Table Of Content.A.
Algorithms and Combinatorics 11
Editorial Board
R. L. Graham, Murray Hill B. Korte, Bonn
L. Lovasz, Budapest
Klaus Voss
Discrete Images,
Objects,
and Functions in Zn
With 100 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Klaus Voss
Friedrich-Schiller-Universitat Jena
Mathematische Fakultat
Lehrstuhl Digitale Bildverarbeitung
UHH, 17. OG
0-6900 Jena, Deutschland
Mathematics Subject Classification (1991): 68T10, 68U10, 68RlO
ISBN-13: 978-3-642-46781-3 e-ISBN-13: 978-3-642-46779-0
DOl: 10.1007/978-3-642-46779-0
Library of Congress Cataloging-in-Publication Data
Voss. K. (Klaus): Discrete images. objects, and functions in Z" 1 Klaus Voss. p. cm. -
(Algorithms and combinatorics ; 11) Includes bibliographical references and index.
ISBN-l3: 978-3-642-46781-3
I. Image processing-Mathematics. 2. Topology. I. Title. II. Series. TA1632.V6714 1993
621.36'1'OI51-dc20 92-45827
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Preface
Man wird dem einzelnen nicht gerecht,
wenn man es gesondert ins Auge jaftt, ohne
seinen Zusammenhang mit dem Ganzen zu
beachten und dem Beziehungssystem
Rechnung zu tragen, in dem es steht.
Thomas Mann
Science in general, as well as in each of its individual fields, is a part of human
culture. In that sense, this book aims to contribute to uncovering a small part of
the connections and relationships which bind image processing, categorized in
informatics and technology, with the knowledge accumulated over the years on
discrete structures.
How does one consider problems, models, mathematical methods and prac
tical applications? How does the search for ideas and the endeavour for know
ledge in the original work of scientists find expression? Is there something to be
learnt from science to date for future developments? Such questions have shaped
the content and style of this book.
Substantial impetus to the discrete theory of image processing was afforded
by the work of Rosenfeld and colleagues. Other fruitful sources of ideas con
sidered here are number theoretical problems (GauB, Minkowski) and integral
geometric investigations (Blaschke, Santalo). Since the beginning of the 1980s
I have strived to build upon these ideas a unified mathematical representation of
discrete image processing working together with R.K1ette and P.Hufnagl.
In a series of lectures, in many individual projects and in a monograph I have
on the one hand dealt with special problems, and on the other hand I have
striven to form a whole from the puzzling parts.
In the end the following scheme developed: at the beginning is a discrete
topology (instead of the classical set theoretical topology) in the form of the
largely combinatoric theory of the n-dimensional incidence structures. On this
foundation the various mathematical disciplines of set theory (n=O), graph
theory (n=1), and the theory of oriented neighborhood structures (n=2) can be
built upon one another.
Finally, especially fruitful for image processing is the consideration of the
n-dimensional number lattice Zn as a special incidence structure and the investi
gation of discrete objects, i.e. of subsets of Z ",
VI Preface
The introduction of an Euclidean metric into the number lattice links number
theoretical methods with geometrical methods (digital straight lines, convex
objects, digitalization effects etc.). Because image processing and computer
graphics come into very close contact in this field, the discreteness of the
number lattice can also lead to new algorithms of computational geometry.
If one assigns numerical values to the points of Z n, one obtains discrete
functions whose investigation forms the last part of the book. Instead of the
symbols J and e-+O of classical analysis, we use here the symbols ~ and .1k= 1.
Several examples will show that the discrete viewpoint also allows the develop
ment of new algorithms for practical image processing.
Set theory, graph theory, geometry, combinatorics, and algebra provide the
necessary mathematical tools. Certainly, these disciplines are only partially
handled during university studies. But since there are sufficient textbooks on
these subjects, I have endeavoured to be concise in the compilation of the
necessary formulas and laws. However, this book should not be seen primarily
as a mathematical textbook since the actual purpose of each theory lies in
imparting new insights into the subject and providing new methods for practical
application.
The book is a textbook as well as a monograph. Therefore the most
important results are cited once more at the end in the form of a compendium.
Many problems and connections are only briefly mentioned -partly to stimulate
the imagination of the reader but also because the questions can still not be
answered today.
