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Efficient NURBS surface fitting via GA with SBX for free-form representation

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Abstract

An accurate technique to perform NURBS surface fitting via genetic algorithms is presented. In this technique, the initial NURBS surface is generated by using object points as control points. Then, the genetic algorithm computes the weights and control points to obtain the NURBS surface fitting. The genetic algorithm is implemented through an objective function, which is deduced from NURBS surface and object points. The objective function is minimised by means of simulated binary crossover. This procedure is carried out based on the initial NURBS surface and NURBS surface constructed by employing the object height average as control point. Thus, the genetic algorithm provides the weights and control points of the NURBS surface that represent the object shape. The proposed algorithm improves the accuracy and speed of the NURBS fitting, which is created via genetic algorithms and gradient methods. It is because the proposed algorithm calculates the weights and control points from a known search space, which is produced by NURBS surfaces. Thus, the genetic algorithm minimises the objective function in fast form with high accuracy. The contribution of the proposed method is corroborated by an evaluation based on accuracy and speed of the traditional genetic algorithms and gradient methods.

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... J. Apolinar Mu Muñoz Rodr Rodríguez et al. and Xiaoqiang Tian et al. used a genetic algorithm and BP neural network-based on particle swarm optimization to optimize the control point and knot sequence of the NURBS surface, respectively. There is improved accuracy and speed of fitting NURBS surfaces [26,27]. Jinho Song et al. divide unorganized points into boundary points and internal points using a deep neural network, in order to facilitate more explicit parameterization of boundary points during NURBS modeling [28]. ...
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This paper presents a computer aided process planning (CAPP) system for numerical control tool path generation of complex shoe molds. This CAPP system includes both the automation of auxiliary boundary curve generation and machining strategies. The automation of auxiliary boundary curve generation and machining strategies make tool path generation more accurately and efficiently. Traditional shoe mold making is a very tedious process. Even with the utilization of computer-aided design and computer-aided manufacturing (CAD/CAM), the CAM process requires long hours of tool path programming and debugging. It would also take a long time to calculate (sometimes several hours) the tool path for complex athletic footwear. In order to reduce the tool path editing and programming time, this paper proposes the use of CAPP to reduce processing time and increase efficiency. It is difficult, if not impossible, to develop a generic CAPP system that can generate a process plan to solve general production problems. However, it is quite possible to capture the domain knowledge of a certain production process and embed that knowledge into a CAPP system. We prove, by using such a system, that a very complicated process planning problem can be overcome by a knowledge-based CAPP approach. With such an approach, the traditional manufacturing process of shoe molds can be converted to an automatic manufacturing process with the CAPP system. In fact, shoe molds for real production have been created using the developed CAPP system, demonstrating the effectiveness of this approach. In this paper, we show that several complex and different shoe molds and their machining strategies were automatically planned by the proposed CAPP system. The result of a comparison between the CAPP system with the traditional approach is presented and discussed. KeywordsCAPP–Shoe mold–Machining–CAM–NC tool path generation
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Curve fitting is commonly used in reverse engineering for the reconstruction of curves from measured points, and it is critically important to provide various kinds of curve-fitting algorithms to acquire curves that satisfy different constraint conditions. We divide the curve-fitting problem into unconstrained and constrained types. For the unconstrained type, three curve-fitting algorithms are investigated: general, smooth and extended curve fitting. The general curve fitting considers only the accuracy of the fitted curve; the smooth curve fitting can control both the accuracy and the fairness of the fitted curve, while the extended curve fitting can acquire a curve longer than the range of the measured points. For the constrained type, we propose three curve-fitting conditions: fixed end-points, closed curve and continuity to adjacent curves. Detailed discussion for each of the above cases is presented. Associated examples are also provided to illustrate the feasibility of the proposed algorithms.
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Reverse engineering is an approach for constructing a CAD model from a physical part through dimensional measurement and a surface model. Different from conventional methods, this paper develops a new algorithm by which a desired fitted surface is obtained with less computation. Let selected m×n measured points be control points to construct B-spline or NURBS surface, then modify this constructed surface by using all the measured points and least squares minimization. A new algorithm for parameterization for measured points is also presented in this paper. The effectiveness and efficiency of these proposed algorithms are demonstrated.
