Reduced Multilinear Constraints: Theory and Experiments

• Authors: Anders Heyden
• Venue: International Journal of Computer Vision
• Year: 1998

This paper deals with the problem of reconstructing the locations ofn points in space from m different images without camera calibration. We will show how these reconstruction problems for different n andm can be put into a similar theoretical framework. This will be done using a special choice of coordinates, both in theobject and in the images, called reduced affine coordinates. Thischoice of coordinates simplifies the analysis of the multilineargeometry and gives simpler forms of the multilinear tensors.In particular, we will investigate the cases, which can be solved bylinear methods, i.e., ≥8 points in 2 images, ≥7 points in 3images and ≥6 points in 4 images.A new concept, the reduced fundamental matrix, is introduced, whichgives bilinear expressions in the image coordinates. It has six nonzero elements, whichdepend on just four parameters and can be used to make reconstructionfrom 2 images.We also introduce the concept of the reduced trifocal tensor, which givestrilinear expressions in the image coordinates in 3 images.It has 15 nonzero elements and depends on nine parameters and can be used to make reconstructionfrom 3 images.Finally, the reduced quadfocaltensor is introduced, which describes the relations between points in4 images and gives quadlinear expressions in the imagecoordinates. This tensor has 36 nonzero elements which depend on 14independent parameters and can be used to makereconstruction from 4 images.These tensors give the possibility to calculate linearsolutions from ≥8 points in 2 images, ≥7 points in 3 images andalso from ≥6 points in 4 images.Furthermore, a canonical form of the camera matrices in a sequence ispresented and it is shown that the quadlinear constraints can becalculated from the trilinear ones, and that in general the trilinearconstraints can be calculated from the bilinear ones.