مسئلهٔ غیرخطی تبدیل داده‌های سه‌بعدی و الگوریتم پروکروستس

A special nonlinear problem is the three-dimensional datum transformation solved by the Procrustes Algorithm . A definition of the three-dimensional datum transformation with the coupled unknowns of type dilatation unknown , also called scale factor , translation and rotation unknown follows afterwards. Abstract: A special nonlinear problem is the three-dimensional datum transformation solved by the Procrustes Algorithm . A definition of the three-dimensional datum transformation with the coupled unknowns of type dilatation unknown , also called scale factor , translation and rotation unknown follows afterwards cover.jpg 1 sdarticle.pdf 2 sdarticle_001.pdf 3 sdarticle_002.pdf 8 Preface to the Third Edition 0 sdarticle_003.pdf 10 Preface to the Second Edition 0 sdarticle_004.pdf 12 Preface to the First Edition 0 sdarticle_005.pdf 15 The Nonlinear Optical Susceptibility 15 Introduction to Nonlinear Optics 15 Descriptions of Nonlinear Optical Processes 18 Formal Definition of the Nonlinear Susceptibility 31 Nonlinear Susceptibility of a Classical Anharmonic Oscillator 35 Properties of the Nonlinear Susceptibility 47 Time-Domain Description of Optical Nonlinearities 66 Kramers-Kronig Relations in Linear and Nonlinear Optics 72 Problems 77 References 79 sdarticle_006.pdf 82 Wave-Equation Description of Nonlinear Optical Interactions 82 The Wave Equation for Nonlinear Optical Media 82 The Coupled-Wave Equations for Sum-Frequency Generation 87 Phase Matching 92 Quasi-Phase-Matching 97 The Manley-Rowe Relations 101 Sum-Frequency Generation 104 Second-Harmonic Generation 109 Difference-Frequency Generation and Parametric Amplification 118 Optical Parametric Oscillators 121 Nonlinear Optical Interactions with Focused Gaussian Beams 129 Nonlinear Optics at an Interface 135 Problems 141 References 145 sdarticle_007.pdf 147 Quantum-Mechanical Theory of the Nonlinear Optical Susceptibility 147 Introduction 147 Schrödinger Calculation of Nonlinear Optical Susceptibility 149 Density Matrix Formulation of Quantum Mechanics 162 Perturbation Solution of the Density Matrix Equation of Motion 170 Density Matrix Calculation of the Linear Susceptibility 173 Density Matrix Calculation of the Second-Order Susceptibility 182 Density Matrix Calculation of the Third-Order Susceptibility 192 Electromagnetically Induced Transparency 197 Local-Field Corrections to the Nonlinear Optical Susceptibility 206 Problems 213 References 216 sdarticle_008.pdf 219 The Intensity-Dependent Refractive Index 219 Descriptions of the Intensity-Dependent Refractive Index 219 Tensor Nature of the Third-Order Susceptibility 223 Nonresonant Electronic Nonlinearities 233 Nonlinearities Due to Molecular Orientation 240 Thermal Nonlinear Optical Effects 247 Semiconductor Nonlinearities 252 Concluding Remarks 259 References 263 sdarticle_009.pdf 265 Molecular Origin of the Nonlinear Optical Response 265 Nonlinear Susceptibilities Calculated Using Time-Independent Perturbation Theory 265 Semiempirical Models of the Nonlinear Optical Susceptibility 271 Model of Boling, Glass, and Owyoung 272 Nonlinear Optical Properties of Conjugated Polymers 274 Bond-Charge Model of Nonlinear Optical Properties 276 Nonlinear Optics of Chiral Media 280 Nonlinear Optics of Liquid Crystals 283 Problems 285 References 286 sdarticle_010.pdf 288 Nonlinear Optics in the Two-Level Approximation 288 Introduction 288 Density Matrix Equations of Motion for a Two-Level Atom 289 Steady-State Response of a Two-Level Atom to a Monochromatic Field 296 Optical Bloch Equations 304 Rabi Oscillations and Dressed Atomic States 312 Optical Wave Mixing in Two-Level Systems 324 Problems 337 References 338 sdarticle_011.pdf 340 Processes Resulting from the Intensity-Dependent Refractive Index 340 Self-Focusing of Light and Other Self-Action Effects 340 Optical Phase Conjugation 353 Optical Bistability and Optical Switching 370 Two-Beam Coupling 380 Pulse Propagation and Temporal Solitons 386 Problems 394 References 399 sdarticle_012.pdf 402 Spontaneous Light Scattering and Acoustooptics 402 Features of Spontaneous Light Scattering 402 Microscopic Theory of Light Scattering 407 Thermodynamic Theory of Scalar Light Scattering 413 Acoustooptics 424 Problems 438 References 439 sdarticle_013.pdf 440 Stimulated Brillouin and Stimulated Rayleigh Scattering 440 Stimulated Scattering Processes 440 Electrostriction 442 Stimulated Brillouin Scattering (Induced by Electrostriction) 447 Phase Conjugation by Stimulated Brillouin Scattering 459 Stimulated Brillouin Scattering in Gases 464 Stimulated Brillouin and Stimulated Rayleigh Scattering 466 Problems 479 References 481 sdarticle_014.pdf 483 Stimulated Raman Scattering and Stimulated Rayleigh-Wing Scattering 483 The Spontaneous Raman Effect 483 Spontaneous versus Stimulated Raman Scattering 484 Stimulated Raman Scattering Described by the Nonlinear Polarization 489 Stokes-Anti-Stokes Coupling in Stimulated Raman Scattering 498 Coherent Anti-Stokes Raman Scattering 509 Stimulated Rayleigh-Wing Scattering 511 Problems 518 References 518 sdarticle_015.pdf 520 The Electrooptic and Photorefractive Effects 520 Introduction to the Electrooptic Effect 520 Linear