Wavelets. Toh Kim Chuan National University of Singapore

Transcription

1 Wvelets Toh Kim Chun Ntionl University of Singpore A wvelet bsis is sequence of functions tht is generted from single function.,p, clled mother wvelet, by tking combintions of trnsltes nd diltes of t/j. More specificlly, it hs the form where, b > 0 re fixed constnts. As n exmple, for = 2, b = 1 nd t/j given by t/j(x) = { ~ ~-X x<o O~x~1 1~x~2 X> 2, the sequence { ~ t/j(2m - n)}m,ne:ll is wvelet bsis. Like Fourier nlysis, wvelet bsis llows decomposition of functions into coefficients. We will discuss lter the dvntges of wvelet decomposition over the conventionl Fourier decomposition. Wvelets were introduced in Frnce in the erly 1980s by Jen Morlet, geophysicist nd Alexnder Grossmn, mthemticl physicist, to nlyse seismic signl. The mthemticl theory of wvelets took off in 1985 when Yves Meyer, lso in Frnce, constructed the first orthogonl 13

2 system of smooth wvelets such tht their fourier trnsform hve compct support. In 1986, Meyer nd Stephne Mllt developed the theory of multiresolution nlysis tht provides nturl frmewprk for the theory of wvelet pproximtions nd construction of orthonorml wvelet bsis. I. Dubechies, in 1987, constructed orthogonl systems of compctly supported wvelets (the size of the support grows linerly with the degree of smoothness). Since then, this re hs flourished. The potentil pplictions of the wvelet theory in mthemtics, engineering nd physics explins why it hs ttrcted so much ttention. The theory hs shown gret promises in the res of pure nd pplied mthemtics like pproximtion theory, hrmonic nlysis, opertor theory, numericl prtil differentil equtions, etc. Much of the interest in the engineering side is in the pplictions to signl processing (rnging from imge processing, coustic signl, seismic signl to synthetic music) where wvelets seem to hold gret promise for detection of edges nd singulrities, providing efficient decomposition nd reconstruction lgorithm for signls, nd dt compression. Not to forget, physicists re lredy using wvelets in quntum mechnics nd quntum field theory. Why wvelets come into the scene t ll, nturl question one would sk. It is well known tht the representtion of signl f(t) (coustic, electricl, etc) by mens of its spectrum (or Fourier trnsform) is essentil to solve mny problems in engineering nd mthemtics. In fct, the spectrl behviour of the signl (i.e. j( w)) in the frequency domin is the ctul dt tht one hs in prctice. However, F. T. techniques hs very serious deficiency in tht the time evolution of the frequencies is not reflected in this representtion s it requires informtion of the signl in the entire time-domin!"'( ) 1 [oooo f(t)e- itw dt. w = V2-i One cn see tht, if f(t) is perturbed by n impulse t timet= t 0, f(w) would chnge correspondingly but it does not tell us when is f(t) being perturbed. It is importnt to know this if one is to edit out this unwnted perturbtion in f(t) such s n ttck of musicl note. Noticing this deficiency, D Gbor, in his 1946 pper, introduced time-frequency locliztion method (clled short-time Fourier trnsform, STFT) by introducing window function g to "window" the Fourier inte- 14

3 grl Gf(w, t) =I: f(t')e-iwt' g(t'- t)dt' =I: i(w')e-i(w-w')tg(w'- w)dw'. From these two integrls, we see tht G(w, t) depends essentilly on f(t') fort' E [t- u,t + uj nd j(w') for w' E [w- uil,w + uil] in the time nd frequency domin respectively. We hve chosen g to be rel-vlued fu:r;1ction such tht I: I: lg( t) 1 2 dt = 1, tlg(t) 1 2 dt = o nd ug, uil re respectively the stndrd devition of g nd g, 2_ (Til We ssumed tht the dependent of Gf(w,t) on f(t') nd j(w') is significnt only for t' within stndrd devition of g from t nd w' within stndrd devition of g from w respectively. The Fourier trnsform off evluted t w(i.e.j(w)), mesures the mplitude of the sinuosodil wve component of frequencey w. Likewise, Gf(w,t) mesures loclly, round timet, the mplitude of the sinuosoidl wve component of frequency w, depending essentilly on the time-frequency window [t - u, t + U] x [w - uil, w + uil ]. Of course, the size of the window is limited by uncertinty principle which sys tht frequency ~~, 15 time

