Deconvolution methods based on φ HL regularization for spectral recovery

Transcription

1 Research Article Vol. 54, No. 4 / May 25 / Applied Optics 4337 Deconvolution methods based on φ HL regularization for spectral recovery HU ZHU, LIZHEN DENG,, *XIAODONG BAI, MENG LI, 2 AND ZHAO CHENG 2 College of Telecommunication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 23, China 2 Institute of Manned Space System Engineering, China Academy of Space Technology, Beijing 94, China *Corresponding author: alicedenglzh@gmail.com Received 5 February 25; revised 2 April 25; accepted 2 April 25; posted 3 April 25 (Doc. ID ); published 5 May 25 The recorded spectra often suffer noise and band overlapping, and deconvolution methods are always used for spectral recovery. However, during the process of spectral recovery, the details cannot always be preserved. To solve this problem, two regularization terms are introduced and proposed. First, the conditions on the regularization term are analyzed for smoothing noise and preserving detail, and according to these conditions, φ HL regularization is introduced into the spectral deconvolution model. In view of the deficiency of φ HL under noisy condition, adaptive φ HL regularization (φ AHL ) is proposed. Then semi-blind deconvolution methods based on φ HL regularization (SBD-HL) and based on adaptive φ HL regularization (SBD-AHL) are proposed, respectively. The simulation experimental results indicate that the proposed SBD-HL and SBD-AHL methods have better recovery, and SBD-AHL is superior to SBD-HL, especially in the noisy case. 25 Optical Society of America OCIS codes: (37) Spectra; (334) Spectroscopy, infrared; (3.3) Absorption; (256) Optical data processing. INTRODUCTION The spectra data recorded by a spectrometer often suffer band overlapping caused by broadening effect of the instrumental response function, so that the measured spectra are always smoothed and degraded []. Therefore, spectral recovery is necessary and important for spectral analysis and assignment [2]. Mathematically, the process of spectral recovery is an inverse problem, and deconvolution has become one of the widely used methods [3]. However, deconvolution is an instable process and even is ill-posed in the case of low signal-to-noise ratios (SNRs). The result obtained by direct model is not always a satisfying solution. Some different strategies have been proposed to relieve the instability and ill-posedness of the deconvolution solution, such as Fourier deconvolution [4], the deconvolution of combination with linear prediction [5,6], and the deconvolution method based on maximum likelihood restoration [7]. One of the widely used approaches is adding the regularization term to the deconvolution mode, and the regularized solutions sometimes can represent a rather estimation for the recorved signal [8 ]. When using deconvolution method for spectral recovery, instrumental response function [also called point spread function (PSF)] [], which collects the line-shape function and the instrumental broadening (so it is also called blur kernel), is a very important factor. According to whether the PSF is known or not, the deconvolution methods can be divided into three typical forms: blind deconvolution (BD) for PSF is unknown, non-blind deconvolution (NBD) for the PSF is known, and semi-blind deconvolution (SBD) for the form of PSF is known. Blind deconvolution methods must estimate the PSF and the spectrum simultaneously from the measured data, as the highorder statistical Gauss Newton algorithm proposed by Yuan and Hu [2]. However, the unknown PSF makes the blind deconvolution very complicated, and the noise will make the problem more serious. Non-blind deconvolution method, such as the maximum Burg entropy (MaxEntD) method [3], has good performance in the deconvolution of absorption spectra if the PSF is exactly known, especially in the case of low SNR. However, the PSF cannot be exactly known in practice, and the mismatch between the assumed PSF and the true one will result in poor recovery. Semi-blind deconvolution method, such as spectral SBD with adaptive Tikhonov regularization [4], can recover the spectrum and estimate the parameter of the blur kernel. Compared with NBD and BD methods, SBD methods neither need to estimate the spectrum and the PSF simultaneously nor need to know blur kernel accurately. The main contributions of this paper are as follows. First, the conditions of details-preserving in spectral deconvolution are analyzed, and under these conditions, φ HL regularization [5] is introduced and used in the spectral deconvolution model, which has good effects on preserving the spectral details in the steep regions. Second, considering noise has some X/5/ $5/$5. 25 Optical Society of America

