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

Springer Science & Business Media, Sep 30, 2009 - Technology & Engineering - 255 pages
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 Copyright