Variation And Box Dimension Of Recurrent Fractal Interpolation Surface

Box dimension can be used to describe the roughness of fractal graphics, variation is a parameter under different scales in describing function image, variation is used to study box dimension of recurrent fractal interpolation surface. In order to get some properties of the variation of bivariate Recurrent Fractal Interpolation Function( RFIF)on the rectangular domain, the connection matrix is used to deal with the complex mapping relation in the bivariate RFIF, and the estimation of the variation of RFIF is given. Combined with the relation between the eigenvalue and the eigenvector, by the relation of recurrence the rank of the bivariate RFIF variation is estimated.And then using the relation between the box dimension of the graph of continuous function and its variation, we obtain the formula of box dimension of the general recurrent fractal interpolation surface. Finally, an example of the calculation of the box dimension of the recurrent fractal interpolation surface and image simulation is given.This thesis consists of four chapters. The first chapter is a brief introduction of the research background, the current research as well as the main research contents and innovative points of this paper are presented.In Chapter 2, the concept of the box-counting dimension is discussed. And the properties of Iterated Function System(IFS)、Fractal Interpolation Function(FIF) are recalled. The last more general definition and theorem of Recurrent Iterated Function System(RIFS) and Recurrent Fractal Interpolation Function are explained.In Chapter 3, firstly, the concept and properties of the continuous function oscillation and variation are discussed, based on the basis of the variation properties of the FIF and unary RFIF, with the correlation matrix, some properties of bivariate RFIF are proved, and the rank of the bivariate recurrent fractal interpolation function are received.In Chapter 4, we give the concepts of the nonnegative matrix、directed graph and strongly connected component. And the Perron-Frobenius theorem that widely used in the dimension calculation of fractal geometry is discussed. Based on the connection matrix, the rank of the Recurrent Fractal Interpolation Function variation is estimated.Using the relation between the box dimension of the graph of continuous function and its variation, the dimension formula of the bivariate recurrent fractal interpolation function can be proved. Finally, we give an example for using the above dimension formula, and the calculation of the box dimension and image simulation of the recurrent fractal interpolation surface are given in this chapter.In Chapter 5, the summary and the prospect of this thesis are presented.