2014 Adaptive total variation-based spectral deconvolution with the split Bregman method

Adaptive total variation-based spectral deconvolution with the split Bregman methodHai Liu,Sanya Liu,Zhaoli Zhang,*Jianwen Sun,and Jiangbo ShuNational Engineering Research Center for E-Learning,Central China Normal University,Wuhan430079,China*Corresponding author:zl.zhang@Received3July2014;revised17September2014;accepted18September2014;posted4November2014(Doc.ID216309);published3December2014Spectroscopic data often suffer from common problems of band overlap and noise.This paper presents amaximum a posteriori(MAP)-based algorithm for the band overlap problem.In the MAP framework,thelikelihood probability density function(PDF)is constructed with Gaussian noise assumed,and the priorPDF is constructed with adaptive total variation(ATV)regularization.The split Bregman iteration al-gorithm is employed to optimize the ATV spectral deconvolution model and accelerate the speed of thespectral deconvolution.The main advantage of this algorithm is that it can obtain peak structure infor-mation as well as suppress noise simultaneity.Simulated and real spectra experiments manifest thatthis algorithm can satisfactorily recover the overlap peaks as well as suppress noise and are robust to theregularization parameter.©2014Optical Society of AmericaOCIS codes:(140.3510)Lasers,fiber;(300.6450)Spectroscopy,Raman;(100.3190)Inverse problems;(040.2235)Far infrared or terahertz./10.1364/AO.53.0082401.IntroductionIn recent years,spectroscopic data acquisition and processing has led to significant developments,due to the high demand for its application in target detec-tion,rapid identification of chemicals,and analytical techniques[1].However,it often suffers from the common problems of band overlap and noise[2]. These disadvantages limit the development of spec-troscopy in industrial and commercial applications. During the past two decades,various approaches for spectral decomposition have been proposed. The existing algorithms can be generally classified into three major categories:source separation method[3],curve-fitting method[4,5],and spectral deconvolution methods[6–10].Source separation at-tempts to recover overlap spectra from their linear mixtures[11].The curve-fitting method[4]considers that the measured spectra are combined by the Gaussian and Lorentzian functions.In other words, the measured spectra can be fitted by those functions.However,the drawback is that the optimi-zation of this method is not straightforward.Decon-volution is a mathematical technique that removes the broadening effect of the instrumental response function.The deconvolution method recovers the degraded spectrum with different regularizations, such as homomorphic filtering[12],high-order sta-tistics(HOS)[6],Tikhonov regularization[8],Burg entropy[9],Huber deconvolution[13],and the reversible jump Markov chain Monte Carlo method (RJMCMC)[14].These methods have achieved great success for specific spectral decomposition.In recent years,total variation regularization, because of its edge-preserving property by not over-penalizing discontinuities while imposing smooth-ness[15],has enjoyed increasing popularity.It has been central to the solution of ill-posed inverse problems in a wide range of applications,such as1559-128X/14/358240-09$15.00/0©2014Optical Society of America8240APPLIED OPTICS/Vol.53,No.35/10December2014image denoising and deblurring.,Inspired by these,an attempt was made to use total variation for spec-tral deconvolution.As shown in Fig.1,the spectroscopic data include an all-frequency component.The spectral data can be decomposed into flat,noise,and steep regions.The noise includes all-frequencies noise,as indicated by the arrows in Fig.1.The points of the steep region are often next to the impulse peaks.The intensity of the steep points changes dramatically and has the discontinuity property .TV regularization imple-mented through shrinkage would be sufficient to sta-bilize this;thus,an adaptive