Muutke küpsiste eelistusi

Elements Of Digital Geometry, Mathematical Morphology, And Discrete Optimization [Kõva köide]

(Uppsala Univ, Sweden)
  • Formaat: Hardback, 488 pages
  • Ilmumisaeg: 22-Feb-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 981124829X
  • ISBN-13: 9789811248290
Teised raamatud teemal:
Elements Of Digital Geometry, Mathematical Morphology, And Discrete OptimizationSuurem pilt
  • Formaat: Hardback, 488 pages
  • Ilmumisaeg: 22-Feb-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 981124829X
  • ISBN-13: 9789811248290
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9789811248290.html
The author presents three distinct but related branches of science in this book: digital geometry, mathematical morphology, and discrete optimization. They are united by a common mindset as well as by the many applications where they are useful. In addition to being useful, each of these relatively new branches of science is also intellectually challenging.The book contains a systematic study of inverses of mappings between ordered sets, and so offers a uniquely helpful organization in the approach to several phenomena related to duality.To prepare the ground for discrete convexity, there are chapters on convexity in real vector spaces in anticipation of the many challenging problems coming up in digital geometry. To prepare for the study of new topologies introduced to serve in discrete spaces, there is also a chapter on classical topology.The book is intended for general readers with a modest background in mathematics and for advanced undergraduate students as well as beginning graduate students.
Preface v
Acknowledgments ix
List of Figures xxi
List of Tables xxiii
1 Introduction 1(20)
1.1 Why digital geometry?
1(10)
1.2 Why mathematical morphology?
11(2)
1.3 How is image processing at all possible?
13(3)
1.4 Why discrete optimization?
16(1)
1.5 Notes
17(2)
1.6 Exercises
19(2)
2 Sets, mappings, and order relations 21(18)
2.1 Introduction
21(1)
2.2 Sets
21(1)
2.3 Notation for sets of numbers
22(1)
2.4 Counting with infinities
22(1)
2.5 The floor and ceiling functions
23(1)
2.6 Relations and mappings
24(1)
2.7 Preorders
25(2)
2.8 Mappings between preordered sets
27(2)
2.9 Epigraphs and hypographs
29(1)
2.10 Calculating with sets and functions
30(1)
2.11 Cleistomorphisms and anoiktomorphisms
31(3)
2.12 How to measure distribution of sizes?
34(1)
2.13 Notes
34(2)
2.14 Exercises
36(3)
3 Morphological operations: Set-theoretical duality 39(28)
3.1 Groups and semigroups
39(2)
3.2 Dilations and erosions: Set-theoretical duality
41(5)
3.3 Convolution
46(1)
3.4 Infimal convolution
47(3)
3.5 Approximation using infimal convolution
50(1)
3.6 Iterated infimal convolutions
50(1)
3.7 A more general infimal convolution
51(1)
3.8 Restricted Minkowski addition
52(1)
3.9 Combining dilations and erosions
53(2)
3.10 Commuting with translations
55(1)
3.11 Matheron's structural theorems
56(5)
3.12 Notes
61(2)
3.13 Exercises
63(4)
4 Complete lattices 67(14)
4.1 Introduction
67(2)
4.2 Definitions and examples
69(4)
4.3 Moore families and dual Moore families
73(4)
4.4 Dilations and erosions in complete lattices
77(1)
4.5 Notes
78(1)
4.6 Exercises
79(2)
5 Inverses and quotients of mappings 81(34)
5.1 Introduction
81(1)
5.2 Defining inverses of mappings
82(1)
5.3 First properties of inverses
83(3)
5.4 Left inverses
86(1)
5.5 Right inverses
87(3)
5.6 Inverses of inverses
90(1)
5.7 Special cases of inverses
91(5)
5.8 Examples of upper and lower inverses
96(9)
5.9 Quotients of mappings
105(3)
5.10 Pullbacks and pushforwards
108(4)
5.11 Notes
112(1)
5.12 Exercises
113(2)
