Computer Vision: Three-dimensional Reconstruction Techniques

✍ Scribed by Andrea Fusiello

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 348
• Edition: 1st ed. 2024
• Category: Library

✦ Synopsis

From facial recognition to self-driving cars, the applications of computer vision are vast and ever-expanding. Geometry plays a fundamental role in this discipline, providing the necessary mathematical framework to understand the underlying principles of how we perceive and interpret visual information in the world around us.

This text explores the theories and computational techniques used to determine the geometric properties of solid objects through images. It covers the basic concepts and provides the necessary mathematical background for more advanced studies. The book is divided into clear and concise chapters covering a wide range of topics including image formation, camera models, feature detection and 3D reconstruction. Each chapter includes detailed explanations of the theory as well as practical examples to help the reader understand and apply the concepts presented.

The book has been written with the intention of being used as a primary resource for students on university courses in computer vision, particularly final year undergraduate or postgraduate computer science or engineering courses. It is also useful for self-study and for those who, outside the academic field, find themselves applying computer vision to solve practical problems. The aim of the book is to strike a balance between the complexity of the theory and its practical applicability in terms of implementation. Rather than providing a comprehensive overview of the current state of the art, it offers a selection of specific methods with enough detail to enable the reader to implement them.

✦ Table of Contents

Foreword
Preface
Acknowledgements
Contents
Acronyms
Listings
1 Introduction
1.1 The Prodigy of Vision
1.2 Low-Level Computer Vision
1.3 Overview of the Book
1.4 Notation
References
2 Fundamentals of Imaging
2.1 Introduction
2.2 Perspective
2.3 Digital Images
2.4 Thin Lenses
2.4.1 Telecentric Optics
2.5 Radiometry
References
3 The Pinhole Camera Model
3.1 Introduction
3.2 Pinhole Camera
3.3 Simplified Pinhole Model
3.4 General Pinhole Model
3.4.1 Intrinsic Parameters
3.4.1.1 Field of View
3.4.2 Extrinsic Parameters
3.5 Dissection of the Perspective Projection Matrix
3.5.1 Collinearity Equations
3.6 Radial Distortion
Problems
References
4 Camera Calibration
4.1 Introduction
4.2 The Direct Linear Transform Method
4.3 Factorisation of the Perspective Projection Matrix
4.4 Calibrating Radial Distortion
4.5 The Sturm-Maybank-Zhang Calibration Algorithm
Problems
References
5 Absolute and Exterior Orientation
5.1 Introduction
5.2 Absolute Orientation
5.2.1 Orthogonal Procrustes Analysis
5.3 Exterior Orientation
5.3.1 Fiore's Algorithm
5.3.2 Procrustean Method
5.3.3 Direct Method
Problems
References
6 Two-View Geometry
6.1 Introduction
6.2 Epipolar Geometry
6.3 Fundamental Matrix
6.4 Computing the Fundamental Matrix
6.4.1 The Seven-Point Algorithm
6.4.2 Preconditioning
6.5 Planar Homography
6.5.1 Computing the Homography
6.6 Planar Parallax
Problems
References
7 Relative Orientation
7.1 Introduction
7.2 The Essential Matrix
7.2.1 Geometric Interpretation
7.2.2 Computing the Essential Matrix
7.3 Relative Orientation from the Essential Matrix
7.3.1 Closed Form Factorisation of the Essential Matrix
7.4 Relative Orientation from the Calibrated Homography
Problems
References
8 Reconstruction from Two Images
8.1 Introduction
8.2 Triangulation
8.3 Ambiguity of Reconstruction
8.4 Euclidean Reconstruction
8.5 Projective Reconstruction
8.6 Euclidean Upgrade from Known Intrinsic Parameters
8.7 Stratification
Problems
References
9 Non-linear Regression
9.1 Introduction
9.2 Algebraic Versus Geometric Distance
9.3 Non-linear Regression of the PPM
9.3.1 Residual
9.3.2 Parameterisation
9.3.3 Derivatives
9.3.4 General Remarks
9.4 Non-linear Regression of Exterior Orientation
9.5 Non-linear Regression of a Point in Space
9.5.1 Residual
9.5.2 Derivatives
9.5.3 Radial Distortion
9.6 Regression in the Joint Image Space
9.7 Non-linear Regression of the Homography
9.7.1 Residual
9.7.2 Parameterisation
9.7.3 Derivatives
9.8 Non-linear Regression of the Fundamental Matrix
9.8.1 Residual
