Noise Reduction Algorithm Based on Complex Wavelet Transform of Digital Gamma Ray Spectroscopy

Transcription

1 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( Noise Reductio Algorithm Based o Complex Wavelet rasform of Digital Gamma Ray Spectroscopy Mohamed S. El_okhy Electrical Egieerig Departmet, College of Egieerig, Aljouf Uiversity, Aljouf, KSA egtokhy@gmail.com Imbaby I. Mahmoud Egieerig Departmet, NRC, Atomic Eergy Authority, Ishas, Cairo, Egypt Abstract his paper ivestigates the use of complex wavelets i gamma ray spectroscopy sigals. I this paper a algorithm for oise elimiatio of the detected gamma ray spectroscopy sigals is studied. his algorithm is based o the complex wavelet trasform. Recostructio of the origial detected sigal is obtaied by applyig the iverse complex wavelet trasform to the trasformed complex wavelet trasform sigal. Five differet cases are studied with differet five levels of the complex wavelet trasform. Cosequetly, comparisos betwee these levels are cosidered i terms of maximum umber of peak heights, executio time, ad peak sigal to oise ratio (PSNR). Moreover, compariso betwee differet sigal recostructio with respect to differet complex wavelet trasform levels, size of the trasformed sigal i each level, ad umber of coefficiet i each subbad for certai level. Oe of the mai advatages of this algorithm that discussed i the previous literature is that its filters do ot have serious distributed bumps i the wrog side of the power spectrum ad, simultaeously, they do ot itroduce ay redudacy to the origial sigal. he obtaied result cofirms the high accuracy of the cosidered algorithm over traditioal algorithms for both oise elimiatio ad sigal recostructio. Keywords Complex Wavelet rasform; Peak Sigal-to-Noise Ratio; Liear Filters I. INRODUCION HE Scitillatio detectio experimets idicate that the electroic oise i II photodetectors was the domiat source of resolutio broadeig [1]. Hece, the electroic oise behavior of the II detectors was ivestigated to determie the magitude of various oise compoets i the detectors. hese existig oise models are used to aalyze the electroic oise i II detectors. he electroic oise is expected to arise from several sources i II detectors [1]. hese sources are the parallel thermal oise due to the detector leakage curret (also commoly referred to as shot oise). Secodly, the series thermal oise that geerated i the chael of the iput JFE of the pre-amplifier. Fially, the 1/f oise has bee obtaied from the detector pre-amplifier assembly. Sice the PM aode sigal is very oisy ad timig features highly deped o the sigal at specific times, a de-oisig algorithm is required. here exist differet digital de-oisig ad smoothig methods (e. g. movig average filters) depedig o the applicatio [2]. A de-oisig algorithms based o the Wavelet rasform (W) is implemeted to reduce the effect of oise itroduced by the oisy aalog chael ad by the photomultiplier tube as i [2]. I this applicatio, liear smoothig filters are ot appropriate because the sigal cotais a sharp portio associated with the iteractio i the first layer (fast compoet). Wavelet de-oisig which is a o-liear filterig operatio aalyzes the sigal at differet time resolutio levels ad the removes the oise compoets by thresholdig sigal compoets i oe or more levels. Depedig o the applicatio, the level ad threshold should be modified to remove the oise while keepig the importat high-frequecy compoets of the sigal [2]. Complex wavelet trasforms has sigificat advatages over real wavelet trasform for certai sigal processig problem [3]. Complex wavelet trasforms, i which the real Page 565

