Adaptive Approach Based on Curve Fitting and Interpolation for Boundary Effects Reduction

Transcription

1 Adapive Approach Based on Curve Fiing and Inerpolaion for Boundary Effecs Reducion HANG SU, JINGSONG LI School of Informaion Engineering Wuhan Universiy of Technology 122 Loushi Road, Wuhan CHINA Absrac: - Boundary effecs are caused by incomplee daa in he boundary regions when he analysis window ges closer o he edge of a signal. Various exension schemes have been developed o handle he boundaries of finie lengh signals o reduce he boundary effecs. Zero padding, periodic exension and symmeric exension are some basic exension mehods. However, i is well nown ha all of hese soluions may have drawbacs. In his paper, we consider he problem of handling he boundary effecs due o improper exension mehods in he wavele ransform. An exension algorihm based on curve fiing wih properies ha mae i more suiable for boundary effecs reducion is presened here. This exension algorihm could preserve he ime-varying characerisics of he signals and be effecive o reduce disorions appearing a he boundary. Then, an inerpolaion approach is used in he boundary effecs region o furher alleviae he disorions. Procedures for realizaion of hese wo algorihms and relaive issues are presened. Several experimenal ess conduced on synheic signals exhibiing linear and nonlinear laws are shown ha he proposed algorihms are confirmed o be efficien o alleviae he boundary effecs in comparison o he exising exension mehods. Key-Words: - Finie-lengh Signals, Convoluion, Wavele Transform, Boundary Effecs, Fourier Series Exension, Inerpolaion 1 Inroducion Wavele ransform analysis has been presened as a ime-frequency analysis and processing mehod for over he pas wo decades [1], [2]. Bu i has sill received increased aenion in recen years [3], [6], [7]. Wavele ransform analysis has been widely used for he purpose of denoising, daa compression, feaure recogniion, sysem nonlineariies deecion and so on [4]-[7]. The wavele ransform is calculaed as shifing he wavele funcion in ime along he inpu signal and calculaing he convoluion of hem. In mos pracical applicaions, he signals of ineres have finie suppor. As he wavele ges closer o he edge of he signal, compuing he convoluion requires he non-exisen values beyond he boundary [8]- [1]. This creaes boundary effecs caused by incomplee daa in he boundary regions. Since he analysis wavele exends ino a region wih no available daa a boh boundaries of he signal. Thus, he resuls of wavele ransform in hese boundary effecs regions have quesionable accuracy. Acually, he paricular impacs of boundary effecs become increasingly significan for some sysems ha may possess longer period sequence and hus require higher frequency resoluions. To deal wih boundary effecs, he boundaries should be reaed differenly from he oher pars of he signal. If no properly made, disorion would appear a he boundaries [3]. Two alernaives o deal wih boundary effecs can be found. The firs one is o accep he loss of daa and runcae hose unfavorable resuls a boundaries afer convoluion beween signal and wavele. Bu simply neglec hese regions in analysis yields o a considerable loss of daa which is no allowed in many siuaions where he edges of he signal conain criical informaion. The oher one is arificial he exension a boundaries before processing signals. In fac, here is anoher approach ha employs he usual wavele filers for he inerior of he signal and consrucs differen boundary waveles a he ends of he signal. This mehod has been shows o be merged ino he class of signal exension [1]. Various exension schemes have been developed o deal wih he boundaries of finie lengh signals [11]-[14]. Zero padding, periodic exension and symmeric exension are basic exension mehods. I E-ISSN: Issue 2, Volume 9, April 213

