A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve

Abstract

We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.

Manopt.jl: Optimization on Manifolds in Julia

“…Another example are fixed rank matrices, appearing in matrix completion (Vandereycken, 2013). Working on these data, for example doing data interpolation and approximation (Bergmann & Gousenbourger, 2018), denoising Lellmann et al, 2013), inpainting (Bergmann, Chan, et al, 2016), or performing matrix completion (Gao & Absil, 2021), can usually be phrased as an optimization problem Minimize f (x) where x ∈ M, where the optimization problem is phrased on a Riemannian manifold M.…”

“…Based on this theory and algorithm, a higher-order algorithm was introduced in Diepeveen & Lellmann (2021). Optimized examples from Bergmann & Gousenbourger (2018) performing data interpolation and approximation with manifold-valued Bézier curves are also included in Manopt.jl.…”

Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

“…With (D∇)( • )[ • ]: T M d 1 ×d 2 → T N given by Jacobi fields, its adjoint can be computed using the so-called adjoint Jacobi fields, see e. g., [11,Sect. 4.2].…”

Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines

“…its starting point p in direction X ∈ T p M is given by J X , i.e., d p γ(t; •, q)(X) = J X (t) for all t ∈ [0, 1]. Furthermore, since γ(t; p, q) = γ(1 − t; q, p), endpoint variations are given analogously [6,Sec. 3.1].…”

“…Therefore, the introduction of flexible, intrinsic splines is a key contribution of this work. Our model features closed-form, numerically stable and efficient expressions for the gradient of the regression objective in terms of concatenated adjoint Jacobi fields [6]. In particular, we derive an algorithm that only requires basic Riemannian operations: the exponential and logarithmic map as well as certain Jacobi fields.…”