Distributed Bundle Adjustment Based on Penalty Function Method

Bundle Adjustment (BA) aims to estimate the camera poses and build maps utilizing the nonlinear optimization algorithm. The update step of the optimization is obtained by solving a linear system, which is the bottleneck of the BA efficiency. Many works perform bundle adjustment in a distributed manner to reduce the computational cost of solving the linear system and achieve outstanding performance. However, the distributed methods implement the constrained optimization method that involves the computation of the Lagrange multiplier, causing extra overhead. To improve the efficiency of BA, we propose a novel distributed algorithm based on the penalty method, dubbed PenBA, that formulates an unconstrained problem and hence saves the computation of extra variables. To solve the unconstrained linear system with low cost, we design a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/><bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed Preconditioned Conjugate Gradient (PCG)</b><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> which has complexity bounded by hyper-parameters. It can be proved that, given the parameter update strategy of the optimization method, the penalty linear system can be solved with fast convergence. Experimental results in the BAL dataset illustrate that our penalty method outperforms the Lagrange multiplier method and conventional BA. Compared with the most accurate conventional PCG-based BA solver, PenBA decreases the objective values by 3.8<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula>, the time per iteration by 17<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and the maximum memory consumption by 24<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula>. Experimental results in the SfM and SLAM datasets prove that PenBA applies to both computer vision and robotics contexts.