Statistical Optimization for Geometric Computation Theory and Practice

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780486443089 | 0486443086
Cover: Paperback
Copyright: 7/26/2005

Rent

(Recommended)

$21.09

Term

Due

Price

Semester

Dec 12

$22.21

Quarter

Aug 13

$21.09

Short Term

Jul 9

$19.98

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

In Stock Usually Ships in 24-48 Hours

$28.43

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.

1 Introduction

(26)

1.1 The Aims of This Book

(5)

1.1.1 Statistical optimization for image and sensor data

(1)

1.1.2 What are the issues?

(2)

1.1.3 Why is a new statistical theory necessary?

(2)

1.2 The Features of This Book

(3)

1.2.1 Theoretical accuracy bound

(1)

1.2.2 Evaluation of reliability and testing of hypotheses

(1)

1.2.3 Geometric models as manifolds

(1)

1.2.4 Numerical schemes for optimization

(1)

1.3 Organization and Background

(14)

1.3.1 Fundamentals of linear algebra

(1)

1.3.2 Probabilities and statistical estimation

(1)

1.3.3 Representation of geometric primitives

(2)

1.3.4 Geometric correction

(1)

1.3.5 3-D computation by stereo vision

(1)

1.3.6 Parametric fitting

(1)

1.3.7 Optimal filter and renormalization

(1)

1.3.8 Applications of geometric estimation

(2)

1.3.9 3-D motion analysis

(3)

1.3.10 3-D interpretation of optical flow

(2)

1.3.11 Information criterion for model selection

(1)

1.3.12 General theory of geometric estimation

(1)

1.4 The Analytical Mind: Strength and Weakness

(4)

1.4.1 Criticisms of statistical approaches

(1)

1.4.2 Gaussian noise assumption and outlier detection

(1)

1.4.3 Remaining problems

(2)

2 Fundamentals of Linear Algebra

(34)

2.1 Vector and Matrix Calculus

(10)

2.1.1 Vectors and matrices

(2)

2.1.2 Determinant and inverse

(2)

2.1.3 Vector product in three dimensions

(3)

2.1.4 Projection matrices

(1)

2.1.5 Orthogonal matrices and rotations

(2)

2.2 Eigenvalue Problem

(8)

2.2.1 Spectral decomposition

(2)

2.2.2 Generalized inverse

(1)

2.2.3 Rayleigh quotient and quadratic form

(1)

2.2.4 Nonsingular generalized eigenvalue problem

(2)

2.2.5 Singular generalized eigenvalue problem

(1)

2.2.6 Perturbation theorem

(1)

2.3 Linear Systems and Optimization

(9)

2.3.1 Singular value decomposition and generalized inverse

(2)

2.3.2 Linear equations

(2)

2.3.3 Quadratic optimization

(1)

2.3.4 Matrix inner product and matrix norm

(3)

2.3.5 Optimal rotation fitting

(1)

2.4 Matrix and Tensor Algebra

(7)

2.4.1 Direct sum and tensor product

(1)

2.4.2 Cast in three dimensions

(3)

2.4.3 Linear mapping of matrices in three dimensions

(3)

3 Probabilities and Statistical Estimation

(34)

3.1 Probability Distributions

(4)

3.1.1 Mean, variance, and covariance

(2)

3.1.2 Geometry of probability distributions

(2)

3.2 Manifolds and Local Distributions

(3)

3.2.1 Manifolds and tangent spaces

(2)

3.2.2 Local distributions

(1)

3.2.3 Covariance matrix of a 3-D rotation

(1)

3.3 Gaussian Distributions and Χ² Distributions

(8)

3.3.1 Gaussian distributions

(4)

3.3.2 Moment generating functions and moments

(1)

3.3.3 Χ² distributions

(1)

3.3.4 Mahalanobis distance and Χ² test

(2)

3.4 Statistical Estimation for Gaussian Models

(7)

3.4.1 Maximum likelihood estimation

(2)

3.4.2 Optimization with linear constraints

(1)

3.4.3 Maximum a posteriori probability estimation

(2)

3.4.4 Kalman filter

(2)

3.5 General Statistical Estimation

(4)

3.5.1 Score and Fisher information matrix

(2)

3.5.2 Unbiased estimator and Cramer-Rao lower bound

(2)

3.6 Maximum Likelihood Estimation

(3)

3.6.1 Maximum likelihood estimator and the exponential family

(2)

3.6.2 Asymptotic behavior

(1)

3.7 Akaike Information Criterion

(5)

3.7.1 Model selection

(1)

3.7.2 Asymptotic expression for the expected residual

(4)

4 Representation of Geometric Objects

(36)

4.1 Image Points and Image Lines

(7)

4.1.1 Representation of image points

(2)

4.1.2 Representation of image lines

(2)

4.1.3 Incidence, intersections, and joins

(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

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Statistical Optimization for Geometric Computation Theory and Practice

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Statistical Optimization for Geometric Computation Theory and Practice

Summary

Table of Contents

Supplemental Materials

Digital License