Visual Motion of Curves and Surfaces

by Roberto Cipolla , Peter Giblin

• ISBN13: 9780521632515
• ISBN10: 052163251X
• Format: Hardcover
• Copyright: 2000-01-13
• Publisher: Cambridge University Press

Summary

Computer vision aims to detect and reconstruct features of surfaces from the images produced by cameras, in some way mimicking the way in which humans reconstruct features of the world around them by using their eyes. In this book the authors describe research in computer vision aimed at recovering the 3D shape of surfaces from image sequences of their 'outlines'. They provide all the necessary background in differential geometry (assuming knowledge of elementary algebra and calculus) and in the analysis of visual motion, emphasising intuitive visual understanding of the geometric techniques with computer-generated illustrations. They also give a thorough introduction to the mathematical techniques and the details of the implementations and apply the methods to data from real images using the most current techniques.

Table of Contents

Preface	vii
Introduction	1	(4)
Differential Geometry of Curves and Surfaces	5	(49)
Curves and their tangents	5	(2)
Surfaces: the parametric form	7	(6)
Monge form	13	(2)
Implicit form	15	(3)
First fundamental form for surfaces	18	(3)
Curvature of curves	21	(4)
Three surface types	25	(2)
Second fundamental form and curvatures: parametrized surfaces	27	(12)
Second fundamental form and curvatures: Monge form of surface	39	(2)
Special Monge form	41	(4)
Second fundamental form: implicit form of surface	45	(1)
Special curves on a surface	45	(3)
Contact	48	(6)
Views of Curves and Surfaces	54	(25)
Camera models: parallel (orthographic) projection	54	(1)
Perspective projection	55	(4)
Opaque vs. semi-transparent surfaces	59	(1)
Static properties of contour generators and apparent contours	59	(2)
Properties: orthogonal projection	61	(5)
Properties: perspective projection	66	(6)
Methods of proof: Monge-Taylor proofs	72	(1)
Monge--Taylor proofs: orthogonal projection	72	(2)
Monge--Taylor proofs: perspective projection	74	(2)
Vector proofs: orthogonal projection	76	(1)
Vector proofs: perspective projection	77	(1)
Methods of proof: pure geometric proofs	78	(1)
Dynamic Analysis of Apparent Contours	79	(35)
Orthogonal projection	80	(4)
Epipolar parametrization: orthogonal case	84	(1)
Perspective projection	85	(4)
Epipolar parametrization: perspective case	89	(5)
Surface curvatures using the epipolar parametrization	94	(1)
Degeneracies of the epipolar parametrization	95	(1)
Visual events: swallowtail, lips and breaks	96	(1)
Frontiers (epipolar tangencies)	97	(8)
Following cusps	105	(1)
Formulae for K and H by following cusps	105	(3)
Image velocity of a cusp point	108	(1)
Envelopes of surfaces and apparent contours	109	(5)
Reconstruction of Surfaces from Profiles	114	(25)
Localization and tracking of apparent contours	114	(5)
Camera model for perspective projection onto image plane	119	(4)
Camera model for weak perspective and orthographic projection	123	(1)
Camera calibration	124	(2)
Epipolar geometry	126	(3)
Epipolar geometry from projection matrices	129	(2)
Reconstruction of surfaces	131	(8)
Recovery of Viewer Motion from Profiles	139	(34)
The fundamental matrix from point correspondences	139	(3)
Recovery of the projection matrices and viewer motion	142	(2)
Recovery of the projection matrices for uncalibrated cameras	144	(3)
Frontier points and epipolar tangencies	147	(2)
Recovery of motion under pure translation	149	(2)
General motion	151	(4)
Weak perspective	155	(3)
Circular motion	158	(7)
Envelope of apparent contours under circular motion	165	(8)
Afterword	173	(1)
Bibliography	174	(5)
Index	179