Abstract
B-spline functions are widely used in many industrial applications such as computer graphic representations, computer aided design, computer aided manufacturing, computer numerical control, etc. Recently, there exist some demands, e.g. in reverse engineering (RE) area, to employ B-spline curves for non-trivial cases that include curves with discontinuous points, cusps or turning points from the sampled data. The most challenging task in these cases is in the identification of the number of knots and their respective locations in non-uniform space in the most efficient computational cost. This paper presents a new strategy for fitting any forms of curve by B-spline functions via local algorithm. A new two-step method for fast knot calculation is proposed. In the first step, the data is split using a bisecting method with predetermined allowable error to obtain coarse knots. Secondly, the knots are optimized, for both locations and continuity levels, by employing a non-linear least squares technique. The B-spline function is, therefore, obtained by solving the ordinary least squares problem. The performance of the proposed method is validated by using various numerical experimental data, with and without simulated noise, which were generated by a B-spline function and deterministic parametric functions. This paper also discusses the benchmarking of the proposed method to the existing methods in literature. The proposed method is shown to be able to reconstruct B-spline functions from sampled data within acceptable tolerance. It is also shown that, the proposed method can be applied for fitting any types of curves ranging from smooth ones to discontinuous ones. In addition, the method does not require excessive computational cost, which allows it to be used in automatic reverse engineering applications.
Highlights
Piecewise polynomial functions are extensively used in many applications, such in the approximation of a complex function, data regression or data compression and in computing technology due to its simplicity and good properties
We present a new method for B-spline fitting based on the combination of the knot insertion for identifying coarse knots and a local non-linear optimization to optimally identify the knot positions and continuity levels
In order to reduce the computational cost while keeping the optimal knots in the B-spline fitting process, this paper presents a method that will combine the traditional bisecting method for coarsely identifying the knots location and deterministic optimization process based on Gauss-Newton method for solving the optimal knots by the local algorithm
Summary
Piecewise polynomial (pp) functions are extensively used in many applications, such in the approximation of a complex function, data regression or data compression and in computing technology due to its simplicity and good properties. There are a few ways to represent a piecewise polynomial function from an explicit to an implicit form in Bezier or B-spline curve. Non-uniform B-spline identification most well-known piecewise polynomial function is, perhaps, in a spline form. The spline, especially in the form of B-spline, can capture various functions from continuous curves to discontinuous ones. The use of piecewise polynomial to approximate or to fit a complex function or a given data set became a popular research topic in 1970s to 1990s. There are some needs in reverse engineering applications to employ pp functions for representing smooth curves, and curves with non-trivial cases, i.e. curves with discontinuous points, kink points, cusps or turning points from the measured data. The most common way to represent a curve with non-trivial points is by using a Bspline function
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