A Robust O(n) Solution to the Perspective-n-Point Problem

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 34; no. 7; pp. 1444 - 1450
• Main Authors: Li, Shiqi, Xu, Chi, Xie, Ming
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.07.2012; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Artificial intelligence; augmented reality; camera pose estimation; Cameras; Computation; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Cost function; Derivatives; Exact sciences and technology; Iterative algorithms; Iterative methods; Mathematical models; Optimization; Pattern analysis; Pattern recognition. Digital image processing. Computational geometry; Perspective-n-point problem; Polynomials; Robustness; Software; Studies; Theoretical computing; Three dimensional; Three dimensional displays; Vectors; Singularity; User interface; Redundancy; camera pose estimation; Virtual reality; Derivative; Iterative method; Computational complexity; Optimization; Augmented reality; Perspective-n-point problem; Posture
• ISSN: 0162-8828; 1939-3539; 2160-9292; 1939-3539
• DOI: 10.1109/TPAMI.2012.41

We propose a noniterative solution for the Perspective-n-Point ({\rm P}n{\rm P}) problem, which can robustly retrieve the optimum by solving a seventh order polynomial. The central idea consists of three steps: 1) to divide the reference points into 3-point subsets in order to achieve a series of fourth order polynomials, 2) to compute the sum of the square of the polynomials so as to form a cost function, and 3) to find the roots of the derivative of the cost function in order to determine the optimum. The advantages of the proposed method are as follows: First, it can stably deal with the planar case, ordinary 3D case, and quasi-singular case, and it is as accurate as the state-of-the-art iterative algorithms with much less computational time. Second, it is the first noniterative {\rm P}n{\rm P} solution that can achieve more accurate results than the iterative algorithms when no redundant reference points can be used (n\le 5). Third, large-size point sets can be handled efficiently because its computational complexity is O(n).