Curve or surface reconstruction is a challenging problem in the fields of engineering design, virtual reality, film making and data visualization. Non-uniform rational B-spline (NURBS) fitting has been applied to curve and surface reconstruction for many years because it is a flexible method and can be used to build many complex mathematical models, unlike certain other methods. To apply NURBS fitting, there are two major difficult sub-problems that must be solved: (1) the determination of a knot vector and (2) the computation of weights and the parameterization of data points. These two problems are quite challenging and determine the effectiveness of the overall NURBS fit. In this study, we propose a new method, which is a combination of a hybrid optimization algorithm and an iterative scheme (with the acronym HOAAI), to address these difficulties. The novelties of our proposed method are the following: (1) it introduces a projected optimization algorithm for optimizing the weights and the parameterization of the data points, (2) it provides an iterative scheme to determine the knot vectors, which is based on the calculated point parameterization, and (3) it proposes the boundary-determined parameterization and the partition-based parameterization for unorganized points. We conduct numerical experiments to measure the performance of the proposed HOAAI with six test problems, including a complicated curve, twisted and singular surfaces, unorganized data points and, most importantly, real measured data points from the Mashan Pumped Storage Power Station in China. The simulation results show that the proposed HOAAI is very fast, effective and robust against noise. Furthermore, a comparison with other approaches indicates that the HOAAI is competitive in terms of both accuracy and runtime costs.

Iteration and optimization scheme for the reconstruction of 3D surfaces based on non-uniform rational B-splines

Fitting of data points by parametric curves and surfaces is demanded in many scientific fields. In this paper we review and analyze existing least squares orthogonal distance fitting techniques in a general numerical optimization framework. Two new geometric variant methods (GTDMand CDM) are proposed. The geometric meanings of existing and modified optimization methods are also revealed.

/pdf/a-revisit-to-least-squares-orthogonal-distance-fitting-of-1s34us048q.pdf

A revisit to least squares orthogonal distance fitting of parametric curves and surfaces

Fashion Design is an industry closely connected to our daily life. Computer-aided fashion design can enhance the efficiency of product development, in which a 3D mannequin links the development and application of the entire design system, thus rendering the 3D mannequin data required for fashion design very important. This study proposed a systematic method for surface reconstruction of 3D mannequins based on feature curves. First of all, the study applied reverse engineering methods to scan a mannequin model commonly used in the fashion design profession by extracting grid points that represent the shape features from the scanned data after the segmentation of the mannequin model through feature surfaces. Then, the shape of the entire 3D mannequin is reconstructed using B-spline surfaces. Simultaneously, the continuity among the connected B-spline surfaces is adjusted with tangent vector adjustment methods based on the minimum energy required for improving the quality of the shaped surfaces. Finally, the 3D mannequin is applied to the computer-aided fashion design system being developed to achieve product development using synchronous design methods.

A study of surface reconstruction for 3D mannequins based on feature curves

With the development of a new function of computer numerical control controllers, nonuniform rational B-spline (NURBS) interpolation, NURBS tool path generation for sculptured surface machining is under extensive research. The common procedures of the current NURBS tool path planning methods are as follows: first, to find a group of cutter contact points on a sculptured surface; second, to calculate their corresponding cutter locations (CLs); then, to fit a NURBS tool path to the CLs within a prescribed tolerance; and finally, to inspect the tool path for possible gouge by the tool and delete the invalid path segments, if any. However, the NURBS tool path has the following problems: (a) although it passes through the discrete CLs of the theoretical CL path, the deviation along the two paths could be larger than the tolerance; (b) its parameter is not the arc length of the path; and (c) it is difficult to detect gouge along the NURBS path and to remove the invalid segments from it. Consequently, NURBS tool paths generated with the current methods of commercial computer-aided design/computer-aided manufacturing (CAD/CAM) software cannot be used to make smooth and accurate surfaces. To address these problems, this work proposes a new approach to generating arc length parameterized NURBS tool paths with high accuracy in terms of the theoretical CL paths and without gouge and interference. Two practical examples in this work clearly demonstrate the feasibility of this approach and the advantages of the generated NURBS tool paths. Therefore, this approach can be implemented into the CAD/CAM software to promote NURBS machining in industry.

A new approach to generating arc length parameterized NURBS tool paths for efficient three-axis machining of smooth, accurate sculptured surfaces

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.

Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method

Total least squares fitting of Bézier and B-spline curves to ordered data

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Total least squares fitting of Bézier and B-spline curves to ordered data