## Visual Motion of Curves and SurfacesThe 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 |

179 | |

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### Common terms and phrases

3-space algorithm apparent contour asymptotic curve asymptotic direction axis of rotation B-spline B-spline snake calibration camera centre Chapter Cipolla and Blake circular motion Computer Vision cone conjugate coordinate system corresponding epipolar lines cusp point cylinder defined Definition derivatives Differentiating epipolar constraint epipolar curve epipolar geometry epipolar parametrization epipolar tangencies epipoles equation example Faugeras Figure flecnodal frontier points fundamental matrix Gauss curvature given gives II(a image contours image plane image points image sequence image sphere intersection linear Monge form nonzero Note optical axis origin orthogonal projection parabolic point parallel perpendicular perspective projection point correspondences principal directions projection matrices projective transformation Proof of Property quadric radius reconstruction recovered second fundamental form sectional curvature singular smooth space curve spatio-temporal surface normal surface point tangent plane tangent vector uncalibrated unit vector view direction viewer motion visual ray weak perspective world coordinates x-axis