We investigate spline-based continuous-time pose trajectory estimation using non-linear explicit motion priors. Current regularization priors either linearize the orientation, rely on the implicit regularization obtained from the used spline basis function, or use sampling based regularization schemes. The latter is a special case of a Riemann sum approximation, and we demonstrate when and why this can fail, and propose a way to avoid these issues. In addition we provide a number of novel practically useful theoretical contributions, including requirements on knot spacing for orientation splines, new basis functions for constant velocity extrapolation, and a generalization of the popular P-Spline penalty to orientation. We analyze the properties of the proposed approach using synthetic data. We validate our system using the standard task of visual-inertial calibration, and apply it to stereo visual odometry where we demonstrate real-time performance on KITTI.

In this paper we derive and test a probability-based weighting that can balance residuals of different types in spline fitting. In contrast to previous formulations, the proposed spline error weighting scheme also incorporates a prediction of the approximation error of the spline fit. We demonstrate the effectiveness of the prediction in a synthetic experiment, and apply it to visual-inertial fusion on rolling shutter cameras. This results in a method that can estimate 3D structure with metric scale on generic first-person videos. We also propose a quality measure for spline fitting, that can be used to automatically select the knot spacing. Experiments verify that the obtained trajectory quality corresponds well with the requested quality. Finally, by linearly scaling the weights, we show that the proposed spline error weighting minimizes the estimation errors on real sequences, in terms of scale and end-point errors.

We propose a technique for joint calibration of a wide-angle rolling shutter camera (e.g. a GoPro) and an externally mounted gyroscope. The calibrated parameters are time scaling and offset, relative pose between gyroscope and camera, and gyroscope bias. The parameters are found using non-linear least squares minimisation using the symmetric transfer error as cost function. The primary contribution is methods for robust initialisation of the relative pose and time offset, which are essential for convergence. We also introduce a robust error norm to handle outliers. This results in a technique that works with general video content and does not require any specific setup or calibration patterns. We apply our method to stabilisation of videos recorded by a rolling shutter camera, with a rigidly attached gyroscope. After recording, the gyroscope and camera are jointly calibrated using the recorded video itself. The recorded video can then be stabilised using the calibrated parameters. We evaluate the technique on video sequences with varying difficulty and motion frequency content. The experiments demonstrate that our method can be used to produce high quality stabilised videos even under difficult conditions, and that the proposed initialisation is shown to end up within the basin of attraction. We also show that a residual based on the symmetric transfer error is more accurate than residuals based on the recently proposed epipolar plane normal coplanarity constraint, and that the use of robust errors is a critical component to obtain an accurate calibration.