Resonance Paper1 Mar2004

8/3/2019 Resonance Paper1 Mar2004

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GENERAL I ARTICLE

Wavelet TransformA New Mathematical Microscope

Sachin P Nanavati and Prasanta K Panigrahi

KeywordsFourier transform, discretewavelet transform (DWT),Haar wavelets.

In the last decade, a new mathematical microscopehas allowed scientists and engineers to view thedetails of time varying and transient phenomena,in a manner hitherto not possible through con-ventional tools. This invention, which goes by thename of wavelet. transform, has created revolu-tionary changes in the areas of signal processing,image compression, not to speak about the ba-sic sciences. This novel procedure enables one toachieve the so called time-frequency localizationand multi-scale resolution, by suitably focussing andzooming around the neighborhood of one's choice.Wavelets are of very recent origin; their con-struction, properties and applications are sub-jects of intense current research. In this arti-cle, we explain with illustrations the working ofthis transform and its advantages vis-a-vis theFourier transform. In two companion articles,we describe the procedure to construct waveletbasis sets and their applications to data analysisand image compression.IntroductionReducing objects into their basic constituents has beenthe preferred route of investigation in science and engi-neering. At an elementary level, we are taught to an-alyze vectors in terms of their components. In threedimensional Euclidean space, a vector A is decomposedin terms of its components as A = Axi + AyJ + Ai;;.Here, i, J and k are the positive unit vectors in theCartesian coordinate system, pointing along the orthog-

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GENERAL IARTiClEonal axes x, y and z, respectively. The fact that thesevectors form an orthonormal set, i.e.,

~2 ~2 ~ 2 ~ ~ ~ ~ ~ ~i = j = k = 1 and i.j = j.k = k.i = 0, (1)which is complete, allows an arbitrary vector to be ex-panded in terms of these basic unit constituents. Itshould be mentioned that any two of the above unitvectors, although orthonormal, do not form a completebasis in three dimensions. Using the above orthonormal-ity conditions, one can find the components of vectorsthrough suitable projections: Ax = i.A, Ay = 3.A andAz = k.A. Apart from aiding in visualization and facil-itating calculations involving vectors, these componentscan be used to compare two vectors for the purpose ofidentifying similarities and differences. It is worthwhileto note that these Euclidean unit vectors, by no means,provide an unique basis set. Often, it is convenient todecompose vectors in terms of other unit vectors; forexample, circular motion can be better described in aspherical polar coordinate system.Unlike the constant vector A encountered above, the de-composition of a function or a signal f(t), which takesdifferent values at different points of time t, needs morecare. Although we have used time t as a continuousvariable, one can as well replace it by any other contin-uous variable like a space coordinate x. If the functionis periodic, say with period L (J (t + L) = f (t)) and has.finite energy over one period i.e.,itO+L If(t)12dt < 00,to (2)then one can decompose the function f(t) in terms ofsine and cosine waves, such that

ao 00[ (

27rrt)

.

(27rrt

)](t) = 2+?; arcos L +brsin L . (3)RESONANCEI March 2004 ~ 51


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GENERAL I ARTICLEThe well-known

Fourier series hasfound tremendous

application indiverse areas,

particularly afterthe discovery ofthe fast Fouriertransform (FFT).

This is the well-known Fourier series which has foundtremendous application in diverse areas, particularly af-ter the discovery of the fast Fourier transform (FFT).Here ao, ar and br are constant coefficients representingthe average of the function and the amplitudes of thecosine and sine waves, respectively. Much like the unitvectors in Euclidean space, the cosine and sine wavesalso form an orthogonal basis set, as seen from the fol-lowing equations:

l to+L . (21rPt) (21rrt)m - cos - dt=Oto L Lfor all p and r, (4)l to+L (21rPt)os - costo L (21rrt)dt=

{L for P = r = 0,

~L for p = r > 0,0 for p =Fr,(21rrt) ,-dt=L{

0 for p = r = 0,~L for p = r > 0,0 for p =Fr, (6)

(5)

l to+L . (21rPt)m -to L sinwhere integers p and r 2::O. Using the above properties,the coefficients can be easily extracted as

ar - 2 1 to+L (21rrt )j(t)cos - dt,L to L2 1 to+L (21rrt )j(t)sin - dt.L to L (8)

