Whether they are stored digitally in computer memories,or they travel over the Internet, images take up a lot of space.

Fortunately, it is possible to ``condense''them without changing the quality!

a digital image can be compressed, just asorange juice can be reduced to a few gramsof concentrated powder. This is not by sleightof hand, but by mathematical and computerscience techniques making it possible to reducethe amount of space occupied by an image incomputer memory, or in communicationcables. Nowadays, these techniques are essen-tial for storing information, or for transmit-ting it by Internet, telephone, satellite or anyother means.

The compression of an image amounts torepresenting it using a smaller number ofparameters, by eliminating redundancies. Aexaggerated example will help in under-standing the basic principle: in the case of auniformly white image, it is unnecessary toexplicitly specify for each one of its points thegrey level at that point; that would take muchmore space than to simply state: ̀ `all the pointsof the image are white''. The problem of rep-resentation is central in mathematics, and its

Waveletsfor compressing images

Figure 1. These three images illustrate the power of current compression methods. The original image (A) consists of 512 x 512 points, each ofwhich has a certain level of gray taken from a palette of 256 levels. Image (B) is the result of a compression by a factor 8, obtained by reducing thelevels of gray to 2 possible values only (black or white). Image (C) was obtained from (A) upon compressing by a factor 32 by using a wavelet basis.The difference in quality from the initial image is hardly perceptible. (Illustration by the author)

A B C

applications go well beyond data compres-sion. During the last ten years, considerableprogress has been made thanks to the devel-opment of the theory of wavelets. In the fieldof image processing, this progress has led tothe adoption of the new standard of com-pression JPEG-2000. This is a meandering tale,which well illustrates the role of mathemat-ics in the modern scientific, or technologicallandscape.

Thirty-two times less space thanks towavelets

Let us consider an image such as that ofFigure 1A. It consists of 512 x 512 points, whoselevels of grey can vary from 0 (black) to 255(white). Each of the 256 possible levels of greycan be represented by a byte, i.e., a binary num-ber made up of 8 bits (a byte is thus simply asuccession of 8 bits 0 or 1, for example11010001).

One thus needs 512 x 512 x 8 = 2097152bits to encode a single image of this kind -which is a lot! The first idea which comes tomind to reduce the number of bits is to decreasethe number of levels of grey, for example, bylimiting oneself to white and black, as in Figure1B. The two possible values of the level of greyare encoded with only one bit (either 0 or 1),and one has thus decreased the number of bitsby a factor of 8. Obviously, the quality of theimage has deteriorated quite a bit. Now lookat the image of Figure 1C.

It has been encoded with 32 times fewerbits than the original image, by a method usingthe theory of wavelets; the deterioration ishardly perceptible! Why? Because instead ofreducing the precision, it is the manner of rep-resenting the information which was changed.

It all started with the analysis ofJoseph Fourier...

As we have said, a digital image is definedby 512 x 512 numbers which specify the lightintensity at each point. One can thus think ofthis image as a point in a space of 512 x 512dimensions - in the same way that a point ona surface, a two dimensional space, can belocated by two co-ordinates - and to ask whichco-ordinate axes are best adapted for repre-senting such a point. A system of axes (of amore abstract nature here than the familiaraxes of elementary geometry) defines whatone calls a basis.

A first fundamental advance was made bythe mathematician-physicist Joseph Fourier in1802, in his report to the the Académie desSciences on the propagation of heat, a subjectwhich is a priori unrelated to our problem.Fourier showed, in particular, that to repre-sent in a compact and convenient way the func-tion f(x) (from a mathematical point of view,such a function is a point in a space havinginfinitely many dimensions) one can use ̀ `axes''made up of an infinite set of sinusoidal func-tions. More precisely: Fourier showed that onecan represent a function f(x) as a sum of infi-nitely many sine and cosine functions of theform sin(ax) or cos(ax), each one carrying a cer-tain coefficient.