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Fast and accurate fitting of non-uniform rational B-spline (NURBS) curves and surfaces through large sets of measured data is an important problem in applications such as reverse engineering and geometric modelling. This paper presents a method for realising significant improvements in the computational efficiency of this task. The basic idea is that the sparsity structures of the relevant matrices that are specific to the problem of NURBS fitting can be precisely defined and that full exploitation of these structures leads to significant savings in both computational and storage requirements. These savings allow for a large number of control points to be used in order to define the surface and consequently to improve the accuracy of shape representation. The achieved computational complexity is linear in both the number of measured points and the number of control points while the storage requirements of the algorithm are linear with the number of control points only. The complexity analysis, as well as the analysis of actual running times is presented. The results demonstrate that, using this approach, highly complex shapes may be modelled accurately with a single NURBS surface. KeywordsReverse engineering–Geometric modelling–NURBS
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Due to the fact that the cutting occurs around the cutter contact (CC) point, the efficiency and quality of CNC machining can be improved significantly if the CC velocity along the surface is kept costant. Conventional approaches to machining mainly maintain a constant cutter location (CL) velocity, so that the CC velocity along the surface is often not constant and usually results in non-uniform machining and unsatisfactory quality. To overcome this difficulty, this paper presents a novel NURBS surface interpolator that is capable of real-time generation of CL motion command for ball-end milling of NURBS surfaces and maintaining a constant CC velocity along the CC path and its intervals. For performance evaluation, a three-axis servomechanism driven by three servomotors is controlled to track segments represented by NURBS surfaces. Experimental results verify the effectiveness of the proposed method.
Article
NURBS surfaces are among the most commonly used parametric surfaces in CAGD and Computer Graphics. This paper investigates shape modification of NURBS surfaces with geometric constraints, such as point, normal vector, and curve constraints. Two new methods are presented by constrained optimization and energy minimization. The former is based on minimizing changes in control net of surfaces, whereas the latter is based on strain energy minimization. By these two methods, we change control points and weights of an original surface, such that the modified surface satisfies the given constraints. Comparison results and practical examples are also given.
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An approach to automatic foot measurement using 3D scanned data is proposed in this paper. The proposed approach was evaluated through comparisons of simulated measurements (SM) of eleven male and nine female participants with manual measurements (MM) and with the output of a commercially available automated foot measuring system (CP). The registration procedure for the measurements and unambiguous definitions for each measurement were first established. Eighteen dimensions of each foot were calculated from the scanned data that comprised point clouds and selected landmarks. Two operators manually measured each participant's foot twice. These MM showed high inter- and intra-operator reliability (ICC>0.84). Ten of the 18 dimensions obtained from the three measurement methods, SM, CP, and MM, were subjected to an ANOVA and eight of the measurements showed significant differences among the three methods. After establishing a linear correction to adjust for systematic errors, there were no significant differences between the SM and MM methods for 17 of the 18 foot dimensions; and the single exception was the heel width dimension. The differences among the three methods, correction procedures and their significance are discussed.Relevance to industryMeasuring feet to obtain the relevant dimensions that characterize feet can be quite tedious and the measurement may be dependent on the measurer. Automatic measurement with scanned data, on the other hand, can give replicable information even though the measurements depend on the intricacies of the scanning system and the computational algorithms adopted. The proposed definitions and algorithms provide a means to automate foot measurements for customized footwear.
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In the duplication of a physical part such as die and mold, aerospace part and so on, most of patches are trimmed surfaces resulting from Boolean manipulations. Direct generation of tool paths from the practical parts is a fundamental problem. This paper presents an efficient method for generating NC tool paths from some trimmed surfaces. Three types of control points are determined to construct an underlying NURBS surfaces. NC tool paths are then generated based on these surfaces. The method can deal efficiently with parts composed of trimmed surfaces. It can be considered as a tool for reverse engineering software integration.
