Reconstruction from Minimal Data

Abstract

It is shown in Sect. 2.2.1 that reconstruction from image correspondences is algebraic in that it can be expressed as the problem of finding the common zeros of a set of polynomial equations. This fact has profound consequences, especially if the number n of image correspondences is small. The algebraic nature of reconstruction is particularly apparent in the minimal case when n is just sufficient to ensure that only a finite number of reconstructions are possible. In reconstruction with known camera calibration five pairs of corresponding points are sufficient to ensure at most a finite number of reconstructions. The minimal number for reconstruction up to a collineation is seven. Reconstruction from image velocities is also an algebraic problem and the minimal number of image velocity vectors is five. The term ‘minimal data’ is used to refer either to point correspondences or to image velocities, depending on the context. The algebraic properties of reconstruction are important in applications because they have to be taken into account by any algorithm for reconstruction which makes full use of the available information.