Efficient Bundle Adjustment for Coplanar Points and Lines

Abstract

Bundle adjustment (BA) is a well-studied fundamental problem in the robotics and vision community. In man-made environments, coplanar points and lines are ubiquitous. However, the number of works on bundle adjustment with coplanar points and lines is relatively small. This paper focuses on this special BA problem, referred to as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi-\mathbf{BA}$</tex> . For a point or a line on a plane, we derive a new constraint to describe the relationship among two poses and the plane, called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -constraint. We distribute <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -constraints into different groups. Each group is called a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -factor. We prove that, with some simple preprocessing, the computational complexity associated with a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -factor in the Levenberg-Marquardt (LM) algorithm is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(1)$</tex> , independent of the number of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -constraints packed into the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -factor. In <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi-\mathbf{BA}, \pi$</tex> -factors replace original reprojection errors. One problem is how to divide <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -constraints into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -factors. Different strategies may result in different numbers of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> -factors, which in turn affects the efficiency. It is difficult to get the optimal division. We present a greedy algorithm to overcome this problem. Experimental results verify that our algorithm can significantly accelerate the computation.