9780486443089

Statistical Optimization for Geometric Computation Theory and Practice

by
  • ISBN13:

    9780486443089

  • ISBN10:

    0486443086

  • Format: Paperback
  • Copyright: 2005-07-26
  • Publisher: Dover Publications

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Summary

This text discusses the mathematical foundations of statistical inference for building 3-dimensional models from image and sensor data that contain noise a task involving autonomous robots guided by video cameras and sensors. The text employs a theoretical accuracy for the optimization procedure, which maximizes the reliability of estimations based on noise data. 1996 edition.

Table of Contents

1 Introduction
1(26)
1.1 The Aims of This Book
1(5)
1.1.1 Statistical optimization for image and sensor data
1(1)
1.1.2 What are the issues?
2(2)
1.1.3 Why is a new statistical theory necessary?
4(2)
1.2 The Features of This Book
6(3)
1.2.1 Theoretical accuracy bound
6(1)
1.2.2 Evaluation of reliability and testing of hypotheses
6(1)
1.2.3 Geometric models as manifolds
7(1)
1.2.4 Numerical schemes for optimization
8(1)
1.3 Organization and Background
9(14)
1.3.1 Fundamentals of linear algebra
9(1)
1.3.2 Probabilities and statistical estimation
9(1)
1.3.3 Representation of geometric primitives
10(2)
1.3.4 Geometric correction
12(1)
1.3.5 3-D computation by stereo vision
13(1)
1.3.6 Parametric fitting
14(1)
1.3.7 Optimal filter and renormalization
14(1)
1.3.8 Applications of geometric estimation
15(2)
1.3.9 3-D motion analysis
17(3)
1.3.10 3-D interpretation of optical flow
20(2)
1.3.11 Information criterion for model selection
22(1)
1.3.12 General theory of geometric estimation
22(1)
1.4 The Analytical Mind: Strength and Weakness
23(4)
1.4.1 Criticisms of statistical approaches
23(1)
1.4.2 Gaussian noise assumption and outlier detection
24(1)
1.4.3 Remaining problems
25(2)
2 Fundamentals of Linear Algebra
27(34)
2.1 Vector and Matrix Calculus
27(10)
2.1.1 Vectors and matrices
27(2)
2.1.2 Determinant and inverse
29(2)
2.1.3 Vector product in three dimensions
31(3)
2.1.4 Projection matrices
34(1)
2.1.5 Orthogonal matrices and rotations
35(2)
2.2 Eigenvalue Problem
37(8)
2.2.1 Spectral decomposition
37(2)
2.2.2 Generalized inverse
39(1)
2.2.3 Rayleigh quotient and quadratic form
40(1)
2.2.4 Nonsingular generalized eigenvalue problem
41(2)
2.2.5 Singular generalized eigenvalue problem
43(1)
2.2.6 Perturbation theorem
44(1)
2.3 Linear Systems and Optimization
45(9)
2.3.1 Singular value decomposition and generalized inverse
45(2)
2.3.2 Linear equations
47(2)
2.3.3 Quadratic optimization
49(1)
2.3.4 Matrix inner product and matrix norm
50(3)
2.3.5 Optimal rotation fitting
53(1)
2.4 Matrix and Tensor Algebra
54(7)
2.4.1 Direct sum and tensor product
54(1)
2.4.2 Cast in three dimensions
55(3)
2.4.3 Linear mapping of matrices in three dimensions
58(3)
3 Probabilities and Statistical Estimation
61(34)
3.1 Probability Distributions
61(4)
3.1.1 Mean, variance, and covariance
61(2)
3.1.2 Geometry of probability distributions
63(2)
3.2 Manifolds and Local Distributions
65(3)
3.2.1 Manifolds and tangent spaces
65(2)
3.2.2 Local distributions
67(1)
3.2.3 Covariance matrix of a 3-D rotation
67(1)
3.3 Gaussian Distributions and Χ Distributions
68(8)
3.3.1 Gaussian distributions
68(4)
3.3.2 Moment generating functions and moments
72(1)
3.3.3 Χ distributions
73(1)
3.3.4 Mahalanobis distance and Χ test
74(2)
3.4 Statistical Estimation for Gaussian Models
76(7)
3.4.1 Maximum likelihood estimation
76(2)
3.4.2 Optimization with linear constraints
78(1)