6 Structure theorems for mappings 115(10)
6.1 Set-theoretical representation of dilations, erosions, cleistomorphisms, and anoiktomorphisms
115(1)
6.2 Anoiktomorphisms as quotients
116(1)
6.3 Increasing mappings as suprema of elementary erosions
117(1)
6.4 Anoiktomorphisms as suprema of elementary anoiktomorphisms
118(1)
6.5 Strong ethmomorphisms
119(4)
6.6 Notes
123(1)
6.7 Exercises
123(2)
7 Digitization 125(16)
7.1 Introduction
125(3)
7.2 Serra's four principles on gathering information
128(1)
7.3 Distances and metric spaces
129(2)
7.4 Discrete sets and uniformly discrete sets
131(1)
7.5 To discretize a function
132(1)
7.6 Discretization by balayage
132(1)
7.7 Voronoi cells
132(3)
7.8 Difficulties in logic
135(1)
7.9 Comparing differences and derivatives
136(3)
7.10 Notes
139(1)
7.11 Exercises
140(1)
8 Digital straightness and digital convexity 141(26)
8.1 Introduction
141(3)
8.2 Digital lines and Rosenfeld's chord property
144(5)
8.3 Difference operators
149(2)
8.4 Real-valued functions on the integers
151(2)
8.5 Functions taking real values on Zn
153(1)
8.6 Characterization of straightness: The chord property
154(2)
8.7 Characterization of straightness: Balanced words
156(2)
8.8 Hyperplanes in the sense of Reveilles
158(1)
8.9 Refined digital hyperplanes
158(2)
8.10 Extending rectilinear segments
160(3)
8.11 Notes
163(2)
8.12 Exercises
165(2)
9 Convexity in vector spaces 167(36)
9.1 Introduction
167(1)
9.2 Defining convex sets
167(1)
9.3 Properties of convex sets
168(1)
9.4 The convex hull
169(1)
9.5 The Hahn-Banach theorem
170(2)
9.6 Supporting hyperplanes
172(1)
9.7 Caratheodory's theorem
173(1)
9.8 Approximation of the convex hull
173(2)
9.9 Defining convex functions
175(1)
9.10 Properties of convex functions
176(1)
9.11 Strict and strong convexity
177(1)
9.12 Functions taking the value minus infinity
178(1)
9.13 The convex envelope
178(2)
9.14 Approximation of the convex envelope
180(4)
9.15 Separating hyperplanes
184(2)
9.16 Normed spaces
186(2)
9.17 Duality in convex analysis
188(6)
9.18 Duality of infimal convolution and addition
194(1)
9.19 Comparing two convolution operations
194(2)
9.20 Coppel's axioms
196(1)
9.21 Three fundamental properties
196(3)
9.22 Notes
199(2)
9.23 Exercises
201(2)
10 Discrete convexity 203(22)
10.1 Introduction
203(1)
10.2 Restrictions and extensions of functions
203(1)
10.3 Convexity with respect to a subset of a vector space
204(1)
10.4 The integer neighborhood and the canonical extension
205(2)
10.5 Properties of the canonical extension
207(2)
10.6 Integral convexity
209(1)
10.7 Lateral convexity: Definitions
209(2)
10.8 Lateral convexity: Morphological aspects
211(2)
10.9 Lateral convexity: Examples
213(1)
10.10 Representations in terms of elementary convex functions
214(3)
10.11 Separating partially discretized sets
217(2)
10.12 Separating completely discretized sets
219(1)
10.13 Possible future studies
220(1)
10.14 Notes
220(2)
10.15 Exercises
222(3)
11 Discrete convexity in two dimensions 225(34)
11.1 Jensen's inequality in the discrete case
225(3)
11.2 Discrete convexity and the chord property
228(2)
11.3 Submodular and separable functions
230(1)
11.4 The piecewise separately affine extension
231(1)
11.5 Rhomboidal convexity
232(2)
11.6 Conditions for rhomboidal convexity
234(5)
11.7 Independence of the conditions for rhomboidal convexity
239(2)
11.8 The maximal set of pairs which defines rhomboidal convexity
241(1)
11.9 (Z x R)-convexity and separate (Z x R)-convexity
242(1)
11.10 Smooth rhomboidally convex functions
242(1)
11.11 Extending functions from integer points
243(2)
11.12 Conditions for digital straightness
245(2)
