ol 46 No 6 SCIENCE IN CHINA (Series F) December 3 A ew iterative algorithm for recostructig a sigal from its dyadic wavelet trasform modulus maxima ZHANG Zhuosheg ( u ), LIU Guizhog ( q) & LIU Feg ( ) Departmet of Iformatio Sciece, Xi a Jiaotog Uiversity, Xi a 749, Chia; Departmet of Iformatio ad Commuicatio Egieerig, Xi a Jiaotog Uiversity, Xi a 749, Chia Correspodece should be addressed to Zhag Zhuosheg (email: zszhag@mailxtueduc) Received November 8, Abstract A ew algorithm for recostructig a sigal from its wavelet trasform modulus maxima is preseted based o a iterative method for solutios to mootoe operator equatios i Hilbert spaces The algorithm s covergece is proved Numerical simulatios for differet types of sigals are give The results idicate that compared with Mallat s alterate proectio method, the proposed algorithm is simpler, faster ad more effective Keywords: dyadic wavelet trasform, modulus maxima, sigal recostructio, mootoe operator DOI: 36/yf8 Sharp variatio poits or sigularity poits of a sigal carry importat iformatio of the sigal Represetig ad recostructig a sigal by its sigularity poits have attracted wide attetio Loga did iitial research i this aspect He proved that the zero crossigs of a arrowbad sigal ca completely characterize the sigal [] But recostructig the sigal from its zero crossigs is usteady, ad it is difficult to apply i practice So may researchers have doe deep studies i this subect [] Sigularity poits of a sigal s wavelet trasform are closely related with the locatios of its sharp variatio poits Represetig ad recostructig a sigal by its wavelet trasform sigularity poits are a importat applicatio field of wavelet trasforms [3 6] Mallat, i ref [4], represeted a sigal from its wavelet trasform modulus maxima ad proposed a alterate proectio algorithm for recostructig the sigal A large umber of umerical experimets show that the algorithm ca be used to obtai a good approximatio of the origial sigal However, i geeral, Meyer proved that complete recostructio is impossible [7] He foud families of sigals whose dyadic wavelet trasforms have the same modulus maxima However, the sigals with the same wavelet trasform modulus maxima differ from each other oly slightly, which explais the success of umerical recostructio If the sigal, especially, is bad-limited ad if the wavelet has a compact support i the frequecy regio, the Kicey proved that wavelet trasform modulus maxima defie a complete ad stable sigal recostructio [8] Sigal represetatio based o the wavelet trasform modulus maxima have widely bee applied to may fields of sigal processig, such as the deoisig of sigals, the detectio of edges
No 6 RECONSTRUCTING A SIGNAL FOR ITS WAELET TRANSFORM MODULUS MAXIMA 4 ad patter recogitio [9 ] For may practical applicatios, the alterate proectio algorithm has the drawbacks of high computatioal complexity ad slow covergece speed I this paper, we study the problem of recostructig a sigal from its wavelet trasform modulus maxima A simple, fast ad stable recostructio algorithm is give based o the iterative method for solutios of mootoe operator equatios i Hilbert spaces Geeral priciple Let defied by ψ () t L ( R) be a dyadic wavelet The the dyadic wavelet trasform for a sigal f is x t W f() t = f( x) dx, Z For scale, if we deote all the modulus maxima positios of ψ R () {(, ( t W ) ), f t I Z} W f() t by { }, t I the () is called the dyadic