9.8.2 Parameterisation
9.8.3 Derivatives
9.9 Non-linear Regression of Relative Orientation
9.9.1 Parameterisation
9.9.2 Derivatives
9.10 Robust Regression
Problems
References
10 Stereopsis: Geometry
10.1 Introduction
10.2 Triangulation in the Normal Case
10.3 Epipolar Rectification
10.3.1 Calibrated Rectification
10.3.2 Uncalibrated Rectification
Problems
References
11 Features Points
11.1 Introduction
11.2 Filtering Images
11.2.1 Smoothing
11.2.1.1 Non-linear Filters
11.2.2 Derivation
11.3 LoG Filtering
11.4 Harris-Stephens Operator
11.4.1 Matching and Tracking
11.4.2 Kanade-Lucas-Tomasi Algorithm
11.4.3 Predictive Tracking
11.5 Scale Invariant Feature Transform
11.5.1 Scale-Space
11.5.2 SIFT Detector
11.5.3 SIFT Descriptor
11.5.4 Matching
References
12 Stereopsis: Matching
12.1 Introduction
12.2 Constraints and Ambiguities
12.3 Local Methods
12.3.1 Matching Cost
12.3.2 Census Transform
12.4 Adaptive Support
12.4.1 Multiresolution Stereo Matching
12.4.2 Adaptive Windows
12.5 Global Matching
12.6 Post-Processing
12.6.1 Reliability Indicators
12.6.2 Occlusion Detection
References
13 Range Sensors
13.1 Introduction
13.2 Structured Lighting
13.2.1 Active Stereopsis
13.2.2 Active Triangulation
13.2.3 Ray-Plane Triangulation
13.2.4 Scanning Methods
13.2.5 Coded-Light Methods
13.3 Time-of-Flight Sensors
13.4 Photometric Stereo
13.4.1 From Normals to Coordinates
13.5 Practical Considerations
References
14 Multi-View Euclidean Reconstruction
14.1 Introduction
14.1.1 Epipolar Graph
14.1.2 The Case of Three Images
14.1.3 Taxonomy
14.2 Point-Based Approaches
14.2.1 Adjustment of Independent Models
14.2.2 Incremental Reconstruction
14.2.3 Hierarchical Reconstruction
14.3 Frame-Based Approach
14.3.1 Synchronisation of Rotations
14.3.2 Synchronisation of Translations
14.3.3 Localisation from Bearings
14.4 Bundle Adjustment
14.4.1 Jacobian of Bundle Adjustment
14.4.2 Reduced System
References
15 3D Registration
15.1 Introduction
15.1.1 Generalised Procrustes Analysis
15.2 Correspondence-Less Methods
15.2.1 Registration of Two Point Clouds
15.2.2 Iterative Closest Point
15.2.3 Registration of Many Point Clouds
References
16 Multi-view Projective Reconstruction and Autocalibration
16.1 Introduction
16.1.1 Sturm-Triggs Factorisation Method
16.2 Autocalibration
16.2.1 Absolute Quadric Constraint
16.2.1.1 Solution Strategies
16.2.2 Mendonça-Cipolla Method
16.3 Autocalibration via H∞
16.4 Tomasi-Kanade's Factorisation
16.4.1 Affine Camera
16.4.2 The Factorisation Method for Affine Camera
Problems
References
17 Multi-view Stereo Reconstruction
17.1 Introduction
17.2 Volumetric Stereo in Object-Space
17.2.1 Shape from Silhouette
17.2.2 Szeliski's Algorithm
17.2.3 Voxel Colouring
17.2.4 Space Carving
17.3 Volumetric Stereo in Image-Space
17.4 Marching Cubes
References
18 Image-Based Rendering
18.1 Introduction
18.2 Parametric Transformations
18.2.1 Mosaics
18.2.1.1 Alignment
18.2.1.2 Blending
18.2.2 Image Stabilisation
18.2.3 Perspective Rectification
18.3 Non-parametric Transformations
18.3.1 Transfer with Depth
18.3.2 Transfer with Disparity
18.3.3 Epipolar Transfer
18.3.4 Transfer with Parallax
18.3.5 Ortho-Projection
18.4 Geometric Image Transformation
Problems
References
A Notions of Linear Algebra
A.1 Introduction
A.2 Scalar Product
A.3 Matrix Norm
A.4 Inverse Matrix
A.5 Determinant
A.6 Orthogonal Matrices
A.7 Linear and Quadratic Forms
A.8 Rank
A.9 QR Decomposition
A.10 Eigenvalues and Eigenvectors
A.11 Singular Value Decomposition
A.12 Pseudoinverse
A.13 Cross Product
A.14 Kronecker's Product
A.15 Rotations
A.16 Matrices Associated with Graphs
References
B Matrix Differential Calculation
B.1 Derivatives of Vector and Matrix Functions
B.2 Derivative of Rotations
B.2.1 Axis/Angle Representation
B.2.2 Euler Representation
References
C Regression
C.1 Introduction
C.2 Least-Squares
C.2.1 Linear Least-Squares
C.2.2 Non-linear Least-Squares
C.2.2.1 Gauss-Newton Method
C.2.3 The Levenberg-Marquardt Method
C.3 Robust Regression
C.3.1 Outliers and Robustness
C.3.2 M-Estimators
C.3.3 Least Median of Squares
C.3.4 RANSAC
C.4 Propagation of Uncertainty
C.4.1 Covariance Propagation in Least-Squares
References
D Notions of Projective Geometry
D.1 Introduction
D.2 Perspective Projection
D.3 Homogeneous Coordinates
D.4 Equation of the Line
D.5 Transformations
Reference
E Matlab Code
Index