Chapter 1.
Introduction - A Tour of Multi-View Geometry
__1.1 Introduction - Projective Geometry Everywhere
__1.2 Camera projection
__1.3 Reconstruction from multiple viewpoints
__1.4 Triple View Geometry
__1.5 Quadruple View Geometry and n-Scene Reconstruction
__1.6 transmission
__1.7 Euclidean reconstruction
__1.8 Auto Correction
__1.9 Outcome I: 3D Graphics Model
__1.10 Outcome II: Video Augmentation


Part 0.
Background: Projective geometry, transformations, and approximation
Chapter 2.
Two-dimensional projective geometry and transformations
__2.1 Plane Geometry
__2.2 Two-dimensional projective plane
__2.3 Projective transformation
__2.4 Transformation Layer
__2.5 One-dimensional projective geometry
__2.6 Topology of Projective Surfaces
__2.7 Restoring Affine Transformation and Distance Properties in Images
__2.8 Additional properties of cones
__2.9 Fixed points and fixed lines
__2.10 On the way out


Chapter 3.
3D projective geometry and transformations
__3.1 Points and Projective Transformations
__3.2 Representation and transformation of planes, lines, and quadratic surfaces
__3.3 Twisted cubic curve
__3.4 Transformation Layer
__3.5 Infinite surface
__3.6 Absolute Cone
__3.7 Absolute Dual Quadratic Curve
__3.8 On the way out


Chapter 4.
Estimation of two-dimensional projective transformations
__4.1 Direct Linear Transformation (DLT) Algorithm
__4.2 Various cost functions
__4.3 Statistical cost function and maximum likelihood estimation
__4.4 Transformation Invariance and Normalization
__4.5 How to minimize repetitions
__4.6 Experimental Comparison of Algorithms
__4.7 Solid Estimation
__4.8 Automatic calculation of single-response mapping
__4.9 On the way out


Chapter 5.
Algorithm Evaluation and Error Analysis
__5.1 Performance Limits
__5.2 Covariance of the estimated transformation
__5.3 Monte Carlo Estimation of Covariance
__5.4 On the way out


Part 1.
Camera geometry and single-view geometry

Chapter 6.
Camera model
__6.1 Finite Camera
__6.2 Projection Camera
__6.3 Infinite Camera
__6.4 Other camera models
__6.5 On the way out


Chapter 7.
Computation of the camera matrix ??
__7.1 Basic Equations
__7.2 Geometric error
__7.3 Limited Camera Estimation
__7.4 Radial Distortion
__7.5 On the way out


Chapter 8.
Additional single-point geometry
__8.1 Behavior of projective cameras on planes, lines, and cones
__8.2 Smooth surface image
__8.3 Behavior of the Projective Camera for Quadratic Surfaces
__8.4 The Importance of Camera-Centricity
__8.5 Camera Calibration and Absolute Cone Images
__8.6 Vanishing points and vanishing lines
__8.7 Affine 3D Measurement and Reconstruction
__8.8 Camera calibration at a single point in time?? Determining
__8.9 Single-point reconstruction
__8.10 Correction cone
__8.11 On the way out


Part 2.
Dual viewpoint geometry

Chapter 9.
Entropy geometry and fundamental matrices
__9.1 Ascension Geometry
__9.2 Basic matrix ??
__9.3 Fundamental matrices arising from special exercises
__9.4 Geometric representation of elementary matrices
__9.5 Finding the camera matrix
__9.6 Essential Matrices
__9.7 On the way out

Chapter 10.
3D reconstruction of the camera and structure
__10.1 Overview of restoration methods
__10.2 Ambiguity of Reconstruction
__10.3 Projective Reconstruction Theorem
__10.4 Hierarchical Reconstruction
__10.5 Direct reconstruction using the correct answer
__10.6 On the way out


Chapter 11.
Calculation of the basic matrix ??
__11.1 Basic Equations
__11.2 Normalized 8-point algorithm
__11.3 Algebraic Minimization Algorithm
__11.4 Geometric distance
__11.5 Experimental Evaluation of the Algorithm
Automatic calculation of __11.6 ??
__11.7 ?? Special case of calculation
__11.8 Correspondence to other objects
__11.9 Degeneration
__11.10 ?? Geometric interpretation of calculations
__11.11 The envelope of the coronation lines
__11.12 Image Correction
__11.13 On the way out


Chapter 12.
structural calculations
__12.1 Problem Description
__12.2 Linear triangulation
__12.3 Geometric error cost function
__12.4 Sampson approximation (first-order geometric correction)
__12.5 Optimal solution
__12.6 Probability distribution of estimated 3D points
__12.7 Straight line reconstruction
__12.8 On the way out


Chapter 13.
Scene plane and single-response mapping
__13.1 Simplex mapping of a given plane and its opposite
__13.2 Plane induced by a simplex mapping when ?? and image correspondence are given
__13.3 Calculation of ?? in the simplex map induced by the plane
__13.4 Infinite single-response map ??∞
__13.5 On the way out


