Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

Abstract

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve C 0 -and C 1 -continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C 1 -continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group and two Riemannian shape spaces.Keywords: Composite Bézier surface, Riemannian manifold, differentiability conditions, bending energy.Mathematics Subject Classification (2010): 65D05, 65D07, 53C22, 65K10Figure 1: Differentiable Bézier spline surface on the Riemannian space of shells (Section 6.4) interpolating the red shapes. The gray shapes are points on the Bézier surface driven by the control points in green. Their location indicates where in the R 2 domain they are achieved.

B/Surf: Interactive Bézier Splines on Surfaces

“…We adopt the same outer structure of the algorithm of [Morera et al 2008] for curve tracing with the recursive De Casteljau bisection. [Absil et al 2016] defines Bézier curves both with the De Casteljau algorithm and with the Riemannian center of mass, and show that they may produce different results.…”

“…This construction was mentioned, but not developed further, in [Absil et al 2016;Panozzo et al 2013].…”

Lifting Methods for Manifold-Valued Variational Problems

“…Consider a variational image processing or general data analysis problem of the form min u:Ω→M F(u) (1) with Ω ⊂ R d open and bounded. In this chapter, we will be concerned with problems where the image u takes values in an s-dimensional manifold M. Problems of this form are wide-spread in image processing and especially in the process- x start…”

Smooth interpolation of key frames in a Riemannian shell space

“…The concept of Bézier curves can easily be transferred to Riemannian manifolds and in particular to shape spaces. To this end the de Casteljau algorithm with linear interpolation has to be replaced by geodesic interpolation [49,28,1], compare also Effland et al [23] for a corresponding tool on the shape space of images and Brandt et al [6] for this method on the space of triangular shell surfaces using the variational time discretization [34]. Below we will discuss the de Casteljau algorithm on the space of subdivision surfaces to prepare the discussion of Hermite interpolation and cardinal splines.…”