## Exploring Three-dimensional Objects by Controlling the Point of Observation |

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### Contents

Dependence of the visible rims connectivity changes on the initial | 15 |

A Framework for Visual Exploration of Surface Geometry | 22 |

Tangential Viewpoint Control | 44 |

viewpoint | 56 |

A configuration of three rotational axes for object reorientation | 59 |

Geometry of object reorientation | 60 |

Reorienting a pipeshaped object | 61 |

Objects used in the experiments | 64 |

selected points on the occluding contour of the two tori | 145 |

The point being tracked for surface curvature estimation | 146 |

Curvature variation with viewpoint for the rotating toy experiment | 147 |

Viewpoints corresponding to the global minima and maxima of the curvature measurements | 148 |

Viewpoints corresponding to the global curvature minima and maxima for a different run of the tracking process | 149 |

Global Surface Reconstruction | 150 |

Difficulties in reconstructing the surface of a pipeshaped object | 156 |

The epipolar parameterization | 159 |

Tracking a visible rim point while moving on its tangent plane | 66 |

The window used for point tracking | 67 |

Application of the point tracking system to a different object | 68 |

Detecting point disocclusion | 69 |

Occluding Contour Detection | 71 |

Point correspondences induced by the epipolar geometry | 78 |

Distinguishing stationary from nonstationary points | 84 |

Bounding the distance between qt and qt | 87 |

Changing viewing directions on the selected motion plane | 88 |

Polyhedral model of a bottle and its visible rim | 98 |

Detecting the occluding contour of a bottle | 99 |

Two images of a rotating toy | 100 |

Detecting the occluding contour of a toy | 101 |

Detecting the stationarity of the m curve | 104 |

Detecting the nonstationarity of the right arm curve | 106 |

Recovering Local Surface Shape | 109 |

Aligning the viewing direction with a principal direction on TP S | 120 |

Finding the principal directions | 121 |

Determining the complete visibility of rim points | 124 |

Selecting points for surface recovery | 125 |

Recovering the shape of surfaces of revolution | 127 |

Removing p from the rim | 131 |

Models of a candlestick and two tori used for the simulations | 134 |

A sequence of 120 frames used in our experiments | 135 |

Snapshots of the occluding contour of a candlestick model as the view ing direction changes | 142 |

Snapshots of the occluding contour of two tori as the viewing direction changes | 143 |

Variation of the absolute curvature with respect to viewpoint at the selected points on the occluding contour of the candlestick model | 144 |

The visual events | 160 |

Reconstructing a region around an ordinary visible rim point on a torus | 168 |

Forcing a visible rim point to become ordinary | 170 |

Reconstructing a region around a degenerate point on the torus | 171 |

The visibility arcs | 175 |

The reconstructible regions on some surfaces studied by Petitjean et al and Koenderink | 176 |

The reconstructible regions for a pipeshaped surface | 177 |

Difficulties involved in globally reconstructing a dimpleshaped surface | 180 |

Geometry of the reconstruction of a curve segment drawn on an objects surface | 184 |

Moving to the middle of the visibility arc for a point on a pipeshaped object by tangential viewpoint control | 186 |

Changing viewpoint on the normal plane of a point on a pipeshaped object | 187 |

Finding the middle of the visibility arc of a point on a different pipe shaped object | 188 |

Strategies used to accomplish global surface reconstruction | 193 |

Three views of the region reconstructed on the pipes interior surface | 196 |

Reconstructing the surface of a pipeshaped object | 201 |

Two views of the path traced by the moving viewpoint during global reconstruction | 218 |

Conclusions and Future Work | 222 |

A Proofs of Chapter 4 Theorems | 229 |

B Proofs of Chapter 5 Theorems | 233 |

The effects of global occlusion | 236 |

Extent of Viewing Direction Adjustments for Local Shape Recovery | 237 |

The Darboux Frame | 239 |

Changing directions on the TN plane | 244 |

Visual Events and their Associated Visual Event Curves | 245 |

The visual events | 249 |

Inducing the visibility of points in a neighborhood of an ordinary hy perbolic point | 251 |

Representing the configurations of the asymptotes and bitangent lines through a visible rim point | 255 |