Parametric Curves And Surfaces With Implicit Domains And Image Zooming Based On PHT Splines

Geometric modeling is one of the key technologies in the fields of computer aided geometric design and computer graphics. Its main research topics include represen-tation, design, display, and analysis of curves and surfaces. With the progress of science and technology in recent years, research relevant to the processing of curves and surfaces in industry, film and entertainment is required, which promotes con-tinuous development of geometric modeling techniques. This thesis addresses the representation of curves and surfaces and its applications.Firstly, the two major representations of curves and surfaces in CAD/CAM systems, which are parametric and implicit presentation, are briefly recalled. The inherent advantages and disadvantages of both representations are detailed. Re-search work of conversion between them and data reconstructions with them is summarized.In order to better take advantage of these two representations and give a con-venient way for the conversion between them, we present a new representation of curves and surfaces in chapter2, named parametric curves with implicit domains and parametric surfaces with implicit domains. According to the differential geom-etry properties of parametric curves and surfaces, we give the relevant differential geometry concepts of the new representation forms. And the conversion between the new representation and any of the other two representations which are parametric and implicit is provided. Finally examples are provided to show its application in representation of curves or surfaces with complex topology and the local control of curves or surfaces for the parametric curves with implicit domains or parametric surfaces with implicit domains with the tensor-product B-splines forms.PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is convenient and effi-cient. Since the conversion from implicit form to PCID form is difficult and impor-tant, this is the main problem to be solved in the paper. In chapter3, we propose a framework model for constructing the approximate PCID of given implicit curve. It can map given points to the implicit domain homeomorphically. The resulting map is smooth and overlapping-free. The model is also extended to three-dimensional space, constructing approximate parametric surfaces with implicit domains of im-plicit surfaces. In chapter4, we extend the model to more general cases, which constructs the mapping between two groups of given sample points. And we also successfully apply the proposed models to several different fields of digital image processing.In chapter5, image zooming with intermittent features is studied based on the theory of PHT-spline, namely data reconstruction and data resampling using PHT-spline with mixed orders of continuity. According to the characteristics of in-terpolation Hermitian information at basis vertices for PHT-spline, the image data reconstruction can achieve the intermittent effect by using different Hermitian in-formation at the intermittent basis vertices. Since this method can just handle the data whose intermittent characteristic line is in the horizontal and vertical position, a parametric model combined optimization method is provided for general inter-mittent data, which can map the intermittent curves to horizontal or vertical lines. Based on the parameterization result, reconstruction image is got. This method solves the image smoothing problem of general spline functions due to the overall continuity. Reconstruction image can not only maintain a higher resolution in some locations, but also have good smoothness in other locations. The image scaling will not produce blurring in the feature lines, and experimental results show the effectiveness of this method.In the end, we conclude our thesis with future research problems.