In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape paramet...In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape parameter have most properties of Bernstein basis and the Bézier curves. Moreover the shape parameter can adjust the curves’ shape with the same control polygon. As the increase of the shape parameter, the Bézier curves with shape parameter approximate to the control polygon. In the last, the Bézier surface with shape parameter is also constructed and it has most properties of Bézier surface.展开更多
Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an...Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.展开更多
The Bézier curve is one of the most commonly used parametric curves in CAGD and Computer Graphics and has many good properties for shape design. Developing more convenient techniques for designing and modifying B...The Bézier curve is one of the most commonly used parametric curves in CAGD and Computer Graphics and has many good properties for shape design. Developing more convenient techniques for designing and modifying Bézier curve is an im- portant problem, and is also an important research issue in CAD/CAM and NC technology fields. This work investigates the optimal shape modification of Bézier curves by geometric constraints. This paper presents a new method by constrained optimi- zation based on changing the control points of the curves. By this method, the authors modify control points of the original Bézier curves to satisfy the given constraints and modify the shape of the curves optimally. Practical examples are also given.展开更多
Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive man...Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.展开更多
A class of spline curves with four local shape parameters, which includes the quartic spline curves with three local shape parameters given in Han [Xuli Han. A class of general quartic spline curves with shape paramet...A class of spline curves with four local shape parameters, which includes the quartic spline curves with three local shape parameters given in Han [Xuli Han. A class of general quartic spline curves with shape parameters. Comput. Aided Geom. Design, 28：151-163 （2011）], is proposed. Without solving a linear system, the spline curves can be used to interpolate sets of points with C2 continuity partly or entirely. The shape parameters have a predictable adjusting role on the sp[ine curves.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10371110) and the National Basic Research Program (973)of China (No. G2002CB12101)
文摘In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape parameter have most properties of Bernstein basis and the Bézier curves. Moreover the shape parameter can adjust the curves’ shape with the same control polygon. As the increase of the shape parameter, the Bézier curves with shape parameter approximate to the control polygon. In the last, the Bézier surface with shape parameter is also constructed and it has most properties of Bézier surface.
文摘Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.
基金Project (No.10471128) supported by the National Natural ScienceFoundation of China
文摘The Bézier curve is one of the most commonly used parametric curves in CAGD and Computer Graphics and has many good properties for shape design. Developing more convenient techniques for designing and modifying Bézier curve is an im- portant problem, and is also an important research issue in CAD/CAM and NC technology fields. This work investigates the optimal shape modification of Bézier curves by geometric constraints. This paper presents a new method by constrained optimi- zation based on changing the control points of the curves. By this method, the authors modify control points of the original Bézier curves to satisfy the given constraints and modify the shape of the curves optimally. Practical examples are also given.
基金Supported by Natural Science Foundation of China(No.10871208,No.60970097)
文摘Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.
基金Supported by the National Natural Science Foundation of China(No.10871208,No.60970097)Graduate Students Scientific Research Innovation Project of Hunan Province(No.CX2012B111)+1 种基金the Postdoctoral Science Foundation of China(No.2015M571931)the Fundamental Research Funds for the Central Universities(No.2017MS121)
文摘A class of spline curves with four local shape parameters, which includes the quartic spline curves with three local shape parameters given in Han [Xuli Han. A class of general quartic spline curves with shape parameters. Comput. Aided Geom. Design, 28：151-163 （2011）], is proposed. Without solving a linear system, the spline curves can be used to interpolate sets of points with C2 continuity partly or entirely. The shape parameters have a predictable adjusting role on the sp[ine curves.