11.13 What is the result when we digitize a Euclidean line?
247(2)
11.14 Discretizations of a function
249(5)
11.15 Extending convex extensible functions
254(1)
11.16 Possible future studies
255(1)
11.17 Notes
255(2)
11.18 Exercise
257(2)
12 Three problems in discrete optimization 259(28)
12.1 Introduction
259(1)
12.2 Three fundamental problems
259(3)
12.3 Solution to Problem 1: Marginal functions
262(4)
12.4 Solution to Problem 2: Local minima
266(3)
12.5 Solution to Problem 3: Separating hyperplanes
269(6)
12.6 The marginal function of a function of integer variables
275(1)
12.7 Convolution and convex extensibility
276(1)
12.8 The set where the infimum is attained
277(3)
12.9 Lateral convexity of marginal functions
280(4)
12.10 Necessity of lateral convexity
284(1)
12.11 Possible future studies
285(1)
12.12 Notes
286(1)
12.13 Exercises
286(1)
13 Duality of convolution operators 287(18)
13.1 Introduction
287(1)
13.2 Properties of convolution
287(4)
13.3 Families of functions under duality
291(6)
13.4 Duality between classes of functions and second-order difference operators
297(1)
13.5 More general convolution operators
298(2)
13.6 Discrete convexity defined by convolution
300(2)
13.7 Possible future studies
302(1)
13.8 Notes
303(1)
13.9 Exercises
304(1)
14 Topology 305(12)
14.1 Introduction
305(1)
14.2 Mappings
305(2)
14.3 Defining topologies
307(2)
14.4 Transporting topologies
309(1)
14.5 Continuous mappings
310(1)
14.6 Connectedness
310(2)
14.7 Quotient spaces
312(1)
14.8 Separation axioms
313(2)
14.9 Continuous mappings vs. increasing mappings
315(1)
14.10 Matheron's hit-or-miss topology
315(1)
14.11 Notes
315(1)
14.12 Exercises
315(2)
15 The Khalimsky topology 317(28)
15.1 Introduction
317(1)
15.2 Smallest-neighborhood spaces
317(3)
15.3 Continuity for the Khalimsky topology
320(4)
15.4 Digitization of straight lines in the Khalimsky plane
324(1)
15.5 Fixed-point theorems
325(6)
15.6 Jordan curve theorems
331(11)
15.7 Notes
342(1)
15.8 Exercises
343(2)
16 Distance transformations 345(36)
16.1 Defining distances
345(1)
16.2 Translation-invariant distances
345(2)
16.3 Distance transforms
347(5)
16.4 Distance transforms and sublevel sets
352(3)
16.5 Finitely generated distances
355(5)
16.6 Comparing distances
360(3)
16.7 The calculus of balls
363(9)
16.8 Distance transforms in normed vector spaces
372(3)
16.9 Notes
375(2)
16.10 Exercises
377(4)
17 Skeletonizing 381(12)
17.1 Introduction
381(1)
17.2 Defining skeletons, medial axes, and nervures
381(2)
17.3 Comparing medial axes, nervures, and skeletons
383(2)
17.4 Existence of skeletons
385(2)
17.5 Properties of skeletons
387(2)
17.6 Notes
389(2)
17.7 Exercises
391(2)
18 Solutions 393(26)
Bibliography 419(30)
Author Index 449(4)
Subject Index 453