(7)b =

Just as a prism splits white light into its constituentcolours, the Fourier transform breaks up a time depen-dent function into its frequency components. Hence, itis called a mathematical prism. To obtain the informa-tion about a particular frequency component, we inte-grate the function j(t), modulated with a cosine or sinefunction, over one period in the time domain, as in the

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GENERAL I ARTICLE

above expressions. Now .the time information of the orig-inal function is spread over the entire frequency domain.Practically, it is difficult to retrieve them. For seeing thispoint clearly, let us imagine doing a Fourier transformof a 60 minute musical concert, where a tuning fork offrequency 440 Hz was played for 20 minutes. Althoughin the frequency domain, a clear cut peak at 440 Hz willbe present, it will be difficult to specify when exactlythis tuning fork was played in the time span of the con-cert. This is because the time inform~tion is stored inrelative phases (i. e., angles between Fourier coefficientsbr 's and a~s) of the basis functions. This phase has tobe calculated with a precision of f'V 440x~ox60 = 52looo .Computationally, it is impossible to calculate with thisrequired precision! Because of the finite word length ofthe computer, one can write a real number only up toa certain precision, much lower than the above require-ment [1],If the function is not periodic but decreases fast enoughat infinity, then its Fourier transform can also be defined.It transforms a function f(t), that depends on time, intoa new function j (w) depe.nding on frequency w whichtakes continuous values:

j(w) = i: f(t)e-iuJtdt (9)This new function is called the Fourier transform (FT)of the original function. To obtain the time informationof the signal back, the inverse Fourier transform of j (w)is taken as

1 1 00 ~f(t) = - . f(w)eiuJtdw27r -00 (10)One faces difficulties in using the Fourier transform,while dealing with signals or functions which have sharpchanges (transient phenomena) or which are rapidly vary-ing in time. This can be better appreciated with theexample of the box function and its Fourier transform,

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Just as a prismsplits white lightinto its constituentcolours, theFourier transformbreaks up a timedependent functioninto its frequencycomponents.


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GENERAL IARTICLE

-) -2 -,

Figure 1.Box function (left)and its Fourier transform,the sinc function (right).

Figure 2. Box function re-constructed after removalof a handful of Fouriercoefficients with small val-ues. The overshoots andundershoots at the edgesare clearly visible.

b-

-20 0 20 40

the well-known sinc function Ci:W). The box function,as shown in Figure 1, has zero value everywhere, ex-cept between t = -1 and t = 1, where it equals unity.One notices immediatety that the sinc function is an os-cillatory function which decays slowly in the frequencydomain, as Iwl increases. Trying to reconstruct the boxfunction (the inverse transform) after neglecting a fewfrequency components having very small amplitudes, of-ten necessary in practical applications, leads to distor-tions at the sharp edges as seen in Figure 2. This is

12

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OA

02

~2-1.5

54

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-1 -c.5 0-OJ 0.5 1.5~ RESONANCE I March 2004

,

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10.80.60.4-0.2

0-0.2

-40


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GENERAL I ARTICLE

the well-known Gibbs' phenomenon [4]. The reason forthis difficulty can be explained as follows. To producethe box function, an infinite number of sine and cosinewaves, with appropriate amplitudes, are needed to in-terfere in a manner which is constructive only at thesite of the box but destructive elsewhere. It should benoted that the sine and cosine waves themselves extendperiodically in the time domain from -00 to 00, Hence,any disturbance in this delicate interference, as is doneby removing a handful of coefficients having very smallvalues, leads to the overshoots and undershoots at thelocations of the sharp changes, One possible solution tothis problem is to have a basis set with elements whichare themselves localized in time. It will be still betterif the basis functions are of a similar form as the func-tion itself; after all, use of one thorn for the removal ofanother is an ancient wisdom! As we will soon discover,wavelet transform relies precisely on this wisdom.Wavelet TransformA wavelet is a small wave which oscillates and decays inthe time domain. As discerning readers must have no-ticed, the logo of Resonance resembles a wavelet quitewell. Unlike the Fourier transform, wavelets can haveinfinite varieties which are fundamentally different fromeach other. The ones which have strictly finite extent inthe time domain, are known as discrete wavelets, other-wise they go by the name of continuous wavelets. Al-though the most elementary discrete wavelet, named af-ter Haar, was known since'191O, the non-trivial onesand the theory behind wavelet transform are of recentorigin. In the following, we will only concentrate on thediscrete wavelet transform.A wavelet basis set starts with two orthogonal func-tions: the scaling function or father wavelet


