Bézier Curves in the Space of Images

Abstract

Bezier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bezier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model [MY01]. Bezier curves are then computed via the Riemannian version of de Casteljau’s algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bezier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bezier approach for the modulation of face interpolation and shape animation via image sketches is presented.