wavelet trasform modulus maxima represetatio of f We set The () may be rewritte as t t ψ ψ, () t =, I, Z {( t f, ) I Z},, ψ, (4) Hece recostructig a sigal f from its wavelet trasform modulus maxima represetatio requires fidig a approximatio f of f, satisfyig the followig two coditios: (i) f, ψ, = f, ψ,, I, Z ; (ii) for ay scale, the modulus maxima of W are exactly located at { } f() t t I The mai difficulty comes from the o-liearity ad the o-covexity of the costrait coditio (ii) To recostruct a approximatio sigal with a simple ad fast algorithm, this costrait is replaced by a miimizatio of the sigal orm Thus the problem is trasformed ito that of fidig a sigal f such that (3) where f = arg mi g, (5) g K { ψ, ψ, } K = g L ( R) g, = f,, I, Z
4 SCIENCE IN CHINA (Series F) ol 46 The miimizatio geerally ca deduce that We set W takes its modulus maxima at { } f() t { ψ } = spa, () t I, Z t I The followig lemma shows that the recostructio sigal f is the orthogoal proectio of f o the subspace Lemma Let f K The f = arg mi g if ad oly if f is the orthogoal proectio P f of f oto the subspace g K Proof For all g K, we have g, ψ, = f, ψ,, Z, I It follows that g P f, ψ =, Z, I This implies that g P f, where is the orthogoal, complemet of subspace Hece we get if, where f = P f we obtai f = { g g K} [4,9] g = P f + g P f ƒ P f Sice P f K, The proof of the sufficiet coditio is ow complete Next, we give the proof of the ecessary coditio First, we prove that f Otherwise, we have f = f + f, where f, f ad f This follows that f, ψ = f, ψ = f, ψ with Z ad I, which implies that f K,,, We arrive at a cotradictio sice f > f ad { } f = if g g K Thus we have f Secod, we prove that f = P f From f, P f K, we deduce that f P f, ψ, =, Z, I This implies f = P f It follows from Lemma that fidig f is equivalet to solvig P f from the wavelet trasform modulus maxima represetatio of f Settig Tg = g, ψ, ψ,, g, (6) we have the followig lemma Lemma If T : Z, I Proof For all g, g, g g, we have Tg Tg, g g is defied by (6), the T is a strictly mootoe operator
No 6 RECONSTRUCTING A SIGNAL FOR ITS WAELET TRANSFORM MODULUS MAXIMA 43 = g, ψ ψ g, ψ ψ, g g,,,, Z, I Z, I g g, ψ ψ, g g = =,, Z, I Z, I g, g, ψ Sice g g ad g, g, there exists Z ad I such that g g, ψ, This deduces that Tg Tg, g g >, that is, T is a strictly mootoe operator Theorem If T : is defied by (6), the f = T f, ψ, ψ, Z, I Proof By Lemma, T is a strictly mootoe operator The T : Thus P f = T P f, ψ, ψ, Z, I Sice f, ψ, = P f, ψ, with Z ad I, we have P f = T f, ψ, ψ, Z, I (7) is oe-to-oe This together with Lemma implies that (7) holds By Theorem, for ay f, the problem of solvig f is equivalet to fidig the solutio to the followig mootoe operator equatio Tg = f, ψ, ψ,, g (8) Z, I I the ext sectio, we will give a iterative algorithm for solutios to mootoe operator equatios Iterative method for solutios of mootoe operator equatios I this sectio, we propose a regularizig iteratio scheme for solutios of mootoe operator equatio i Hilbert spaces, ad prove its covergece For the detailed cotet see ref [] Let H be a Hilbert space, ad T : D( T) H H T is said to be mootoe, if T is said to be strictly mootoe, if Tx Ty, x y ƒ, xy, DT ( ) Tx Ty, x y >, x, y D( T ), x y T is said to satisfy the liear growth coditio, if there exists a costat C > such that
