Mathematical Analysis of the Multisolution Phenomenon in the P3P Problem

The perspective 3-point problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis and robotics. One line of activity involves formulating it mathematically in terms of finding the solution to a quartic equation. However, in general, the equation does not have a unique solution, and in some situations there are no solutions at all. Here, we present a new approach to the solution of the problem; this involves closer scrutiny of the coefficients of the polynomial, in order to understand how many solutions there will be for a given set of problem parameters. We find that, if the control points are equally spaced, there are four positive solutions to the problem at 25 % of all available spatial locations for the control-point combinations, and two positive solutions at the remaining 75 %.

[1]  Michael Q. Rieck.  Handling Repeated Solutions to the Perspective Three-point Pose Problem , 2010, VISAPP.

[2]  M HaralickRobert,et al.  Review and analysis of solutions of the three point perspective pose estimation problem , 1994 .

[3]  Long Quan,et al.  Linear N-Point Camera Pose Determination , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Michael Q. Rieck.  A Fundamentally New View of the Perspective Three-Point Pose Problem , 2013, Journal of Mathematical Imaging and Vision.

[5]  G. Volkart.  Das Rückwärtseinschneiden im Raum , 1933 .

[6]  Michael Q. Rieck.  An Algorithm for Finding Repeated Solutions to the General Perspective Three-Point Pose Problem , 2011, Journal of Mathematical Imaging and Vision.

[7]  Zhanyi Hu,et al.  A General Sufficient Condition of Four Positive Solutions of the P3P Problem , 2005, Journal of Computer Science and Technology.

[8]  Robert C. Bolles,et al.  Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography , 1981, CACM.

[9]  Daniel Lazard,et al.  Quantifier Elimination: Optimal Solution for Two Classical Examples , 1988, J. Symb. Comput..

[10]  Larry S. Davis,et al.  Pose Determination of a Three-Dimensional Object Using Triangle Pairs , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Hu,et al.  Why is the Danger Cylinder Dangerous in the P3P Problem , 2006 .

[12]  Michael J. Magee,et al.  The Perspective View of Three Points , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Lu Yang.  A Simplified Algorithm for Solution Classification of the Perspective-three-Point Problem 1) , 1998 .

[14]  Hu Zhan,et al.  A Probabilistic Study on the Multiple Solutions of the P3P Problem , 2007 .

[15]  E. L. Rees,et al.  Graphical Discussion of the Roots of a Quartic Equation , 1922 .

[16]  Jianliang Tang,et al.  Complete Solution Classification for the Perspective-Three-Point Problem , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Joseph S.-C. Yuan.  A general photogrammetric method for determining object position and orientation , 1989, IEEE Trans. Robotics Autom..