Fractal image approximation and orthogonal bases

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Abstract

We are concerned with the fractal approximation of multidimensional functions in L2. In particular, we treat a position-dependent approximation using orthogonal bases of L2 and no search. 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, the result avoids to solve ill-conditioned linear systems that are usually needed in former fractal approximation techniques. Additionally, using orthogonal bases the most compact representation of the approximation is obtained. We discuss the approximation of gray-scale digital images as a direct application of our approximation scheme.

Introduction

Some years ago it has been shown that deterministic fractal geometry is capable to produce very complex behaviors using apparently simple mathematical models [3]. In particular fractal models appeared suitable to represent real world images 6, 14, 20, 21.

In 1987, Barnsley originally proposed to use deterministic fractal geometry to obtain a compressed representation of digital images. Some years later, one of his students devised the first algorithm capable to partially achieve that goal [10].

The idea of fractal coding is to represent the signal, or better, the function to be approximated, solely by the relations that are present between affinely transformed parts of the signal and the signal itself. Through the removal of ‘self-affine redundancy’, one hopes to obtain a more compact representation than the original one.

Barnsley [4], Jacquin 10, 11, 12 and Jacobs et al. [9] presented different methods for looking for the similarities present in digital images. For simplicity of implementation the search for similarities was performed only between blocks in which the image was initially decomposed. The brightness of a block was being approximated by a linear transformation of the brightness of another bigger block. Among all the bigger candidate blocks the one that best approximated the original was chosen, together with a particular transformation.

The whole image was hence represented through the relationship between blocks and by the coefficients of such brightness transformation. They originally chose linear transformations with a constant translation term with respect to the position inside the block. Although later many other strategies has been proposed (see e.g. [1]) the search process was always computationally very intensive.

Motivated by the desire to reduce substantially the computational cost, Monro and Dudbridge proposed a different approach in which the approximation is applied independently on each single block [16]. The basic method, although simple to implement and very fast, does not perform well. It constrains too strong auto-similarities inside the blocks that are generally not present in real-world images. To obtain a better quality of the approximation the authors propose to substitute the constant translation term with a polynomial in the pixel coordinates. The polynomial approximates the residual error that cannot be captured by the fractal approximation.

Barnsley himself introduced, in the one-dimensional case, a class of fractal interpolation functions which have a self-similarity property [2]. In this paper we want to show that it is possible to reformulate Barnsley’s theory in terms of fractal approximation functions in L2(Rn). In particular, since we are going to treat the problem of image coding (i.e., the approximation of a brightness function) we will consider, without loss of generality, the two-dimensional case.

In that framework we will describe a more general type of position-dependent approximation than the one by Monro and Dudbridge, in which the translation term is a function that belongs to the subspace generated by a particular orthogonal basis. Other techniques that use orthogonal basis, although developed from a different approach, can be found in 18, 19.

The main result of this work is a theorem that builds the fractal approximation from an approximation of the gray-scale function expressed with respect to the same basis. Since the resulting approximation is optimal with respect to the chosen basis we will call it the best fractal orthogonal approximation (BFOA).

In practice, if we suppose to have a ‘classic’ place-dependent approximation the rules of the theorem ‘turn it’ into a fractal approximation. In this way, we avoid using heavy numerical methods to overcome the ill-conditioned problems associated to the type of polynomials used in 15, 16, 17.

We will show some results on the approximation of digital images obtained with cosine and Haar basis. We want to emphasize that the initial approximation can be computed with any algorithm, for example with fast technique like FFT or DWT. However, this work proposes a new approximation model, and not yet a compression technique.

Section 2recalls some notations used in the rest of the paper, while Section 3introduces a theory for fractal approximation in L2(R2) with a variant of the Collage theorem. Section 4presents the main result of the BFOA and an issue on the contractivity of the operator.

The application of BFOA to image approximation is described in Section 5, where we show the results of using different orthogonal bases in a block coding framework. Once we fix the number of parameters, our approach gives a lower reconstruction error than the original Monro and Dudbridge polynomial approximation. In the same section we analyze the best splitting point heuristics as a searching method and we show the advantages given by the utilization of bigger blocks than the ones generally used.

Section snippets

Notations

We briefly recall some notation used in the paper. We consider functions in Lp with the metric d(f,g)=‖f−g‖p,where fp=R2|f(x)|pdμ1/p.

Let f be a function in Lp and U a subspace of Lp. With best approximation of f in U we define the function fLp that satisfies f−fp=infg∈Uf−g‖p.In other words, f is the function that achieves the minimum distance from f with respect to the particular norm chosen.

We recall that if U is a finite-dimensional subspace, then there exists at least one best

Fractal approximation in L2(R2)

We identify a continuous gray-scale image with a function f∈L2 whose domain is a compact set A, attractor of an IFS {A;w1,…,wN} (see [3]) A=i=1NAi=i=1Nwi(A),where the maps wi are affine, contractive and non-overlapping,1 i.e., wi(x)=Lix+τi,x∈R2, and where Li are 2×2 scaling matrices, and τi are translation vectors. The maps wi describe the underlying ‘geometry’ of the domain A of the function f.

A fractal approximation of f is a function f associated with an L2

The best fractal orthogonal approximation

When the functions qi belong to a subspace U generated by an orthogonal basis {u0,u1,…,un}, the operator T can be obtained by fairly simple rules.

The following theorem allows us to construct the function that approximates fwi, i.e., the function that minimizes fwi−(αif+qi)‖2 with respect to qiU, αiR. We call such approximating function the best fractal orthogonal approximation of fwi in U.


Theorem 2. Let f∈L2(A), AR2 be a compact set with μ(A)<+∞ and {w1,…,wN} be non-overlapping contractive

Applications to block image coding

In order to apply the BFOA to image coding we consider the image decomposed in square blocks of 8×8 or 16×16 pixels. A single block becomes the domain A of the brightness function f, the function we want to approximate. We choose w1,w2,w3,w4 as the functions that map a square in its four equal sub-quadrants. Later in this section we will discuss a more general subdivision of blocks.

We first choose Legendre and Chebychev polynomials as orthogonal system {ui}i=0,…,n. However, the best results

Conclusions

We introduced a general theory for the position-dependent fractal approximation of functions in L2(Rn), called the ‘best fractal orthogonal approximation’, that connects IFS and orthogonal bases. Loosely speaking, our method is capable of ‘transforming’ a classic place-dependent approximation into a fractal approximation.

Our approach can be very useful in multidimensional signal processing. In particular, we showed an application to two-dimensional discrete data, and specifically to digital

Acknowledgements

The authors want to thank B. Forte for helpful discussions about this paper.

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