44 SCIENCE IN CHINA (Series F) ol 46 Tx C( + x ), x D( T) T is said to be weak * cotiuous o lie segmets, if, for ay x D( T) ad h H with t> ad x+ th D( T), T( x+ th) coverges weak * to Tx as t For the mootoe operator equatio Tx =, we cosider the regularizig iteratio scheme x+ x λ( θ( x b) Tx), = + =,,, (9) x D( T), where b D( T) We have the followig covergece theorem Theorem Let H be a Hilbert space ad T : D( T) H be a mootoe operator ad T is weak * cotiuous o lie segmets Let D(T) be a closed covex set i H, ad RI ( T) DT ( ) Assume that x D( T) satisfy the followig coditios: (a) ad two real sequeces { λ } ad { θ } λθ = with { } { } ( ] = is mootoously decreasig ad covergestoas ; (b) ( ) + λ, θ,, { θ } θ θ λ θ ad λ θ as If the sequece {x } is defied by (9), the the followig coclusios hold: (i) { θ x } coverges to the miimum orm elemet i RT ( ); (ii) if T φ,the{x } coverges to the elemet of the closest to b i (iii) if T = φ,the lim x = + T ; Proof Let x deote the uique solutio for the equatio θ (x b) +Tx =SiceT satisfies the liear growth coditio, there exists a costat c > such that This implies x θ( x b) + Tx c ( + x x + x b ) + x ( λθ ) x x + λ θ( x b) + Tx ( λθ + cλ ) x x + cλ + cλ x b O the other had, because T is a mootoe operator, we have x x x x + θ ( Tx Tx ) = ( θ θ ) θ x b Hece + + + + + x + x+ + + + + + + x x x x x x x x θ θ+ θ θ+ λθ + cλ + ( λθ ) + c λ x x θ θ
No 6 RECONSTRUCTING A SIGNAL FOR ITS WAELET TRANSFORM MODULUS MAXIMA 45 Sice { x b } + c + c x b + + x b θ λ θ + θ λ θ θ θ + + + θ θ θ θ θ + c + + λ θ is mootoe, we deduce where x + x+ x+ b + x x ( α) + β, + x b θ θ θ θ α λ θ λ ( λ θ ) λ, + + = c c θ θ It is easily proved that This implies θ θ+ θ θ+ θ θ+ β = c + λ + + θ θ θ = α =, lim β α = Thus lim x x ( + x b ) = lim x x ( + x b ) = () We are ow ready to fiish the proof of the theorem (i) Observig x x θx θx = θ( + x b ), + x b we get lim θ x θ x = by () Therefore we coclude from Lemma i ref [] that { θ x } coverges to the miimum orm elemet i RT ( ) (ii) By () ad Lemma i ref [], it ca be easily proved (iii) Suppose that, o the cotrary, lim x = is ot true For simplicity, we ca assume that {x } is bouded It follows from Lemma i ref [] that lim x x ( + x b ) =, which is a cotradictio to () By Theorem, we have: Theorem 3 Uder the coditios of Theorem, the followig statemets are equivalet (i) T φ;
46 SCIENCE IN CHINA (Series F) ol 46 (ii) {x } is bouded; (iii) {x } coverges to the elemet of the closest to b i T We give two examples about the sequeces {λ }ad{θ }, which satisfy coditios (a) ad (b) i Theorem Example α β λ = ( + m), θ = ( + m), =,,,, α >/, β >,α + β <, m, m wz + Example λ = ( + m), θ = (loglog( + m)), =,,,, m, m wz + 3 Iterative algorithm for recostructig a sigal from its wavelet trasform modulus maxima I terms of the discussio i sectio, the problem of recostructig the sigal f (t) from its wavelet trasform modulus maxima is trasformed ito that of fidig the solutio to the mootoe operator equatio Ts = f, ψ, ψ, Z, I Therefore we ca obtai the recostructed sigal f by the iterative scheme (9) I the followig, we preset a fast ad stable recostructio algorithm for a discrete sigal of size N Cosider the sigal f (t), t =,,, N, N = J We assume for simplicity that N t= f() t = Otherwise, we set N f = f N f( t) Thus the algorithm is described as the followig: () Discrete dyadic wavelet decompositio Choosig a suitable wavelet ψ (t), amely, a couple of cougate filter baks {h()} ad {g()}, we calculate the discrete dyadic wavelet decompositio of f by [9] S f() t = S f h (), t =,, J, () W f() t = S f g (), t =,, J, () where S f = f, h ( ) = h ( ), g ( ) = g ( ), ad for ay filter x() wedeotebyx () the filter obtaied by isertig zeros betwee each sample of x() Therefore we get all the wavelet coefficiets for the scales with J Sice N f() t =, t= W f(), t W f(), t, W f() t S f() t is costat ad equal to zero 9] J () Recordig the positios of the wavelet trasform modulus maxima ad the correspodig J [ t=