These ``Fourier bases'' became an essen-tial tool, frequently used in science, becausethey can be used to represent many types offunctions, therefore many physical quantities.In particular, they are also used to representsounds and images. And yet, engineers knowwell that these sinusoids are far from beingideal for signals as complex as images: they donot efficiently represent transitory structuressuch as contours of the image.

Wavelets for compressing images 33

...then came the wavelet transform

Specialists in signal processing were notthe only ones to become aware of the limita-tions of Fourier bases. In the 1970’s, a Frenchengineer-geophysicist, Jean Morlet, realisedthat they were not the best mathematical toolfor underground exploration; this led to oneof the discoveries-in a laboratory of Elf-Aquitaine-of wavelet transform. This mathe-matical method, based on a set of basic func-tions different from the sinusoidal functionsused in Fourier's method, advantageouslyreplaces the Fourier transform in certain situ-ations. In addition, already in the 1930’s, physi-cists had realised that the Fourier bases werenot well-adapted for analysing the states ofan atom. This spurred much work, which lateron contributed much to the theory of wavelets.It was also in the 1930’s that mathematiciansstarted trying to improve the Fourier bases foranalysing localised singular structures, whichopened an important research program stillvery much alive today.

In other words, a multitude of scientificcommunities developed modifications ofFourier bases with the means at their disposal.

In the 1980’s, Yves Meyer, a French mathe-matician, discovered the first orthogonalwavelet bases (orthogonality is a propertywhich considerably simplifies reasoning andcalculations; Fourier bases are also orthogo-nal). This discovery, followed by some unex-pected meetings around photocopiers or cof-fee tables, started a vast multi-disciplinaryscientific movement in France, which has hada considerable impact internationally. Theapplications of the theory and algorithms ofwavelets have made their way not only intomany scientific and technological disciplines,but have also led to the creation of severalcompanies in the United States.

Mathematics of waveletshas played a pivotal rolein a number of fields

Mathematics has played a fundamentalrole here as a catalyst, and in the clarificationand deepening of ideas. By isolating the fun-damental concepts from specific applications,it allowed scientists from very diverse fieldsof physics, such as signal processing, computer

science, etc. to realise that theywere working with the same tool.Modern mathematical work onFourier analysis has now permittedus to go further and to refine thesetools, and to control their per-formance. Finally, this theory pro-vides a standard technique for sci-entific computation (the fastwavelet transform), thanks to a col-laboration between mathemati-cians and specialists in signal pro-cessing. The image of Figure 1C wasthus obtained thanks to the samewavelet bases as those used in sta-

34 L’explosion des mathématiques

Figure 2. The graph of a wavelet used in the compression of images.

tistics, seismology, or scientific computation,with the same fast algorithm. And throughthe international standard JPEG-2000 for thecompression of images, these wavelets havecurrently invaded all fields of imaging, fromthe Internet to digital cameras, and is mov-ing towards satellites.

A bridge remains to be built betweenthe world of wavelets and the worldof geometry

Fourier bases were not well-adapted tothe analysis of transitory phenomena, whereaswavelet bases are. Is this the end of the story?No. In image processing, as in all other fieldswhere wavelets have became a basic tool,everyone currently confronts the same typeof problem: how to exploit geometrical reg-ularities. Indeed, we know that an image,however complex, is remarkably well repre-sented by a simple drawing made up of rela-tively few strokes, and one can often think ofthe contours of the objects appearing in theimage as being made up of rather simple geo-metrical curves. Using profitably these curvesand their regularity should make it possibleto improve considerably the results obtainedup until now; but wavelet theory is not atpresent capable of this. To build this bridgewith the world of geometry poses difficultmathematical problems. However, the scien-tific and industrial stakes being high, one canexpect that it will be built in the coming tenyears. In France?

Stéphane MallatDépartement de mathématiques appliquées,

École polytechnique, Palaiseau

Wavelets for compressing images 35

Some references:

• B. B. Hubbard, The world according to wavelets(AK Peters, 1996).

• S. Mallat, A wavelet tour of signal processing(Academic Press, 1999).• Y. Meyer, Wavelets and Operators(Cambridge University Press, 1996).