# A Second-Order Lower Bound for Globally Optimal 2D Registration

## TL;DR

This paper introduces a novel second-order relaxation bound for a Branch and Bound algorithm to achieve globally optimal 2D point set registration, improving efficiency over existing bounds.

## Contribution

The work presents a new relaxation bound that dominates the cheap bound, with quadratic convergence properties, enhancing the performance of global 2D registration algorithms.

## Key findings

- The relaxation bound outperforms the cheap bound in reducing BnB nodes.
- The relaxation bound's quadratic approximation accelerates convergence.
- Experimental results show improved computational efficiency.

## Abstract

The problem of planar registration consists in finding the transformation that better aligns two point sets. In our setting, the search domain is the set of planar rigid transformations and the objective function is the sum of the distances between each point of the transformed source set and the destination set. We consider a Branch and Bound (BnB) method for finding the globally optimal solution. The algorithm recursively splits the search domain into boxes and computes an upper and a lower bound for the minimum value of the restricted problem. The main contribution of this work is the introduction of a novel lower bound, the relaxation bound, which corresponds to the solution of a concave relaxation of the objective function based on the linearization of the distance. In the BnB we also employ the so called cheap bound, equal to to the sum of the minimum distances between each point of source point set, transformed according to current box, and all the candidate points in the destination point set. We prove, both theoretically and practically, that the novel relaxation bound dominates the cheap bound over small boxes. More precisely, from the theoretical point of view, we prove that the relaxation bound is a second-order approximation of the minimum value, i.e., its distance from the minimum value decreases quadratically with respect to the diameter of the box (see Theorem 1), while the cheap bound is a first-order one (see Proposition 3). From the practical point of view, we show through different computational experiments that the addition of the relaxation bound considerably enhances the performance of the BnB algorithm, compensating the higher cost of its computation with respect to the cheap bound with a strong reduction of the number of BnB nodes to be explored.