3.4.3 Maximum a posteriori probability estimation
79(2)
3.4.4 Kalman filter
81(2)
3.5 General Statistical Estimation
83(4)
3.5.1 Score and Fisher information matrix
83(2)
3.5.2 Unbiased estimator and Cramer-Rao lower bound
85(2)
3.6 Maximum Likelihood Estimation
87(3)
3.6.1 Maximum likelihood estimator and the exponential family
87(2)
3.6.2 Asymptotic behavior
89(1)
3.7 Akaike Information Criterion
90(5)
3.7.1 Model selection
90(1)
3.7.2 Asymptotic expression for the expected residual
91(4)
4 Representation of Geometric Objects
95(36)
4.1 Image Points and Image Lines
95(7)
4.1.1 Representation of image points
95(2)
4.1.2 Representation of image lines
97(2)
4.1.3 Incidence, intersections, and joins
99(3)
4.2 Space Points and Space Lines
102(7)
4.2.1 Representation of space points
102(1)
4.2.2 Representation of space lines
103(4)
4.2.3 Incidence, intersections, and joins
107(2)
4.3 Space Planes
109(4)
4.3.1 Representation of space planes
109(1)
4.3.2 Incidence, intersections, and joins
110(3)
4.4 Conics
113(5)
4.4.1 Classification of conics
113(2)
4.4.2 Canonical forms of conics
115(3)
4.5 Space Conics and Quadrics
118(6)
4.5.1 Representation in three dimensions
118(2)
4.5.2 Polarity and conjugate direction
120(1)
4.5.3 Visualization of covariance matrices
121(3)
4.6 Coordinate Transformation and Projection
124(7)
4.6.1 Coordinate transformation
124(3)
4.6.2 Perspective projection
127(4)
5 Geometric Correction
131(40)
5.1 General Theory
131(13)
5.1.1 Basic formulation
131(2)
5.1.2 Optimal solution
133(3)
5.1.3 Practical considerations
136(2)
5.1.4 A posteriori covariance matrices
138(3)
5.1.5 Hypothesis testing and noise level estimation
141(2)
5.1.6 Linear constraint
143(1)
5.2 Correction of Image Points and Image Lines
144(6)
5.2.1 Optimal correction for coincidence
144(3)
5.2.2 Optimal correction for incidence
147(3)
5.3 Correction of Space Points and Space Lines
150(6)
5.3.1 Optimal correction for coincidence
150(3)
5.3.2 Optimal correction for incidence
153(3)
5.4 Correction of Space Planes
156(7)
5.4.1 Optimal correction for coincidence
156(1)
5.4.2 Optimal incidence with space points
157(3)
5.4.3 Optimal incidence with space lines
160(3)
5.5 Orthogonality Correction
163(5)
5.5.1 Correction of two orientations
163(2)
5.5.2 Correction of three orientations
165(3)
5.6 Conic Incidence Correction
168(3)
6 3-D Computation by Stereo Vision
171(38)
6.1 Epipolar Constraint
171(4)
6.1.1 Camera imaging geometry
171(1)
6.1.2 Epipolar equation
172(2)
6.1.3 Parallel stereo system
174(1)
6.2 Optimal Correction of Correspondence
175(4)
6.2.1 Correspondence detection and optimal correction
175(2)
6.2.2 Correspondence test and noise level estimation
177(2)
6.3 3-D Reconstruction of Points
179(7)
6.3.1 Depth reconstruction
179(2)
6.3.2 Error behavior of reconstructed space points
181(2)
6.3.3 Mahalanobis distance in the scene
183(3)
6.4 3-D Reconstruction of Lines
186(2)
6.4.1 Line reconstruction
186(1)
6.4.2 Error behavior of reconstructed space lines
187(1)
6.5 Optimal Back Projection onto a Space Plane
188(7)
6.5.1 Back projection of a point
188(5)
6.5.2 Back projection of a line
193(2)
6.6 Scenes Infinitely Far Away
195(5)
6.6.1 Space points infinitely far away
195(3)
6.6.2 Space lines infinitely far away
198(2)
6.7 Camera Calibration Errors
200(9)
6.7.1 Errors in base-line
200(2)
6.7.2 Errors in camera orientation
202(2)
6.7.3 Errors in focal length
204(5)
7 Parametric Fitting
209(38)
7.1 General Theory
209(11)
7.1.1 Parametric fitting
209(1)
7.1.2 Maximum likelihood estimation
210(3)
7.1.3 Covariance matrix of the optimal fit
