| In recent years,following the development of aviation,shipbuilding and industrial design and manufacture,Computer Aided Geometric Design(CAGD)becomes a new cross-cover subject.As basic element of CAGD,representation,design,display,analysis and handle of curve and surface is main task of CAGD.The interpolation and the approximation are the basic means to solve these questions.As typical case of nonlinear numerical methods,rational interpolation and approximation has attracted more and more attention in numerical approximation,representation of function,,CAGD and digital image processing.Following results are obtained in this dissertation.First of all,a univariate piecewise linear interpolation algorithm is discussed which produces a monotone interplant to monotone data. It overcomes linear polynomial s un-smoothness and high order polynomial s oscillations.The form of the function is simple and can be determined by only the function s values.The rational bilinear interpolatory surface is an extension of those results.For a given set of monotone data values fi,j arranged over a rectanglar grid,an algorithm is presented that produces a C1piecewise bilinear function P(x,y) which interpolates to the given date and which is monotone.Secondly,weighted averaging of corresponding parts of the input images is often used method to fuse images.How to construct the weight function is a critical problem. For rational cubic interpolating splines having shape preserving properties,we use it to Construct the weight function in image fusion.The experiments show that the proposed method is superior to the multiresolution fused algorithm based on wavelet analysis for The strictly registered multi-focus images.Thirdly,as irregularity of surface in project application,and as randomness and disorder of scatter data points,it’s difficult to express the surface with a sole mathematical formalism.Therefore the piecewise method is used generally in the surface design,finally all the surface pieces are connected smoothly to form a whole surface.The most commonly used surface patches are the triangle and the quadrangle surface patch.As the triangular interpolation method has obvious geometry significance,and the surface constructed is easy for the adjustment,more and more researchers pay their attention to the method.At present one of commonly used methods of constructing interpolated surface can be described simply as follows:the given scattered data points are triangulated into triangle network,the surface patch over each triangle region is formed according to boundary condition of continuity,and each surface patch joins together to form overall interpolated surface with C1continuities. There are two methods for interpolation on triangle network,one is rational interpolation,another polynomial interpolation.Because of the merit of simple structure and easy calculating,polynomial interpolation is used more widely.Now there are many methods to construct polynomial surface to the scattered data points.A new method is developed to generate curve offset using rational cubic interpolation spline with parameters. The interpolation spline is produced using only a few discrete point data and the spline surface can be modified by adjusting the shape parameters under the condition that the interpolation data are not changed. This feature gives users full freedom to control the shape of the design surface in an interactive way and at the same times the resultant surface preserves the data shape such as monotonicity and convexity. |