4 Suppose f(t) is perturbed by n impulse t t = t 0, then this would be reflected in Gf(w,t) fort E [to - uu,to + uu] Thus, the informtion provided by this decomposition is therefore unloclized within intervls of size uu. Similr conclusion holds in the frequency domin. If signl hs discontinuity such s n edge, it is difficult to locte it with precision better thn Uu. In generl, for edge detection, the tim~window must be very nrrow t high-frequency bnd (for the loction of n edge) for ccurcy, nd very wide t low-frequency bnd for efficiency. Becuse the time nd frequency resolution of STFT is constnt (i.e. the size of the time nd frequency windows re independent of t nd w), it is impossible to define n optiml resolution for nlysing signl tht hs importnt fetures of very different sizes. This is prticulrly the cse with imges, for exmple, in the imge of house, the pttern we wnt to nlyse might rnge from the overll structure of the house (low frequency bnd) to the detils on one of the curtins (high frequency bnd) To overcome the inflexibility of fixed time-frequency resolution of STFT, A Grossmn nd J. Mrlet introduced integrl wvelet trnsform (IWT) in W /(, b) = J /_: f(t)t/j((t- b))dt 1 foo.... "(w) = - f(w)e bw t/j - dw, V -oo where t/j is chosen fixed rel-vlued function such tht f.~oo lt/j(t) 1 2 dt = 1, f~oo tlt/j(t)l 2 dt = 0 nd f~oo t/j(t)dt = 0. The first integrl shows tht Wf(,b) depends on f(t) essentilly fort E [b- 7, b + 7), where u! = f~oo t 2 lt/j(t)l 2 00 dt. Let w 0 = f 0 wl~(w)l 2 dw. In prctice, 1/(w)l = 0 for w < 0, the second integrl shows tht W f(, b) depends on /(w) essentilly for we [w 0 - u.p, w 0 + u.p], where u~ = fooo (w- w0 ) 2 ~(w)l 2 dw. The time-frequency locliztion is thus given by [ U.p U.p] b--, b +- X [w 0 - u.p,w 0 + u.p] The significnce of IWT is tht, when the scle is lrge, the resolution is corse in the frequency domin nd fine in the time domin. As the scle decreses, the resolution increses in the frequency domin nd decreses 16

5 in the time domin. This vrition of resolution enbles the IWT to zoom into the detils of function in wy STFT cnnot, (identify with constnt multiple of the frequency) giving shrper time resolution t higher frequencies nd efficiency t low frequencies m rn EJ EJ ---~-----~ ~-----~ Time-frequency locliztion windows of IWT As wvelet decomposition seprtes nd loclize the spectrl informtion in different frequency bnds, hence filtering, detection, dt reduction, enhncement cn be esily implemented before pplying the wvelet reconstruction lgorithm. And it is this cpbility- of wvelets tht hs mde it str in the engineering, physicl nd mthemticl communities. References [1] Chrles K. Chui, An overview of wvelets, pproximtion theory nd functionl nlysis, Acdemic Press, (1990), pp [2] Ingrid Dubechies, Orthonorml bses of compctly supported wvelets, CPAM, 41 (1988), pp [3] Christopher E. Heil nd Dvid F. Wlnut, Continuous nd discrete wvelet trnsforms, SIAM Review, 1989, pp [4] Stephne G. Mllt, Multifrequency chnnel decompositions of imges nd wvelet models, IEEE Trns. on Acoustics, Speech nd Signl Processing, Vol. 37(1989), pp