2 is well nown ha each mehod has is disadvanages [3], [1]. Compuing he wavele ransform of an exension signal is equivalen o using he corresponding boundary waveles. The boundary waveles corresponding o zero padding and periodic exension have no vanishing momens a he boundaries. Therefore, he ransform values behave as if signal were disconinuous a he boundaries. They inroduce a singulariy in he signal. And boundary waveles of symmeric exension have one vanishing momen and avoid he disconinuous a he boundaries. So i inroduces a singulariy in he firs derivaion. However, if he reflecion is symmeric he waveles mus be symmeric o ensure no disorion in he ransform values. I is well nown ha Haar is he only symmeric wavele wih a compac suppor ha has been found so far. One goal of his paper is o see an exension scheme ha preserves he propery of vanishing momens. In addiion, hese basic exension schemes are usual exploied o he applicaion of daa coding which focuses on he procedures of analysis and synhesis using filer bans [15]-[2]. However, when i comes o paricular applicaions ha pu he emphasis on he abiliy o recognize coheren srucure wihin a signal, he above menioned mehods don have he abiliy o recover hose significan feaures. They only mae simple assumpions abou he signal s characerisics ouside he boundaries. Many signals of ineresed could no be easily included in he above hree caegories. So we need a new exension mode appropriae o he requiremens of he applicaion of non-saionary signals analysis. In his paper, a new exension mode based on curve fiing echnique will be inroduced for non-saionary signals analysis. This exension mode exends signal according o he ime-varying characerisics of he signals inside of he boundaries so ha disorions due o improper exensions could be reduced. I should be aware ha feaures appearing near he boundaries of ransform values will conain informaion from ouside he suppor of he signal which is synheic. In oher word, he wavele ransform resuling a he boundaries will be affeced by he adding daa no maer whichever exension mode is employed. Therefore, we will consider he problem from a perspecive way ha is differen from exension mehod o alleviae hese effecs. In he paper, we will employ an inerpolaion processing in he region of he boundary effecs o reduce he disorions. We will show ha improvemen can be obained by such processing. The paper is organized as follows. In he nex secion a brief review of he boundary effecs in he basic exension mehods is given. A general marix formulaion ha is common o all signal exension mehods is also included. In Secion 3, we give deph analysis of he significan imporance of smooh exension and presen he design mehod for adapive smooh exensions wih properies ha mae i more suiable han oher exension for nonsaionary signals analysis. In Secion 4, we develop a new algorihm based on inerpolaion echnique for furher boundary effecs reducion along wih some discuss on he implemen of his echnique. In Secion 5, we presen he mehod of esing and he resuls concerning he performance of performance of he proposed mehods applied on boh linear and nonlinear frequency modulaion signals. The performance of proposed adapive exension mehod based on curve fiing and inerpolaion is shown o be superior o all of he oher mehods. Secion 6 summarizes he resuls obained hroughou he paper. 2 Boundary Effecs in he Timefrequency Signal Analysis using Wavele Scalogram The need for a signal ime-frequency analysis comes from he incomplee of eiher ime domain or frequency domain analysis o fully describe he behavior of non-saionary signals. The imefrequency represenaion of a signal for imefrequency analysis provides informaion abou how he frequency conen varies wih ime, hus providing an ideal approach o examine, analyse and sudy non-saionary signals. Time-frequency represenaion is an image of a wo dimensional ime-frequency represenaion mapped from one signal. A number of mehods have been developed o obain he energy disribuion funcion wih respec o boh he ime and frequency. Wavele ransform is one of mos noably ools. Waveles have he grea advanage of being able o isolae he fine deails in a signal. Very small waveles can be used o idenify very fine deails in a signal, while very large waveles can idenify coarse deails. Wavele heory is capable of revealing aspecs of daa ha oher signal analysis echniques fail o be presen he aspecs lie rends, breadown poins, E-ISSN: Issue 2, Volume 9, April 213

