### Abstract

The chapter presents an approach for the interactive definition of curves and motion paths based on Gaussian mixture model (GMM) and optimal control. The input of our method is a mixture of multivariate Gaussians defined by the user, whose centers define a sparse sequence of keypoints, and whose covariances define the precision required to pass through these keypoints. The output is a dynamical system generating curves that are natural looking and reflect the kinematics of a movement, similar to that produced by human drawing or writing. In particular, the stochastic nature of the GMM combined with optimal control is exploited to generate paths with natural variations, which are defined by the user within a simple interactive interface. Several properties of the Gaussian mixture are exploited in this application. First, there is a direct link between multivariate Gaussian distributions and optimal control formulations based on quadratic objective functions (linear quadratic tracking), which is exploited to extend the GMM representation to a controller. We then exploit the option of tying the covariances in the GMM to modulate the style of the calligraphic trajectories. The approach is tested to generate curves and traces that are geometrically and dynamically similar to the ones that can be seen in art forms such as calligraphy or graffiti art.

### Bibtex reference

@incollection{Berio19MM,
author="Berio, D. and Fol Leymarie, F. and Calinon, S.",
title="Interactive Generation of Calligraphic Trajectories from {G}aussian Mixtures",
booktitle="Mixture Models and Applications",
publisher="Springer",
editor="Bouguila, N. and Fan, W.",
year="2019"
}