# A Robust and Efficient Algorithm for the PnL Problem Using Algebraic Distance to Approximate the Reprojection Distance

### Abstract

This paper proposes a novel algorithm to solve the pose estimation problem from 2D/3D line correspondences, known as the Perspective-n-Line (PnL) problem. It is widely known that minimizing the geometric distance generally results in more accurate results than minimizing an algebraic distance. However, the rational form of the reprojection distance of the line yields a complicated cost function, which makes solving the first-order optimality conditions infeasible. Furthermore, iterative algorithms based on the reprojection distance are time-consuming for a large-scale problem. In contrast to previous works which minimize a cost function based on an algebraic distance that may not approximate the reprojection distance of the line, we design two simple algebraic distances to gradually approximate the reprojection distance. This speeds up the computation, and maintains the robustness of the geometric distance. The two algebraic distances result in two polynomial cost functions, which can be efficiently solved. We directly solve the first-order optimality conditions of the first problem with a novel hidden variable method. This algorithm makes use of the specific structure of the resulting polynomial system, therefore it is more stable than the general Gröbner basis polynomial solver. Then, we minimize the second polynomial cost function by the damped Newton iteration, starting from the solution of the first cost function. Experimental results show that the first step of our algorithm is already superior to the state-of-the-art algorithms in terms of accuracy and applicability, and faster than the algorithms based on Gröbner basis polynomial solver. The second step yields comparable results to the results from minimizing the reprojection distance, but is much more efficient. For speed, our algorithm is applicable to real-time applications.