Chapter 14.
Affine ascension geometry
__14.1 Affine ascension geometry
__14.2 Affine fundamental matrices
__14.3 Estimation of ??A from point correspondences of two images
__14.4 Triangulation
__14.5 Affine Reconstruction
__14.6 Necker's Reversal and Bas-relief
__14.7 Calculating Exercise
__14.8 On the way out


Part 3.
Triple-view geometry

Chapter 15.
Triple focus tensor
__15.1 Basic Geometry of the Trifocal Tensor
__15.2 Trifocal Tensors and Tensor Notation
__15.3 Transmission
__15.4 Basic matrices for three points in time
__15.5 On the way out


Chapter 16.
Computation of the trifocal tensor T
__16.1 Basic Equations
__16.2 Regularized linear algorithm
__16.3 Algebraic Minimization Algorithm
__16.4 Geometric distance
__16.5 Experimental Evaluation of the Algorithm
__16.6 Automatic calculation of T
__16.7 Special cases of T calculations
__16.8 On the way out


Part 4.
N-viewpoint geometry

Chapter 17.
N-linearity and multi-view tensors
__17.1 Bilinear Relationships
__17.2 Trilinear Relationships
__17.3 Quadruple linear relationship
__17.4 Four intersections on the surface
__17.5 Counting Logic
__17.6 Number of independent equations
__17.7 Selecting an Equation
__17.8 On the way out


Chapter 18.
N-point calculation method
__18.1 Projective Reconstruction - Clump Adjustment
__18.2 Affine Reconstruction-Decomposition Algorithm
__18.3 Non-rigid decomposition
__18.4 Projective decomposition
__18.5 Projective reconstruction using a plane
__18.6 Reconstruction from sequence
__18.7 On the way out


Chapter 19.
Auto-correction
__19.1 Introduction
__19.2 Algebraic Systems and Problem Statements
__19.3 Correction using absolute double quadratic surfaces
__19.4 Krupa equation
__19.5 Layered solution
__19.6 Calibration on a rotating camera
__19.7 Automatic correction on the plane
__19.8 Planar motion
__19.9 Single-axis rotation-turntable motion
__19.10 Automatic calibration of stereo equipment
__19.11 On the way out


Chapter 20.
Duality
__20.1 Carlson-Vineshall duality
__20.2 Abbreviated reconstruction
__20.3 On the way out


Chapter 21.
Chirality
__21.1 Quasi-affine transformations
__21.2 Front and back of the camera
__21.3 3D point set
__21.4 Computing Quasi-Affine Reconstruction
__21.5 The Effect of Transformation on Chirality
__21.6 direction
__21.7 Chiral Inequality
__21.8 Points visible from the third point of view
__21.9 Location between points
__21.10 On the way out


Chapter 22.
degenerate composition
__22.1 Camera Rear Church
__22.2 Degeneration at Dual Viewpoint
__22.3 Carlson-Vineshall duality
__22.4 Critical configuration of triple point
__22.5 On the way out

Part 5.
supplement
__A1 tensor notation
__A2 Gaussian (normal) and χ² distributions
__A3 parameter estimation
__A4 Properties and decomposition of matrices
__A5 Least Squares Minimization
__A6 iterative estimation method
__A7 Special Plane Projection Transformation

◈ Structure of this book ◈

It consists of six parts and seven short appendices.
Each section introduces new geometric relationships.
A homography for the background, a camera matrix for a single viewpoint, a fundamental matrix for dual viewpoints, a trifocal tensor for triple viewpoints, and a quadfocal tensor for quadruple viewpoints.
For each case, there is a chapter describing the relationships, properties and applications, and a chapter describing the algorithms that estimate them from image measurements.
Estimation algorithms are described, ranging from simple and inexpensive approaches to optimal algorithms that are currently considered the best.


Part 0: Background.
Part 0 is more of a guidebook than the other parts.
Introduces important concepts of projective geometry in two-dimensional and three-dimensional space (such as ideal points and absolute conic sections).
We explain how to represent, manipulate, and estimate projective geometry, and how it relates to various goals in computer vision, such as rectifying images of planes to remove perspective distortion.


Part 1: Single-view geometry.
We define and explore the architecture of various cameras that model perspective projection from three-dimensional space to two-dimensional images.
We describe the estimation of existing techniques using a calibration target and camera calibration using vanishing points and vanishing lines.


Part 2: Dual View Geometry.
Part 2 describes the epipolar geometry of two cameras, projective reconstruction from point correspondences between images, methods for resolving projective ambiguities, optimal triangulation, and transfer between photographs through a plane.


Part 3: Triple View Geometry.
Describes the tripod geometry of three cameras.
It involves transferring point and line correspondences from two images to a third image, computing shape from point and line correspondences, and retrieving the camera matrix.


Part 4: N-point of view.
The purpose of Part 4 is twofold.
First, we describe an estimation method that (partially) extends the triple-view geometry to quadruple-view and can be applied to N-viewpoints.
We introduce the simultaneous computation of structure and motion from multiple images using the factorization algorithm of Tomasi and Kanade.
And, although we covered this in Part 3, we will cover topics that can be understood in more depth by emphasizing commonalities.
For example, we derive multi-linear view constraints for correspondence, automatic correction, and ambiguity.

supplement.
It covers tensors, statistics, parameter estimation, linear and matrix algebra, iterative estimation, inverse matrices of sparse matrices, and special projective transformations.