213(3)
7.1.4 Hypothesis testing and noise level estimation
216(3)
7.1.5 Linear hypothesis
219(1)
7.2 Optimal Fitting for Image Points
220(6)
7.2.1 Image point fitting
220(2)
7.2.2 Image line fitting
222(4)
7.3 Optimal Fitting for Image Lines
226(4)
7.3.1 Image point fitting
226(2)
7.3.2 Image line fitting
228(2)
7.4 Optimal Fitting for Space Points
230(8)
7.4.1 Space point fitting
230(1)
7.4.2 Space line fitting
231(4)
7.4.3 Space plane fitting
235(3)
7.5 Optimal Fitting for Space Lines
238(4)
7.5.1 Space point fitting
238(1)
7.5.2 Space line fitting
239(2)
7.5.3 Space plane fitting
241(1)
7.6 Optimal Fitting for Space Planes
242(5)
7.6.1 Space point fitting
242(1)
7.6.2 Space line fitting
243(2)
7.6.3 Space plane fitting
245(2)
8 Optimal Filter
247(20)
8.1 General Theory
247(4)
8.1.1 Bayesian approach
247(1)
8.1.2 Maximum a posteriori probability estimation
248(3)
8.2 Iterative Estimation Scheme
251(5)
8.2.1 Optimal update rule
251(3)
8.2.2 Bayesian interpretation
254(2)
8.3 Effective Gradient Approximation
256(4)
8.4 Reduction from the Kalman Filter
260(4)
8.5 Estimation from Linear Hypotheses
264(3)
9 Renormalization
267(28)
9.1 Eigenvector Fit
267(6)
9.1.1 Least-squares approximation
267(2)
9.1.2 Statistical bias of eigenvector fit
269(4)
9.2 Unbiased Eigenvector Fit
273(5)
9.2.1 Unbiased least-squares approximation
273(2)
9.2.2 Analysis of residual
275(3)
9.3 Generalized Eigenvalue Fit
278(5)
9.3.1 Noise level estimation
278(1)
9.3.2 Accuracy of generalized eigenvector fit
279(2)
9.3.3 Analysis of residual
281(2)
9.4 Renormalization
283(3)
9.4.1 Iterations for generalized eigenvalue problem
283(1)
9.4.2 Iterations for weight update
284(2)
9.5 Linearization
286(4)
9.5.1 Linearized algorithm
286(2)
9.5.2 Decomposability condition
288(2)
9.6 Second Order Renormalization
290(5)
9.6.1 Effective value of nonlinear data
290(2)
9.6.2 Second order unbiased estimation
292(3)
10 Applications of Geometric Estimation 295(30)
10.1 Image Line Fitting
295(5)
10.1.1 Optimal line fitting
295(1)
10.1.2 Unbiased estimation and renormalization
296(4)
10.2 Conic Fitting
300(6)
10.2.1 Optimal conic fitting
300(2)
10.2.2 Unbiased estimation and renormalization
302(4)
10.3 Space Plane Fitting by Range Sensing
306(8)
10.3.1 Optimal space plane fitting
306(3)
10.3.2 Noise model of range sensing
309(2)
10.3.3 Unbiased estimation and renormalization
311(3)
10.4 Space Plane Fitting by Stereo Vision
314(11)
10.4.1 Optimal space plane fitting
314(3)
10.4.2 Unbiased estimation and renormalization
317(8)
11 3-D Motion Analysis 325(44)
11.1 General Theory
325(7)
11.1.1 Camera and object motion
325(2)
11.1.2 Optimal estimation of motion parameters
327(2)
11.1.3 Theoretical bound on accuracy
329(3)
11.2 Linearization and Renormalization
332(4)
11.2.1 Linearization
332(2)
11.2.2 Unbiased estimation and renormalization
334(2)
11.3 Optimal Correction and Decomposition
336(5)
11.3.1 Correction of the essential matrix
336(2)
11.3.2 Decomposition into motion parameters
338(3)
11.4 Reliability of 3-D Reconstruction
341(7)
11.4.1 Depth and its variance
341(3)
11.4.2 Effect of errors in the motion parameters
344(4)
11.5 Critical Surfaces
348(3)
11.5.1 Weak critical surfaces
348(2)
11.5.2 Strong critical surfaces
350(1)
11.6 3-D Reconstruction from Planar Surface Motion
351(13)
11.6.1 Optimal solution and planarity test
351(3)
11.6.2 Unbiased estimation and renormalization
354(2)
11.6.3 Computation of surface and motion parameters
356(8)
11.7 Camera Rotation and Information
364(5)
11.7.1 Rotation test
364(3)
11.7.2 Information in motion images
367(2)
