Minimisation of alignment error between a camera and a laser range finder using Nelder-Mead simplex direct search

Presented in this paper is a novel method to calibrate the co-ordinate systems used by two separate sensor devices for the purposes of sensor fusion. In this example the sensors are a camera and a LIDAR device which are observing the same scene from different viewpoints. Using a synthetic set of corresponding 2D image co-ordinates and 3D LIDAR measurements as reference data the task of aligning re-projected measurements with reference measurements was posed as an optimisation problem. The objective of the optimisation is to find a set of calibration parameters (external offsets and internal camera parameters) which minimise the sum of squared errors between the reference image co-ordinates and the re-projected data. The re-projected data is obtained by transforming the reference LIDAR measurements using the calibration parameters and the errors are defined as the straight-line distance between each reference and re-projected pixel pair. Using the Nelder-Mead simplex search method calibration parameters were found in under a second such that the sum of squared errors across a data set of 200 points was less than 0.19 i.e. average error per pixel of 0.031px. The method finds both internal and external calibration factors and makes no assumptions about the model. Furthermore if a second optimisation pass is made the error can be reduced to almost zero using only 4 reference pairs assuming these points are selected correctly.