Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

Continuous-Time Fast Motion of Explosion Fragments Estimated by Bundle Adjustment and Spline Representation Using HFR Cameras

This paper introduces a continuous-time fast motion estimation framework using high frame-rate cameras. To recover the high-speed motions trajectory, we inherent the bundle adjustment using a different frame-rate strategy. Based on the optimized trajectory, a cubic B-spline representation was proposed to parameter the continuous-time position, velocity and acceleration during this fast motion. We designed a high-speed visual system consisting of the high frame-rate cameras and infrared cameras, which can capture the fast scattered motion of explosion fragments and evaluate our method. The experiments show that bundle adjustment can greatly improve the accuracy and stability of the trajectory estimation, and the B-spline representation of the high frame-rate can estimate the velocity, acceleration, momentum and force of each fragments at any given time during its motion. The related estimated result can achieve under 1% error.