2 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( ad imagiary parts of the trasform coefficiets are a approximate Hilbert-trasform pair, offer three sigificat advatages over real wavelet trasforms: shift ivariace, directioality, ad explicit phase iformatio. hese properties eable efficiet statistical models for the coefficiets that are also geometrically meaigful [4]. Complex wavelets have ot bee used widely i sigal processig due to the difficulty i desigig complex filters which satisfy a perfect recostructio property [5]. o overcome this Kigsbury [6] proposed a dual-tree implemetatio of the CW (D CW) which uses two trees of real filters to geerate the real ad imagiary parts of the wavelet coefficiets separately. Eve though the outputs of each tree are dowsampled by summig the outputs of the two trees durig recostructio, the aliased compoets of the sigal is suppressed ad achieved approximate shift ivariace [5]. he complex wavelets are first used to perform aalysis of the sigals [5]. We describe how to extract features to characterize textured sigals ad test this characterizatio by resythesizig textures with matchig features. he term de-oisig is usually referred to removig the white Gaussia oise or thermal oise which is added to the sigal. I our applicatio, this type of oise is mostly itroduced by the oisy aalog read-out system. I our applicatio, a 5-level de-oisig algorithm based o complex wavelet trasform fuctios was used. Also, the rescalig i wavelet decompositio is performed usig level-depedet estimatio of the oise level. his paper is orgaized as follows: Sectio 2 presets the spectroscopy system. he more iterestig characteristics of the studied complex wavelet trasform are represeted i Sectio 3. Results ad discussio are summarized i Sectio 4 ad we termiate our study by a briefly coclusios that we oted from our obtaied results. II. SYSEM CONFIGURAION I this system, the compoets of the system for evaluatio of oise elimiatio usig complex wavelet trasform algorithms are described. Cotais the followig elemets; 137 Cs poit source, scitillatio detector, amplifier, digital system ad coectio to a desktop persoal computer (PC). A 1.5 iches x 7.5 iches NaI(I) scitillatio detector is used to detect the radiatio sigal from Cs137 poit source. his detector is coected to amplifier through coaxial cable which i tur coected to the PC. MALAB eviromet is used to perform oise elimiatio usig complex wavelet trasform evaluatio. I this paper, a algorithm for oise elimiatio usig complex wavelet trasform is studied o digital gamma ray spectroscopy sigals. his algorithm is proposed for multidimesioal sigal processig i [7]. Block diagram showig the algorithm of oise elimiatio evaluatio usig the complex wavelet trasform is illustrated i Fig. 1. Moreover, differet wavelet trasform levels are cosidered. Fig. 1 Block diagram model of the complex wavelet trasform algorithm for oise elimiatio of gamma ray spectroscopy sigal III. 137 Cs/ 6 Co Poit Source Scitillatio Detector Amplifier Digital Oscilloscope Sigal Stored i Excel File Determie otal Number of Levels i the Wavelet ree Applyig the Complex Wavelet rasform to the Iput Sigal Applyig the iverse Complex Wavelet rasform Recostructed Sigal HE COMPLEX WAVELE RANSFORM A. Liear Phase Filters Oe of the most importat properties of the filters which ca be applied to 3-bad filter baks is liearity of the phase. he phase of the sigals which are filtered usig the liear phase filters is ot perturbed, which meas all frequecy compoets of the sigal are shifted equally. he filter h(t), which has the liear phase property, satisfies the followig equatio[7] i e ( 1 ), 1 h h N N (1) where N ad θ is the legth of the filter ad a arbitrary variable betwee zero ad 2π. he followig four special kids of the liear phase filters are used: 1. Real filter with θ =, 2π ad N to be odd, 3. Complex filter with θ =π/2, 3π/2 B. Filter Bak Characteristics Filter baks are widely used i digital sigal processig, ofte itegrated i a multirate scheme, to reduce the implemetatio cost ad to improve algorithmic performace [8]. he ideal ormalized power spectrum of the filters i the filter bak that satisfies all desirable characteristics, such as havig the Hilbert-pairs wavelet Page 566