The ideas presented here have been stimulated in discussions with many
colleagues in the past. Above all I would like to mention U.Eckardt (Hamburg),
E.Hertel (Jena), P.Hufnagl (Berlin), A.Hiibler (Jena), R.Klette (Berlin), H.Sii.Be
(Jena), and W.Wilhelmi (Passau). The manuscript of the book was read
carefully and critically by H.D.Hecker (Jena) and H.Sii.Be (Jena), and it was
linguistically corrected by R.Baker (Bradford). My very special thanks go to all
of them.
Klaus Voss
Jena, Juni 1992
Content
1 Neighborhood Structures
1.1 Finite Graphs
1.1.1 Historical Remarks ............................ 1
1.1.2 Elementary Theory of Sets and Relations . . . . . . . . . . . . . . . 3
1.1.3 Elementary Graph Theory ........................ 4
1.2 Neighborhood Graphs
1. 2.1 Graph Theory and Image Processing . . . . . . . . . . . . . . . . . . 9
1.2.2 Points, Edges, Paths, and Regions ................. , 12
1.2.3 Matrices of Adjacency ......................... 15
1.2.4 Graph Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
1.3 Components in Neighborhood Structures
1.3.1 Search in Graphs and Labyrinths . . . . . . . . . . . . . . . . . .. 19
1.3.2 Neighborhood Search .......................... 20
1.3.3 Graph Search in Images ........................ 22
1.3.4 Neighbored Sets and Separated Sets ................. 25
1.3.5 Component Labeling .......................... 27
1.4 Dilatation and Erosion
1.4.1 Metric Spaces .............................. 30
1.4.2 Boundaries and Cores in Neighborhood Structures ........ 32
1.4.3 Set Operations and Set Operators ................... 35
1.4.4 Dilatation and Erosion ......................... 36
1.4.5 Opening and Closing .......................... 38
2 Incidence Structures
2.1 Homogeneous Incidence Structures
2.1.1 Topological Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.2 Cellular Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.3 Incidence Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1.4 Homogeneous Incidence Structures . . . . . . . . . . . . . . . . . . 50
2.1.5 Zn as Incidence Structure ....................... 53
vrn
Content
2.2 Oriented Neighborhood Structures
2.2.1 Orientation of a Neighborhood Structure .............. 57
2.2.2 Euler Characteristic of a Neighborhood Structure ......... 59
2.2.3 Border Meshes and Separation Theorem .............. 63
2.2.4 Search in Oriented Neighborhood Structures ............ 66
2.2.5 Coloring in Oriented Neighborhood Structures ........... 68
2.3 Homogeneous Oriented Neighborhood Structures
2.3.1 Homogeneity in Neighborhood Structures .............. 72
2.3.2 Toroidal Nets .............................. 73
2.3.3 Curvature of Border Meshes in Toroidal Nets ...... . . . .. 76
2.3.4 Planar Semi-Homogeneous Graphs .................. 78
2.4 Objects in N-Dimensional Incidence Structures
2.4.1 Three-Dimensional Homogeneous Incidence Structures. . . . .. 82
2.4.2 Objects in Zn ............................... 85
2.4.3 Similarity of Objects .......................... 89
2.4.4 General Surface Formulas ....................... 91
2.4.5 Interpretation of Object Characteristics ............... 94
3 Topological Laws and Properties
3.1 Objects and Surfaces
3.1.1 Surfaces in Discrete Spaces .............. . . . . . . . . 99
3.1.2 Contur Following as Two-Dimensional Boundary Detection.. 100
3.1.3 Three-Dimensional Surface Detection .......... . . . .. 102
3.1.4 Curvature of Conturs and Surfaces. . . . . . . . . . . . . . . .. 105
3.2 Motions and Intersections
zn ......................
