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A gradient-descent method for curve fitting on Riemannian manifolds. (English) Zbl 1245.65017
For given data points \(p_0,\ldots,p_N\) on a closed Riemannian manifold of \({\mathbb R}^n\) and time instants \(0=t_0 < t_1 < \ldots < t_N =1\), the authors consider the problem of finding a curve \(\gamma\) on \(M\) that best approximates the data points at the given instants while being as “regular” as possible. They study an optimization problem with two objective functions, namely a fitting function \[ E_d(\gamma) = \frac{1}{2}\, \sum_{i=0}^N d^2(\gamma(t_i), p_i)\,, \] where \(d\) denotes the distance function on \(M\), and a regularity function \(E_s(\gamma)\). In the first case, the regularity function is the mean squared velocity of \(\gamma\), i.e. \[ E_s(\gamma) = \frac{1}{2}\, \int_0^1 \|\dot{ \gamma}(t)\|^2 \, dt\,. \] In the second case, \(E_s(\gamma)\) is the mean squared acceleration of \(\gamma\). Then the authors search for an optimizer of the objective function \(E(\gamma) = E_d(\gamma) + \lambda \, E_s(\gamma)\), where \(\lambda >0\) is a smoothing parameter, using a steepest descent method in a set of curves \(\gamma\) on \(M\). The steepest descent direction, defined in the sense of first order and second order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. The method is illustrated on fitting problems in \(M={\mathbb R}^2\) and the unit sphere.

MSC:
65D10 Numerical smoothing, curve fitting
65K10 Numerical optimization and variational techniques
49M30 Other numerical methods in calculus of variations (MSC2010)
49J15 Existence theories for optimal control problems involving ordinary differential equations
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