1. Introduction
????????????? With the advance of the information age the need for mass information storage and fast communication links grows. Storing images in less memory leads to a direct reduction in storage cost and faster data transmissions. These facts justify the efforts, of private companies and universities, on new image compression algorithms.
Storing an image on a computer requires a? very? large? memory. This? problem? can? be? averted? by ?the? use? of? various? image? compression? techniques. Most? images contain? some? amount? of? redundancy? that? can? be? removed? when? the? image? is? stored? and? then? replaced? when? it? is? reconstructed.
Fractal? image? compression? is? a? recent ?technique? based? on? the? representation? of? an? image. The? self-transformability? property? of? an? image? is? assumed? and? exploited? in? fractal? coding. It provides? high? compression? ratios? and? fast? decoding. Apart? from? this? it? is? also? simple? and? is? an? easily? executable? technique.
Images are stored on computers as collections of bits (a bit is a binary unit of information which can answer ?yes? or ?no? questions) representing pixels or points forming the picture elements. Since the human eye can process large amounts of information (some 8 million bits), many pixels are required to store moderate quality images. These bits provide the ?yes? and ?no? answers to the 8 million questions that determine the image.
Most data contains some amount of redundancy, which can sometimes be removed for storage and replaced for recovery, but this redundancy does not lead to high compression ratios. An image can be changed in many ways that are either not detectable by the human eye or do not contribute to the degradation of the image.
The standard methods of image compression come in several varieties. The current most popular method relies on eliminating high frequency components of the signal by storing only the low frequency components (Discrete Cosine Transform Algorithm). This method is used on JPEG (still images), MPEG (motion video images), H.261 (Video Telephony on ISDN lines), and H.263 (Video Telephony on PSTN lines) compression algorithms.
Fractal Compression was first promoted by M.Barnsley, who founded a company based on fractal image compression technology but who has not released details of his scheme. The first public scheme was due to E.Jacobs and R.Boss of the Naval Ocean Systems Center in San Diego who used regular partitioning and classification of curve segments in order to compress random fractal curves (such as political boundaries) in two dimensions. A doctoral student of Barnsley?s, A. Jacquin, was the first to publish a similar fractal image compression scheme.
The birth of fractal geometry (or rebirth, rather) is usually traced to IBM mathematician Benoit B. Mandelbrot and the 1977 publication of his seminal book ?The Fractal Geometry of Nature?. The book put forth a powerful thesis: traditional geometry with its straight lines and smooth surfaces does not resemble the geometry of trees and clouds and mountains. Fractal geometry, with its convoluted coastlines and detail ad infinitum, does.
This insight opened vast possibilities. Computer scientists, for one, found a mathematics capable of generating artificial and yet realistic looking landscapes, and the trees that sprout from the soil. And mathematicians had at their disposal a new world of geometric entities.
It was not long before mathematicians asked if there was a unity among this diversity. There is, as John Hutchinson demonstrated in 1981, it is the branch of mathematics now known as Iterated Function Theory. Later in the decade Michael Barnsley, a leading researcher from Georgia Tech, wrote the popular book ?Fractals Everywhere?. The book presents the mathematics of Iterated Functions Systems (IFS), and proves a result known as the Collage Theorem. The Collage Theorem states what an Iterated Function System must be like in order to represent an image.
This presented an intriguing possibility. If, in the forward direction, fractal mathematics is good for generating natural looking images, then, in the reverse direction, could it not serve to compress images? Going from a given image to an Iterated Function System that can generate the original (or at least closely resemble it), is known as the inverse problem. This problem remains unsolved.
Barnsley, however, armed with his Collage Theorem, thought he had it solved. He applied for and was granted a software patent and left academia and found Iterated Systems Incorporated (US patent 4,941,193. Alan Sloan is the co-grantee of the patent and co-founder of Iterated Systems.) Barnsley announced his success to the world in the January 1988 issue of BYTE magazine. This article did not address the inverse problem but it did exhibit several images purportedly compressed in excess of 10,000:1. Alas, it was a slight of hand. The images were given suggestive names such as "Black Forest" and "Monterey Coast" and "Bolivian Girl" but they were all manually constructed. Barnsley's patent has come to be derisively referred to as the "graduate student algorithm."
Attempts to automate this process have continued to this day, but the situation remains bleak. As Barnsley admitted in 1988: "Complex color images require about 100 hours each to encode and 30 minutes to decode on the Masscomp [dual processor workstation]." That's 100 hours with a person guiding the process.
Ironically, it was one of Barnsley's PhD students that made the graduate student algorithm obsolete. In March 1988, according to Barnsley, he arrived at a modified scheme for representing images called Partitioned Iterated Function Systems (PIFS). Barnsley applied for and was granted a second patent on an algorithm that can automatically convert an image into a Partitioned Iterated Function System, compressing the image in the process. (US patent 5,065,447. Granted on Nov. 12 1991.) For his PhD thesis, Arnaud Jacquin implemented the algorithm in software, a description of which appears in his landmark paper "Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformations." The algorithm was not sophisticated, and not speedy, but it was fully automatic. This came at price: gone was the promise of 10,000:1 compression. A 24-bit color image could typically be compressed from 8:1 to 50:1 while still looking "pretty good." Nonetheless, all contemporary fractal image compression programs are based upon Jacquin's paper.