3.2.1 Motions of Objects in 109
3.2.2 Count Measures and Intersections of Objects. . . . . . . . . .. 111
3.2.3 Applications of Intersection Formula. . . . . . . . . . . . . . .. 113
3.2.4 Count Formulas .. . . . . . . . . . . . . . . . . . . . . . . . . .. 116
3.2.5 Stochastic Images .. . . . . . . . . . . . . . . . . . . . . . . . .. 120
3.3 Topology Preserving Operations
3.3.1 Topological Equivalence . . . . . . . . . . . . . . . . . . . . . .. 125
3.3.2 Simple Points ............................. 127
3.3.3 Thinning ................................ 131
Content IX
4 Geometrical Laws and Properties
4.1 Discrete Geometry
4.1.1 Geometry and Number Theory ................... 133
4.1.2 Minkowski Geometry . . . . . . . . . . . . . . . . . . . . . . . .. 136
4.1.3 Translative Neighborhood Structures. . . . . . . . . . . . . . .. 139
4.1.4 Digitalization Effects . . . . . . . . . . . . . . . . . . . . . . . .. 143
4.2 Straight Lines
4.2.1 Rational Geometry .......................... 148
4.2.2 Digital Straight Lines in Z2 .................... 151
4.2.3 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . .. 153
4.2.4 Straight Lines in Zn . . . . . . . . . . . . . . . . . . . . . . . . .. 157
4.3 Convexity
4.3.1 Convexity in Discrete Geometry .................. 161
4.3.2 Maximal Convex Objects ...................... 163
4.3.3 Determination of Convex Hull ................... 169
4.3.4 Convexity in Zn . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171
4.4 Approximative Motions
4.4.1 Pythagorean Rotations ........................ 173
4.4.2 Shear Transformations ........................ 175
4.3.3 General Affine Transformations .................. 178
5 Discrete Functions
5.1 One-Dimensional Periodical Discrete Functions
5.1.1 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
5.1. 2 Space of Periodical Discrete Function . . . . . . . . . . . . . .. 181
5. 1.3 LSI-Operators and Convolutions .................. 183
5.1.4 Products of Linear Operators .................... 186
5.2 Algebraic Theory of Discrete Functions
5.2.1 Domain of Definition and Range of Values . . . . . . . . . . .. 188
5.2.2 Algebraical Structures ........................ 190
5.2.3 Convolution of Functions ...................... 195
5.2.4 Convolution Orthogonality. . . . . . . . . . . . . . . . . . . . .. 196
5.3 Orthogonal Convolution Bases
5.3.1 General Properties in OCB's .................... 199
5.3.2 Fourier Transform .......................... , 201
x
Content
5.3.3 Number Theoretical Transforms .................. 203
5.3.4 Two-Dimensional NTT . . . . . . . . . . . . . . . . . . . . . . .. 208
5.4 Inversion of Convolutions
5.4.1 Conditions for Inverse Elements .................. 213
5.4.2 Deconvolutions and Texture Synthesis. . . . . . . . . . . . . .. 216
5.4.3 Approximative Computation of Inverse Elements ........ 218
5.4.4 Theory of Approximative Inversion ................ 220
5.4.5 Examples of Inverse Filters ..................... 221
5.5 Differences and Sums of Functions
5.5.1 Differences of One-Dimensional Discrete Functions ...... 225
5.5.2 Difference Equations and Z-Transform .............. 227
5.5.3 Sums of Functions .......................... 228
5.5.4 Bernoulli's Polynomials ....................... 230
5.5.5 Determination of Moments ..................... 232
5.5.6 Final Comments. . . . . . . . . . . . . . . . . . . . . . . . . . .. 236
6 Summary and Symbols ........................... 237
7 References ................................... 248
8 Index . ...................................... 265
1 Neighborhood Structures
1.1 Finite Graphs
1.1.1 Historical Remarks
Nearly 250 years ago, a small paper was published by Leonhard Euler [Eu36].
This paper can be considered the birth-certificate of graph theory. Euler stated
in a letter from March 1736:
"Es wurde mir einmal eine Aufgabe iiber eine Insel vorgelegt, die in der
Stadt Konigsberg gelegen und von einem FluJ3 umgeben ist, iiber welchen
sieben Briicken flihren, und es wurde gefragt, ob jemand die einzelnen
Briicken in einem zusammenhangenden Laufe so durchwandem konne,
daJ3 jede Briicke nur einmal iiberquert wird. Dabei wurde mir auch
mitgeteilt, daJ3 bisher weder jemand sich flir diese Moglichkeit verbirgt
noch jemand bewiesen habe, daJ3 es unmoglich sei, dies zu tun." [Euler,
K086]
TI)
Fig. 1.1.-1: The seven bridges of Konigsberg
It was a great achievement to abstract from all the details of river, island,
and bridges the simple graph structure on the right of figure 1.1. -1. Today, we
can solve a wide range of such problems using the theorem that an open or
closed Euler line in a graph only then exists when the graph has exactly two or
zero nodes with an odd number of outgoing edges.
Precisely 200 years later, D.Konig had published his book "Theorie der
endlichen und unendlichen Graphen" in Leipzig. The Hungarian author had
collected at that time almost all relevant publications creating the first textbook
Translation of German quotations are given at the end of the reference list.
TI)
