Efficient Second-Order Plane Adjustment

TL;DR

This paper introduces an efficient second-order method for plane adjustment in 3D reconstruction, leveraging Newton's method with closed-form plane solutions to achieve faster convergence than traditional Levenberg-Marquardt approaches.

Contribution

It proposes a novel Newton-based approach for plane adjustment that eliminates planes analytically, reducing variables and improving convergence speed in large-scale 3D reconstruction tasks.

Findings

Our method converges significantly faster than LM algorithm.

Eliminating planes reduces computational complexity.

The approach ensures optimal planes are updated at each iteration.

Abstract

Planes are generally used in 3D reconstruction for depth sensors, such as RGB-D cameras and LiDARs. This paper focuses on the problem of estimating the optimal planes and sensor poses to minimize the point-to-plane distance. The resulting least-squares problem is referred to as plane adjustment (PA) in the literature, which is the counterpart of bundle adjustment (BA) in visual reconstruction. Iterative methods are adopted to solve these least-squares problems. Typically, Newton's method is rarely used for a large-scale least-squares problem, due to the high computational complexity of the Hessian matrix. Instead, methods using an approximation of the Hessian matrix, such as the Levenberg-Marquardt (LM) method, are generally adopted. This paper challenges this ingrained idea. We adopt the Newton's method to efficiently solve the PA problem. Specifically, given poses, the optimal planes have close-form solutions. Thus we can eliminate planes from the cost function, which significantly reduces the number of variables. Furthermore, as the optimal planes are functions of poses, this method actually ensures that the optimal planes for the current estimated poses can be obtained at each iteration, which benefits the convergence. The difficulty lies in how to efficiently compute the Hessian matrix and the gradient of the resulting cost. This paper provides an efficient solution. Empirical evaluation shows that our algorithm converges significantly faster than the widely used LM algorithm.