# Linearly Converging Quasi Branch and Bound Algorithms for Global Rigid Registration

## TL;DR

This paper introduces Quasi BnB, a novel framework that uses quadratic quasi-lower bounds to achieve linear convergence in global rigid registration, outperforming traditional BnB algorithms especially at high accuracy levels.

## Contribution

The paper proposes Quasi BnB, a new approach that replaces linear bounds with quadratic quasi-lower bounds, enabling linear convergence and improved efficiency in global rigid registration.

## Key findings

- Quasi BnB achieves $ ilde{O}( ext{log}(1/ extepsilon))$ time complexity.
- Quasi BnB outperforms state-of-the-art BnB algorithms in efficiency.
- The method is especially effective for high-accuracy registration problems.

## Abstract

In recent years, several branch-and-bound (BnB) algorithms have been proposed to globally optimize rigid registration problems. In this paper, we suggest a general framework to improve upon the BnB approach, which we name Quasi BnB. Quasi BnB replaces the linear lower bounds used in BnB algorithms with quadratic quasi-lower bounds which are based on the quadratic behavior of the energy in the vicinity of the global minimum. While quasi-lower bounds are not truly lower bounds, the Quasi-BnB algorithm is globally optimal. In fact we prove that it exhibits linear convergence -- it achieves $\epsilon$-accuracy in $~O(\log(1/\epsilon)) $ time while the time complexity of other rigid registration BnB algorithms is polynomial in $1/\epsilon $. Our experiments verify that Quasi-BnB is significantly more efficient than state-of-the-art BnB algorithms, especially for problems where high accuracy is desired.