A New Geometric Approach for Faster Solving the Perspective-Three-Point Problem

An Efficient Algebraic Solution to the Perspective-Three-Point Problem

“…solving the generalized P3P resulting into an octic univariate polynomial whose odd monomials vanish for the case of the central P3P. 2 Although Masselli and Zell [18] claim that their algorithm runs faster than Kneip et al's [15], our results (see Section 3) show the opposite to be true (by a small margin). The reason we arrive at a different conclusion is that our simulation randomly generates a new geometric configuration for each run, while Masselli employs only one configuration during their entire simulation, in which they save time due to caching.…”

“…Our algorithm is implemented 6 in C++ using the same linear algebra library, TooN [22], as [15]. We employ simulation data to test our code and compare it to the solutions of [15] and [18]. For each Algorithm 1: Solving for the camera's pose Input: G pi, i = 1, 2, 3 the features' positions; C bi, i = 1, 2, 3 bearing measurements Output: G pC, the position of the camera; C G C, the orientation of the camera 1 Compute k1, k3 using (6) 2 Compute ui and vi using (11), i = 1, 2 3 Compute δ and k 3 using (22) and (23) 4 Compute the fij's using (29)-(36) 5 Compute αi, i = 0, 1, 2, 3, 4 using (39)-(50) 6 Solve (38) to get n (n = 2 or 4) real solutions for cos θ 1 , denoted as cos θ (i)…”

“…• Our algorithm's implementation takes about 40% of the time required by the current state of the art [15]. 2 • Our method achieves better accuracy than [15,18] under nominal conditions. Moreover, we are able to further improve the numerical precision by applying root polishing to the solutions of the quartic polynomial while remaining faster than [15,18].…”

“…In both cases, however, several intermediate terms (e.g., tangents and cotangents of certain angles) need to be computed, which negatively affect the speed and numerical precision of the resulting algorithms. Similar to [15] and [18], our proposed approach does not require first computing the features' distances. Differently though, in our derivation, we first eliminate the camera's position and the features' distances to result into a system of three equations involving only the camera's orientation.…”

“…Section 2 presents the definition of the P3P problem, as well as our derivations for estimating first the orientation and then the position of the camera. In Section 3, we assess the performance of our approach against [15] and [18] in simulation for both nominal and singular configurations. Finally, we conclude our work in Section 4.…”