Short communicationFractal image coding schemes using nonlinear grey scale functions
Introduction
In the last few years many efforts to develop the iterated function systems (IFS) technique have been made by many researchers [1], [2], [4], [5]. IFS is a very attractive technique, achieving good results in terms of objective (and subjective) quality and compression ratio. The main drawback of this approach stems in the computational time, especially in encoding, when compared with other coding techniques such as sub-band-based techniques or the standard JPEG [9]. It is, then, comprehensible that major effort of the scientific community has been – and is actually – devoted to develop techniques that are able to speed up the IFS technique [3], [8].
Recently, some effort has also been made in order to utilize nonlinear methods and, with regard to image coding, an interesting approach has been proposed by Popescu et al. [7], where some nonlinear geometrical transformations have been used.
Moreover, some studies closer to the topic of this paper are found in [2], where place-dependent functions are utilized along with affine ones.
In this paper, we present an approach using maps with nonlinear grey scale functions, i.e. second degree polynomial functions, preserving the structure of the classical fractal transform. Nevertheless, a simplification of the model is required, since the number of constraints for the contractivity of a generic parabola is large. In particular, the centred vertex parabolas (CVPs), pcv(z)=az2+c have been used, because of their low computational effort (quite comparable with using affine functions) and good compression ratio: only two parameters have to be stored. In the following, the two IFS-based models, affine (the classical one) and not, will be, respectively, denoted by IFS-LIN and IFS-CVP.
The results obtained by these two models are quite comparable but, in some cases, there is an improvement in the quadtree partitioning using IFS-CVP, i.e., there are regions split at a lower level than in the linear case. Such an interesting peculiarity has been utilized to build a hybrid model where both the approaches (IFS-LIN and IFS-CVP) have been used.
The remainder of the paper is as follows. Section 2 presents the constraints useful to guarantee the contractivity of a generic parabola (more details are in Appendix A) along with the case relative to CVPs. In Section 3, some experimental results are shown with regard to the objective quality of the restored image, the compression ratio and the computational time, while Section 4 draws the conclusions.
Section snippets
From the classical case to CVPs
IFS theory [1], [2], [5] is based on the concept of contractive function (CF). It can be stated as a function on a complete metric space (X,h) and a constant such that , where h is the metric function and the real number s is the contractivity factor for f. From Banach's fixed point theorem, we know that for a CF f the iteration sequence defined by converges to one and only one point xf∈X, independent of the starting point on which it is
Some experimental results
The proposed model has been applied to several test images. Nevertheless, for purposes of brevity, only the results obtained on 512×512 8-bits Lena and Peppers images will be presented. In the simulations, a four level quadtree has been adopted for both the algorithms along with an exhaustive search to determine the optimal domain block for a given range block. Moreover, the usual subsampling by a factor is used. In Fig. 1, a comparison between IFS-CVP and IFS-LIN, in terms of
Conclusions
In this paper, we have presented a novel approach using nonaffine contractive functions for fractal image coding. After presenting the contractivity constraints for a generic parabola, we have proposed some strategies to simplify the model, in order to reduce the increase in the computational time. To exploit the local structure of the image, a hybrid model using both affine and noncontractive functions has been proposed. The improvement of the IFS performance is evident in terms of PSNR and
Acknowledgements
I would like to thank Dr. D. Dominici for the help to realize this work and the anonymous referees for their input to improve this paper.
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2005, Advances in Imaging and Electron PhysicsCitation Excerpt :In effect, Zhao and Yuan noticed some problems for convergence in decoding using a polynomial function of second degree: g(z) = az2 + bz + c. Relaxing the contractivity constraint of the geometric part, the problem is to prove contractivity of the only massic component. Considering an image with intensity belonging to the range [0, L] (for gray scale it becomes [0, 255]), it has been proved by Vitulano (2001) that constraints for guaranteing contractivity become: Range g([0, L]) has to be included in the domain, and
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2007, IET Conference Publications