3 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( filters ad itroducig o bumps o the wrog side of the power spectrum. his filter bak itroduces o redudacy because oe of the wavelet filters is the complex cojugate of the other, so the resultig coefficiets of oe filter is the complex cojugate of the other filter, ad we ca discard them i further aalysis [7]. I the ideal case, without cosiderig this low-pass filter, the samplig frequecy ca be computed usig Nyquist s theorem. I the o-ideal case, a ati-aliasig filter must be applied before the samplig part. I this case, the samplig frequecy ca be calculated usig the followig equatio [7]: f f 2B (2) tr where f is the Nyquist frequecy ad B tr is the trasiet bad of the ati-aliasig filter. Whe usig a discrete time low-pass filter after a cotiuous-to-discrete coverter, we ca compute aother costrait for avoidig distortio. he samplig rate must be computed i such a way that the ormalized badwidth of the aalog sigal be equal to the pass-bad of the low-pass filter. herefore, the followig costrait is computed [7]: f f s (3) 2B low where B low is the ormalized pass-bad of the low-pass filter. I this case, the amout of samplig frequecy icrease with respect to the commo samplig is [7]: f 2B low R (4) f 2B tr I additio, atural sigals decay very fast with frequecy icrease; therefore if we deviate a little from this samplig rate costrait, the amout of distortio itroduces ito sigal is ot very much. We will see this i the secod part of the simulatio results. C. Filter Bak Desig Procedure A orthogoal filter bak for digital gamma ray spectroscopy which cosists of oe real filter h ()ad two * complex cojugate filters h is studied. It is cosidered [7] h ad c 1 hc h1 ih2 (5) 2 the for havig a complete orthogoal trasform, h (), h 1() ad h 2() must satisfy the shift orthogoal coditio expressed i the ext equatio. I order to have a complete orthogoal trasform, the scalig ad the wavelet fuctios must satisfy the shift orthogoal coditio. he desig cost fuctio is give by [7] c hi Qt hm l m i m l i h M h h1 M re 2 h1 h2 M re 2 h2 2h2 M im 2 h 1 where α is the weightig coefficiet ad the stopbad power is calculated at those frequecies ω1 ad ω2 that maximize it. For this purpose, the stopbad is sampled uiformly ad the stopbad power is evaluated at these sampled frequecies. A gradiet desced algorithm for miimizig the cost fuctio is used. At every iteratio the weightig coefficiet α should decrease such that the optimized filters satisfy the shift orthogoal coditio. his iteratio will termiate whe the cost of the shift orthogoal coditio expressed i the first lie of Equatio 6 becomes sufficietly small ( 1 6 ). IV. RESULS AND DISCUSSION We apply the low-pass filter before the aalysis ad after the sythesis filter baks, therefore we deviate from the perfect recostructio coditio. o observe the amout of distortio itroduced ito the sigal, we apply differet levels of the filter bak with the low-pass filter to the acquired sigal. he origial detected sigal is depicted i Fig. 2. he impulse respose of the filters at levels oe, two, three, four, ad five at differet shifts ad the resultig absolute value of the wavelet coefficiets at these levels are show i Figs. 3-7, respectively. As illustrated i these figures, these filters are highly orieted i, 45, 9, ad 135 ad the real ad complex parts of the complex filter costitute Gabor-like filters as i [7]. As illustrated, the filters eve at the first level are orieted. It is iterestig to test the shift-ivariace performace. It is obvious that the studied filter bak has a excellet shift-ivariace property which is comparable to the dual-tree complex wavelet trasform. For the studied filter bak with the filters of legth 37, Ra is calculated ad is show i able 3. As a illustrative example, we costruct the sigal of oe, two, three, four, ad five levels ad the recostructed sigals are show i Figs Oe ca see that trasformed sigals are shift-ivariat ad free of aliasig. Compariso betwee the differet wavelet levels are depicted i able 1. From this table, the umber of couted peaks decreases with the wavelet level. Also, the umber of couted peaks of the origial sigal is equal to the umber couts of level five complex wavelet trasform. However, the executio time icreases with the level. Also, compariso betwee differet five levels is illustrated i terms of PSNR. From the theoretical results, the PSNR of the recostructed sigals are very high. herefore, the (6) Page 567

