# Cameras, Shapes, and Contours: Geometric Models in Computer Vision

1 WILLOW - Models of visual object recognition and scene understanding
Inria de Paris, DI-ENS - Département d'informatique de l'École normale supérieure
Abstract : This thesis studies mathematical models for describing the geometry of imaging processes in computer vision. Our approach is rooted in the language of projective geometry, which provides the most general setting for studying properties of lines and incidences that are at the heart of geometric vision. We also apply some tools from algebraic geometry, since many of the objects that we encounter are described by polynomial equations. For example, the multi-view geometry of $n$ pinhole cameras (or in fact any type of cameras) can be encoded in the "joint image", that is an algebraic variety in $(\mathbb P^2)^n$ formed by all point correspondences. The Grassmannian of lines ${\rm Gr}(1,\mathbb P^3)$ also plays a central role in our study. In particular, surfaces in the Grassmannian (or "line congruences") can be used to represent abstract cameras, that are mappings from points to viewing lines. In addition to modeling cameras, we investigate the relationship between 3D shapes and their images. For arbitrary sets projecting onto opaque silhouettes, the image is determined by the set of viewing lines that meet the observed object; for smooth surfaces, the "image contour" is determined by the set viewing lines that are tangent to the surface. This perspective is applied to the study of "visual hulls" and "visual events".
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https://hal.inria.fr/tel-01856415
Contributor : Matthew Trager <>
Submitted on : Monday, September 3, 2018 - 5:29:50 PM
Last modification on : Tuesday, January 29, 2019 - 3:05:42 PM
Long-term archiving on : Tuesday, December 4, 2018 - 6:25:45 PM

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• HAL Id : tel-01856415, version 2

### Citation

Matthew Trager. Cameras, Shapes, and Contours: Geometric Models in Computer Vision. Computer Vision and Pattern Recognition [cs.CV]. Ecole Normale Superieure de Paris - ENS Paris, 2018. English. ⟨tel-01856415v2⟩

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