2 4338 Vol. 54, No. 4 / May 25 / Applied Optics Research Article influence on φ HL, an adaptive φ HL regularization is proposed for preserving spectral peak details better under noisy condition. Lastly, two semi-blind spectral deconvolution methods based on φ HL regularization (SBD-HL) and based on adaptive φ HL regularization (SBD-AHL) are proposed for spectral recovery, which not only can recover spectra with better detailspreserving performance, but also can estimate the width of blur kernel adaptively. The remainder of this paper is organized as follows. Sections 2 4 give the details of theory and methods. In Section 5, the performances of the proposed methods are verified by deconvolving simulated IR infrared spectra and a real experimental Roman spectrum. Section 6 draws the final conclusions. 2. SPECTRAL DECONVOLUTION MODEL The spectroscopic data measured by a spectrograph can be expressed as the sum of a random process n v and the linear convolution of a resolved signal f v and the blur kernel h v, g v f h v n v ; () where denotes the convolution operation, which is defined by f h v P kh k f v k. Generally, the resolved spectrum can be deconvolved by minimizing the cost function [4] E f 2 f h g 2 : (2) However, from Eq. () we can get G t f v I H t N t ; (3) H t where G t I g v, H t I h v, and N t I n v, I and I stand for the direct and inverse Fourier transform. It is known that H t is always cut at a limited time, namely, H t when t. Under this condition, N t H t will be an unpredictable value due to the experimental spectrum data always containing noise, so that f v solved by Eq. (3) will be an ill solution. Therefore, we cannot obtain even a poor approximation to the solution f v from g v and h v alone. Then the regularized solution sometimes represents a rather good estimation for f v [3]. Therefore, spectral deconvolution with the regularization term is formulized as E f 2 f h g 2 α XL φ jf i j ; (4) where f is the first derivative of f with respect to variable v, and j j denote 2-norm operator and the absolute value, L is the number of data points of the test spectrum. φ jf i j is the regularization term which is often formed by the first-order difference of spectrum. α is called regularization parameters controlling the smoothness of the resolved spectrum. Minima of Eq. (4), the Euler Lagrange equation with Neumann boundary will always be used: f h g h v α d φ jf j dv jf f ; (5) j where φ denotes the first derivative of φ with respect to variable. Similar with the definition of weighting i function in [6,7], φ is also called weighting function, which controls the diffusion degree of spectra during the spectral deconvolution process. 3. DETAILS-PRESERVING REGULARIZATION A. φ HL Regularization In addition to relieving the instability and ill-posedness of the deconvolution solution, another important effect of the regularization term in the spectral deconvolution model is smoothing noise while preserving the details. Generally, the spectral details usually refer to the information of steep regions near peaks. In order to have a good recover result, the conditions on the weighting function φ for smoothing noise while preserving details are given as follows: (i) φ is continuous and strictly decreasing on ;, which can avoid instabilities. (ii) lim φ M, <M<, which can make the regularization term do the smoothing in the flat regions. (iii) lim φ, which ensures that the details in steep regions can be preserved. Due to this term, we introduce φ HL proposed by Hebert and Leahy [5] in the spectral deconvolution model. φ HL regularization term is defined as φ HL ln 2 : (6) It is obvious that φ HL satisfies the conditions (i) (iii), namely, it can smooth the noise in the flat regions while it preserves the spectral details in the steep regions. The main reason we use φ HL as the regularization term for spectral recovery is that there are some similar