term is constructed.To accomplish these results,the proposed method has two main contributions.First,a new adaptive operator is introduced to stably create total variation regularization.The ATV can preserve the peak detail information and suppress noise simultaneity .Second,the split Bregman method is introduced to optimize the proposed model.To our knowledge,ATV is introduced to spectroscopic data deconvolution for the first time.2.Proposed ModelMost spectroscopic data measured by spectropho-tometer can be mathematically modeled as a convo-lution of the true spectrum with an instrument response function (IRF).In this paper,the degrada-tion process is described asg Hf n;(1)where f and g denote the actual spectrum and mea-sured spectrum,and H represents the IRF (also called the blur kernel),which collects the intrinsic line-shape and the instrumental broadening.The corresponding symbols are discriminated asgg 1;g 2; g N T;H 2666664h 11h 12 h 1Nh 21h 21h 2N .........h N 1h N 1 h NN 3777775N ×N;ff 1;f 2; f N T ;n n 1;n 2; n N T ;where N is the length of the spectrum.MAP estima-tor is a commonly used approach to estimate the original spectrum f ,given the degraded spectrumg .If H is known in advance,the latent spectrum ˆfcan be resolved by the Wiener filtering method [16].MAP estimator maximizes the conditional probability of actual spectrum f ,given the degraded spectrum g .That is,ˆfarg max fP f j g :(2)Applying Bayes ’formula P f j g ∝P g j f P f ,and using the monotonic logarithm function,Eq.(2)can be expressed asˆfarg min ff −log P g j f −log P f g :(3)It can be seen that two probability density func-tions need to be constructed.Considering the caseswhere the data are contaminated by Gaussian noise,2004006008000.20.40.60.81wavenumber (1/cm)R a m a n I n t e n s i t y Fig.1.Raman spectrum characteristics analysis.Raman data include an all-frequency component.For the spectral data,it can be decomposed into flat,noise,and steep regions.For the noise,it includes all-frequency noise.0.010.020.030.040.010.020.030.0400.010.020.030.040.0050.0100.015T V a n d A T V v a l u eSpectral dynamical value D f i Spectral dynamical value D f iFig.2.Effect of the adaptive operator.(a)Difference between TV and ATV .(b)ATV regularization with K 0.01and 0.025for different noise levels.10December 2014/Vol.53,No.35/APPLIED OPTICS8241the intensity of each pixel g i in the observed spec-trum is a random variable that follows an indepen-dent Gaussian distribution.Hence,the likelihood can be written asP g j fY Ni 112πpσiexp− Hf−g 2i2σ2i;(4)whereσand N denote the noise standard deviation and spectral length.Thus,Eq.(4)can be rewritten as P g j f ∝exp −‖Hf−g‖22 ;(5) where the symbol‖·‖is the Euclidean norm.For the prior probability P f in Eq.(3),TV prior is selected as the prior probability.And the expression P f can be written asP f ∝exp −λ‖Df‖1 ;(6) where Df i f i 1−f i ∕2,called the spectral differ-ence.Submit Eqs.(5)and(6)to Eq.(3)and the MAP estimation is then equivalent to minimizing the func-tional E f −log p f j g ,i.e.,to minimizeE f 1‖Hf−g‖22 λ‖Df‖1 l f≥0;(7)where the first term‖Hf−g‖22is the data term,λis the regularization parameter,‖Df‖1acts as the regularization term,the role of l f≥0is to impose the non-negative constraint on the estimation spec-trum f,and the matrix D is the difference matrix of spectrum.TV regularization is effective for piecewise-constant reconstruction and allows the abrupt and intense bands in the recovered spectrum.But for the noisy spectrum,the noise will be amplified with the bands and produce incorrect spectral peaks.For better noise suppression and detail preservation,an adaptive operator should be added to control the TV regularization.The choice of the regularization term depends on the dynamics of the spectral data and of the noise.The operator should be a non-negative, monotonically decreasing function.Only such an operator,combined with TV regularization,can suppress all noise in all frequencies and preserve the spectral structure.Thus,the adaptive operator takes the formw Df 14;(8)where K is a constant.Figure2(a)shows the differ-ence between TV and ATV.It can be seen that ATV has the strongest constraint for the noise region.The setting of parameter K will be discussed in Section4. Substituting the TV term in Eq.