Geometry and Critical Configurations of Multiple Views

This thesis is concerned with one of the core problems in computer vision, namely to reconstruct a real world scene from several images of it. The interplay between the geometry of the scene, the cameras and the images is analyzed. The framework is based on projective geometry, which is the natural language for describing the geometry of multiple views. The affine and Euclidean geometries are regarded as special cases of projective geometry. First, the projection of several different geometric primitives in multiple views is described and analyzed. The analysis includes points, lines, quadrics and curved surfaces. The cameras are assumed to be uncalibrated and both the perspective/projective and the affine camera model are considered. Several new reconstruction methods are developed. Some features of these methods include the possibility of handling: (i) missing data, (ii) several different primitives simultaneously and (iii) minimal cases. Then, focus is turned to the process of obtaining a Euclidean reconstruction of the scene from uncalibrated images. This problem is known as auto-calibration. A reconstruction method which imposes regularity constraints on the camera motion is introduced which makes the auto-calibration problem more stable. The last part of the thesis is devoted to a theoretical study of necessary and sufficient conditions for obtaining a unique scene reconstruction. One classical result is that for two images of a 3D scene this happens if and only if the scene points and the camera centres do not lie on a ruled quadric. This is generalized to any number of views. Furthermore, analogous critical configurations for the 1D camera are derived. In auto-calibration, it is shown that the critical configurations depend only on the camera motion. Complete classifications of such critical motions which lead to ambiguous reconstructions are given under different settings