Abstract
We derive a variational model to fit a composite B\'ezier 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 B\'ezier 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.
Highlights
This papers addresses the problem of fitting a smooth curve to data points d0, . . . , dn lying on a Riemannian manifold M and associated with real-valued parameters t0, . . . , tn
The advantage to work with such objects, compared to classical approaches, are that (i) the search space is drastically reduced to the so-called control points of the Bézier curves and (ii) it is very simple to impose differentiability for the optimal curve, which is appreciated in several of the above-mentioned applications
For Bézier curves γ (t) = B(t) we obtain for the regularizer in (1) the discretized mean squared acceleration (MSA) A(b) that depends on the control points b and reads
Summary
This papers addresses the problem of fitting a smooth curve to data points d0, . . . , dn lying on a Riemannian manifold M and associated with real-valued parameters t0, . . . , tn. The advantage to work with such objects, compared to classical approaches, are that (i) the search space is drastically reduced to the so-called control points of the Bézier curves (and this leads to better time and memory performances) and (ii) it is very simple to impose differentiability for the optimal curve, which is appreciated in several of the above-mentioned applications While obtaining such an optimal curve reduces directly to solving a linear system of equations for data given on a Euclidean space, there is up to now no known closed form of the optimal Bézier curve for manifold valued data. We derive a gradient descent algorithm to compute a differentiable composite Bézier curve B : [t0, tn] → M that satisfies (1), i.e., such that B(t) has a minimal mean squared acceleration, and fits the set of n + 1 manifold-valued data points at their associated time-parameters.
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