No 6 RECONSTRUCTING A SIGNAL FOR ITS WAELET TRANSFORM MODULUS MAXIMA 47 maxima For the wavelet coefficiets W f(), t if the W f() t > W f( t ) ad W f() t > W f( t + ), W f() t takes its modulus maxima at the t If we deote all the positios of those modulus maxima by the the correspodig maxima are respectively t, t,, t N, W f( t ), W f( t ),, W f( t N ) (3) Recostructig the sigal f We set =spa{ψ, (t) J, N } ad defie T : by J N Ts = s, ψ ψ, s = =,, The the iterative scheme, which recostructs the sigal f from the wavelet trasform modulus maxima, is f+ = f λ( θ f + ( Tf f )), =,,,, (3) where f ad J N f Tf f, ψ ψ ( t) = = = =,, We ca drive a fast filter baks algorithm for computig Tf m, m =,,,, with the same procedure as i the proof of the fast algorithm of discrete dyadic wavelet trasform [9], as follows Set ad W f ( ), if m t t = t W f (),,, m t = = J, if t t a () t = a h () t + W f g (), t J, + + m where aj ( t ) = The we have Tfm () t = a () t 4 Numerical experimets I this sectio, we preset some umerical results The performace of the proposed algorithm is compared with Mallat s method by computer simulatios We deote by f the reco-
48 SCIENCE IN CHINA (Series F) ol 46 structed sigal, ad the sigal-to-oise ratio (SNR) of the recostructed sigal is defied by N f() t t= SNR = log N f () t f() t t= wavelet is used I the itera- I all the followig umerical experimets, Daubechies db8 tive scheme (3), we choose f =,ad 5 46 θ (4) λ = ( m + ), = ( m + ), =,,, First, we take a sigal with sharp variatio, which is similar to that costructed by Mallat i ref [4], ad Heavisie sigal as test sigals Figs ad show the recostructed sigals ad the recostructio SNR for the two sigals usig the proposed algorithm ad Mallat s method, respectively From figs (d) ad (d) we see that, after 4 iteratios, the recostructio SNR by the proposed algorithm are larger tha those by Mallat s algorithm Figs (b) ad (b) are the recostructed sigals usig the proposed algorithm for 8 iteratios, ad figs (c) ad (c) give the recostructed sigals usig Mallat s method for iteratios As ca be see i figs ad, the Fig Recostructed sigals ad their SNR for the sharp variatio sigal usig the proposed algorithm ad Mallat s method (a) Origial sigal, (b) sigal recostructed with 8 iteratios usig the proposed algorithm, (c) sigal recostructed with iteratios usig Mallat s method, (d) SNR of the recostructed sigals usig the proposed algorithm (real lie) ad Mallat s method (dot lie)
