Visual Motion of Curves and Surfaces

Front Cover
Cambridge University Press, 2000 - Computers - 184 pages
The world is full of objects, many of which are visible to us as surfaces. Examples are people, cars, machines, computers and bananas. Exceptions are such things as clouds and trees, which have a more detailed, fuzzy structure. Computer vision aims to detect and reconstruct features of surfaces from the images produced by cameras, in some ways mimicking the way in which humans reconstruct features of the world around them by using their eyes. This book describes how the 3D shape of surfaces can be recovered from image sequences of outlines. Cipolla and Giblin provide all the necessary background in differential geometry (assuming knowledge of elementary algebra and calculus) and in the analysis of visual motion, and emphasizes 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.
 

Contents

Introduction
1
Differential Geometry of Curves and Surfaces
5
the parametric form
7
23 Monge form
13
24 Implicit form
15
25 First fundamental form for surfaces
18
26 Curvature of curves
21
27 Three surface types
25
45 Surface curvatures using the epipolar parametrization
94
46 Degeneracies of the epipolar parametrization
95
swallowtail lips and beaks
96
48 Frontiers epipolar tangencies
97
49 Following cusps
105
411 Image velocity of a cusp point
108
412 Envelopes of surfaces and apparent contours
109
Reconstruction of Surfaces from Profiles
114

parametrized surfaces
27
Monge form of surface
39
210 Special Monge form
41
implicit form of surface
45
213 Contact
48
Views of Curves and Surfaces
54
32 Perspective projection
55
33 Opaque vs semitransparent surfaces
59
orthogonal projection
61
perspective projection
66
MongeTaylor proofs
72
perspective projection
74
orthogonal projection
76
perspective projection
77
pure geometric proofs
78
Dynamic Analysis of Apparent Contours
79
41 Orthogonal projection
80
orthogonal case
84
43 Perspective projection
85
perspective case
89
52 Camera model for perspective projection onto image plane
119
53 Camera model for weak perspective and orthographic projection
123
54 Camera calibration
124
55 Epipolar geometry
126
56 Epipolar geometry from projection matrices
129
57 Reconstruction of surfaces
131
Recovery of Viewer Motion from Profiles
139
62 Recovery of the projection matrices and viewer motion
142
63 Recovery of the projection matrices for uncalibrated cameras
144
64 Frontier points and epipolar tangencies
147
65 Recovery of motion under pure translation
149
66 General motion
151
67 Weak perspective
155
68 Circular motion
158
69 Envelope of apparent contours under circular motion
165
Afterword
173
Bibliography
174
Index
179
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