(7)with ATV in Eq.(8),we introduce the cost functional:E f12‖Hf−g‖22 λw Df ‖Df‖1 l f≥0:(9)We call the model Eq.(7)spectral deconvolution with total variation(SDTV)and model Eq.(9)spec-tral deconvolution with adaptive total variation (SDATV).Summarizing,three good features of ATV can be drawn in Fig.2(a):(i)for the flat region points be-cause Df i is close to zero and w Df i is close to1, which means a small ATV regularization strength is enforced at these points;(ii)for the steep region because Df i is very large and w Df i is small and nearly zero,and this weakens the TV regularization strength enforced to these points,so the steep points will be well preserved;and(iii)for the noise region, the large ATV regularization strength,and then the noise will be well suppressed.Figure2clearly illus-trates the difference between TV and ATV.That is, SDATV can adjust the regularization adaptively.3.OptimizationRecently,a well-performing optimization method called the split Bregman algorithm was developed by Goldstein and Osher to solve the L1norm-based regularization[17].In this paper,the split Bregman method is used to optimize the SDATV model in Eq.(9).The basic idea of this optimization algorithm can be stated as follows.First,two auxiliary variables,d1,d2,are intro-duced into the optimization process and then added to Eq.(9)in the following way:E f;d1;d2 min1‖Hf−g‖22 λw‖d1‖1 l d2≥0d2 subject to∶d1 Df;d2 f:(10) The constrained problem in Eq.(10)can be changed into an unconstrained problem with the Bregman iteration as follows:E f;d1;d2 min1‖Hf−g‖22 λw‖d1‖1 l d2≥0d212α‖d1−Df−b1‖22 ‖d2−f−b2‖22 ;(11)whereαis the penalty parameter.The minimization of Eq.(11)can be performed alternately with the fol-lowing three subproblems:ˆf arg min12f‖Hf−g‖221‖d1−Df−b1‖22 ‖d2−f−b2‖22 ;(12)ˆd1arg mind1λw‖d1‖11‖d1− Df −b1‖22;(13)8242APPLIED OPTICS/Vol.53,No.35/10December2014andˆd 2 arg mind2l d2≥0d212α‖d2−f−b2‖22:(14)In the aforementioned equations,to solve the f subproblem,the following equation must be solved: f k 1 αH T H D T D I −1 αH T g D T d k1−b k1d k2−b k2 :(15)For the linear function in Eq.(15),because the sys-tem is strictly diagonal,the most natural choice is the Gauss–Seidel method.The d subproblem equation in Eqs.(13)and(14) can be solved using a shrinkage operator as follows:d k 1 1 max fj Df k 1 b k1j−λw kα;0gDf k 1 b k1j Df k 1 b k1j;(16)d k 12max f f k 1 b k2;0g;(17)where the d1,d2are solved by the soft thresholding formulation proposed in[18].Finally,the Bregman variables b1and b2are updated as follows:b k 1 1 b k1Df k 1−d k 11;(18)b k 1 2 b k2f k 1−d k 12:(19)From the earlier introduction,it can be seen that the split Bregman iteration optimization method mainly composites the solution of the two subpro-blems:the f and d subproblems.The advantage of the split Bregman method is that the difficult optimi-zation problem in Eq.(9)is split into the aforemen-tioned three subproblems,which are very easy to optimize.According to the aforementioned analysis, the corresponding split Bregman algorithm is given as follows:Algorithm I.SDATV Method1.Initialization:The measured spectrum g is normalized to[0,1], compute noise level,and set K value.Set f0 g,d01 d02 0,b01 b02 0.2.While( j f k 1−f k j∕f k >ε1)repeati)fix d,solve f using the Gauss–Seidel iteration;ii)fix f,solve d1,d2using the soft threshold formulation;iii)Update b k 11 b k1Df k 1−d k 11b k 12b k2f k 1−d k 12;Iteration number k k 1.End3.Output:the latent Raman spectrum f.Here,ε1is a small positive constant between10−9 and10−7.4.Experimental ResultsTo verify the performance of the proposed SDATV model,the HOS[6]and spectral deconvolution with Tikhonov regularization(SDTR)[19]method are compared.Two parameters in the split Bregman iter-ation