Methods Of Rational Interpolating Splines And Their Applications In Digital Image Processing

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. On the other hand, with the development of digital image processing, interpolation and approximation have been widely used. As typical case of nonlinear numerical methods, rational interpolation and approximation has attacted more and more attention in numerical approximation, representation of function, CAGD and digital image processing. For these reasons, this dissertation focuses on the study of various piecewise rational interpolation splines and its applications in the 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 interpolant to monotone data. It overcomes linear polynomial's unsmoothness 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 f_i,j arranged over a rectanglar grid, an algorithm is presented that produces a C¹ piecewise bilinear function P(x,y) which interpolates to the given data and which is monotone.Secondly, a rational cubic spline function involving four families of parameters is presented, which is a rational polynomial with cubic numerator and cubic denominator. The scheme has been used to obtain a C¹ curve and at the same time preserves monotonic. Data dependent shape constraints are derived on two families of shape parameters to ensure preserving the shape of the data while the other two families of parameters are left free for user to control and modify the shape of the given data further in the desired region of curves. The algorithm for constructing s(x) proposed in this dissertation is simple and little computation-consuming, and numerical experiments indicate that the produced curves are smooth and keep the inherent properties of the given data.Thirdly, constructing a piecewise rational function of the quartic/linear type (with quartic numerator and linear denominator), the quartic spline function involving two families of parameters produces a monotonic interpolant to a given monotonic data set and at the same time the curve is C² smooth. The algorithm for constructing s(x) proposed in this dissertation is simple and little computation-consuming. The parametersα_i andβ_i, can be freely chosen to adjust the shape of the interpolating curve to obtain a better approximation to the original function from which the data come, and numerical experiments indicate that the method produces visually pleasant curves. The method under consideration in this dissertation has the features that the curve has an explicit representation and the curve reaches C2 continuity through choosing approprivate d_i.Fourthly, 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.Finally, the shape preserving rational spline is adopted for the first time to process image zooming. Using the biquartic splines, we construct a continous model, and using it, which can be resampled for image enlargement. The calculation only relates to several neighbor points, the algorithm is local and stable.What follows are the main results achieved in this dissertation.1. Constructing rational spline functions of the quartic/linear type (with quartic numerator and linear denominator) and discussing the conditions with one and two order continuities and the conditions preserving monotonic. We can adjust the shapes of the curves using the two families of parametersα_i andβ_i of the function. By means of the given appropriate values of the derivatives d_i, the curve is C² smooth and one does not need to solve a system of nolinear equations as in other literatures.2. Using the quartic rational spline function, we can construct a net of feature curves passing through the four corner points P_i,j which situate at the rows (U-direction) and columns (V-direction) of a m×n rectangular control mesh. Then blending its four boundary curves, the surface patch is produced, and the sufficient monotone-preserving conditions are discussed.3. Constructing rational splines of the cubic/linear type (with cubic numerator and linear denominator) and discussing the conditions with preserving monotonic and convex. Using this spline and the standard cubic Hermite interpolant, we can incorporate in tensor-product-like manner to yield four kinds of bivarite interpolation schemes.4. 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.5. The shape preserving rational spline is adopted for the first time to process image zooming. Using the biquartic spline, we construct a continous model and using this model, we can resample data from it for image enlargement. The algorithm is local, i.e., a change in one of the data or the addition of a new date point will only affect the shape of the surface in a small neighbourhood of this point. The calculation is economical and stable.