 3),
min(3, 8),min(8, 10))
= 10 + 8 = 18.
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Hyper-Octagonal Distances
These neighbourhoods are path-dependent and keep onchanging along the path.
Example
Two paths from (0,0) to (9,5) using octagonal distance. Note|Π($)|=15 and |Π∗(#)|=10. Along a path, O(1)- andO(2)-neighbour alternates. Clearly |Π∗| has the minimal length.
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Hypersurface
S(N(·); r) is the Hypersurface of radius r in n-D forNeighborhood Set N(·). It is the set of n-D grid pointsthat lie exactly at a distance r , r ≥ 0, from the originwhen 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 ahypersurface S(N(·); r) is defined as the number of pointsin S(N(·); r).
In the digital space surf (N(·); r) often is a polynomial in rof degree n − 1 with rational coefficients.
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Hypersheres
H(N(·); r) is the Hypersphere of radius r in n-D forNeighborhood Set N(·). It is the set of n-D grid pointsthat lie within at a distance r , r ≥ 0, from the origin whend(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 hypersphereH(N(·); r) is defined as the number of points inH(N(·); r).
In the digital space vol(N(·); r) often is a polynomial in rof degree n with rational coefficients.
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Octagonal Disks
Example
Distance Vertices Perimeter / Area /Surface Area Volume
City Block {(±r, 0), (0,±r)} 4r 2r2 + 2r + 1
Chessboard {(±r,±r)} 8r 4r2 + 4r + 1
Digital Circles of 2D Octagonal Distances. (a) {4} (b){4,8} (c) {4,4,8} (d) {4,4,4,8} (e) {4,8,8} (f) {8}
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Knight’s Disks
Example
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Knight’s Disks
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Spheres in 3D
Example
Distance Vertices Perimeter / Area /Surface Area Volume
Lattice {(±r, 0, 0), (0,±r, 0), (0, 0,±r)} 24r2 + 2 18r3 + 12r2 + 6r + 1
d18 {(±r,±r, 0), (±r, 0,±r), (0,±r,±r)} 20r2 − 4r + 2 203r3 + 8r2 + 10
3r + 1
Grid {(±r,±r,±r)} 4r2 + 2 43r3 + 2r2 + 8
3r + 1
Sphere of d6 for radius = 6
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Spheres in 3D
Example
Sphere of a non-metric Distance
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Computations
Approximations of Euclidean Distance by Digital Distance
Distance Transforms
Medial Axis Transforms
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Conclusion
The World IS Digital
Source: https://www.youtube.com/watch?v=0fKBhvDjuy0
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References
Reinhard Klette and Azriel Rosenfeld (2004)
Digital Geometry: Geometric Methods for Digital Picture Analysis
Morgan Kaufmann.
Jayanta Mukhopadhyay, Partha Pratim Das, Samiran Chattopadhyay,Partha Bhowmick, Biswa Nath Chatterji (2013)
Digital Geometry in Image Processing
CRC Press.
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The End