3 and disconinuiies in higher derivaives and selfsimilariy. Bu as menioned in he previous secion, wavele ransform suffers from boundary effecs lie oher signal analysis echniques which involve convoluion operaion. The boundary effecs would lead o serious disorion a boh boundaries of signal which maes i hard o disrac he righ informaion paricularly on he sar and he end of signals. Therefore, his secion will firs explore he effec of basic exension mehods, which include zero padding, periodic exension and symmeric exension, on he wavele ransform in order o design a suied exension mehod ha is able o minimize boundary effecs. We sar wih a general formula of various exension modes. We denoe vecors by bold lower case leers. Subscrip (superscrip) l, c and r represen lef, cenral and righ respecively. Marices are denoed by bold upper case leers. We use subscrip, such as M N, o denoe he size of a marix. A finie signal wih lengh N is s (n), n,1, N. Then we can express his signal in anoher form as s= [ s, s, s ] (1) Τ Τ Τ Τ l c r where s l and s r are vecors consising of he firs and las M componens of he signal. s c is he cenral par. Denoe he exension vecor of s (n) as s = [ s, s, s ] (2) Τ Τ Τ Τ e el, er, where similarly, s el, and s er, are he lef and righ exension vecors of lengh M. We use subscrip o denoe he size of marix. Generalized expression for signal exension mehods is given by s = H s (3) e (2 M+ N) N where H is he exension marix. The basic exension mehods are all linear exension. Hence (3) can be wrien in form l H se = IN s (4) r H where I N is an N N ideniy marix; H l r and H are respecive lef and righ exension marices. For he zero padding exension, he exension marix is H M N = I N M N (5) where M N is an M N zero marix. Since for he periodic exension s el, = s r and ser, = s l he exension marices of he periodic exension are H I M ( N M) M = IN I M M ( N M) (6) Similar resul is available for he exension marix of he symmeric exension J M ( or J N H J M M ( N M) = JN J M ( N M) M (7) where ) is an exchange marix where he 1 elemens reside on he counerdiagonal and all oher elemens are zero. In order o illusrae he boundary effecs of various basic mehods, we consider a linear frequency modulaion signal s( ) wih consan ampliude and frequency varied wih ime from.1 o.4(normalized frequency). The sampling frequency used is f s = 1Hz s wih 3 daa poins. We perform differen exension mehods on he es daa and exrac he insananeous frequency from wavele ransform of exension daa. The esimaion error of he insananeous frequency obained from hree basic exension mehods is shown in Fig. 1. The symmeric exension performs beer han zero and periodic mehods. This is due o symmeric exension have one vanishing momen and zero padding and periodic exension have no vanishing momens. This paper is o see an exension scheme ha preserves he propery of vanishing momens. Define he momens of wavele funcion as m = ψ ( )d (8) E-ISSN: Issue 2, Volume 9, April 213

4 As a consequence of he Fourier ransform properies, we can obain Error(dB) m d Ψ( ω) = ( j) (9) dω ω= -8 Zero Periodic Symmeric -9 Time(s) Fig. 1. Comparison of boundary effecs for linear FM signal of hree basic mehods. The values on he verical axes are normalized o he adoped sample rae. where Ψ( ω) is he Fourier ransform of ψ (). If Ψ ( ω) has p order muliple zeros a ω =, ha is ( ) p Ψ ω = ω Ψ ( ω ), Ψ ( ω ) = (1) hen we can find ω m = ψ ( )d =, =,1, p 1 (11) If a wavele funcion ψ () saisfies(11), hen we say his wavele funcion has p vanishing momens. Assume signal s () is a polynomial of degree p 1, which is given by p 1 s () = α (12) = where α, α1,, αp 1 are consan coefficiens. Addiional, we assume ψ () has p vanishing momens. Equaion (11) indicaes ha s (), ψ () = (13) In oher words, he wavele ransform of s () is idenical o zero. If s () can be expanded ino a high-order polynomial of degree N wih N > p, hen he erms of he polynomial wih degree lower han p conribue nohing o he wavele ransform which only reflecs he erms wih degree higher han p (high frequency componen). Such a wavele has he advanages o capure he high frequency componen and breapoins of signals. Therefore, ψ () is required o have an as high as possible vanishing momens so ha Ψ ( ω) is smooh a ω = o possess a saisfied band-pass propery. 3 Boundary Effecs Reducion via Adapive Smooh Exension I has been shown ha every basic exension mehod has is own drawbacs. We should see a mehod represening he feaure of signal. Moreover, smooh exension is also criical o he reducion of boundary effecs. 3.1 Design of Adapive Exension Mehod In he following, we will invesigae a new exension mode which could characerize signal beer. On he one hand, he signal used in previous secion is comprised by many harmonic oscillaions, and on he oher hand, i is very common o use Fourier series o represen such harmonic oscillaions. Thus, Fourier series can be consider as a new mode o exend signal o preserve he harmonic oscillaions. The Fourier series model is given by m y( ) = a + a cos( ω) + b cos( ω) i i i i (14) i= 1 where a is a consan erm in he signal, boh a i, b i and ω i are parameers ha need o be esimaed by he fi, m is he number of harmonics in he daa. In summary, he following are he seps of he proposed adapive algorihm for signal exension: 1) Iniialize he number of harmonics, for example, se m = 3. 2) Main Ieraion: Incremen m by 1, and apply hese seps: Perform daa ransformaions o obain a linear or simple model. Find he above model parameers o minimize he summed square of residual defined as he difference beween he real dae value s and he E-ISSN: Issue 2, Volume 9, April 213

