The Study Of Several Issues On The Third-order Trigonometric Spline Functions

A trigonometric spline is a piecewise smooth trigonometric polynomial.It can generate and approximate a variety of curves,and has played an important role in theoretical and applied fields due to its simple structures and numerical stability.This paper studies the structure and dimension theory of the third-order trigonometric spline functions,and we also apply the third order trigonometric spline functions to the problem of image interpolation.The first chapter introduces the development history of trigonometric spline functions,and presents several different definitions of third order trigonometric spline functions.The second chapter introduces the structure of third order trigonometric spline functions.We first propose the definition of cubic uniform trigonometric spline functions.This kind of splines have seven parameters in each interval,and thus provide more flexibilities in practical applications.To investigate the structures of proposed splines,we consider the constraint of second order continuity imposed on two adjacent segments of the trigonometric function.This constraint leads to our structure theorem.This theorem demonstrates that the spline expression in each interval can be expressed as the sum of the previous cubic trigonometric function and one specific trigonometric function.This form can provide a more convenient approach in the design of trigonometric splines.Furthermore,we propose a dimension theorem of this kind of spline function space with finite nodes.Finally,we give several related examples according to the structure theorem and discuss the relationship between basis functions simply.The third chapter applies third-order trigonometric spline functions to the problem of image interpolation.The purpose of image interpolation is to resize the image and achieve a good image quality at the same time.It has widely applications in many fields such as medicine,meteorology,animation.We construct a convolution kernel function from cubic trigonometric splines.To construct the convolution kernel function,we set the cubic trigonometric splines to meet second order continuous,interpolation condition and partition of unity.These constraints lead to three free parameters.To derive better free parameters,we appromate Sinc function in frequency domain.Then we get several comparably good groups of parameters.Our numerical results demonstrate that our new kernels perform better than cubic Catmull-Rom in image interpolation.