Article
A non-deterministic evolutionary approach for approximating the outlines of planar shapes has been developed. Non-uniform Rational B-splines (NURBS) have been utilized as an underlying approximation curve scheme. Simulated Annealing heuristic is used as an evolutionary methodology. In addition to independent studies of the optimization of weight and knot parameters of the NURBS, a separate scheme has also been developed for the optimization of weights and knots simultaneously. The optimized NURBS models have been fitted over the contour data of the planar shapes for the ultimate and automatic output. The output results are visually pleasing with respect to the threshold provided by the user. A web-based system has also been developed for the effective and worldwide utilization. The objective of this system is to provide the facility to visualize the output to the whole world through internet by providing the freedom to the user for various desired input parameters setting in the algorithm designed.
Article
One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k-means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).
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The unprecedented success of the iterative closest point (ICP) method for registration in geometry processing and related fields can be attributed to its efficiency, robustness, and wide spectrum of applications. Its use is however quite limited as soon as the objects to be registered arise from each other by a transformation significantly different from a Euclidean motion. We present a novel variant of ICP, tailored for the specific needs of production engineering, which registers a triangle mesh with a second surface model of arbitrary digital representation. Our method inherits most of ICP’s practical advantages but is capable of detecting medium-strength bendings i.e. isometric deformations. Initially, the algorithm assigns to all vertices in the source their closest point on the target mesh and then iteratively establishes isometry, a process which, very similar to ICP, requires intermediate re-projections. A NURBS-based technique for applying the resulting deformation to arbitrary instances of the source geometry, other than the very mesh used for correspondence estimation, is described before we present numerical results on synthetic and real data to underline the viability of our approach in comparison with others.
Article
NURBS surfaces are commonly used in CAD/CAM software systems to represent the complex shapes of mechanical parts. Well-planned tool paths for machining the surfaces can significantly increase cutting efficiency and improve part quality. The steepest ascent tool-path pattern has been proposed for making sculptured surfaces in a 3-axis finish milling operation, and it has been proven that a steepest ascent tool-path is inherently more efficient in removing material to make these surfaces within tolerances than a tool path of any other type. However, the mathematical representation of steepest ascent paths on NURBS surfaces has not been addressed. In our work, simplified formulae of these paths are derived, and a comprehensive, efficient algorithm to plan steepest ascent tool-paths on compound NURBS surfaces is developed. To verify its validity and efficiency, this innovative approach is applied to a complicated compound surface. Furthermore, a comparison between the steepest ascent and CATIA tool-paths on two NURBS surfaces is conducted to demonstrate the advantages of the steepest ascent tool-paths for NURBS surface part production.
Article
Surface reconstruction is a very challenging problem arising in a wide variety of applications such as CAD design, data visualization, virtual reality, medical imaging, computer animation, reverse engineering and so on. Given partial information about an unknown surface, its goal is to construct, to the extent possible, a compact representation of the surface model. In most cases, available information about the surface consists of a dense set of (either organized or scattered) 3D data points obtained by using scanner devices, a today’s prevalent technology in many reverse engineering applications. In such a case, surface reconstruction consists of two main stages: (1) surface parameterization and (2) surface fitting. Both tasks are critical in order to recover surface geometry and topology and to obtain a proper fitting to data points. They are also pretty troublesome, leading to a high-dimensional nonlinear optimization problem. In this context, present paper introduces a new method for surface reconstruction from clouds of noisy 3D data points. Our method applies the genetic algorithm paradigm iteratively to fit a given cloud of data points by using strictly polynomial B-spline surfaces. Genetic algorithms are applied in two steps: the first one determines the parametric values of data points; the later computes surface knot vectors. Then, the fitting surface is calculated by least-squares through either SVD (singular value decomposition) or LU methods. The method yields very accurate results even for surfaces with singularities, concavities, complicated shapes or nonzero genus. Six examples including open, semi-closed and closed surfaces with singular points illustrate the good performance of our approach. Our experiments show that our proposal outperforms all previous approaches in terms of accuracy and flexibility.
NURBS Fitting Optimization Based on Ant Colony Algorithm
  • R Xiao
  • J Shang
  • H Liu