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GENERAL 1ARTICLE

A recent use ofHaar wavelet wasin the extraction ofthe characteristicsof individual thumbimpressions by

Federal Bureau ofInvestigation (FBI)

of USA.

a complete basis set.The scaling and wavelet functions, respectively, satisfy

i: if>(t)dt = A andi: 'ljJ(t)dt = 0 (11)where A is a constant. The energies of these functionsare finite, which meansi: 1if>(t)12dt < 00 and i: 1"p(t)12dt< 00 . (12)The scaling function and the mother wavelet are orthog-onal to each other:i: if>*(t)"p(t)dt = 0 . (13)From (11), it is apparent that "p(t) resembles a wavewhich is localized in time, in other words it is a smallwave or a wavelet. IIi a given wavelet basis set, there isonly one scaling function; the rest of the elements are thewavelets. Starting from the mother wavelet, one derivesthe thinner daughter wavelets by appropriate amount ofscaling. Scaling is an operation which makes a givenobject thicker or thinner, by the choice of a parameter.When combined with translation, by amounts commen-surate with the size of the wavelets at various scales, oneobtains a complete orthogonal basis set, where each ele-ment has a finite size. Below we illustrate these points,through the Haar wavelets, the grand old basis.Haar WaveletIt is the simplest one to visualize and has found manyuseful applications. A recent use of Haar wavelet was inthe extraction of the characteristics of individual thumbimpressions by Federal Bureau of Investigation (FBI) ofUSA. In this case, the scaling function is given by thebox function, while the mother wavelet is an oscillatingfunction of the same maximum height and width. Al-though we have chosen the height and width as unity

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fGENERAL I ARTICLE

here, in principle it is arbitrary, as long as (11-13) aresatisfied.It is clear that for the above choice, J 4J(t)dt= 1 andJ1f;(t)dt = O. Also J 14J2(t)ldt = J 11f;2(t)ldt = 1 andJ4J*(t)1f;(t)dt= O. It should be noted that, in the presentcase, all the basis functions are real. Let us now under-stand the action of the two operations, translation andscalingTranslation by one unit of the scaling function 4J(t)pro-duces 4J(t -1) VIhich starts at t = 1. The reason to choosethe translation by unity, same as the width of 4J(t), isto maintain orthogonality between 4J(t)and 4J(t - 1). Inthe same way, one can get 4J(t- k) and 1f;(t- k), wherek can take integer values, in the range -00 to 00. It iseasy to convince oneself that all these functions are or-thogonal to each other. In the literature, it is customaryto denote 4J(t - k) by 4Jk(t) and 1f;(t - k) by 1f;k(t).Although we have now an infinite number of these smallbuilding blocks, these are still not complete. For exam-ple, ,tP(2t) given by an oscillatory function, which takesvalues +1 for 0 :::; t < 1/4 and' -1 for 1/4 :::; t < 1/2and its translations, in units of half, are orthogonal toboth 4Jk and 'tPk for any value of k. Note that 1f;(2t)is half as thin as 1f;(t) and moves in steps half as wideas that of the mother wavelet. The translation step iscommensurate with the width. This thinner identicallooking wavelet is called a daughter wavelet. Figure 3depicts the father, mother, daughter wavelets and theircorresponding translations. One can repeat tpis processto obtain progressively thinner and thinner daughterwavelets, which are orthogonal to the scaling functionand all the wavelets prior to them. Note that, in eachcase, the width of the wavelets gets reduced by a fac-tor of half. All these wavelets can be translated, intheir commensurate steps, to cover the entire time axis.The scaling function, the mother wavelet and all these

~ESONANCE I March 2004 57


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GENERAL I ARTICLE'Figure 3. (a) Haar scalingfunction orthe father wave-let; (t) (0 ~ t


