Iteratively Reweighted Midpoint Method for Fast Multiple View Triangulation

Abstract

The classic midpoint method for triangulation is extremely fast, but usually labelled as inaccurate. We investigate the cost function that the midpoint method tries to minimize, and the result shows that the midpoint method is prone to underestimate the accuracy of the measurement acquired relatively far from the three-dimensional (3-D) point. Accordingly, the cost function used in this letter is enhanced by assigning a weight to each measurement, which is inversely proportional to the distance between the 3-D point and the corresponding camera center. After analyzing the gradient of the modified cost function, we propose to do minimization by applying fixed-point iterations to find the roots of the gradient. Thus, the proposed method is called the iteratively reweighted midpoint method. In addition, a theoretical study is presented to reveal that the proposed method is an approximation to the Newton's method near the optimal point and, hence, inherits the quadratic convergence rate. Finally, the comparisons of the experimental results on both synthetic and real datasets demonstrate that the proposed method is more efficient than the state-of-the-art while achieves the same level of accuracy.