Hidden Variable Fractal Interpolation Curve And Its Box-counting Dimension

In recent years,the theory of fractal interpolation has obtained fruitful results in the study of construction,dimension and analytical properties.Due to the limitation of the fractal interpolation function,there are still many problems to be solved for the complex and irregular phenomenon.Hidden variable fractal interpolation was proposed to improve the diversity and flexibility of fractal interpolation.Hidden variable fractal interpolation function(HVFIF)is the projection in the plane of vector-valued fractal interpolation function.Those hidden variables can be used to adjust the shape and the fractal dimension of image.Therefore this dissertation conduct related research on the basis of the structure of HVFIF and concrete content are as follows:First,according to the method of hidden variable fractal interpolation function,we get fractal interpolation curve by computer simulation.By changing parameters of value,there are different hidden variable fractal interpolation curve for the same data set.The properties are also changed.At the same time,the operability is verified of hidden variables fractal interpolation theory.Second,article gives emphasis on the dimension of hidden variable fractal interpolation curve(HVFIC).Dimension is important content of fractal interpolation theory.It explicity reflects the property of the image.The complexity of the image is given quantitatively by dimension.Thus this paper studies the influence of hidden variables for the dimension and determines the relationship between the dimension of HVFIC and free parameters.Due to the structure of the hidden variable fractal interpolation function is relatively complex than general fractal interpolation function.It has more degrees of freedom.So this article discusses two kind of situations,which estimation range of the box number is given respectively.Finally,the dimension formula is obtained of the corresponding HVFIC,which shows the HVFIF dimension of different types related to different free parameters.