characters between the gradient of the image and the derivative of the spectrum, and φ HL regularization has been used in image reconstruction for edgepreserving. The similar characters show mainly in the following aspects: () in homogeneous areas, the gradients of the image and the derivatives of the spectra are small; (2) the gradients of the image are large around edges and the derivatives of the spectra are large in steep regions. B. Adaptive φ HL Regularization From Eq. (6) we can find that the regularization term φ HL is a function of, which is greatly affected by the noise level. For the case of noise-free, in the steep regions is always larger than that in the flat regions. But in the noisy case, it is often not, especially when noise is strong, as it can be seen clearly from Fig.. Under this condition, using φ HL as regularization cannot preserve details well. In view of the deficiency of φ HL regularization, an adaptive φ HL regularization is proposed: φ AHL ln w 2 ; (7) where w is a adaptive coefficient, which is defined as w k2 k 2 f 2 : (8) Figure 2 gives comparison results of weighting functions corresponding to φ HL and φ AHL. It can be found that in the case of noise-free, both weighting functions can distinguish flat and steep regions. It is worth mentioning that in the steep

3 Research Article Vol. 54, No. 4 / May 25 / Applied Optics 4339 intensity f (a.u) intensity f (a.u) 2 5 wavenumber(cm ) wavenumber(cm ) (d) f..5 f wavenumber(cm ) wavenumber(cm ) Fig.. Example of spectra and the first derivative. Noise-free spectrum, noisy spectrum, the first derivative of noise-free spectrum, and (d) the first derivative of noisy spectrum. 5 regions φ AHL means that φ AHL is smaller obviously than φ HL, which will have much smaller diffusion in the steep regions. For the noisy case, φ HL flat and steep regions, but φ AHL cannot distinguish can distinguish them obviously. Therefore, for both cases of noise-free and noise, φ AHL will get better effects than φ HL, just the effects are much more obvious in the noisy case. It is noted that the adaptive coefficient only applies to the absorption spectra. Fig. 2. Effect of noise on weighting function. φ HL corresponding to Fig. ; and (d) φ AHL corresponding to Fig.. corresponding to Fig. ; φ HL corresponding to Fig. ; φ AHL

4 434 Vol. 54, No. 4 / May 25 / Applied Optics Research Article 4. SPECTRAL SEMI-BLIND DECONVOLUTION BASED ON φ HL REGULARIZATION From Eq. (4) we can find that, in order to solve f, the blur kernel h should be known prior. However, the blur kernel cannot be known exactly in practice, and the wrong kernel will result in a poor solution. According to the some experimental results and the prior knowledge about the instruments, the blur kernel resolved from measurement spectrum always approximates a Gaussian-like shape [8]. Due to this term, the blur kernel can be approximated as the form of Gaussian [9], Lorentzian, and so on. In order to be feasible for deconvolving the spectrum with varying degrees of degradation, the width of the blur kernel can be seen as an unknown parameter, and the parametric Gaussian blur kernel is modeled as h σ v ffiffiffiffiffi p exp v2 2π σ 2σ 2 ; (9) where σ is an unknown parameter, and the half-width pffiffiffiffiffiffiffiffiffiffiffiffi at halfmaximum (HWHM) of the Gaussian kernel is 2ln2σ. Then the energy function of spectral semi-blind deconvolution with φ HL regularization (SBD-HL) is given as E HL f;h σ 2 f h σ g 2 α XL φ HL jf i j βxl jhσij i i () and that with φ AHL regularization (SBD-AHL) is given as E AHL f;h σ 2 f h σ g 2 α XL φ AHL jf i j βxl jhσij: i i () Equations () and () can be solved using the same method. For simplicity, E HL and E AHL can be written as E. Minimizing E to find the resolved spectrum f and the blur kernel h σ can be converted into minimizing E [see Eq. (2)] and E 2 [see Eq. (3)] using the iterative alternate minimization (AM) approach [2]: E f 2 f h σ g 2 α XL i φ jf i j ; (2) E 2 h σ 2 f h σ g 2 β X jh σj; (3) where φ jf i j denotes φ HL jf i j or φ AHL jf i j. First, we fix h σ and minimize Eq. (2) for solving f. It can be found that Eq. (2) has the same form with Eq. (4), so we can solve Eq. (2) using the Euler Lagrange equation of Eq. (5). Here, for simplicity of computation, when computing the partial derivative with respect to f in the Euler