Realization theory and design of digital images

Thismonographisconcernedwithdescriptionanddesignfortwo-dimensional and three-dimensional images; it will be of special interest to researchers and graduate students who specialized in image processing and system theory. From the data in digital images, mathematical models will be constructed. Then new systems which describe faithfully any two-dimensional or thr- dimensional digital images will be proposed. Using the systems thus allows description to be treated as realization problem and design. By virtue of this approach, this monograph provides new results and their extensions which are designing of two-dimensional and three-dimensional images. Some actual design examples will be also shown. In usual image processing today, two-dimensional images are transformed into one-dimensional signals, then which are analyzed by means of various established methods in signal processing theory. Likewise, three-dimensional images are transformed into two-dimensional signals and these signals are analyzedbyestablishedmethodsintwo-dimensionalsignalprocessingtheory. Another common processing procedure employs tree structures such as qu- trees for two-dimensional images and oct-trees for three-dimensional ones.

1 Introduction	p. 1
2 Two-Dimensional Images and Three-Dimensional Images	p. 9
2.1 Two-Dimensional Images and Input/Output Relations	p. 9
2.2 Three-Dimensional Images	p. 10
2.3 Historical Notes and Concluding Remarks	p. 11
3 Realization Theory of Two-Dimensional Images	p. 13
3.1 2-Commutative Linear Representation Systems	p. 14
3.2 Definite Examples of Two-Dimensional Images Generated by Finite-Dimensional 2-Commutative Linear Representation Systems	p. 18
3.3 Finite-Dimensional 2-Commutative Linear Representation Systems	p. 19
3.4 Partial Realization Theory of Two-Dimensional Images	p. 27
3.5 Historical Notes and Concluding Remarks	p. 37
Appendix to Chapter 3	p. 39
3-A Realization Theorem	p. 39
3-A.1 Linear State Structure: {{[alpha], [beta]}}-Actions	p. 39
3-A.2 Pointed {{[alpha], [beta]}}-Actions	p. 43
3-A.3 {{[alpha], [beta]}}-Actions with a Readout Map	p. 45
3-A.4 2-Commutative Linear Representation Systems	p. 46
3-A.5 Sophisticated 2-Commutative Linear Representation System	p. 47
3-B Finite-Dimensional 2-Commutative Linear Representation Systems	p. 49
3-B.1 Finite-Dimensional {{[alpha], [beta]}}-Actions and Pointed {{[alpha], [beta]}}-Actions	p. 49
3-B.2 Finite-Dimensional {{[alpha], [beta]}}-Actions with a Readout Map	p. 54
3-B.3 Finite-Dimensional 2-Commutative Linear Representation Systems	p. 57
3-B.4 Existence Criterion for 2-Commutative Linear Representation Systems	p. 58
3-B.5 Realization Procedure for 2-Commutative Linear Representation Systems	p. 60
3-C Partial Realization Theory	p. 60
3-C.1 Pointed {{[alpha], [beta]}}-Actions	p. 60
3-C.2 {{[alpha], [beta]}}-Actions with a Readout Map	p. 61
3-C.3 Partial Realization Problem	p. 62
4 Structures of 2-Commutative Linear Representation Systems	p. 71
4.1 Structure Theory of 2-Commutative Linear Representation Systems	p. 72
4.2 Structure Theory and a Coding Theory of Two-Dimensional Images	p. 74
4.3 Historical Notes and Concluding Remarks	p. 79
Appendix to Chapter 4	p. 80
5 Design for Two-Dimensional Images	p. 89
5.1 2-Commutative Linear Representation Systems for Design	p. 90
5.2 Design Methods for Geometrical Patterns	p. 99
5.3 Historical Notes and Concluding Remarks	p. 100
6 Realization Theory of Three-Dimensional Images	p. 101
6.1 3-Commutative Linear Representation Systems	p. 102
6.2 Definite Examples of Images Generated by Finite-Dimensional 3-Commutative Linear Representation Systems	p. 104
6.3 Finite-Dimensional 3-Commutative Linear Representation Systems	p. 107
6.4 Partial Realization of Three-Dimensional Images	p. 117
6.5 Historical Notes and Concluding Remarks	p. 130
Appendix to Chapter 6	p. 131
6-A Proof of the Realization Theory of Three-Dimensional Images	p. 131
6-A.1 Linear State Structure: {{[alpha], [beta], [gamma]}}-Actions	p. 131
6-A.2 Pointed {{[alpha], [beta], [gamma]}}-Actions	p. 133
6-A.3 {{[alpha], [beta], [gamma]}}-Actions with a Readout Map	p. 134
6-A.4 3-Commutative Linear Representation System	p. 136
6-A.5 Sophisticated 3-Commutative Linear Representation System	p. 137
6-B Finite-Dimensional 3-Commutative Linear Representation Systems	p. 139
6-B.1 Finite-Dimensional {{[alpha], [beta], [gamma]}}-Actions	p. 139
6-B.2 Finite-Dimensional Pointed {{[alpha], [beta], [gamma]}}-Actions	p. 140
6-B.3 Finite-Dimensional {{[alpha], [beta], [gamma]}}-Actions with a Readout Map	p. 144
6-B.4 Finite-Dimensional 3-Commutative Linear Representation Systems	p. 145
6-B.5 Existence Criterion for Finite-Dimensional 3-Commutative Linear Representation Systems	p. 145
6-B.6 Realization Procedure for Finite-Dimensional 3-Commutative Linear Representation Systems	p. 147
6-C Partial Realization Theory	p. 147
6-C.1 Pointed {{[alpha], [beta], [gamma]}}-Actions	p. 147
6-C.2 {{[alpha], [beta], [gamma]}}-Actions with a Readout Map	p. 148
6-C.3 Partial Realization Problem	p. 150
7 Structures of 3-Commutative Linear Representation Systems	p. 159
7.1 Structure Theory of 3-Commutative Linear Representation Systems	p. 160
7.2 Structure Theory and a Coding Theory of Three-Dimensional Images	p. 163
7.3 Historical Notes and Concluding Remarks	p. 176
Appendix to Chapter 7	p. 177
8 Design for Three-Dimensional Images	p. 201
8.1 3-Commutative Linear Representation Systems for Design	p. 202
8.2 Design Methods for Geometrical Patterns	p. 213
8.3 Historical Notes and Concluding Remarks	p. 214
References	p. 215
Index	p. 223

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