5 fied response value y, producing resul a i, b i and ω i. Updae he fied response value y using a i, b i and ω. If i 2 s y is smaller han some predeermined hreshold, sop. Oherwise, apply ieraion. 3) Exend he producing Fourier series o define he daa beyond he borders. 3.2 Properies of Smooh Exension From he perspecive of convoluion operaion, he wavele ransform of a signal could be inerpreed as he oupu of a sysem whose uni impulse response is he scaled wavele funcion ψ () 1 a = a ψ a (15) where a is scale. Le s consider a low-pass(smooh) funcion θ (). Se (1) (2) 2 2 ψ ( ) = d θ( )/d, ψ ( ) = d θ( )/d (16) We use ψ (1) () and ψ (2) () as he moher waveles. Then compuing he firs derivaive of a signal afer smoohing is equivalen o processing his signal using he firs derivaive of he smooh funcionψ (1) (). Similarly, compuing he second derivaive of a signal afer smoohing is equivalen o processing his signal using he second derivaive of he smooh funcion ψ (2) (). This resul can be generalized o he higher order. Mahemaically, a poin of a funcion wih zero firs derivaive corresponds o exreme value while zero second derivaive corresponds o inflecion poin. Hence, he wavele ransform is able o reflec he exreme and inflecion poins of a signal if he wavele is original from a smooh funcion. An improper exension maybe resuls in exra ransien componen referred o singular poins which is defined as poins wih derivaive on he righ and he derivaive on he lef exis wih differen signs, ha is, he poins a which is derivaive is disconinuous or no defined and finding he ampliudes of he jumps. In oher words, singular poin represens he exreme and inflecion poins presen in he signal. I is easy o obain ha he singular poins of signal are indicaed by he ampliudes of is wavele ransform, i.e., zero- crossing poins or maximum poins of he ransform. In he case of signal exension, an exreme poin due o exension would lead o zero poin or very small value in he wavele ransform a he corresponding locaion. More ordinary case is ha exension inroduces a sep a he boundary leading o very large wavele ransform ampliude. For example, he resul ha wavele ransform of signal sn ( ) using funcion ψ (1) () is very large indicaes he inflecion poin of sn ( ). An unsmooh exension a s() or sn ( ) leads o wavele ransform modulus maximum a he same poins which is he reason of disorion. Hence, we should selec an exension mode ha is as smooh as possible a he boundaries o avoid disorion. 4 Boundary Effecs Reducion via Inerpolaion Whichever exension mehod is employed o reduce he boundaries disorion phenomenon, he exension pars would definiely affec he analysis resuls which are deermined by boh original and exension signals. If he exension pars do no properly reflec he rend of he original signal, i will fail o produce saisfacory or perfec resuls. Neverheless, i is well nown ha he signals in he applicaion of ime-frequency analysis are usually random and i is difficul o esimae he pas and fuure of he signals based on he presen daa. Hence, his problem should be seen from a perspecive ha is broader han devising a convenien exension for he signal. Apar from he Fourier series exension mehod, an addiional goal of his paper is o propose an approach o shoren he widh of he boundary effec region defined in he above secion. This approach is based on inerpolaion in he boundary effec region o reduce he boundary effecs. Fig. 2 explains he principle of reducion of boundary effecs using inerpolaion mehod. Wihou inerpolaion, he convoluion is compued beween wavele and daa wih lengh N, from s () o s ( N 1). Afer inerpolaion, he convoluion is sill compued beween wavele and daa. However, he end poin of hese daa has become o s ( N 2 1) if N is odd. As shown in Fig. 2(b), he lengh of wavele is he half of ha before inerpolaion and becomes shorer compared o he original signal. Based on he discussion of Secion 4, i is easy o show ha he boundary effecs region which is E-ISSN: Issue 2, Volume 9, April 213

