Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method
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Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method A. Ebrahimi1 · G. Barid Loghmani1
Received: 27 September 2017 / Revised: 11 April 2018 / Accepted: 8 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this paper, we consider the problem of fitting the B-spline curves to a set of ordered points, by finding the control points and the location parameters. The presented method takes two main steps: specifying initial B-spline curve and optimization. The method determines the number and the position of control points such that the initial B-spline curve is very close to the target curve. The proposed method introduces a length parameter in which this allows us to adjust the number of the control points and increases the precision of the initial B-spline curve. Afterwards, the scaled BFGS algorithm is used to optimize the control points and the foot points simultaneously and generates the final curve. Furthermore, we present a new procedure to insert a new control point and repeat the optimization method, if it is necessary to modify the fitting accuracy of the generated B-spline fitting curve. Associated examples are also offered to show that the proposed approach performs accurately for complex shapes with a large number of data points and is able to generate a precise fitting curve with a high degree of approximation. Keywords Geometric modeling · Curve fitting · Initial B-spline curve · Optimization method · Scaled BFGS method
1 Introduction Geometric modeling contributes a momentous role in several scientific and technological domains such as Computer Aided Design (CAD), Computer Aided Geometric Design
G. Barid Loghmani
[email protected] A. Ebrahimi [email protected] 1
Computer Geometry and Dynamical Systems Laboratory, Faculty of Mathematical Sciences, Yazd University, Yazd, Iran
Multimed Tools Appl
(CAGD), font designing, data compression and object recognition [6, 11, 34, 36, 37, 40, 44, 47, 49, 51, 52, 56, 62, 63]. In the geometric modeling and computer graphics, the Bspline functions are the most powerful tools due to their efficacious geometric properties. The B-spline basis functions are the piecewise polynomial functions defined by parameters and knot vectors. The B-spline curve has several useful attributes such as local effects of coefficient changes, great flexibility, smoothness and numerical stability of computations [15, 45]. This paper proposes a novel algorithm to fit the B-spline curve to a set of ordered points. The algorithm is presented in two phases to optimize the control points and foot points simultaneously. In first phase, the initial position of the control points will be detected. The second phase is used to hit the target position very precisely. A B-spline curve C(t) of order (degree + 1) k in R2 is defined as C(t) =
m
Pi Ni,k (t)
i=1
where Ni,k are the B-spline basis functions defined on the knot vectors ⎧ ⎨ u1 = u2 = . . . = uk = 0, um+1 = um+2 = . . . = um+k = 1, ⎩ j , j
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