digital geometry an introduction

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

[email protected]

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?

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

Agenda

History

Digital World

Fundamentals

Tessellation

Neighbourhood

Picture

Distances

nD Geometry

Metric Spaces

nD Graph

Hypersheres

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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

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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

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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.


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


DigitalGeometry

Partha PratimDas

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Digital World

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Components in a Digital Picture

Consider the digital picture below:

How many components does it have?


DigitalGeometry

Partha PratimDas

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As an (8, 4) digital picture it has:

3 8-components and 3 4-components.


DigitalGeometry

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Digital World

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As a (4, 8) digital picture it has:

5 4-components and 2 8-components.


DigitalGeometry

Partha PratimDas

Agenda

History

Digital World

Fundamentals

Tessellation

Neighbourhood

Picture

Distances

nD Geometry

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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


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

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Paths

Example

2D 3D

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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

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Digital World

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nD Geometry

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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

DigitalGeometry

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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

Partha PratimDas

Agenda

History

Digital World

Fundamentals

Tessellation

Neighbourhood

Picture

Distances

nD Geometry

Metric Spaces

nD Graph

Hypersheres

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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

Partha PratimDas

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History

Digital World

Fundamentals

Tessellation

Neighbourhood

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Distances

nD Geometry

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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

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.

DigitalGeometry

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Digital World

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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

DigitalGeometry

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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

DigitalGeometry

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Digital World

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Tessellation

Neighbourhood

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Distances

nD Geometry

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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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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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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.

DigitalGeometry

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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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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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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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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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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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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

digital geometry an introduction

Engineering

digital digital geometry

history of geometry

digital geometry deals

digital geometry partha

digital outline

fact digital images

introduction partha

nonuniform scalings