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GENERAL I A~ICLEconstant. Note that apart from the translation index k,we have introduced a scaling index j, to properly cap-ture the diadic process of thinning, narrated above inwords. From the defining equation of 'l/Jj,k,t is clear thatj = 0 corresponds to the mother wavelet,translated byk units. For values of j ~ 1, one obtains the subsequentdaughter wavelets. Increasing values of j produce thin-ner and thinner daughters! We further note that thenormalized, thinner daughter wavelet (one correspond-ing to j = 1) is taller by the amount V2, as compared tothe mother wavelet. The subsequent generations followthe same rule. Hence, as j ~ 00, the wavelets becomeextremely thin with large heights. As will become clearbelow, the scaling function captures the average of a partof the signal, in an interval determined by its width; thelocations of the scaling function window are given by thevalues of k. The wavelets capture the differences. Themother wavelet obtains the differences of the signal ata scale similar to the scaling function and the thinnerdaughter wavelets probe the differences at progressivelyfiner scales. This is the reason why wavelet transformis called multi-resolution analysis. The fact that thewavelet transform uses a finite size window enables it tocapture the local nature of the function or a signal in amuch more efficient way than the Fourier transform.Using the complete orthonormal set of basis functions,the wavelet transform of a function f (t) can be writtenin the form

00 00 00

f(t) = L Ck(Pk(t) + L L dj,k'l/Jj,k(t)k=-oo k=-ooj=O (15)Note that, as compared to Fourier transform, discretewavelet transform involves two indices. Here the coef-ficients Ck 'sand dj,k 's represent the discrete wavelettransform (DWT) of the function f(t). Ck 'scapture theaverage parts, while dj,k 's represent the variations, atdifferent scales, present in the function or signal. These

~ 59ESONANCEI March 2004

The fact that thewavelet transformuses a finite sizewindow enables itto capture the localnature of thefunction or a signalin a much moreefficient way thanthe Fouriertransform.


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GENERAL I ARTICLE

features are common to all the wavelets. Explicitly, Ck '8and dj,k '8 are given, respectively, by Ck = J f(t)k(t)dtand dj,k = J f(t)'ljJj;k(t}dt.Let us understand the meaning of these coefficients inthe simple Haar basis. Consider Co,given by Co= J f (t)(t)dt;ince (t)appens to be the Haar father wavelet,a box function of unit width and height, located at ori-gin, Co is the average of the values of 'the function inthe time interval 0 to 1. This represents the averagepart of the signal ill that interval. Similarly, do,o is thedifference in the averages of the function in the two in-tervals, 0 and 0.5 and 0.5 and 1. Other dj,Q '8 scan thefunction for still finer variations starting from the origin.The other locations in the time domain are reached bychanging values of k.Because of the presence of two indices j and k, thereare several ways to display the wavelet coefficients. Webriefly describe here the most popular representationamongst them. Keeping practical applications in mind,instead of a continuous function, we will consider a dis-crete finite data set f(t), where t takes integral values ina suitable range, for example, 0,1,2,3,... First of all,one needs to have N = 2n data points, n being a positiveinteger. In case of inadequate data points, there are sev-eral methods to augment the data set. to the nearest nvalue, the simplest one being padding with zeros or someother constant numbers. With such a data set, one cango up to maximum of n levels of decomposition. Waveletcoefficients at various levels capture the variations in thefunction at corresponding scales. As is suggestive, thescaling function coefficients (Ck '8) are called average orlow pa88 coefficients and the wavelet coefficients (dj,k '8)are called detail or high pa8S coefficients. In any type ofwavelet transform, the total number of Ck '8 and dj,k '8equals the number of data points, as it should be.Let us now understand the freedom of deciding the num-

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GENERAL IARTICLESuggested Readinger of levels of decomposition. As has been mentioned

earlier, the choice of the width of the scaling functionis a free parameter, depending upon which the motherand daughter wavelets' sizes are determined. For ex-ample, one can take scaling function of width equallingthe length of the data set, resulting in only one averagecoefficient Co. In case of Haar, this corresponds to thesum of all the data points. The mother wavelet finds thedifference between the averages of the first and secondhalves of the function. The daughter wavelets explorethe differences at finer scales, until the maximum reso-lution is reached due to the discrete nature of the data.In the present case, the thinnest daughter wavelet willfind the nearest neighbour differences. In the above, wehave performed a full n level decomposition of the data.One could have started with a scaling function having asize half that of the data set, in which case the two av-erage' coefficients Coand Cl, would have represented theaverages of the first and second half respectively. A mo-ment's reflection will reveal that the corresponding twomother wavelet coefficients are equal to the first daugh-ter wavelet coefficients in the earlier case. This examplecorresponds to a (n - l)th level of decomposition of thedata set.Let us see for ourselves the display of a Haar wavelettransform result of a multibox function, after a one leveldecomposition. From the above two examples, it is clearthat the scaling function now extends only upto two datapoints. The wavelet and the scaling coefficients are halfthe size of data-points. It is immediately apparent thataverage coefficients are similar to the original data, afeature common to all wavelets. The locations and thedegree of variations of the function are present in thedetail coefficients, of which only four are non-zero! Thedifferences between the discrete wavelet transform andFourier transform are clearly visible in this example.