12 3-D Interpretation of Optical Flow 369(46)
12.1 Optical Flow Detection
369(5)
12.1.1 Gradient equation
369(2)
12.1.2 Reliability of optical flow
371(3)
12.2 Theoretical Basis of 3-D Interpretation
374(4)
12.2.1 Optical flow equation
374(2)
12.2.2 Epipolar equation for optical flow
376(1)
12.2.3 3-D analysis from optical flow
377(1)
12.3 Optimal Estimation of Motion Parameters
378(5)
12.3.1 Optimal estimation
378(2)
12.3.2 Theoretical bound on accuracy
380(3)
12.4 Linearization and Renormalization
383(3)
12.4.1 Linearization
383(1)
12.4.2 Unbiased estimation and renormalization
384(2)
12.5 Optimal 3-D Reconstruction
386(8)
12.5.1 Optimal correction and decomposition
386(3)
12.5.2 Optimal correction of optical flow
389(3)
12.5.3 Computation of depth
392(2)
12.6 Reliability of 3-D Reconstruction
394(6)
12.6.1 Effect of image noise
394(1)
12.6.2 Effect of errors in the motion parameters
395(5)
12.7 Critical Surfaces for Optical flow
400(2)
12.7.1 Weak critical surfaces
400(1)
12.7.2 Strong critical surfaces
401(1)
12.8 Analysis of Planar Surface Optical Flow
402(6)
12.8.1 Optical flow equation for a planar surface
402(1)
12.8.2 Estimation of the flow matrix
403(2)
12.8.3 Planarity test
405(1)
12.8.4 Computation of surface and motion parameters
406(2)
12.9 Camera Rotation and Information
408(7)
12.9.1 Rotation estimation
408(2)
12.9.2 Rotation test
410(1)
12.9.3 Information in optical flow
411(1)
12.9.4 Midpoint flow approximation
412(3)
13 Information Criterion for Model Selection 415(36)
13.1 Model Selection Criterion
415(5)
13.1.1 Model estimation
415(3)
13.1.2 Minimization of expected residual
418(2)
13.2 Mahalanobis Geometry
420(3)
13.2.1 Mahalanobis projection
420(2)
13.2.2 Residual of model fitting
422(1)
13.3 Expected Residual
423(5)
13.3.1 Evaluation of the expected residual
423(2)
13.3.2 Accuracy of parametric fitting
425(3)
13.4 Geometric Information Criterion
428(6)
13.4.1 Model selection by AIC
428(3)
13.4.2 Model comparison by AIC
431(3)
13.4.3 Model selection vs. testing of hypotheses
434(1)
13.5 3-D Reconstruction by Stereo Vision
434(4)
13.5.1 General model
434(2)
13.5.2 Planar surface model
436(1)
13.5.3 Infinity model
437(1)
13.5.4 Model comparison
437(1)
13.6 3-D Motion Analysis
438(8)
13.6.1 General model
438(2)
13.6.2 Planar surface model
440(1)
13.6.3 Rotation model
441(1)
13.6.4 Model comparison
442(4)
13.7 3-D Interpretation of Optical Flow
446(5)
13.7.1 General model
446(2)
13.7.2 Planar surface model
448(1)
13.7.3 Rotation model
449(1)
13.7.4 Model comparison
450(1)
14 General Theory of Geometric Estimation 451(34)
14.1 Statistical Estimation in Engineering
451(1)
14.2 General Geometric Correction
452(8)
14.2.1 Definition of the problem
452(1)
14.2.2 The rank of the constraint
453(2)
14.2.3 Cramer-Rao lower bound for geometric correction
455(1)
14.2.4 Proof of the main theorem
456(4)
14.3 Maximum Likelihood Correction
460(7)
14.3.1 Maximum likelihood estimator
460(2)
14.3.2 Geometric correction of the exponential family
462(1)
14.3.3 Computation of maximum likelihood correction
463(3)
14.3.4 Locally Gaussian model
466(1)
14.4 General Parametric Fitting
467(8)
14.4.1 Definition of the problem
467(1)
14.4.2 The rank of the hypothesis
468(2)
14.4.3 Cramer-Rao lower bound for parametric fitting
470(2)
14.4.4 Proof of the main theorem
472(3)
14.5 Maximum Likelihood Fit
475(10)
14.5.1 Maximum likelihood estimator
475(3)
14.5.2 Fitting of the exponential family
478(2)
14.5.3 Computation of maximum likelihood fit
480(5)
References 485(16)
Index 501

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