are these settingλ 0.04,α 10∕λ.For the parameter K setting,the formula of Median Absolute Differences(MAD)[20]is employed.This formula can estimate the noise levelδ,which is de-fined asδ 1.4826∕2pmedian fj g i−g i−1j;i 2;…;n g. Because the spectrum intensities are normalized to the range[0,1],theδvalue can reflect the noise level at the same scale.The stronger the intensity of the spectral noise,the larger the noise levelδ.The value of K is set as K 10δ.In Fig.2(b),two cases of the ATV regularization for different noise levels are shown.The normalized mean square error(NMSE)‖f−ˆf‖2∕‖f‖2and weighted correlation coefficient (WCC)[21]are used to evaluate the similarity between the ground-truth spectrum f and the recov-ered spectrumˆf.A.Simulation ExperimentsUsing a computer,we simulated degraded spectra based on the experimental infrared(IR)spectrum. Figure3shows the IR spectrum of Malononitrile (C3H2N2)from400to4000cm−1.We convolved it with a Gaussian IRF with a widthσof13cm−1(top-right corner in Fig.3(c)),to generate the degraded spectrum(Fig.3(c)).The degraded spectrum becomes much smoother and less resolved,with the bands becoming wider and lower.For example,it is hard to distinguish the peak2931.8cm−1from2966.3cm−1. Afterward,for an investigation of the robustness to noise of the proposed methods,Gaussian noise was added to the degraded spectrum with different noise levels(SNR 200,100).The case noise level equal to 200is shown in Fig.3(d).To measure the peak posi-tion at a finer resolution,the Gaussian function(dis-credited as a finer grid0.1cm−1)is used to fit the spectral bands.For example,the peak at1394cm−1 in the original spectrum is finely measured at 1394.1cm−1.The recovered spectral band positions are also detected in this format.parison with Other MethodsFigure4shows the deconvolution results for the noise-free degraded spectrum in Fig.3(c).All of the deconvoluted spectra are clearly resolved.Visual analysis of the deconvolution spectrum clearly shows that the SDATV method splits the overlap peaks very well.In Fig.4(d),the peak at2960.4cm−1is dis-tinctly separated to2931and2966.5cm−1,respec-tively.While in Fig.4(a)–4(c),the overlapping bands are split slightly.Thus,the SDATV produces a much narrower spectrum with more details than the compared methods.Furthermore,the band distortions were investi-gated for the spectra deconvoluted using the HOS and SDTR methods.In the ground-truth spectrum (Fig.3(a)),the four peaks at1394.1,2273.4,2931.8, and2966.3cm−1are taken as the references.Table1 lists the position and height distortions of these10December2014/Vol.53,No.35/APPLIED OPTICS8243peaks between the recovered (Fig.4)and ground-truth spectra (Fig.3(a)).The root of mean square error (RMSE)of the distortions is calculated.It shows that the proposed SDATV method achieves the smallest RMSE value.C.Effect of ATV TermFig.5presents the deconvoluted results with noise level equal to 200.The results shown in Figs.5(a)–5(d)are SDTR,HOS,SDTV,and SDATV ,respectively.Compared with Figs.5(c)and 5(d),they seem to have the same noise suppression effect in the noise region.But for the steep regions,the Fig.5(d)produces more details.Furthermore,in Fig.5(c),the spectral struc-tures are also smoothed although the noise has been suppressed.We suggest that the SDATV method can suppress noise well in the flat region and preserve details in the steep region.For the SDTV and SDATV methods,the NMSE of the restored spectra versus varying regularization10002000300000.20.40.60.8(a)1394.12273.42931.82966.310002000300000.20.40.60.8(c)1392.82273.12960.4(b)IRF 10002000300000.20.40.60.8(d))m c /1(r e b m u n e v a w )m c /1(r e b m u n e v a w i n t e n s i t y (a .u .)i n t e n s i t y (a .u .)13601380140014200.20.40.60.8Spectral Band Gauss curveFig.3.Simulated degraded spectrum.(a)Ground-truth IR spectrum (Malononitrile C 3H 2N 2,resolution:1cm −1).(b)Curve fitting the spectral band (1394.1)with Gauss curve (resolution:0.1cm −1).(c)Convolute (a)with the IRF (top-right corner:Gaussian function σ 13cm −1).