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[1] B B Hubbard, The WorldAccording to Wavelets, 2ndEdition, Universities Press(India),Hyderabad, 2003.[2] C S Burrus, R A Gopinathand H Guo, IntTOductWn toWavelets and WaveletTransforms-A Primer,Prentice-Hall, New Jersey,USA, 1998.

[3] G BFolland, From calculusto wavelets: A New Math-ematical Technique,Reso-nance, Vol.2, No.4, pp.25-37,1997.

[4] S Thangavelu, Fourier Se-ries, Resonance, Vol.l,No.10, pp.44-55, 1996.

[5] A Sitaram and S Thanga-velu, From Fourier seriesto Fouriertransforms,Reso-nance, Vo1.3,No.10,pp.3-5,1998.

[6] http://www.wavelets.org.Wavelet Digest, a portalwhich keeps the informa-tion oflatest happenings inthis field. Could be sub-scribed through mail.

[7] http://engineering. rowan.edu/ -polikar/W AVE-LETS/WTtutorial.html;An introductory tutorial onwavelets by R Polikar.


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GENERAL I ARTICLE

1.2

0'~ 0.6

0,,1

0.2

ilL0 100 :!IX) 300 .tOO !iOO IlOO 7GO 800 1100 1IlOO1iH(/I

Figure 4.Multi box function(left), with its one level dis-crete wav~/et transform(right).

The enterprisingreader can perform

a multi-leveldecomposition andreconstruction todiscover that the

problems ofovershoots and

undershootsplaguing the Fouriertransform areabsent in discretewavelet transform.

l4oeIUIIII- l_-Q.!,Q2t

~'

~,.,.,0 111 31 !II) 0 l1li ! !II)

The enterprising reader can perform a multi-level de-composition and reconstruction, after removal of coef-ficients having small values, to discover that the prob-lems of overshoots and undershoots plaguing the Fouriertransform are absent in discrete wavelet transform.To see the mathematical microscope nature of wavelettransform, let us consider the example of Doppler func-tion (Figure 5), containing N = 211= 2048 data points,where variations at different scales and locations arepresent. Although eleven levels of decomposition arepossible, we have performed a five level decomposition,with the Haar wavelets. As is familiar by now, the lowpass coefficients at the fifth level, containing ~ = 64data points, capture the gross features of the Dopplerfunction. The level one detail coefficients zoom on tothe finest of the variations present in the initial partof the data; the mother wavelet coefficients (level fivedetail coefficients), keep track of the broader variationspresent in the latter part of the data. The other daugh-ter wavelets localize the variations present at scales in

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I

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, " ,

2

1 I.1

2

J


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GENERAL I ARTICLE

...

...L. 601> - - ....

LM.I1doIBICI:.a:iInII LMfdolllllXllllcMnlll

~EJ.:EJ45- -t-t - ~0 !lOG 1000 15011 0 D 400 fiOIILM 3dItaiICI:.a:iInII 1.Ni14d1b1C1111111:18n1a:EJ t:EJI D .~ ~~ -to0 ~ ~ ~ 0 ~ ~ ~B~~ ~ ro ~ 0 ~ ~ ~ ~....between the mother and the thinnest daughter wavelet.The neighbourhood to be observed at various levels ofresolution is fixed by the size of the scaling function.This is the functioning of the mathematical microscope.The fact that the average coefficients resemble the orig-inal function (~ith lesser data points), makes wavelettransform an ideal tool for a number of applications,a very important one being image compression! Imagesare two dimensional data sets, analysis of which needstwo dimensional wavelet transform. For motivating thereaders to a subsequent article, dealing with applicationsof wavelet transform to image compression', we demon-strate below a one level decomposition of an academicmodel, popular with the workers of wavelet community(Figure 6). For decomposition, we have chosen a waveletfrom the Daubechies' family, the acknowledged queen inthe domain of wavelet transform, who in the last decadeand a half contributed significantly to the understandingand construction of these microscopes.

RESONANCEMarch 2004 ~ 63

Figure 5. Doppler function(left), with Its five level dis-crete wavelet transform(right).


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GENERAL I ARTICLE

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