Robust Algorithms for Multiple View Geometry: Outliers and Optimality

Abstract: This thesis is concerned with the geometrical parts of computer vision, or more precisely, with the three-dimensional geometry. The overall aim is to extract geometric information from a set of images. Most methods for estimating the geometry of multiple views rely on the existence of robust solvers for a set of basic problems. Such a basic problem can be estimating the relative orientation of two cameras or estimating the position of a camera given a model of the scene. The first part of this thesis presents a number of new algorithms for attacking different instances of such basic problems. Normally methods for these problems consist of two parts. First, interest points are extracted from the images and point-to-point correspondences between images are determined. In the second step these correspondences are used to estimate the geometry. A major difficulty lies in the existence of incorrect correspondences, often called outliers. Not modelling these outliers will result in very poor accuracy. Hence, the algorithms in this thesis are designed to be robust to such outliers. In particular, focus lies on obtaining optimal solutions in presence of outliers. For example, it is shown how optimal solutions can be found in cases when the residuals are quasiconvex functions. Moreover, optimal algorithms for the non-convex problems of calibrated camera pose estimation, registration and relative orientation estimation are presented. The second part of the thesis discusses how the solutions from these basic problems can be combined and refined to form a model of the whole scene. Again, robustness is crucial. In this case robustness with respect to incorrect solutions from the basic problems. In particular, a complete system for estimating muliple view geometry is proposed. Often this problem has been solved in a sequential manner, but recently there has been a large interest in non-sequential methods. The results in this thesis show the advantages of a non-sequential approach based on graph methods as well as convex optimization. The last chapter of the thesis is concerned with structure from motion for the special case of one-dimensional cameras. It is shown how optimal solutions can be obtained using linear programming.