Group-Theoretical Methods in Image Understanding
Published on 6. December 2012
XII, 459 pages
978-3-642-61275-6 (ISBN)
Description
Image understanding is an attempt to extract knowledge about a 3D scene from 20 images. The recent development of computers has made it possible to automate a wide range of systems and operations, not only in the industry, military, and special environments (space, sea, atomic plants, etc.), but also in daily life. As we now try to build ever more intelligent systems, the need for "visual" control has been strongly recognized, and the interest in image under standing has grown rapidly. Already, there exists a vast body of literature-ranging from general philosophical discourses to processing techniques. Compared with other works, however, this book may be unique in that its central focus is on "mathematical" principles-Lie groups and group representation theory, in particular. In the study of the relationship between the 3D scene and the 20 image, "geometry" naturally plays a central role. Today, so many branches are inter woven in geometry that we cannot truly regard it as a single subject. Neverthe less, as Felix Klein declared in his Erlangen Program, the central principle of geometry is group theory, because geometrical concepts are abstractions of properties that are "invariant" with respect to some group of transformations. In this text, we specifically focus on two groups of transformations. One is 20 rotations of the image coordinate system around the image origin. Such coordi nate rotations are indeed irrelevant when we look for intrinsic image properties.
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Ken-ichi Kanatani
Group-Theoretical Methods in Image Understanding
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03/1990
Springer
€85.55
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Content
1. Introduction.- 1.1 What is Image Understanding?.- 1.2 Imaging Geometry of Perspective Projection.- 1.3 Geometry of Camera Rotation.- 1.4 The 3D Euclidean Approach.- 1.5 The 2D Non-Euclidean Approach.- 1.6 Organization of This Book.- 2. Coordinate Rotation Invariance of Image Characteristics.- 2.1 Image Characteristics and 3D Recovery Equations.- 2.2 Parameter Changes and Representations.- 2.3 Invariants and Weights.- 2.4 Representation of a Scalar and a Vector.- 2.5 Representation of a Tensor.- 2.6 Analysis of Optical Flow for a Planar Surface.- 2.7 Shape from Texture for Curved Surfaces.- Exercises.- 3. 3D Rotation and Its Irreducible Representations.- 3.1 Invariance for the Camera Rotation Transformation.- 3.2 Infinitesimal Generators and Lie Algebra.- 3.3 Lie Algebra and Casimir Operator of the 3D Rotation Group.- 3.4 Representation of a Scalar and a Vector.- 3.5 Irreducible Reduction of a Tensor.- 3.6 Restriction of SO(3) to SO(2).- 3.7 Transformation of Optical Flow.- Exercises.- 4. Algebraic Invariance of Image Characteristics.- 4.1 Algebraic Invariants and Irreducibility.- 4.2 Scalars, Points, and Lines.- 4.3 Irreducible Decomposition of a Vector.- 4.4 Irreducible Decomposition of a Tensor.- 4.5 Invariants of Vectors.- 4.6 Invariants of Points and Lines.- 4.7 Invariants of Tensors.- 4.8 Reconstruction of Camera Rotation.- Exercises.- 5. Characterization of Scenes and Images.- 5.1 Parametrization of Scenes and Images.- 5.2 Scenes, Images, and the Projection Operator.- 5.3 Invariant Subspaces of the Scene Space.- 5.4 Spherical Harmonics.- 5.5 Tensor Expressions of Spherical Harmonics.- 5.6 Irreducibility of Spherical Harmonics.- 5.7 Camera Rotation Transformation of the Image Space.- 5.8 Invariant Measure.- 5.9 Transformation of Features.- 5.10 Invariant Characterization of a Shape.- Exercises.- 6. Representation of 3D Rotations.- 6.1 Representation of Object Orientations.- 6.2 Rotation Matrix.- 6.3 Rotation Axis and Rotation Angle.- 6.4 Euler Angles.- 6.5 Cayley-Klein Parameters.- 6.6 Representation of SO(3) by SU(2).- 6.7 Adjoint Representation of SU(2).- 6.8 Quaternions.- 6.9 Topology of SO(3).- 6.10 Invariant Measure of 3D Rotations.- Exercises.- 7. Shape from Motion.- 7.1 3D Recovery from Optical Flow for a Planar Surface.- 7.2 Flow Parameters and 3D Recovery Equations.- 7.3 Invariants of Optical Flow.- 7.4 Analytical Solution of the 3D Recovery Equations.- 7.5 Pseudo-orthographic Approximation.- 7.6 Adjacency Condition of Optical Flow.- 7.7 3D Recovery of a Polyhedron.- 7.8 Motion Detection Without Correspondence.- Exercises.- 8. Shape from Angle.- 8.1 Rectangularity Hypothesis.- 8.2 Spatial Orientation of a Rectangular Corner.- 8.3 Interpretation of a Rectangular Polyhedron.- 8.4 Standard Transformation of Corner Images.- 8.5 Vanishing Points and Vanishing Lines.- Exercises.- 9. Shape from Texture.- 9.1 Shape from Texture from Homogeneity.- 9.2 Texture Density and Homogeneity.- 9.3 Perspective Projection and the First Fundamental Form.- 9.4 Surface Shape Recovery from Texture.- 9.5 Recovery of Planar Surfaces.- 9.6 Numerical Scheme of Planar Surface Recovery.- Exercises.- 10. Shape from Surface.- 10.1 What Does 3D Shape Recovery Mean?.- 10.2 Constraints on a $$2\frac{1}
{2}\text{D}$$ Sketch.- 10.3 Optimization of a $$2\frac{1}{2}\text{D}$$ Sketch.- 10.4 Optimization for Shape from Motion.- 10.5 Optimization of Rectangularity Heuristics.- 10.6 Optimization of Parallelism Heuristics.- Exercises.- Appendix. Fundamentals of Group Theory.- A.1 Sets, Mappings, and Transformations.- A.2 Groups.- A.3 Linear Spaces.- A.4Metric Spaces.- A.5 Linear Operators.- A.6 Group Representation.- A.7 Schur's Lemma.- A.8 Topology, Manifolds, and Lie Groups.- A.9 Lie Algebras and Lie Groups.- A.10 Spherical Harmonics.
