The Application Of Wu-Ritt's Decomposition Method In PnP Problem

The perspective-n-point(PnP) problem originated from the ancient problem of photogrammetry. But in recent two decades, it has been reintroduced in the form of the determination of the location of the camera in many fields, such as image analysis, automated cartography, computer vision, machine vision, robotics and photogrammetry. In 1981,Fishier and Bolles transformed the LDP problem which relies on the feature points into the mathematical form, which is the so-called PnP problem. And subsequently, a series of discussion about the problem were ignited. All the discussion can be categorized into two groups: numerical solution and analytic solution.However, the most troublesome difficulty is the multisolution of the PnP problem. In real-time system or in the field of cartoon-synthesizing, the improvement of the feature points will also lead to the improvement of the expenditure. In addition, the more feature points we choose, the more complexity the arithmetic will have. Simultaneously the speed of the computation will also decline. And so, for many years, one of the emphasisses on the research of the PnP problem is to find a simple, practical and effective arithmetic which based on as less feature points as possible. Thus the P3P problem and the P4P problem is the first choice. In this paper, the analytic method is mainly used to get the method of discrimination.In chapter 2, the Wu-Ritt's decomposition method is used to decompose the equations the P3P problem at first and some character-sets are obtained. According to the degree of the polynomials, the character-sets are categorized, and then the complete classification of the P3P problem is obtained.In chapter 3, a smart method of grouping decomposition, based on which the difficulty that too many restriction lead to is conquered, is used to decompose the super-restriction equations of the P4P problem. After getting the character-set of each group, an integrative analyse is performed and a series of sufficient condition to decide the sum of the solution of the P4P problem are obtained. And moreover, it is proved that the conclusion in [2] is the specification of the foregoing conclusion in this paper. Based on the presented condition of discrimination in this paper, those unreliable positions can be easily avoided, or some decisions can be quickly made. Therefore, the stability and accuracy of the system are improved.