Input and Output Manifold Constrained Gaussian Process Regression for Galvanometric Setup Calibration

Abstract

Data-driven techniques are finding their way into the calibration procedure of galvanometric setups. However, they bypass the underlying physical or mathematical model completely. Recent work has shown that a simple assumption about an underlying truth can improve the predictions: laser beams leaving the device follow a straight line. In this paper we take that approach a step further. Both the inputs (the pairs of rotations of the two mirrors) and outputs (the straight lines) lie on a manifold. We can incorporate this prior knowledge in the model via constraints built in the formulation of a covariance function. We propose two constrained models: one in which a linear constraint on the direction vector is written as a differential equation and one in which a quadratic constraint is imposed by a reparametrization of the line coordinates. We compare them to data-driven and unconstrained model-based approaches. We show that enforcing constraints improves the quality of the predictions significantly and thus the accuracy of the calibration. We validate our findings against real world data by predicting points on validation planes, calculating line segment distances, considering the training times for the models and assessing how much a predicted line resembles an actual straight line.

On the Angular Control of Rotating Lasers by Means of Line Calculus on Hyperboloids

“…The synthetic experiments are executed by means of a virtual 2M-GLS that simulates real world hardware. We observe a very accurate and precise performance, as well as a favourable comparison with the data-based calibration of [14]. The method of the latter article is based on the training of a Gaussian process (GP).…”

“…to the observed world. The calibration of this mapping is established by a data-driven procedure, requiring the availability of sufficiently large datasets that enable interpolation [6,8], or look-up tables [9], or the training of neural networks or Gaussian processes [10][11][12][13][14]. An important issue of this approach is that it requires world point clouds with reliable coordinates, which significantly cover the work space.…”

“…Our model presents lasers as perfect lines and the mirrors as mathematical planes, containing the axis of rotation, and enabling perfect reflections. If the deficiencies of the generated laser or the reflecting mirrors imply a significant deviation from the mathematical assumptions, the method of [14] appears to perform better than the proposed method, since the trained Gaussian processes take the sensor defects into account.…”

“…Once this threshold is exceeded, the described algorithm in the Appendix A for fitting hyperboloids of revolution to noisy lines might result into an over-idealized model. In these cases, the method of [14] is preferred. This is due to the fact that a Gaussian process can be seen as a universal smoother, filtering out noise.…”

Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry

“…This approach is taken in [40]. Another way to ensure the Grassmann-Plücker relation is given in [41], where constraints are built in the kernel functions of the Gaussian process themselves, although at a considerable extra computational cost. A representation that does not suffer from this hurdle is the stereographic projection of a line [24].…”