(d)Gaussian noise (SNR 200)added.10002000300000.20.40.60.8(a)10002000300000.20.40.60.8(c)10002000300000.20.40.60.8(d)1000200030002468x 10−3(b))m c /1(r e b m u n e v a w )m c /1(r e b m u n e v a w i n t e n s i t y (a .u .)i n t e n s i t y (a .u .)Fig.4.Noise-free degraded spectrum restored by (a)SDTR,(b)HOS [6],(c)SDTV ,and (d)SDATV .8244APPLIED OPTICS /Vol.53,No.35/10December 2014parameter λin Fig.3(d)is plotted in Fig.6.The SDATV method appears more robust with the change of the regularization parameter at a wide range from 0.005to 0.17.However,the change of the regu-larization parameter λhas a very large effect on the performance of SDTV.In particular,the NMSE valueincreases abruptly when the regularization param-eter becomes large.The NMSE versus the iteration number of the four methods on the SNR 200spec-tra is plotted in Fig.7,where the convergence of SDATV is also highlighted.Table 1.Band Distortions in Deconvoluted Spectra by HOS,SDTR,SDTV,and SDATV in Fig.4Band Position a 1394.12273.42931.82966.3RMSE b PositionSDTR −1.3 1.1 1.2 2.1 1.455HOS 1.5−2.4 1.4 2.9 2.272SDTV 0.3−2.2−1.1 1.2 1.502SDATV 0.1−0.9−0.8 0.20.580HeightSDTR −0.249−0.089−0.171−0.1720.065HOS —————SDTV −0.199−0.073−0.124−0.1190.052SDATV−0.103−0.034−0.107−0.0950.034a In cm −1,obtained from the band maximum.bRoot mean square error in the determination of the position.1000200030000.20.40.60.8(a)100020003000510x 10−3(b)10002000300000.20.40.60.8(c)10002000300000.20.40.60.8(d))m c /1(r e b m u n e v a w )m c /1(r e b m u n e v a w i n t e n s i t y (a .u .)i n t e n s i t y (a .u .)parison results with noise level equal to 200.(a)SDTR.(b)HOS [6].(c)SDTV .(d)SDATV .0.050.10.150.0150.0200.0250.0300.0350.040 SDATV SDTVN M S EλFig.6.NMSE versus regularization parameter λof SDTV and SDATV .10203040500.0150.0200.0250.030HOS SDTR SDTV SDATVNumber of iterationsN M S EFig.7.NMSE versus the iteration number of the four methods for the simulated spectrum.10December 2014/Vol.53,No.35/APPLIED OPTICS8245Table 2shows the values of NMSE and WCC for each method on simulated data with different SNRs.It is clear that SDATV is superior to HOS and SDTR in all SNR conditions.The best restored spectrum is selected to be the one with the lowest NMSE when the regularization parameter changes.It can be seen that the SDATV method has achieved the smallest NMSE among the four methods.The NMSE of the SDATV methods is considerably smaller than that of the degraded spectrum for different noise levels.Therefore,we may conclude that the proposed SDATV method recovers more spectral details andsuppresses more noise than the HOS and SDTR methods.Moreover,to investigate the algorithm speed between our method and the compared methods,we test three more simulated IR spectra,which are all from 400to 4000cm −1at 1cm −1resolution.The total CPU time (unit:second)is provided in Table 3.The SDATV took much less time than SDTR and HOS be-cause the SDATV is accelerated using split Bregman.D.Real ExperimentsMoreover,two real Raman spectra are tested in our experiments.Gaussian function is often chosen as the IRF [10].Because the existing noise has a signifi-cant impact on the extraction of the spectral features (peak position,peak area,FWHM,etc.),the noisy spectra need to be denoised.The experiments are set with the same parameter setting λ 0.04,δ 0.0021,and K 0.02.Fig.8(a)shows 600cm −1length Raman spectral data of D(+)-Glucopyranose (C 6H 12O 6)[22]from 2600to 3200cm −1at 1cm −1resolution.These spec-tral data suffer from heavy overlap,especially from 2850to 2900cm −1.Figure 8presents the results of the HOS (Fig.8(b)),SDTR method (Fig.8(c))and SDATV (Fig.8(d)).They all perform well at the slightly over-lapped peaks 2946cm −1.However,for the heavily overlapping peak at 2892cm −1,only two peaks can be split by the compared methods.Three peaks can be split by SDATV ,such as 2879.2,2892.5,and 2903.3cm −1(shown in Fig.8(d)).It seems that the val-leys in Fig.8(d)are deeper than those in Figs.8(b)and 8(c).Compared