This text for graduate students discusses the mathematical foundations of statistical inference for building three-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. The numerous mathematical prerequisites for developing the theories are explained systematically in separate chapters. These methods range from linear algebra, optimization, and geometry to a detailed statistical theory of geometric patterns, fitting estimates, and model selection. In addition, examples drawn from both synthetic and real data demonstrate the insufficiencies of conventional procedures and the improvements in accuracy that result from the use of optimal methods.


1. Introduction. 1.1. The aims of this book. 1.2. The Features of this book. 1.3. Organization and background. 1.4. The analytical mind: Strength and weakness. 2. Fundamentals of Linear Algebra. 2.1. Vector and matrix calculus. 2.2. Eigenvalue problem. 2.3. Linear systems and optimization. 2.4. Matrix and tensor algebra. 3. Probabilities and Statistical Estimation. 3.1. Probability distributions. 3.2. Manifolds and local distributions. 3.3. Gaussian distributions and &khgr;2 distributions. 3.4. Statistical estimation for gaussian models. 3.5. General statistical estimation. 3.6. Maximum likelihood estimation. 3.7. Akaike information criterion. 4. Representation of Geometric Objects. 4.1. Image points and image lines. 4.2. Space points and space lines. 4.3. Space planes. 4.4. Conics. 4.5. Space conics and quadrics. 4.6. Coordinate transformation and projection. 5. Geometric Correction. 5.1. General theory. 5.2. Correction of image points and image lines. 5.3. Correction of space points and space lines. 5.4. Correction of space planes. 5.5. Orthogonality correction. 5.6. Conic incidence correction. 6. 3-D Computation by Stereo Vision. 6.1. Epipolar constraint. 6.2. Optimal correction of correspondence. 6.3. 3-D reconstruction of points. 6.4. 3-D reconstruction of lines. 6.5. Optimal back projection onto a space plane. 6.6. Scenes infinitely far away. 6.7. Camera calibration errors. 7. Parametric Fitting. 7.1. General theory. 7.2 Optimal fitting for image points. 7.3. Optimal fitting for image lines. 7.4. Optimal fitting for space points. 7.5. Optimal fitting for space lines. 7.6. Optimal fitting for space planes. 8. Optimal Filter. 8.1. General theory. 8.2. Iterative estimation scheme. 8.3. Effective gradient approximation. 8.4. Reduction from the kalman filter. 8.5. Estimation from linear hypotheses. 9. Renormalization. 9.1. Eigenvector fit. 9.2. Unbiased eigenvector fit. 9.3. Generalized eigenvalue fit. 9.4 Renormalization. 9.5. Lincarization. 9.6. Second order renormalization. 10. Applications of Geometric Estimation.10.1. Image line fitting. 10.2. Conic fitting. 10.3. Space plane fitting by range sensing. 10.4. Space plane fitting by stereo vision. 11. 3-D Motion Analysis. 11.1. General theory. 11.2. Lincarization and renormalization. 11.3. Optimal correction and decomposition. 11.4. Reliability of 3-D reconstruction. 11.5. Critical surfaces. 11.6. 3-D reconstruction from planar surface motion. 11.7. Camera rotation and information. 12. 3-D Interpretation of Optical Flow. 12.1. Optical flow detection. 12.2. Theoretical basis of 3-D interpretation. 12.3. Optimal estimation of motion parameters. 12.4. Lincarization and renormalization. 12.5. Optimal 3-D reconstruction. 12.6. Reliability of 3-D reconstruction. 12.7. Critical surfaces for optical flow. 12.8. Analysis of planar surface optical flow. 12.9. Camera rotation and information. 13. Information Criterion for Model Selection. 13.1. Model selection criterion. 13.2. Mahalanobis geometry. 13.3. Expected residual. 13.4. Geometric information criterion. 13.5. 3-D reconstruction by stereo vision. 13.6. 3-D motion analysis. 13.7. 3-D interpretation of optical flow. 14. General Theory of Geometric Estimation. 14.1. Statistical estimation in engineering. 14.2. General geometric correction. 14.3. Maximum likelihood correction. 14.4. General parametric fitting. 14.5. Maximum likelihood fit. References. Index.