Lagrange equation, w is considered a fixed constant. Then we fix f and minimize Eq. (3) for solving h σ. Due to h σ being determined by σ, Eq. (3) can be minimized using E 2 σ, namely, X hσ f h σ g σ f β jh σj ; (4) σ where hσ denotes the first derivative of h σ with respect to variable v. The numerical solution procedure of the alternate minimization approach to find the resolved spectrum f and the blur kernel h σ is illustrated as follows: Set α, β and the maximal iteration number N max ; Initialization: n, f g, σ σ, σ k ; while (jσ σ k j > ε or n<n max ), where ε is a small positive constant, and n is the iteration number. do { (i) Let σ k σ, n n ; (ii) Fix h σ, solve Eq. (5) using gradient descent flow method [2] and update f : f n f n Δt δe ; (5) δf where Δt is the step size, n is the iteration number. (iii) Fix f, solve Eq. (4) using bisection method [22] and update h σ. } end It is noted that in the iterative process, the first and second derivative are defined as f f i f i 2 and f f i 2f i f i, respectively. 5. EXPERIMENTAL RESULTS A. Deconvolution of Simulated Spectra. Simulated Spectra To verify the performance of the proposed deconvolution methods, we first give the simulated IR spectra, shown as Fig. 3. Figure 3 is the IR spectrum of,7-diaminoheptane (C 7 H 8 N 2 ) from 4 cm to 4 cm at cm resolution. Figure 3 is the simulated degraded spectrum obtained by convolving Fig. 3 with a Gaussian blur kernel h σ with σ 24 cm. The simulated noisy degraded spectrum with SNR is shown as Fig. 3, which is obtained by wavenumber(cm ) Fig. 3. Simulated spectrum. Original infrared spectrum. Degraded spectrum with noise-free. Degraded spectrum with SNR 72 84

5 Research Article Vol. 54, No. 4 / May 25 / Applied Optics 434 adding white Gaussian noise to Fig. 3. It is obvious that the degraded spectra become much smoother than the original one, especially the peaks at 2924 cm and 2852 cm meet together, and some bands such as between 3364 cm and 3284 cm and between 464 cm and 338 cm have distortions, which make it difficult to distinguish them Deconvolution Results of the Simulated Spectra For comparison, the FSD and MaxEntD methods are used here to deconvolve the degraded spectra. Figures 4 and 5 show the best deconvolution results of Figs. 3 and 3 using FSD, MaxEntD, SBD-HL, and SBD-AHL methods by changing the parameters. It can be found that in the noise-free case, SBD-HL and SBD-AHL can separate the peaks more clearly than FSD and MaxEntD, especially at 2924 cm and at 3284 cm.in the case of SNR, the ability of SBD-HL and SBD-AHL is superior to FSD and MaxEntD on splitting the peaks, while it is a little inferior to MaxEntD on suppressing noise. It is worth mentioning that in the test, the Gaussian kernel used in FSD and MaxEntD is the same as the accurate value used in the degraded simulation, while the ones used in SBD-HL and SBD-AHL are estimated adaptively, and the convergence processes of the kernel parameter σ in the cases of noise-free and SNR are shown in Fig. 6. It can be found that for both noise-free and SNR cases, σ approximately converges to 24, which is the same as the one used in the degraded simulation. Since the kernel is not known accurately in practice, FSD and MaxEntD would not get reliable results if the inaccurate kernel is used. 3. Quantitative Evaluation To measure the merits of the deconvolved spectra, quantitative performance evaluations, such as the root of mean square error (d) wavenumber(cm ) Fig. 5. Deconvolution results for case of SNR. Deconvolved by FSD method, MaxEntD method, SBD- HL method, and (d) SBD-AHL method. (RMSE), correlation coefficient (CC), and self-weighted correlation coefficient (WCC) are tested. They are defined as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P Li RMSE ˆf i f i 2 ; (6) L P Li CC ˆf i ˆf i f i f i r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h PLi ˆf i ˆf i 2ih P Li f i f i ; (7) i 2 2 P Li w WCC i ˆf i ˆf i f i f i r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h PLi w i ˆf i ˆf i 2ih P Li w i f i f i; (8) i σ value iteration number 3 (d) wavenumber(cm ) 5 5 Fig. 4. Deconvolution results for case of noise-free. Deconvolved by FSD method, MaxEntD method, SBD-HL method, and (d) SBD-AHL method. σ value 2 Noise free SNR= iteration number Fig. 6. Convergence process of σ. SBD-HL method and SBD-AHL method.

