A Construction of the Best Fractal Approximation

In this paper we present a method for constructing the continuous best fractal approximation in the space of bounded functions. We construct the finite-dimensional subspace of the space of bounded functions whose base consists of the continuous fractal functions, and propose how to find the best approximation of given continuous function by element of the constructed space.Comment: 9 page

Fractal Image Approximation and Orthogonal Bases

We are concerned with the fractal approximation of multidimensional functions in L². In particular, we treat a position-dependent approximation with no search using orthogonal bases of L². We describe a framework that establishes a connection between the classic orthogonal approximation and the fractal approximation. The main theorem allows easy and univocal computation of the parameters of the approximating function. From the computational perspective, we can avoid to solve linear systems often suffering from ill conditioning and needed in former fractal approximation techniques. Moreover, using orthogonal bases we obtain the most compact representation of the approximation. As a direct application we show some results on the compression of gray scale digital images