Supplementary Material for Camera Pose Estimation using Implicit Distortion Models

1. Overview of Supplementary Material The supplementary material contains the following: • Implementation details (Section 2) • Detailed results from calibration experiment (Table 4) • Detailed results for the BA experiment (Section 3) • Evaluation with GT principal point (Section 4) • Dataset details (Table 1) • Additional qualitative results (Figure 4) 2. Implementation Details Neighborhood selection. To construct the residuals in the regularizer, we select the k = 2m closest neighbours symmetrically around each point. To avoid the instability from the denominator vanishing, a minimum distance δrmin to the closest neighbor is enforced. Further, since the locally linear assumption fails when the distance between two neighbors is too large, δrmax is set to ensure the distance of that falls below a threshold. Parameters. In our experiments, δrmin is set to be 1 px, while δrmax is set to be 100 px. The number of neighbors is chosen to be 4 since it provides a good trade-off between redundancy in the fitting and the sparsity pattern. This is evaluated in the ablation study in the main paper. Outlier filtering. Given an initial pose estimate we propose to filter outliers in a similar fashion as in [1], using a sliding median filter on the computed fi. Points where fi deviate significantly from the median of the local neighbourhood (n = 4) are removed. The threshold is set as k = 3 × 1.4826 times the median error, which correspond to the 99.7% confidence interval assuming the deviations are normally distributed. 3. Evaluation of Bundle Adjustment The proposed bundle adjustment method optimizes a surrogate problem in each iteration to enable use of the Schur complement trick. However, this inner optimization problem does not minimize the true cost. Figure 1 and Figure 2 show the cost across iterations for bundle adjustment with and without groundtruth principal point. One can observe spikes and energy steps due to the imperfection of the approximation, but still, the surrogate problem is a good approximation and the cost for outer iterations converges almost monotonically as shown in the plots. 4. Impact of Estimated Principal Point For all the experiments in the paper we estimate the principal point by first assuming it is in the center of the image. The principal point is then refined by minimizing the radial reprojection error (see Section 4 in the main paper). For comparison we here show the results using the ground truth principal point (as used in the reference SfM model). See Table 2 and Table 3. Experiments for optimization with estimated principal point are also listed for reference. Similarly, the proposed method consistently outperforms the the Camposeco et al. [1] when ground truth principal point is used. The improvement for estimation of pose for dataset Kazan and Doge Palace using ground truth principal point is significant, which is due to the poor estimation of principal point. In Figure 1 we show the cost across iterations for bundle adjustment using ground truth principal point, and that with estimated principal point can be found in Figure 2. Cumulative reprojection error plots for bundle adjustment with / without groundtruth principal points can be found in Figure 3. Generally, optimizations with groundtruth principal point give better error, and the difference is more significant when the estimation of principal point is less accurate. Empirically we found that the principal point refinement works slightly better if points close to the center are downweighted in the optimization (as the radial reprojection error is very sensitive to principal point shifts for these correspondences). Checkerboard calib. images [3] github.com/ylochman/babelcalib Doge Palace and Kazan [4] www.maths. lth.se/matematiklth/personal/calle/dataset/dataset.html Grossmunster and Kirchenge [2] github.com/vlarsson/radialsfm Aachen Day-Night [5, 6] https://www.visuallocalization. net/datasets/ InLoc [7] https://www.visuallocalization. net/datasets/ Table 1. Datasets used in the experimental evaluation and where to obtain them. Kirchenge [2] (369) Grossmunster [2] (373) Kazan [4] (282) Doge Palace [4] (241) rot (deg.) pos (cm.) rot (deg.) pos (cm.) rot (deg.) pos (cm.) rot (deg.) pos (cm.) Single image Proposed w/ 0.014 0.4 0.014 0.4 0.005 1.9 0.010 2.4 w/o 0.023 0.6 0.038 0.9 0.386 8.7 0.338 11.7 Camposeco et al. [1] w/ 0.024 0.8 0.031 3.7 0.007 2.7 0.016 91.1 w/o 0.027 0.8 0.044 3.8 0.403 19.8 0.364 121.2 Multiple images Proposed w/ 0.012 0.3 0.016 0.3 0.004 0.4 0.006 0.4 w/o 0.020 0.4 0.036 0.5 0.379 3.4 0.328 3.4 Camposeco et al. [1] w/ 0.019 0.4 0.027 0.7 0.007 0.6 0.013 0.7 w/o 0.027 0.8 0.044 3.8 0.403 19.8 0.364 121.2 Table 2. Average rotation error (in degree) and camera position error (in centimeters) with COLMAP reconstruction result as pseudo groundtruth and optimized with / without groundtruth principal point. The number of images for each dataset is shown in the bracket. Single image optimization and multiple images optimization are presented separately and both compared with method proposed in [1]. Error for principal point estimation for Kirchenge, Grossmunster, Kazan and Doge Palace in px are 0.830, 1.361, 16.456 and 15.377 respectively.