An approach to image coding based on a fractal theory of iterated contractive transformations defined piecewise is described. The main characteristics of this approach are that it relies on the assumption that image redundancy can be efficiently captured and exploited through piecewise self-transformability on a block-wise basis, and it approximates an original image by a fractal image, obtained from a finite number of iterations of an image transformation called a fractal code. This approach is referred to as fractal block coding. The general coding-decoding system is based on the construction, for an image to be encoded, of a fractal code-a contractive image transformation for which the original image is an approximate fixed point-which, when applied iteratively on any initial image of the decoder, produces a sequence of images which converges to a fractal approximation of the original. The design of a system for the encoding of monochrome digital images at rates below 1 b/pixel is described. Ideas and extensions from the work of other researchers are presented. >

Fractal image coding: a review

Fractal image compression is a technique based on the representation of an image by a contractive transform, on the space of images, for which the fixed point is close to the original image. This broad principle encompasses a very wide variety of coding schemes, many of which have been explored in the rapidly growing body of published research. While certain theoretical aspects of this representation are well established, relatively little attention has been given to the construction of a coherent underlying image model that would justify its use. Most purely fractal-based schemes are not competitive with the current state of the art, but hybrid schemes incorporating fractal compression and alternative techniques have achieved considerably greater success. This review represents a survey of the most significant advances, both practical and theoretical, since the publication of Jacquin's (1990) original fractal coding scheme.

A review of the fractal image coding literature

Practical data compression in wireless sensor networks: A survey

This paper presents a new adaptive search approach to reduce the computational complexity of fractal encoding. A simple but very efficient adaptive necessary condition is introduced to exclude a large number of unqualified domain blocks so as to speed-up fractal image compression. Furthermore, we analyzed an unconventional affine parameter that has better properties than the conventional luminance offset. Specifically, we formulated an optimal bit allocation scheme for the simultaneous quantizations of the usual scaling and the aforementioned unconventional affine parameter. Experiments on standard images showed that our adaptive search method yields superior performance over conventional fractal encoding.

Fast fractal image encoding based on adaptive search

In fractal image compression the encoding step is computationally expensive. A large number of sequential searches through a list of domains (portions of the image) are carried out while trying to find the best match for another image portion. Our theory developed here shows that this basic procedure of fractal image compression is equivalent to multi-dimensional nearest neighbor search. This result is useful for accelerating the encoding procedure in fractal image compression. The traditional sequential search takes linear time whereas the nearest neighbor search can be organized to require only logarithmic time. The fast search has been integrated into an existing state-of-the-art classification method thereby accelerating the searches carried out in the individual domain classes. In this case we record acceleration factors from 1.3 up to 11.5 depending on image and domain pool size with negligible or minor degradation in both image quality and compression ratio. Furthermore, as compared to plain classification our method is demonstrated to be able to search through larger portions of the domain pool without increased the computation time.

https://cms.uni-konstanz.de/fileadmin/informatik/ag-saupe/Webpages/fractal2/pdf/Saup95a.pdf.lic

Accelerating fractal image compression by multi-dimensional nearest neighbor search

A method for block coding of images based on a least squares fractal approximation by a self-affine system (SAS) is presented. The computational cost of the approximation is linear in the number of pixels in the image. The approximation to a rectangularly tiled block involves evaluating various low-order moments over the block, and solving a system of four linear equations for each tile. The method is applied to a standard test image and the effects of various optimizations are shown. A quantitative comparison with the adaptive discrete cosine transform at 8:1 compression is made. The fidelity of the fractal method shows promise and its greater speed and simplicity compared to other fractal transforms suggest immediate applications such as interactive browsing of remote image archives or image representation in multimedia systems. >

Fractal approximation of image blocks

We study the fidelity/compression performance of fast fractal image coding. The Bath fractal transform (BFT) is a generalization of earlier methods with a range of implementation options for the order of approximation and degree of searching. All fractal transforms are fast to decode, and the zero searching variants are nearly as fast to code. We examine BFTs of zero order (flat), first order (bilinear) and second order (biquadratic). The degradation introduced by Lloyd-Max (1982) quantization of the fractal coefficients is evaluated. Further compression is achieved by exploiting correlation in one of the fractal coefficients through differential coding. The other coefficients are quantized directly, and Huffman coding of each gives a compressed image. Possibilities for further compression are discussed. >

Fractal image compression without searching

This paper reports improvements in compression of both inter- and intra-frame images by the matching pursuits (MP) algorithm. For both types of image, applying a 2D wavelet decomposition prior to MP coding is beneficial. The MP algorithm is then applied using various separable ID codebooks. MERGE coding with precision limited quantization (PLQ) is used to yield a highly embedded data stream. For inter-frames (residuals) a codebook of only 8 bases with compact footprint is found to give improved fidelity at lower complexity than previous MP methods. Great improvement is also achieved in MP coding of still images (intra-frames). Compared to JPEG 2000, lower distortion is achieved on the residual images tested, and also on intra-frames at low bit rates.

Improved matching pursuits image coding

A method of video motion estimation is described for determining the dominant motion in a video image. The dominant motion is defined by a parametric transform, for example a similarity transform. In the preferred embodiment, selected pairs of blocks in one frame are traced by a block matching algorithm into a subsequent frame, and their change in position determined. From that information, an individual parameter estimate is determined. The process is repeated for many pairs of blocks, to create a large number of parameter estimates. These estimates are then sorted into an ordered list, the list is preferably differentiated, and the best global value for the parameter is determined from the differentiated list. One approach is to take the minimum value of the differentiated list, selected from the longest run of values which fall below a threshold value. Alternatively, the ordered list may be examined for flat areas, without explicit differentiation. The technique is particularly suited to low complexity, low bit rate multimedia applications, where reasonable fidelity is required without the computational overhead of full motion compensation.

Motion estimation and compensation in video compression

Methods of reducing the complexity of the matching pursuits algorithm with minimal loss of fidelity when coding displaced frame difference (DFD) images in video compression are investigated. A full search using 2D basis functions is used as a benchmark. The use of separable 1D bases greatly reduces the complexity, and significant further reductions are achieved by using only a 1D inner product search to locate the atom position, followed by a further 1D inner product search in the opposite direction to identify the second 1D basis function. To avoid ignoring significant structures orthogonal to the search direction, it is proposed to alternate the initial search direction between horizontal and vertical scanning. This produces a modest increase in distortion compared to the full 2D search, with a complexity reduction in excess of an order of magnitude.

D.M. Monro

Papers

Fractal approximation of image blocks

Fractal image compression without searching

Improved matching pursuits image coding

Motion estimation and compensation in video compression

Low complexity separable matching pursuits