9780521632515

Visual Motion of Curves and Surfaces

by
  • ISBN13:

    9780521632515

  • ISBN10:

    052163251X

  • Format: Hardcover
  • Copyright: 2000-01-13
  • Publisher: Cambridge University Press
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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

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