with the other two methods,the SDATV method obtains the most resolved peaks.Figure 9(a)shows almost 1000-length Raman spectral data of 6-O-(N-heptylcarbamoyl)-a-D-glucopyranoside (C 15H 29NO 7)[23]from 20toTable 2.Figures of Merit for the Performance of the HOS,SDTR,andSDATV Methods at Different Noise Levels aMethodsMerits Noise Level (SNR)SDTR [19]HOS [6]SDATV NMSENoise-free0.01950.02160.01052000.02030.02240.01241000.03580.04070.0159WCCNoise-free0.99940.99920.99962000.99890.99850.99911000.99770.99780.9984aThe larger the value of WCC,the higher the spectral quality .The lowest NMSE indicates the best spectral quality .Table 3.Comparison of the Computing Time between the SDTR,HOS,and SDATV MethodsIR Spectra SDTR [19](s)HOS [6](s)SDATV (s)Malononitrile 23.12520.156 1.945Cyclohexanol 20.36517.524 1.726Propyl acetate 22.34519.972 1.885Mesityl oxide21.86118.5641.8040.20.40.6(a)28922946270028002900300031000.20.40.6(c)2878.82892.12946.62961.1510x 10−42942.1(b)2873.12887.92958.427002800290030003100)m c /1(r e b m u n e v a w )m c /1(r e b m u n e v a w R a m a n I n t e n s i t yR a m a n I n t e n s i t ypared results for real Raman spectrum.(a)Observed spectrum of D(+)-Glucopyranose (C 6H 12O 6).(b)HOS.(c)SDTR [19].(d)Proposed method.8246APPLIED OPTICS /Vol.53,No.35/10December 20141000cm −1at 1cm −1.The real spectrum suffers from heavy noise from 20to 200cm −1,caused by the spec-trometer.The noise level is estimated as δ 0.0028,and K 0.03.The noise from 20to 200cm −1is sup-pressed well in Figs.9(c)and 9(d);on the contrary,the noise is enhanced in Fig.9(b).Moreover,the peak at 511cm −1cannot be split in Fig.9(c),but it can be split into two peaks at 511and 527cm −1in Fig.9(b)and Fig.9(d),respectively .Fig.9(d)can split more peaks than Fig.9(c),such as the peak at 527cm −1.The introduced method also has some intrinsic limitations.It is hard to assume the IRF width is accurate for the real Raman spectrum because differ-ent interferometers may produce different degraded Raman spectra,which need different-width IRFs.In this paper,the problem that we discuss and study is “what prior knowledge (adaptive total variation)is more suitable for spectral noise suppression and overlap band recovery.”In the future,we plan to re-search accurate IRF estimation and computation.5.ConclusionIn this paper,we proposed a MAP-based method to restore the degraded Raman spectrum,which can ob-tain peak structure information as well as suppress noise simultaneity.In this method,the likelihood probability P g ∕f is constructed with Gaussian noise assumed,and prior probability P f is con-structed as an ATV function.ATV regularization can make total variation regularization stable and distinguish the flat,noisy ,and steep regions.To re-duce the high computation load,the split Bregman iteration algorithm is employed to optimize the pro-posed model.Simulated and real experimental re-sults manifest that the proposed method can satisfactorily produce a deconvolution result with noise well suppressed.The recovered Raman spectraare easily utilized for extracting the spectral feature and interpreting the unknown chemical mixture.The authors thank the anonymous reviewers for their valuable comments.This research was partially funded by the Project of the Program for New Cen-tury Excellent Talents in University (NCET-11-0654),the National Key Technology Research and Development Program (2013BAH72B01),the Na-tional Key Technology Research and Development Program (2013BAH18F02),the Scientific R &D Project of State Education Ministry and China Mo-bile (MCM20121061),the National Social Science Fund of China (14BGL131),and the New Ph.D.Researcher Award from the Chinese Ministry of Education.The authors thank Prof.Jinghe Yuan for providing the HOS MATLAB source code.References1.A.L.Gallego,A.R.Guesalaga,E.Bordeu,and A.S.González,“Rapid measurement of phenolics compounds in red wine using Raman spectroscopy ,”IEEE Trans.Instrum.Meas.60,507–512(2011).2.P . 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