Optimal Spline Approximation via ℓ₀‐Minimization

Abstract

Splines are part of the standard toolbox for the approximation of functions and curves in R d . Still, the problem of finding the spline that best approximates an input function or curve is ill-posed, since in general this yields a "spline" with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an 0 -regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B-splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.

Synthesizing Character Animation with Smoothly Decomposed Motion Layers

“…The motion primitives were then utilized as bases to generate or synthesize new motions. Selecting sparse key postures from a given dense motion has proven to be feasible for use with spline‐based interpolation techniques [LT01, BSH15]. The extraction of sparse motion bases in the quaternion space [ZSD12] or the extraction of high‐level motion bases from large motion databases [XFJ*15] has also been explored.…”

“…As an alternative to motion primitives, the direct processing or modulation of raw motion signals has been widely adapted to achieve a general solution for various types of motions. Due to its simplicity and computational advantages, a parametric model is used most often for the decomposition of a single motion [RCB98, LS01, BSH15]. Though the parametric model is well suited for use with spline‐based interpolation techniques, it is also associated with limited expressiveness that leads to re‐parameterizations, corrections or sophisticated manual interventions for further extensions.…”

Generalization of one algorithm for constructing recurrent splines

“…In paper [6], in order to establish coordinates of the knots where spline links are joined, it is proposed first to form their redundant sequence followed by chopping using optimization methods. The framework of this approach includes the results obtained by authors of article [7].…”

Compressed vibration modes of elastic bodies

“…minimizing it under soft or hard constraints). More examples, where 1 or 0 regularization has been employed for geometry processing tasks, surface smoothing [29] and optimal spline approximation [30,31]. For a recent survey on compressed sensing for geometry processing, we refer to [32].…”