◈ Author's Note ◈

A fundamental problem in the field of computer vision is to recreate a real-world scene using multiple given images.
Projective geometry and photogrammetry can be used to solve this problem.
This book covers geometric principles, camera projection matrices, fundamental matrices, and algebraic representations using trifocal tensors.
We reconstruct scenes from given multiple images and demonstrate these theories and computational methodologies through practical examples.
The revised edition provides more detailed explanations of important concepts through up-to-date case studies and appendices.
And added major studies that appeared after the first edition.


This book also provides detailed background knowledge required to read the book, so if you know linear algebra and basic numerical analysis, you will be able to understand and implement projective geometry and estimation algorithms yourself.

◈ Translator's Note ◈

When I was in college, I used to commute by train between Daejeon, where my school was located, and my hometown, Daegu.
The Gyeongbu Line train passed over the rugged Chupungnyeong Pass, where even clouds rest.
The only view from the window was of mountains.
But when I looked closely, I noticed that the mountains closer to me seemed to be moving backward faster, while the mountains farther away seemed to be moving forward.
After much thought, I realized that this phenomenon was similar to feeling like the full moon following me as I move on a dark night.
Since things that are close go back and things that are far go forward, I thought there would be a fixed point somewhere in the middle, and I thought about how to calculate this.


The answer to this problem is projective (shadow) geometry (the answer to this problem is in this book, and the fixed point that was a concern does not exist).
The basic principles of projective geometry were established during the Renaissance as painters actively studied perspective to more realistically express three-dimensional objects on a two-dimensional plane, but they were not fully established until the early 19th century.
After that, Professor Heo Jun, who recently won the Fields Medal, developed his specialty into algebraic geometry.


Projection has long roots in European thought.
Before the advent of Kant's epistemology, the idea that what we can observe in Plato's allegory of the cave is merely a projection of reality was widely held.
Although this is only a metaphysical analogy, something similar has been found in the real world.
In quantum mechanics, the exact location of an object is unknown, and what can be observed is a probabilistic projection of the actual object.
In fact, Dirac, a British physicist who predicted antimatter and won the Nobel Prize in Physics, mentioned that projective geometry, which he had briefly dabbled in during his undergraduate years, was very helpful when studying quantum mechanics.
This book takes this question one step further.
How can we reconstruct a 3D shape from multiple images of the same object taken simultaneously by multiple cameras? Given the low cost of digital cameras, the potential applications for developing effective algorithms are limitless, making this a field actively being researched.

This book begins with the basics of projective geometry and progresses to dual-view, triple-view, and multi-view geometry.
It is expected to be of great help to researchers developing computer vision-related engines, as it provides a thorough explanation of the fundamentals of geometry and algorithms for dealing with the inevitable noise that arises in numerical calculations.
I majored in mathematics in college, but at the time there were no lectures on non-Euclidean geometry, so I learned about projective geometry, which has such a wide range of applications, while translating a book.
While translating, I was able to experience the power of geometry and work with excitement.

◈ Recommended Articles ◈

“Research in computer vision has achieved remarkable success both practically and theoretically.

As a practical example, the potential of using computer vision technology to guide vehicles such as cars and trucks on public roads or in rough terrain has been demonstrated in Europe, the United States, and Japan for several years.
This requires very sophisticated real-time 3D dynamic scene analysis capabilities.
Today, automotive companies are increasingly using these capabilities. Some remarkable advances have been made in what could theoretically be called geometric computer vision.
This includes a description of how the appearance of an object changes when viewed from different viewpoints as a function of object shape and camera parameters.
These achievements could not have been achieved without the use of very sophisticated mathematics, including classical and modern theories of geometry.
This book deals particularly with the complex and beautiful geometric relationships that exist between images of objects in the world.
Analyzing these relationships is important in itself.
This is because one of the goals of science is to explain appearance.
Analysis is also important because the applications that can be developed through this understanding are limitless. The authors of this book are pioneers and experts in the field of geometric computer vision.
They have successfully accomplished a very challenging task.
“It conveys the mathematics necessary to understand the fundamental geometric concepts in a simple and accessible way, includes a wide range of results obtained by these researchers and others around the world, analyzes the interaction of geometry with noisy image measurements, presents these theoretical results in an algorithmic form that can be easily converted into computer code, and provides numerous practical examples that illustrate the concepts and show the range of applications of the theory.”
- Olivier Faugeras

“The authors present a clear and consistent account of important classical and modern techniques in the mainstream of multi-view geometry.
“I highly recommend this book to readers interested in the theoretical background of traditional and modern techniques in multi-view geometry.”
- Computing Review

“This book clearly explains difficult concepts and introduces the formulas needed to use them.
Explains the theory and its results in a way that is easy for readers to understand and implement.
The material is well organized, the references are easy to find, and there are good examples (which would be even more useful if there were a few more), useful notes, and practice problems at the end of each chapter.
“With its clear approach and explanations, it is ideal for use in lectures or as a reference for researchers interested in this field.”
- BMVA News