No 6 RECONSTRUCTING A SIGNAL FOR ITS WAELET TRANSFORM MODULUS MAXIMA 49 Fig Recostructed sigals ad their SNR for Heavisie sigal usig the proposed algorithm ad Mallat s method (a) Origial sigal, (b) sigal recostructed with 8 iteratios usig the proposed algorithm, (c) sigal recostructed with iteratios usig Mallat s method, (d) SNR of the recostructed sigals usig the proposed algorithm (real lie) ad Mallat s method (dot lie) recostructed sigals by the proposed algorithm resemble the origial sigals much more tha the oes by Mallat s method Next, to further examie the computatioal efficiecy ad the recostructio quality of the proposed algorithm, we use Bumps sigal, Blocks sigal ad the above two sigals as test sigals Table shows the SNR of the recostructed sigals ad the CPU times spet for recostructig the sigals usig the proposed algorithm with 8 iteratios ad Mallat s method with iteratios respectively, which was performed o the Petium -667 computer with Matlab6 From table, we see that the recostructio SNR obtaied with the proposed algorithm are superior to the oes with Mallat s method, ad the CPU times are about oe-third of those with Mallat s method 5 Coclusio I this paper, we study the problem of recostructig a sigal f from its wavelet trasform modulus maxima First we trasform the problem ito fidig the orthogoal proectio of f oto a subspace, which is spaed by all the wavelets correspodig to the wavelet trasform modulusmaxima positios The we prove that the orthogoal proectio is the solutio to a mootoe op-
43 SCIENCE IN CHINA (Series F) ol 46 Table SNR of the recostructed sigals ad CPU times spet for recostructig the sigals Algorithms usig the proposed algorithm ad Mallat s method The proposed algorithm Mallat s method Sigals SNR/dB Time/s SNR/dB Time/s Sharp variatio 345887 768 34474 5 Heavisie 4755 763 497 86 Bumps 98736 767 964 3 Blocks 9634 774 6988 57 erator equatio By proposig a iterative method for solutios of mootoe operator equatios, we give a ew algorithm for recostructig a sigal from its wavelet trasform modulus maxima Compared with Mallat s alterate proectio method, this algorithm is simpler, faster ad more effective Numerical experimets demostrate that, for differet types of sigals, a good quality of the recostructed sigals ca be obtaied for oly a few iteratios Ackowledgemets This work was supported by the Natioal Natural Sciece Foudatio of Chia (Grat No 677) ad the Natural Sciece Foudatio of Xi a Jiaotog Uiversity (Grat No 6) Refereces Loga, B, Iformatio i the zero-crossigs of bad pass sigals, J Bell Syst Tech, 977, 56: 5 53 Hummel, R, Moiot, R, Recostructio from zero crossig i scale space, IEEE Tras Acoust, Speech, Sigal Processig, 989, 37(6): 3 3 Mallat, S, Zero-crossigs of a wavelet trasform, IEEE Tras Iformatio Theory, 99, 37(4): 9 33 4 Mallat, S, Zhag, S, Characterizatio of sigal from multiscale edges, IEEE Tras Patter Aalysis ad Machie Itelligece, 99, 4(7): 7 73 5 Cvetkovic, Z, etterli, M, Discrete-time wavelet extreme represetatio: desig ad cosistet recostructio, IEEE Tras Sigal Processig, 995, 43(3): 68 693 6 Liu, G Z, Zhag, Z M, Iterative shapig recostructio algorithm based o the modules maxima of sigals dyadic wavelet trasform, Progress i Natural Sciece,, (8): 66 664 7 Meyer, Y, Wavelet: Algorithm ad Applicatios, Philadelphia: SIAM, 993 8 Kicey, C J, Leard, C J, Uique recostructio of bad-limited sigals by a Mallat-Zhag wavelet trasform algorithm, Fourier Aalysis ad Appl, 997, 3(): 63 8 9 Mallat, S, A Wavelet Tour of Sigal Processig, Bosto: Academic, 998 Yag, F S, Egieerig Aalysis of Wavelet Trasform ad Its Applicatios (i Chiese), Beiig: Academic Press, Zhag, Z S, Liu, G Z, Liu, F, Costructio of a ew adaptive wavelet etwork ad its learig algorithm, Sciece i Chia (Series F),, 44(): 93 3 Zhag, Z S, Liu, C Y, Iterative solutio for a class of equatios with mootoe operators (i Chiese), J Xi a Jiaotog Uiversity, 996, 3(8): 4 9