# Reduced Multilinear Constraints: Theory and Experiments

### Abstract

This paper deals with the problem of reconstructing the locations of n points in space from m different images without camera calibration. We will show 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 coordinates. This choice of coordinates simplifies the analysis of the multilinear 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., ≥8 points in 2 images, ≥7 points in 3 images and ≥6 points in 4 images. A new concept, the reduced fundamental matrix, is introduced, which gives bilinear expressions in the image coordinates. It has six nonzero elements, which depend on just four parameters and can be used to make reconstruction from 2 images. We also introduce the concept of the reduced trifocal 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 quadfocal tensor is introduced, which describes the relations between points in 4 images and gives quadlinear expressions in the image coordinates. This tensor has 36 nonzero elements which depend on 14 independent parameters and can be used to make reconstruction from 4 images. These tensors give 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.

Furthermore, a canonical form of the camera matrices in a sequence is presented and it is shown that the quadlinear constraints can be calculated from the trilinear ones, and that in general the trilinear constraints can be calculated from the bilinear ones.