Digital geometry - An introduction
- 1. 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 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
- 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
- 3. 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.
- 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
- 6. 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? Projective Geometry
- 7. 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? Projective Geometry Fuzzy Geometry
- 8. 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? Projective Geometry Fuzzy Geometry Fractal Geometry
- 9. 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? Projective Geometry Fuzzy Geometry Fractal Geometry Digital Geometry
- 10. 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? Projective Geometry Fuzzy Geometry Fractal Geometry Digital Geometry Computational Geometry
- 11. 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 Pioneer Geometers Euclidean Astronomy Euclidean Cartesian Euclid Aryabhata Brahmagupta Descartes 325-265 BC 476-550 597-668 1596-1650 Algebraic Digital Computational Fractal Coxeter Rosenfeld Edelsbrunner Mandelbrot 1907-2003 1931-2004 1958- 1924-
- 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
- 18. 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 11 Archimedean Lattices All polygons are regular and each vertex is surrounded by the same sequence of polygons. For example, (34, 6) means that every vertex is surrounded by 4 triangles and 1 hexagon. Source: http://en.wikipedia.org/wiki/Percolation threshold
- 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
- 22. 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 Adjacency between a Point and a Set A point p is said to be adjacent to a set of points S if p is adjacent 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
- 23. 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 Simple Closed Curve A connected curve that does not cross itself and ends at the same point where it begins. Simple Closed Curve Non-Simple Closed Curve
- 24. 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 • Let C be a Jordan (Simple Closed) Curve in the plane R2. Then its complement, R2 − C, consists of exactly two connected components. One of these components is bounded (interior) and the other is unbounded (exterior), and the curve C is the boundary of each component.
- 25. 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 • Let C be a Jordan (Simple Closed) Curve in the plane R2. Then its complement, R2 − C, consists of exactly two connected components. One of these components is bounded (interior) and the other is unbounded (exterior), and the curve C is the boundary of each component.
- 26. 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 • Let C be a Jordan (Simple Closed) Curve in the plane R2. Then its complement, R2 − C, consists of exactly two connected components. One of these components is bounded (interior) and the other is unbounded (exterior), and the curve C is the boundary of each component.
- 27. 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 Simple Closed Curve - Digital A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbours in X. 4 Curve 8 Curve Not 4 Curve Not 8 Curve Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
- 28. 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. (4,4) Adjacency (8,8) Adjacency Source: http://www.esiee.fr/ info/gt/SibTut01c.ppt
- 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
- 32. 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 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
- 33. 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 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
- 34. 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 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
- 35. 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 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
- 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
- 38. 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 Example 2D 3D
- 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
- 40. 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 Distance Geometry Generalize Digital Geometry to n dimensions based on notions of Distance Distance Function: d : Rn × Rn → R is a function of two points in a space measuring their separation or dissimilarity. Digital Distance Function: d : Zn × Zn → P
- 41. 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 Examples of Distance Function Example For u ≡ (u1, u2, · · · , un), v ≡ (v1, v2, · · · , vn) ∈ Rn Lp(u, v) = ( n i=1 |ui − vi |p) 1 p L1(u, v) = n i=1 |ui − vi | L2(u, v) = En(u, v) = n i=1 |ui − vi |2 L∞(u, v) = maxn i=1 |ui − vi |
- 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
- 44. 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 Any distance function d : X × X → R over a set X is called a Metric if it satisfies the following properties:
- 45. 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 Any distance function d : X × X → R over a set X is called a Metric 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)
- 46. 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 Any distance function d : X × X → R over a set X is called a Metric 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.
- 47. 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
- 48. 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
- 49. 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
- 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 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 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 Partha Pratim 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.
- 54. 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 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)}
- 55. 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 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)}
- 56. 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 3D Example Face (6) Edge (18) Corner (26)
- 57. 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 • 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 Partha Pratim 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 Partha Pratim 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 Partha Pratim Das Agenda History Digital World Fundamentals Tessellation Neighbourhood Picture Distances nD Geometry Metric Spaces 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 Partha Pratim Das Agenda History Digital World Fundamentals Tessellation Neighbourhood Picture Distances nD Geometry Metric Spaces nD Graph Hypersheres Computations 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 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 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 Partha Pratim Das Agenda History Digital World Fundamentals Tessellation Neighbourhood Picture Distances nD Geometry Metric Spaces nD Graph Hypersheres Computations World IS Digital 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 Partha Pratim Das Agenda History Digital World Fundamentals Tessellation Neighbourhood Picture Distances nD Geometry Metric Spaces nD Graph Hypersheres Computations 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 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 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.
- 67. 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 Example of Digital Paths in 2D Example O(2) or 8-paths between two points u = 0 and v = (9,5) in 2-D. The paths Π1 (marked by ’*’) and Π2 (marked by ’#’) are both minimal while the path Π (marked by ’$’) is not minimal. Note that |Π∗ 1|=|Π∗ 2|=9 and |Π|=14.
- 68. 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 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.
- 69. 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-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|)
- 70. 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-Neighbour Distance Theorem ∀m, n ∈ N, dn m is a metric over Zn. Lemma ∀m, n ∈ N, m > n and ∀x ∈ Zn, dn m(x) = dn n (x) Corollary There exists exactly n number of m-neighbor distance functions in n-D space Zn given by dn m(u, v) = max(dn n (u, v), dn 1 (u,v) m ) for 1 ≤ m ≤ n.
- 71. 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-Neighbour Distance Lemma ∀u ∈ Zn, dn r (u) ≥ dn s (u), ⇐⇒ r ≤ s Lemma ∀x, y ∈ Zn, x and y are r-neighbors iff dn r (x, y) = 1 and dn s (x, y) > 1, ∀s, s < r Corollary ∀x, y ∈ Zn are O(r)-adjacent neighbors iff dn r (x, y) = 1 Theorem ∀u, v ∈ Zn, dn m(u, v) = |Π∗(u, v; m : n)|
- 72. 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 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.
- 73. 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 Hyper-Octagonal Distances These neighbourhoods are path-dependent and keep on changing 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)- and O(2)-neighbour alternates. Clearly |Π∗| has the minimal length.
- 74. 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 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.
- 76. 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 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}
- 77. 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 Knight’s Disks Example
- 78. 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 Knight’s Disks Example
- 79. 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 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 20 3 r3 + 8r2 + 10 3 r + 1 Grid {(±r, ±r, ±r)} 4r2 + 2 4 3 r3 + 2r2 + 8 3 r + 1 Sphere of d6 for radius = 6
- 80. 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 Spheres in 3D Example Sphere of a non-metric Distance
- 81. 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 Computations Approximations of Euclidean Distance by Digital Distance Distance Transforms Medial Axis Transforms
- 82. 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 Conclusion The World IS Digital Source: https://www.youtube.com/watch?v=0fKBhvDjuy0
- 83. 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 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.
- 84. 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 The End