Ebook: Group-Theoretical Methods in Image Understanding

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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This book presents the mathematics relevant to image understanding by computer vision and gives examples of actual applications. Group representation theory, Lie groups and Lie algebras, the theory of invariance, tensor calculus, differential geometry and projective geometry are used for three-dimensional shape and motion analysis from images, making use of techniques such as shape from motion, shape from texture, shape from angle and shape from surface. Although the mathematics itself may be well known to mathematicians, people working in areas related to computer science, image understanding, computer vision and image processing have usually never studied such mathematics, and so may be surprised to learn that abstract mathematical concepts can be of enormous help in building intelligent computer vision systems.

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This book presents the mathematics relevant to image understanding by computer vision and gives examples of actual applications. Group representation theory, Lie groups and Lie algebras, the theory of invariance, tensor calculus, differential geometry and projective geometry are used for three-dimensional shape and motion analysis from images, making use of techniques such as shape from motion, shape from texture, shape from angle and shape from surface. Although the mathematics itself may be well known to mathematicians, people working in areas related to computer science, image understanding, computer vision and image processing have usually never studied such mathematics, and so may be surprised to learn that abstract mathematical concepts can be of enormous help in building intelligent computer vision systems.
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
Introduction....Pages 3-20
Coordinate Rotation Invariance of Image Characteristics....Pages 21-60
3D Rotation and Its Irreducible Representations....Pages 61-102
Algebraic Invariance of Image Characteristics....Pages 103-146
Characterization of Scenes and Images....Pages 147-196
Representation of 3D Rotations....Pages 197-235
Front Matter....Pages 237-237
Shape from Motion....Pages 239-277
Shape from Angle....Pages 278-326
Shape from Texture....Pages 327-355
Shape from Surface....Pages 356-397
Back Matter....Pages 436-459

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This book presents the mathematics relevant to image understanding by computer vision and gives examples of actual applications. Group representation theory, Lie groups and Lie algebras, the theory of invariance, tensor calculus, differential geometry and projective geometry are used for three-dimensional shape and motion analysis from images, making use of techniques such as shape from motion, shape from texture, shape from angle and shape from surface. Although the mathematics itself may be well known to mathematicians, people working in areas related to computer science, image understanding, computer vision and image processing have usually never studied such mathematics, and so may be surprised to learn that abstract mathematical concepts can be of enormous help in building intelligent computer vision systems.
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
Introduction....Pages 3-20
Coordinate Rotation Invariance of Image Characteristics....Pages 21-60
3D Rotation and Its Irreducible Representations....Pages 61-102
Algebraic Invariance of Image Characteristics....Pages 103-146
Characterization of Scenes and Images....Pages 147-196
Representation of 3D Rotations....Pages 197-235
Front Matter....Pages 237-237
Shape from Motion....Pages 239-277
Shape from Angle....Pages 278-326
Shape from Texture....Pages 327-355
Shape from Surface....Pages 356-397
Back Matter....Pages 436-459
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