An Atlas for the Pinhole Camera

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.Comment: 47 pages with references and appendices, final versio

Affine Structure and Motion from Points, Lines and Conics

Abstract. In this paper several new methods for estimating scene structure and camera motion from an image sequence taken by a ne cameras are presented. All methods can incorporate both point, line and conic features in a uni ed manner. The correspondence between features in di erent images is assumed to be known. Three new tensor representations are introduced describing the viewing geometry for two and three cameras. The centred a ne epipoles can be used to constrain the location of corresponding points and conics in two images. The third order, or alternatively, the reduced third order centred a ne tensors can be used to constrain the locations of corresponding points, lines and conics in three images. The reduced third order tensors contain only 12 components compared to the 16 components obtained when reducing the trifocal tensor to a ne cameras. A new factorization method is presented. The novelty lies in the ability to handle not only point features, but also line and conic features concurrently. Another complementary method based on the so-called closure constraints is also presented. The advantage of this method is the ability to handle missing data in a simple and uniform manner. Finally, experiments performed on both simulated and real data are given, including a comparison with other methods