Non–linear fractal interpolating functions of one and two variables

In this paper we consider two non–linear generalizations of such fractal interpolating functions. The first concerns how to extend the linear interpolation of Eq. (4) to higher–degree interpolations. The second generalization arises when one considers the construction of fractal interpolating functions for functions of two (or more) variables – here, even a linear interpolation of the form of Eq. (4), when applied to each variable, will result in a non–linear interpolating function. This case has an obvious application to the problem of how to represent a two– dimensional image in terms of an iterated function system; these two–dimensional interpolating functions (as a function of the pixel coordinates) can be used to represent a black–and–white image (using a Boolean function), a gray–scale image (using a scalar function), or a colour image (using a vector–valued function of the three rgb [red, green, blue] values). This problem has been examined extensively in the context of image compression [4–7]; in the last section we consider a related problem of using these iterated function systems to rescale images, or portions thereof.