| Introduction | p. 1 |
| Motivation | p. 1 |
| Objective of the Thesis | p. 3 |
| Reconstructions of Polyhedral Objects | p. 4 |
| Object, Sensor, and Image Models | p. 4 |
| Representation of Polygons and Polyhedra | p. 6 |
| Automated Reconstruction | p. 7 |
| Geometric Reasoning | p. 8 |
| Reasoning and Object Reconstruction | p. 9 |
| Reasoning with in Geometry | p. 10 |
| Uncertain Reasoning | p. 11 |
| Previous Work | p. 11 |
| Projective Geometry | p. 11 |
| Statistical Geometry and Reasoning | p. 13 |
| Polyhedral Object Reconstruction | p. 15 |
| Overview of Thesis | p. 17 |
| Representation of Geometric Entities and Transformations | p. 19 |
| Projective Geometry | p. 19 |
| Projective Space and Homogeneous Coordinates | p. 20 |
| Hyperplanes | p. 23 |
| Representation of Geometric Entities | p. 24 |
| Points and Lines in 2D | p. 25 |
| Points and Planes in 3D | p. 28 |
| Lines in 3D | p. 30 |
| Plucker Coordinates | p. 32 |
| Basic Geometric Transformations | p. 34 |
| Homography | p. 34 |
| Projective Camera | p. 35 |
| Fundamental Matrix | p. 36 |
| Conditioning of Homogeneous Entities | p. 38 |
| Duality Principle | p. 40 |
| Dual Entities | p. 41 |
| Dual of Point Transformations | p. 44 |
| Geometric Reasoning Using Projective Geometry | p. 47 |
| Unique Constructions of Entities | p. 48 |
| Join and Intersection | p. 49 |
| Transformation of Points, Lines, and Planes | p. 56 |
| Fundamental Matrix | p. 59 |
| Inverse Projective Camera with Respect to a Plane | p. 59 |
| Construction Matrices and Their Interpretation | p. 60 |
| Canonical Entities in Construction Matrices | p. 61 |
| Reduction of Construction Matrices | p. 67 |
| Nullspaces of Construction Matrices | p. 69 |
| Relations between Entities | p. 70 |
| Projective Relations | p. 71 |
| Affine and Similarity Relations | p. 77 |
| Distance Relations | p. 79 |
| Checking Geometric Relations | p. 79 |
| General Construction of Entities | p. 80 |
| Relations as Constraints | p. 81 |
| Minimizing Algebraic Distance | p. 83 |
| Enforcing Plücker Constraint | p. 84 |
| Estimating Projective Transformations | p. 85 |
| Collinearity Equations | p. 85 |
| Coplanarity Equations | p. 88 |
| Simultaneous DLT with Points and Lines | p. 89 |
| Simultaneous DLT Algorithms for Homographies | p. 90 |
| Estimating Constrained Transformations | p. 92 |
| Conditioning of Entities for Minimization | p. 93 |
| Generic Construction Algorithm | p. 95 |
| Statistical Geometric Reasoning | p. 97 |
| Representation of Uncertain Geometric Entities | p. 98 |
| General Representation | p. 98 |
| Approximate Representation | p. 102 |
| Transformation of Uncertain Homogeneous Vectors | p. 104 |
| First Order Error Propagation | p. 104 |
| Transfer to Homogeneous Coordinates | p. 105 |
| Normalization to Euclidean Coordinates | p. 109 |
| Normalization to Spherical Coordinates | p. 110 |
| Changing Nullspaces Using Orthogonal Projections | p. 110 |
| Construction | p. 111 |
| Errors in Approximated Uncertainty Representation | p. 113 |
| Second Moments and Gaussian Assumption | p. 113 |
| Bias in Scalar Multiplication | p. 114 |
| Bias in Bilinear Constructions | p. 117 |
| Bias in Normalization | p. 120 |
| Construction of Entities | p. 122 |
| A Statistical Approach to Join and Intersection | p. 122 |
| Validation of the Statistical Approach | p. 125 |
| Construction Using Geometric Transformations | p. 128 |
| Testing Geometric Relations | p. 129 |
| Hypothesis Testing | p. 129 |
| Properties of the Approximated Test-Value T | p. 132 |
| A Statistical Algorithm for Testing Geometric Relations | p. 135 |
| Validation of Hypothesis Tests | p. 135 |
| Further Improvements | p. 137 |
| Optimal Geometric Estimation | p. 139 |
| Statistical Model | p. 139 |
| Iterative Estimation | p. 143 |
| Generic Algorithm for Optimal Geometric Estimation | p. 144 |
| Example: Estimating Entities of a Cube | p. 146 |
| SUGR: a Library for Statistical Uncertain Geometric Reasoning | p. 147 |
| Polyhedral Object Reconstruction | p. 149 |
| Principle Workflow | p. 150 |
| Feature Extraction | p. 150 |
| Acquiring the Camera Parameters | p. 151 |
| Enhancing User Assisted Reconstruction Systems | p. 153 |
| Existing User Assisted Systems | p. 153 |
| User Assisted Constructions of Building Edges | p. 155 |
| Grouping 3D Line Segments to Surface Patches | p. 159 |
| Automated Reconstruction | p. 161 |
| Matching of Corresponding Line Segments | p. 161 |
| Examples | p. 164 |
| Discussion | p. 167 |
| Effect of Topological Selection to Matching | p. 169 |
| Combining Interaction and Automation | p. 170 |
| Summary | p. 172 |
| Conclusions | p. 173 |
| Summary | p. 173 |
| Contribution of Thesis | p. 174 |
| Outlook | p. 176 |
| Notation | p. 179 |
| Linear Algebra | p. 183 |
| Ranks and Nullspaces | p. 183 |
| Orthogonal Projections | p. 184 |
| Kronecker Product and vec(·) Operator | p. 184 |
| Statistics | p. 187 |
| Covariance Matrices for 2D Lines | p. 187 |
| Uncertainty of a 2D Line | p. 187 |
| Euclidean Interpretation of Homogeneous Covariances | p. 188 |
| Gauss Helmert Estimation | p. 190 |
| General Gauss Helmert Model | p. 191 |
| General Gauss Helmert Model with Block Structure | p. 194 |
| References | p. 197 |
| Table of Contents provided by Publisher. All Rights Reserved. |