6 decided by he lengh of wavele also becomes shorer in consequence of he inerpolaion procedure. Therefore he boundary effecs are alleviaed by exploiing inerpolaion. In pracical applicaions, we only require o employ inerpolaion in he boundary effecs region o obain good resuls wihou heavy compuaional burden. Inerpolaion can be considered as an expansion of he exension mehod owards he inerior of signal. Compare wih exension mehods, i is easier and more accurae o esimae signal value beween wo poins han predic he daa from he view of probabiliy. 4.1 The Range of Inerpolaion Performing inerpolaion processing a he boundaries could furher reduce boundary effecs. In he implemen, we should consider he range of inerpolaion processing, ha is, how long of he s () s () inerpolaion and hen explore how o deermine i in pracice. Le us assume signal s () has an singular poin a =. I is obvious ha he singular poin a = will no impac he whole ime-scale plane bu only he neighborhood of. We refer i as cone of influence of. The range of inerpolaion depends on he cone of influence. For he sae of simpliciy, suppose he wavele ha we use has a suppor[ CC, ]. Then he scaled wavele ψ a () has suppor [ Ca, + Ca]. We define he cone of influence as he se of poins conaining in he suppor [ Ca, + Ca] from he whole ime-scale plane. Thus, he cone of influence of is Ca (17) In he cone of influence, he performance of wavele ransform is impaced by he singular poin inroduced by exension. We refer he cone of influence as he region of boundary effec where he inerpolaion processing should be performed. I is noice ha he range of inerpolaion is proporional o he scale facor a. Fig. 3. illusraes he lengh of inerpolaion required a differen scale on he imescale plane. s () ψ () (a) 4.2 The Implemen of Inerpolaion Based on he previous discussion, algorihm for he inerpolaion processing for boundary effec reducion can be summarized as follows: 1) Obain he wavele ransform of he signal. 2) We find he scale a corresponding o he maximum ampliude of he wavele ransform a =. 3) The range of inerpolaion processing is deermined by he scale a from he above sep. Fig. 3 shows differen range of inerpolaion processing a differen scale. (b) Fig. 2. Convoluion beween signal and wavele: (a) before inerpolaion; (b) afer inerpolaion. O > Ca > Ca signal should be involved in he inerpolaion processing. A range which is oo long or oo shor would yield an expensive compuaion or an inaccurae resul. We firs discuss he range of a Ca Fig. 3. The range of inerpolaion processing a differen scale. E-ISSN: Issue 2, Volume 9, April 213

7 5 Numerical Examples In order o validae he resuls given in Secion 3 and Secion 5, we presen he numerical examples of he proposed algorihms. The performance of he proposed mehods has been assessed by means of ess on generic synheic signals. The purpose of he es is o esablish he measuremen accuracy of he proposed mehods as well as heir advanages in boundary effecs reducion over he basic mehods. The es consiss of wo pars which involve he proposed exension mehod and inerpolaion preprocessing. Two signals exhibiing linear and nonlinear insananeous frequency laws are used for evaluaing he performance of he algorihms. 5.1 Performance Assessmen of Fourier Series Exension Firs consider he linear FM signal was presened in Secion 2. Some resuls ha illusrae he performance of he Fourier series exension in he insananeous frequency esimaion are shown in Fig. 4. For comparison purposes, he exension algorihm is compared wih symmeric exension which is superior over he oher wo basic mehods. I is apparen ha he resuls provided by he Fourier series exension mehod are in beer agreemen wih he heoreical values in Fig. 4(a). As we have done in he previous secion, he error beween heoreical and esimaed wavele ridge are shown in Fig. 4(b) o illusrae he effec of Fourier series exension. I can be observed ha Fourier series exension has less singulariy appearing a he boundary han symmeric exension. Normalized Frequency Theoreical resul Symmeric Fourier series.95 Time(s) (a) Error(dB) Symmeric Fourier series Time(s) (b) Fig. 4. Comparison of boundary effecs for linear FM signal of symmeric and Fourier series exension. (a) The righ boundary of wavele ridge. (b) Esimaion error. The values on he verical axes are normalized o he adoped sample rae. For he nonlinear case, we consider a logarihmic frequency modulaed signal wih same samples as he linear case and is IF is given by ( ) 1 f f = f( ) (18) f We se f =.1, f 1 =.4, 1 = 3. The oal signal lengh and sample period used are N = 3, T =.1s. The proposed algorihm is successfully applied on his nonlinear FM signal. Fig. 5 illusraes he resuls provided by Fourier series exension and symmeric exension applied on he nonlinear FM signal. I can be seen ha he resuls are similar o he resuls of linear FM signal. Fig. 5(b) shows ha he Fourier series exension is indeed efficien o reduce boundary effecs for complicaed signals wih ime-varying IF laws. 5.2 Performance Assessmen of he Inerpolaion Mehod Several ess have been conduced in order o assess he capabiliy of he proposed mehod inerpolaion preprocessing o furher reduce he boundary effecs. All he basic exension mehods and he Fourier series exension will be considered in his secion. The performance of he inerpolaion preprocessing is examined using he same wo classes of signals. 1 E-ISSN: Issue 2, Volume 9, April 213

