Spline Curve Approximation and Design by Optimal Control Over the Knots

Abstract

In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced. The advantage of Optimal Control over Global Minimization such as in [17] is that it can handle both approximation and interpolation. Moreover a cost function is introduced to implement a design objective (shortest curve, smoothest one etc...). The present work introduces the Optimal Control over the knot vectors of non-uniform B-Splines.An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney condition). This situation is dealt naturally within the Optimal Control framework. A geometric description of the resulting null space is provided as well.

From CT to NURBS: Contour Fitting with B-spline Curves

“…B-spline curves can be fitted to the smoothed data or to the original data [6,7,10,[14][15][16]20,22,34,35]. In many CAD applications curve interpolation followed by knot removal is an effective technique to approximate a large set of smooth point set (right side of the flow chart in Figure 1).…”

Elitist clonal selection algorithm for optimal choice of free knots in B-spline data fitting

“…The general approach to this problem consists of starting with a certain number of knots and iteratively modify such amount by either knot insertion or knot removal to satisfy a prescribed error bound [13,24,37,44,45,53]. However, both methods require human intervention in order to determine (subjectively at certain extent) some needed parameters.…”

“…However, both methods require human intervention in order to determine (subjectively at certain extent) some needed parameters. For instance, the methods in [13,24,44,45,67] require a tolerance error or a smoothing factor, whose determination is often based on subjective factors. Similarly, the method in [37] needs a good initial guess of both knot vector length and knot locations, a task that typically demands a high level of expertise from the user.…”

“…This problem is far from being trivial because it requires to obtain suitable values for the B-spline parameters. In particular, the choice of knots has considerable effect on the shape of a curve [15,24]. Most current methods consider fixed knots, so that the fitting process reduces to a simple linear optimization problem.…”

B-spline based image tracking by detection

“…B-splines can be used to model any probability density functions and probability hypotheses density 4 functions without any assumption regarding the system and measurement noises. 4, 7 A spline approach has been used to solve the nonlinear estimation problem of phase modulation, 8 B-splines using genetic algorithm for the optimization of a cost function discussed by Goldenthan and Bercovier, 9 and monosplines to solve a nonlinear estimation problem. 5 The B-Spline method for single target tracking in a clean environment, i.e., no false alarms or missed detections, was presented in previous work.…”