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The Scientific World Journal
Volume 2014, Article ID 109318, 8 pages
Research Article

A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images

1College of Information Science and Engineering, Northeastern University, Shenyang 110819, China
2Department of Computer Science, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received 14 August 2013; Accepted 14 November 2013; Published 29 January 2014

Academic Editors: J. Shu and F. Yu

Copyright © 2014 YuanBin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points in five images. This paper presents a method to compute projective invariants of 3D points from four uncalibrated images directly. For a set of six 3D points in general position, we choose four of them as the reference basis and represent the other two points under this basis. It is known that the cross ratios of the coefficients of these representations are projective invariant. After a series of linear transformations, a system of four bilinear equations in the three unknown projective invariants is derived. Systems of nonlinear multivariable equations are usually hard to solve. We show that this form of equations can be solved linearly and uniquely. This finding is remarkable. It means that the natural configuration of the projective reconstruction problem might be six points and four images. The solutions are given in explicit formulas.