That is not to say there are many fractal compression programs available. There are not. Iterated Systems sell the only commercial compressor/decompressor, an MS-Windows program called "Images Incorporated." There are also an increasing number of academic programs being made freely available. Unfortunately, these programs are of merely academic quality.
This scarcity has much to do with Iterated Systems tight lipped policy about their compression technology. They do, however, sell a Windows DLL for programming. In conjunction with independent development by researchers elsewhere, therefore, fractal compression will gradually become more pervasive. Whether it becomes all-pervasive remains to be seen.
Historical Highlights:
1977 -- Benoit Mandelbrot finishes the first edition of The Fractal Geometry of Nature.
1981 -- John Hutchinson publishes "Fractals and Self-Similarity."
1983 - Revised edition of The Fractal Geometry of Nature is published.
1985 - Michael Barnsley and Stephen Demko introduce Iterated Function Theory in "Iterated Function Systems and the Global Construction of Fractals."
1987 - Iterated Systems Incorporated is founded.
1988 - Barnsley publishes the book Fractals Everywhere.
1990 - Barnsley's first patent is granted.
1991 - Barnsley's second patent is granted.
1992 - Arnaud Jacquin publishes describes the first practical fractal image compression method.
1993 - The Iterated Systems' product line matures.
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What is Fractal Image Compression?
Imagine a special type of photocopying machine that reduces the image to be copied by half and reproduces it three times on the copy (see Figure 1). What happens when we feed the output of this machine back as input? Figure 2 shows several iterations of this process on several input images. We can observe that all the copies seem to converge to the same final image, the one in 2(c). Since the copying machine reduces the input image, any initial image placed on the copying machine will be reduced to a point as we repeatedly run the machine; in fact, it is only the position and the orientation of the copies that determines what the final image looks like.
The way the input image is transformed determines the final result when running the copy machine in a feedback loop. However we must constrain these transformations, with the limitation that the transformations must be contractive (see contractive box), that is, a given transformation applied to any two points in the input image must bring them closer in the copy. This technical condition is quite logical, since if points in the copy were spread out the final image would have to be of infinite size. Except for this condition the transformation can have any form.In practice, choosing transformations of the form is sufficient to generate interesting transformations called affine transformations of the plane. Each can skew, stretch, rotate, scale and translate an input image.
A common feature of these transformations that run in a loop back mode is that for a given initial image each image is formed from a transformed (and reduced) copies of itself, and hence it must have detail at every scale. That is, the images are fractals. This method of generating fractals is due to John Hutchinson.
Barnsley suggested that perhaps storing images as collections of transformations could lead to image compression. His argument went as follows: the image in Figure 3 looks complicated yet it is generated from only 4 affine transformations.
Each transformation wi is defined by 6 numbers, ai, bi, ci, di, ei, and fi , see eq(1), which do not require much memory to store on a computer (4 transformations x 6 numbers /transformations x 32 bits /number = 768 bits). Storing the image as a collection of pixels, however required much more memory (at least 65,536 bits for the resolution shown in Figure 2). So if we wish to store a picture of a fern, then we can do it by storing the numbers that define the affine transformations and simply generate the fern whenever we want to see it. Now suppose that we were given any arbitrary image, say a face. If a small number of affine transformations could generate that face, then it too could be stored compactly. The trick is finding those numbers.
2.1 Why the name ?Fractal?
The image compression scheme described later can be said to be fractal in several senses. The scheme will encode an image as a collection of transforms that are very similar to the copy machine metaphor. Just as the fern has detail at every scale, so does the image reconstructed from the transforms. The decoded image has no natural size, it can be decoded at any size. The extra detail needed for decoding at larger sizes is generated automatically by the encoding transforms. One may wonder if this detail is ?real?; we could decode an image of a person increasing the size with each iteration, and eventually see skin cells or perhaps atoms. The answer is, of course, no. The detail is not at all related to the actual detail present when the image was digitized; it is just the product of the encoding transforms which originally only encoded the large-scale features. However, in some cases the detail is realistic at low magnifications, and this can be useful in Security and Medical Imaging applications. Figure 4 shows a detail from a fractal encoding of ?Lena? along with a magnification of the original.
2.2 PROPERTIES? OF? FRACTALS
A? set? F? is? said? to? be? a? fractal? if? it? possesses? the? following? poperties.
CONCLUSION
??????????? The? power? of? fractal? encoding? is? shown? by? its? ability? to? outperform? the? DCT, which? forms? the? basis? of? the? JPEG? scheme? of? image? compression. This? method? has? had? the? benefit? of? thousands? of? man hours? of? research, optimization? and? general? tweaking. Thus? the? fractal? scheme? has? won? for? itself? more? than? the? attention? of? the? average? engineer.
??????????? Several new schemes? of? image? compression? are? being? developed? day-by-day. The? most notable? of? these? is? the? fractal? scheme. It provides endless? scope? and? is? under? research. Further new partitioning schemes are? also? being? developed. Another? recent? advance? in? this? field? has? been? through? wavelet? transformations. This? is? a? highly? efficient? method? based? on? the? Fourier? representation? of? an? image. This? scheme? comes? as? a? competitive? development? to? the? fractal? approach. Fractal image compression is a promising new technology but is not without problems. Most critically, fast encoding is required for it to find wide use in multimedia applications. This is now within reach: recent methods are five orders of magnitude faster than early attempts
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