| When using mathematical methods to describe digital images, a two dimensional static graylevel image can be expressed as a surface. After the image is scanned into a one dimensional gray value sequence, it is also can be expressed as a plane curve. So the curve and surface modeling method becomes an effective tool for dealing with digital image processing problems. This dissertation mainly aims on researching the modeling theories and applications of non-polynomial parametric curves and surfaces in image data compression and image enlargement, which focus on the construction and study of some interrelated non-polynomial parametric curve and surface models, and also the study of image data compression and image enlargement which based on the proposed models by respectively using the approximation and interpolation method. Generally, the main contributions of the dissertation are as follows,(1) The related non-polynomial parametric curve models for image data compression are constructed and studied. A quasi quadratic rational Bézier curve with a shape parameter is constructed by constraining the weights of the traditional quadratic rational Bézier curve; A quasi cubic rational Bézier curve with a shape parameter is constructed by reparameterizing the traditional cubic Bézier curve; A quasi cubic trigonometric Bézier curve with a shape parameter is constructed and studied, which is based on the functions space spanned by {1, sint, cost, sin2t}. Those curves not only have similar properties with the corresponding original curves, but also become more suitable for studying the image data compression problems, which further enrich the parametric curve modeling methods and establish the theoretical foundation for the follow-up study of image data compression.(2) Image data compression based on the non-polynomial parametric curves approximation is studied. An image data compression algorithm based on the quasi quadratic rational Bézier curve approximation is presented. The optimal quasi quadratic rational Bézier curves for approximating the image scanning data are obtained by computing the best values of the shape parameters, and then the approximate parameters are stored to realize the compression for image data. The use of quasi quadratic rational Bézier curve approximation overcomes the default of quadratic and cubic polynomial Bézier curve approximation, which can only reflect the gradual change of data but without the mutability of data, therefore, the compression ratio and quality of the image data have been greatly improved. Because the quasi quadratic rational Bézier curve cannot represent inflection point when approximating to data, concerning to this problem, an image data compression algorithm based on the quasi cubic rational Bézier curve approximation is presented. When using the quasi rational Bézier curve to approximate the image scanning data, regardless of whether the data has an inflection point or not, the approximation accuracy is higher than the quasi quadratic rational Bézier curves so that it can obtain a better compression quality. Then, image data compression using the quasi cubic trigonometric Bézier curve approximation is investigated. The use of quasi cubic trigonometric Bézier curve approximation algorithm for image data compression can obtain higher compression ratio and quality than the quadratic and cubic polynomial Bézier curve approximation algorithms, and has the equivalent compression effects to the quasi cubic rational cubic Bézier curve approximation algorithm, which extends application fields of the trigonometric parametric curve models and also presents an effective method for the study of image compression problems.(3) The related non-polynomial parametric surface models for image enlargement are constructed and studied. A hyperbolic Coons surface with shape parameters is constructed and studied which is based on the functions space spanned by {1, t, sinht, cosht}; A bicubic rational Coons surface with shape parameters is constructed by reparameterizing the traditional bicubic Coons surface; A class of quasi cubic trigonometric spline surface is constructed and studied which is based on the functions space spanned by {1, sint, cost, sin2 t, cos2t}. Those surfaces not only have similar properties with the corresponding original surfaces, but also are more suitable for studying image enlargement problems, which would further enrich the parametric surface modeling methods and establish the theoretical foundation for the follow-up study of image enlargement.(4) Image enlargement based on the non-polynomial parametric surfaces interpolation is studied. An image enlargement method based on the hyperbolic Coons surface interpolation is presented. The use of hyperbolic Coons surface interpolation for image enlargement can not only overcome the short of the normal interpolation methods in constructing image interpolating surface, but also obtain satisfactory target image by altering the values of the shape parameters of the hyperbolic Coons surface, which provides an effective method for image enlargement. Then, the use of the bicubic rational Coons surface interpolation for image enlargement is studied. Image enlargement using the bicubic rational Coons surface interpolation can maintain clear borders of source images by altering values of the shape parameters, not only the effects are superior to the normal interpolation methods but also the hyperbolic Coons surface interpolation method. Finally, image enlargement using the quasi cubic trigonometric spline interpolation surface is discussed. When using the quasi cubic trigonometric spline interpolation surface to enlarge the image, it can maintain better effects than the normal interpolation methods and the two non-polynomial Coons surface interpolation methods, which extends application fields of the trigonometric parametric spline models and also presents a simple and effective method for the study of image enlargement problems.
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