Invariant recovery of curves in m-dimensional space from sparse data

We address the problem of reconstructing a smooth curve from sparse and noisy information that is invariant to the choice of the coordinate system. Tikhonov regularization is used to form a well-posed mathematical problem statement, and conditions for an invariant reconstruction are given. The resulting functional minimization problem is shown to be nonconvex. Approximations to the invariant functional are often used to form a convex problem that can be solved efficiently. Two common approximations, those of cubic and weighted cubic splines, are detailed, and examples are given to show that the approximations are often invalid. To form a valid approximation to the invariant functional we propose a two-step algorithm. The first step forms a piecewise-linear curve, which is invariant to the coordinate system. This piecewise-linear curve is then used to construct a parameterization of the curve for which we can make a valid approximation to the invariant functional. Examples are given to demonstrate the effectiveness of the algorithm, and two example applications for which the invariant property is important are given.