Reduced multilinear constraints: Theory and experiments

This paper deals with the problem of reconstructing the locations of n poin ts in space from m different images without camera calibration. We will sho w how these reconstruction problems for different n and m can be put into a similar theoretical framework. This will be done using a special choice of coordinates, both in the object and in the images, called reduced affine c oordinates. This choice of coordinates simplifies the analysis of the multi linear geometry and gives simpler forms of the multilinear tensors. In particular, we will investigate the cases, which can be solved by linear methods, i.e., greater than or equal to 8 paints in 2 images, greater than or equal to 7 points in 3 images and greater than or equal to 6 points in 4 images. A new concept, the reduced fundamental matrix, is introduced, whi ch gives bilinear expressions in the image coordinates. It has six nonzero elements, which depend on just four parameters and can be used to make reco nstruction from 2 images. We also introduce the concept of the reduced trif ocal tensor, which gives trilinear expressions in the image coordinates in 3 images. It has 15 nonzero elements and depends on nine parameters and can be used to make reconstruction from 3 images. Finally, the reduced quadfoc al tensor is introduced, which describes the relations between points in 4 images and gives quadlinear expressions in the image coordinates. This tens or has 36 nonzero elements which depend on 14 independent parameters and ca n be used to make reconstruction from 4 images. These tensors give the poss ibility to calculate linear solutions from greater than or equal to 8 point s in 2 images, greater than or equal to 7 points in 3 images and also from greater than or equal to 6 points in 4 images. Furthermore, a canonical form of the camera matrices in a sequence is prese nted and it is shown that the quadlinear constraints can be calculated from the trilinear ones, and that in general the trilinear constraints can be c alculated from the bilinear ones.