8 .11 daa from he boundary par. The number of daa paricipaing o he calculaion is 5 poins. Normalized Frequency.15.1 Theoreical resul Symmeric Fourier series (a) Error (db) Wihou inerpolaion -9 Wih inerpolaion -5 (a) Error (db) Symmeric Fourier series -75 (b) Fig. 5. Comparison of boundary effecs for logarihmic FM signal of symmeric and Fourier series exension. (a) The righ boundary of wavele ridge. (b) Esimaion error. The values on he verical axes are normalized o he adoped sample rae. We plo he comparison resuls of symmeric exension and Fourier exension wih and wihou inerpolaion processing on he linear FM signal and logarihmic FM signal in Fig. 6 and Fig. 7 respecively. I can be clearly observed ha he inerpolaion processing is indeed able o furher reduce he disorion resuled from he boundary effecs for boh signals. To show he effec of he inerpolaion preprocessing, compression es resuls of he linear FM signal for differen exension mehods wih and wihou inerpolaion processing are provided in Table 1. All of he values are calculaed from rue normalized frequency and esimaing normalized frequency. In order o display he performance of boundary effecs reducion, we only calculae he Error (db) Wihou inerpolaion Wih inerpolaion -95 (b) Fig. 6. Esimaion error of wavele ridge of linear FM signal wih and wihou inerpolaion processing for (a) symmeric exension and (b) Fourier series exension. Error (db) Wihou inerpolaion Wih inerpolaion -85 (a) E-ISSN: Issue 2, Volume 9, April 213

9 Error (db) Wihou inerpolaion Wih inerpolaion -85 (b) Fig. 7. Esimaion error of wavele ridge of logarihmic FM signal wih and wihou inerpolaion processing for (a) symmeric exension and (b) Fourier series exension. I can be concluded ha he Fourier series exension wih he inerpolaion processing provides he bes performance among he four mehods menioned in his sudy. Furhermore, he inerpolaion processing is able o produce a beer accuracy of he ime-frequency characerisics esimae no maer which exension mehod is applied. Table 1. Performance comparison of exension mehods wih and wihou inerpolaion preprocessing. Mehod Bias Variance MSE 1 Zero Zero (inerpolaion) Periodic Periodic (inerpolaion) Symmeric Symmeric (inerpolaion) Fourier Fourier (inerpolaion) MSE is he mean square error. 6 Conclusion In his paper, we have invesigaed he problem of dealing wih he boundary effecs ha would arise in he applicaion of ime-frequency analysis. Basic mehods including zero padding, periodic exension and symmeric exension were shown o provide unsaisfied performance o reduce he boundary effecs. We derived a generalized expression for various exension mehods. The relaionship beween smooh and he boundaries effec has been sressed. A smooh exension scheme using Fourier series o avoid disorion appearing a he boundaries was proposed. This exension echnique possesses he propery of preserving he harmonic oscillaions of he ime-vary signal ha maes i more suiable han he oher mehods for he imefrequency analysis applicaion. A new algorihm based on inerpolaion echnique was proposed from new perspecives o furher reduce he boundary effecs. I has been shown ha he range of inerpolaion is deermined by he scale facor maximized he ampliude of he wavele ransform a he boundaries. Some deails on he procedures for implemen of he proposed echnique have been presened. By comparing he resuls of he analysis, i has been shown ha he adapive smooh exension wih he inerpolaion processing provided he bes performance in he sudy. Alhough we have resriced he analysis o he wavele ransform, he proposed mehods can be applied on any imefrequency disribuions ha involve convoluion operaion. References: [1] I. Daubechies, Ten Lecures on Waveles, Philadelphia, PA: SIAM, [2] C. K. Chui, An Inroducion o Waveles, New Yor: Academic, [3] S. Malla, A Wavele Tour of Signal Processing, 3rded, New Yor: Academic, 28. [4] B. Zhang, J. M. Fadili and Sarc, J.-L., Waveles, ridgeles, and curveles for Poisson noise removal, IEEE Trans. Image Processing, vol. 17, no. 7, 28, pp [5] J. Porilla, V. Srela, M. J. Wainwrigh and E. P. Simoncelli, Image denoising using scale mixures of Gaussians in he wavele domain, IEEE Trans. Image Processing, vol. 12, no. 11, 23, pp [6] H. Ngan, G.K.H. Pang and N. Yung, Auomaed fabric defec deecion A Review, Image and Vision Compuing, vol. 29, 211, pp E-ISSN: Issue 2, Volume 9, April 213

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