B\'ezier curves in the space of images

TL;DR

This paper extends Bézier curves to the space of images using a Riemannian metric based on image transport and intensity variation, enabling new applications in image interpolation and shape animation.

Contribution

It introduces a Riemannian framework for Bézier curves in the infinite-dimensional image space, generalizing classical methods with a hierarchical geodesic-based computation scheme.

Findings

Qualitative demonstration of the method on test cases

Application to face interpolation and shape animation

Effective hierarchical geodesic-based computation

Abstract

B\'ezier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of B\'ezier 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 by Miller and Younes. B\'ezier 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 B\'ezier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a B\'ezier approach for the modulation of face interpolation and shape animation via image sketches is presented.