Algebraically Approximate and Noisy Realization of Discrete-Time Systems and Digital Images

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Springer Science & Business Media, Sep 30, 2009 - Technology & Engineering - 255 pages
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This monograph deals with approximation and noise cancellation of dyn- ical systems which include linear and nonlinear input/output relationships. It also deal with approximation and noise cancellation of two dimensional arrays. It will be of special interest to researchers, engineers and graduate students who have specialized in ?ltering theory and system theory and d- ital images. This monograph is composed of two parts. Part I and Part II will deal with approximation and noise cancellation of dynamical systems or digital images respectively. From noiseless or noisy data, reduction will be made. A method which reduces model information or noise was proposed in the reference vol. 376 in LNCIS [Hasegawa, 2008]. Using this method will allow model description to be treated as noise reduction or model reduction without having to bother, for example, with solving many partial di?er- tial equations. This monograph will propose a new and easy method which produces the same results as the method treated in the reference. As proof of its advantageous e?ect, this monograph provides a new law in the sense of numerical experiments. The new and easy method is executed using the algebraic calculations without solving partial di?erential equations. For our purpose,manyactualexamplesofmodelinformationandnoisereductionwill also be provided. Using the analysis of state space approach, the model reduction problem may have become a major theme of technology after 1966 for emphasizing e?ciency in the ?elds of control, economy, numerical analysis, and others.
 

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Contents

Introduction
3
InputOutput Map and Additive Noises
11
22 Analysis for Algebraically Approximate and Noisy Realization
13
23 Measurement Data with Noise
14
25 Algebraically Constrained Least Square Method
17
251 Algebraic CLS Method and Analytic CLS Method
19
Algebraically Approximate and Noisy Realization of Linear Systems
21
32 Finite Dimensional Linear Systems
23
65 Algebraically Approximate Realization of Pseudo Linear Systems
120
66 Algebraically Noisy Realization of Pseudo Linear Systems
131
67 Historical Notes and Concluding Remarks
144
Algebraically Approximate and Noisy Realization of Affine Dynamical Systems
146
71 Basic Facts about Affine Dynamical Systems
148
72 Finite Dimensional Affine Dynamical Systems
150
73 Partial Realization Theory of Affine Dynamical Systems
153
74 Algebraically Approximate Realization of Affine Dynamical Systems
155

33 Partial Realization Theory of Linear Systems
25
34 Algebraically Approximate Realization of Linear Systems
26
35 Algebraically Noisy Realization of Linear Systems
36
351 Comparative Table of the Algebraic CLS and AIC Method
48
36 Historical Notes and Concluding Remarks
49
Algebraically Approximate and Noisy Realization of Socalled Linear Systems
53
41 Basic Facts about Socalled Linear Systems
54
42 Finite Dimensional Socalled Linear Systems
55
43 Partial Realization of Socalled Linear Systems
58
44 RealTime Partial Realization of Almost Linear Systems
60
45 Algebraically Approximate Realization of Socalled Linear Systems
61
46 Algebraically Noisy Realization of Socalled Linear Systems
72
47 Historical Notes and Concluding Remarks
83
Algebraically Approximate and Noisy Realization of Almost Linear Systems
86
51 Basic Facts of Almost Linear Systems
88
52 Finite Dimensional Almost Linear Systems
89
53 Algebraically Approximate Realization of Almost Linear Systems
90
54 Algebraically Noisy Realization of Almost Linear Systems
99
55 Historical Notes and Concluding Remarks
108
Algebraically Approximate and Noisy Realization of Pseudo Linear Systems
111
62 Finite Dimensional Pseudo Linear Systems
113
63 Partial Realization of Pseudo Linear Systems
116
64 RealTime Partial Realization of Pseudo Linear Systems
118
75 Algebraically Noisy Realization of Affine Dynamical Systems
170
76 Historical Notes and Concluding Remarks
182
Algebraically Approximate and Noisy Realization of Linear Representation Systems
185
81 Basic Facts about Linear Representation Systems
186
82 Finite Dimensional Linear Representation Systems
187
83 Partial Realization Theory of Linear Representation Systems
190
84 Algebraically Approximate Realization of Linear Representation Systems
191
85 Algebraically Noisy Realization of Linear Representation Systems
201
86 Historical Notes and Concluding Remarks
211
Algebraically Approximate and Noisy Realization of TwoDimensional Images
214
91 Commutative Linear Representation Systems
216
92 FiniteDimensional Commutative Linear Representation Systems
218
93 Partial Realization Theory of TwoDimensional Images
223
94 Measurement Data with Approximate and Noisy Error
226
95 Analyses for Approximate and Noisy Data
227
96 Nonlinear Integer Programming for Digital Images
228
97 Algebraically Approximate Realization of TwoDimensional Images
229
98 Algebraically Noisy Realization of TwoDimensional Images
237
99 Historical Notes and Concluding Remarks
246
References
249
Index
252
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