Reconstruction from Multiple Images using Kinetic Depths

This paper deals with the problem of reconstructing the locations of n points in space from m diierent images without camera calibration. It is assumed that the correspondences between the points in the diierent images are known. We will show how these reconstruction problems for diierent n and m can be put into a similar theoretical framework and will solve the so called minimal cases. When n = 7 and m = 2 there are in general three solutions to the reconstruction problem as well as when n = 6 and m = 3. The solutions are given by a third degree polynomial with coeecients depending on the coordinates of the points in the images. We will also investigate the so called minimal overdetermined cases, which can be solved by linear methods. There are three interesting such cases, obtained by adding one image or one point to the minimal cases described above. A new concept, the reduced fundamental matrix, is introduced, which gives bilinear expressions in the image coordinates. It contains just 4 parameters and can be used to make reconstruction. We also introduce the concept of reduced fundamental tensor, which describes the relations between points in three images and gives trilinear expressions in the image coordinates. It has 15 components and depends on 9 parameters. Finally, the reduced quadrilinear tensor is introduced, which describes the relations between point in four images and gives quadrilinear expressions in the image coordinates. This tensor has 36 components which depends on 14 independent parameters. This gives the possibility to calculate linear solutions from 8 points in 2 images, 7 points in 3 images and also from 6 points in 4 images, by bilinear, trilinear and quadrilinear expressions, respectively, in the image coordinates. Furthermore, a canonical form of the camera matrices in a sequence is presented and it is shown that the quadrilinear constraints follow from the trilinear ones, and that in general the trilinear constraints follow from the bilinear ones.