Electrooptic Effect 521 Electrooptic Modulators 525 Introduction to the Photorefractive Effect 532 Photorefractive Equations of Kukhtarev et al. 535 Two-Beam Coupling in Photorefractive Materials 537 Four-Wave Mixing in Photorefractive Materials 545 Problems 549 References 549 sdarticle_016.pdf 551 Optically Induced Damage and Multiphoton Absorption 551 Introduction to Optical Damage 551 Avalanche-Breakdown Model 552 Influence of Laser Pulse Duration 554 Direct Photoionization 556 Multiphoton Absorption and Multiphoton Ionization 557 Problems 567 References 567 sdarticle_017.pdf 569 Ultrafast and Intense-Field Nonlinear Optics 569 Introduction 569 Ultrashort Pulse Propagation Equation 569 Interpretation of the Ultrashort-Pulse Propagation Equation 575 Intense-Field Nonlinear Optics 579 Motion of a Free Electron in a Laser Field 580 High-Harmonic Generation 583 Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics 587 Nonlinear Quantum Electrodynamics 591 Problem 594 References 594 sdarticle_018.pdf 596 The SI System of Units 596 Further reading 603 The Gaussian System of Units 603 Further reading 607 Systems of Units in Nonlinear Optics 607 Relationship between Intensity and Field Strength 609 Physical Constants 610 sdarticle_019.pdf 612 Index 0 Here we present a nearly complete treatment of the Grand Universe of linear and weakly nonlinear regression models within the first 8 chapters. Our point of view is both an algebraic view as well as a stochastic one. For example, there is an equivalent lemma between a best, linear uniformly unbiased estimation (BLUUE) in a Gauss-Markov model and a least squares solution (LESS) in a system of linear equations. While BLUUE is a stochastic regression model, LESS is an algebraic solution. In the first six chapters we concentrate on underdetermined and overdeterimined linear systems as well as systems with a datum defect. We review estimators/algebraic solutions of type MINOLESS, BLIMBE, BLUMBE, BLUUE, BIQUE, BLE, BIQUE and Total Least Squares. The highlight is the simultaneous determination of the first moment and the second central moment of a probability distribution in an inhomogeneous multilinear estimation by the so called E-D correspondence as well as its Bayes design. In addition, we discuss continuous networks versus discrete networks, use of Grassmann-Pluecker coordinates, criterion matrices of type Taylor-Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overdetermined system of nonlinear equations on curved manifolds. The von Mises-Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter eight is devoted to probabilistic regression, the special Gauss-Markov model with random effects leading to estimators of type BLIP and VIP including Bayesian estimation. The fifth problem of algebraic regression, the system of conditional equations of homogeneous and inhomogeneous type, is formulated. An analogue is the inhomogeneous general linear Gauss-Markov model with fixed and random effects, also called mixed model. Collocation is an example. Another speciality is our sixth problem of probabilistic regression, the model "errors-in-variable”, also called Total Least Squares, namely SIMEX and SYMEX developed by Carroll-Cook-Stefanski-Polzehl-Zwanzig. Another speciality is the treatment of the three-dimensional datum transformation and its relation to the Procrustes Algorithm. The sixth problem of generalized algebraic regression is the system of conditional equations with unknowns, also called Gauss-Helmert model. A new method of an algebraic solution technique, the concept of Groebner Basis and Multipolynomial Resultant is finally presented, illustrating polynomial nonlinear equations. A great part of the work is presented in four Appendices. Appendix A is a treatment, of tensor algebra, namely linear algebra, matrix algebra and multilinear algebra. Appendix B is devoted to sampling distributions and their use in terms of confidence intervals and confidence regions. Appendix C reviews the elementary notions of statistics, namely random events and stochastic processes. Appendix D introduces the basics of Groebner basis algebra, its careful definition, the Buchberger Algorithm, especially the C. F. Gauss combinatorial algorithm. Throughout we give numerous examples and present various test computations. Our reference list includes more than 3000 references, books and papers attached in a CD. This book is a source of knowledge and inspiration not only for geodesists and mathematicians, but also for engineers in general, as well as natural scientists and economists. Inference on effects which result in observations via linear and nonlinear functions is a general task in science. The authors provide a comprehensive in-depth treatise on the analysis and solution of such problems. I wish all readers of this brilliant encyclopaedic book this pleasure and much benefit. Prof. Dr. Harro Walk Institute of Stochastics and Applications, Universität Stuttgart, Germany. Annotation This volume offers a thorough treatment of the 'grand universe' of linear and weakly nonlinear regression models, from an algebraic view as well as a stochastic one. It includes examples and test computations, and a bibliography with over 2000 references