4 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( uderlied filter bak is cosidered as a early perfect recostructio filter bak o atural sigals. Moreover, compariso betwee differet sigal recostructio i terms of differet complex wavelet trasform levels, size of the iput sigal, size of the trasformed sigal i each level, umber of coefficiet i each subbad for certai level, ad total legth of the iput sigal is depicted i able 2. Also, the ormalized amplitude of level 1 complex wavelet trasform is depicted i Fig Fig. 2 Origial detected sigal from the scitillatio detector Fig. 4 he impulse respose of the filters at level two at differet shifts ad the resultig absolute value of the wavelet coefficiets ABLE I COMPARISON BEWEEN DIFFEREN COMPLEX WAVELE RANSFORM LEVELS Number of Peaks Executio ime (s) Level Level Level Level Level ABLE 2 COMPARISON BEWEEN DIFFEREN LEVELS IN ERMS OF DIFFEREN SRUCURE PARAMEERS Subbads Coefficiet Number otal legth Level 1 [482 1] Level 2 [2x2 double] [ ] Level 3 [3x2 double] [ ] Level 4 [4x2 double] [ ] Level 5 [5x2 double] [ ] 269 Fig. 3 he impulse respose of the filters at level oe at differet shifts ad the resultig absolute value of the wavelet coefficiets Page 568

5 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( Fig. 5 he impulse respose of the filters at level three at differet shifts ad the resultig absolute value of the wavelet coefficiets Fig. 6 he impulse respose of the filters at level four at differet shifts ad the resultig absolute value of the wavelet coefficiets Fig. 7 he impulse respose of the filters at level five at differet shifts ad the resultig absolute value of the wavelet coefficiets Fig. 8 Recostructed sigal usig first level of the algorithm Page 569

6 ICI 215 he 7 th Iteratioal Coferece o Iformatio echology doi: /icit ICI 215 ( Fig. 9 Recostructed sigal usig secod levels of the algorithm Fig. 1 Recostructed sigal usig three levels of the algorithm Fig. 11 Recostructed sigal usig four levels of the algorithm Fig. 12 Recostructed sigal usig five levels of the algorithm. Normalized Amplitude Number of Sigals i Subbads x 1 5 Fig. 13 Normalized amplitude of the output sigal. CONCLUSION his paper focuses o oise elimiatio due to both electroic system readout ad coaxial cables betwee scitillatio detector ad amplifier. A algorithm based o complex wavelet trasforms is studied to do this fuctio. he iput sigal is trasformed usig the complex wavelet trasform. he, the iverse complex wavelet trasform is applied to the trasformed sigal. Compariso betwee differet complex wavelet trasform levels is cosidered. his compariso is based o both umber of couted peaks, executio time ad PSNR. his filter bak has liear phase filters. he resultig filters at differet levels do ot produce serious bumps o the wrog side of the frequecy axis. REFERENCES [1] K. S. Shah, P. Beett, L. P. Moy, M. M. Misra, W. W. Moses, "Characterizatio of Idium Iodide Detectors for scitillatio studies", Nuclear Istrumets ad Methods i Physics Research A, Vol. 38, pp , [2] Siavash Yousefi ad Luca Lucchese, "Digital Pulse Shape Discrimiatio i riple-layer Phoswich Detectors Usig Fuzzy Logic", IEEE rasactios o Nuclear Sciece, Vol. 55, No. 5, October 28. [3] H. N. Abdullah, "SAR image deoisig based o dual-tree complex wavelet trasform", Joural of Egieerig ad Applied Sciece, Vol. 3, No. 7, pp , 28. [4] Felix C. A. Ferades,t Michael B. Wakir ad Richard G. Baraiukr, "No-redudat, liear-phase, semi-orthogoal, directioal complex wavelets", ICASSP, IEEE, 24, pp. II 953- II 956. [5] Musoko Victor, Prochazka ales, "Complex wavelet trasform i sigal ad image aalysis", 14 th Iteratioal Scietific - echical coferece o Process Cotrol, Czech Republic, Jue 24. [6] N. G. Kigsbury, Complex wavelets for shift ivariat aalysis ad filterig of sigals, Joural of Applied ad Computatioal Harmoic Aalysis, Vol 1, No 3, pp , 21. [7] Reshad Hosseii, Masur Vafadust, "Almost Perfect Recostructio Filter Bak for No-redudat, Approximately Shift-Ivariat, Complex Wavelet rasforms", Joural of Wavelet heory ad Applicatios, Vol. 2, No. 1, pp. 1 14, 28. [8] Koe. Eema, ad Marc. Mooe, "DF modulated filter bak desig for oversampled subbad systems", Sigal Processig Joural, Vol. 81 No.9, pp , 21. Page 57