6 4342 Vol. 54, No. 4 / May 25 / Applied Optics Research Article where ˆf is the deconvolved spectrum, and f is the original spectrum. ˆf and f i represent the mean of corresponding spectra in CC and the weighted mean in WCC. The weighted mean is defined as ˆf i P w i ˆf i P w i and f i P w i f i P w i, and w is weight array (see [23]). Obviously, for the two compared spectra, the more similar they are, the smaller the RMSE is and the larger the CC and WCC are. It is worth to mention that RMSE and CC evaluate the average similarity between the two compared spectra, and WCC emphasizes the local detailed similarity. The test results of RMSE, CC, and WCC are listed in Table by comparing the simulated spectra [Figs. 3 and 3] and the corresponding deconvolved spectra (Figs. 4 and 5). It can be found that in both cases of noise-free and SNR, SBD-HL and SBD-AHL can achieve smaller RMSE and larger CC and WCC than FSD and MaxEntD. It is worth to mention that in the case of noise-free, SBD- AHL is just a little better than SBD-HL, but obviously better in the case of SNR. This is consistent with the above theory analysis result. In addition, for comparing the recovery results of FSD, MaxEntD, SBD-HL, and SBD-AHL methods in detail, the deviations of bands parameters, such as position, height, and full width of half-maximum (FWHM) between the deconvolved spectra and the original one are investigated and listed in Table 2. The five bands of the original spectrum [Fig. 3] at 2924 cm, 2852 cm, 6 cm, 464 cm, and 84 cm are listed here for reference. The RMSE of these deviations are calculated and listed in the last column. It can be found that in terms of RMSE, SBD-HL and SBD-AHL are smaller than FSD and MaxEntD except for the deviations of position in the noise-free case. Moreover, SBD-AHL has the same deviations with SBD-HL in position, and gets smaller RMSE than SBD-HL in terms of the deviations of height and FWHM. Therefore, it can be concluded that SBD-AHL and SBD-HL have better details-preserving effects than FSD and MaxEntD, and SBD-AHL is also better than SBD-HL. Table. Merits of Spectra Deconvolved by FSD, MaxEntD, SBD-HL, and SBD-AHL Methods Merits SNR FSD MaxEntD SBD-HL SBD-AHL Noise-free RMSE CC WCC SNR RMSE CC WCC Table 2. Deviations of Band Parameters between the Deconvolved Spectra and the Original Spectrum Band Position (cm ) RMSE FSD Position MaxEntD SBD-HL SBD-AHL FSD Noise-free Height MaxEntD SBD-HL SBD-AHL FSD FWHM a MaxEntD SBD-HL SBD-AHL FSD Position MaxEntD SBD-HL SBD-AHL FSD SNR Height MaxEntD SBD-HL SBD-AHL FSD FWHM MaxEntD SBD-HL SBD-AHL Note: + or indicates larger or smaller than the original one, respectively. a For the overlapped bands, FWHM is the FWHM of the overlapped band divided by the number of the peaks included in it.

7 Research Article Vol. 54, No. 4 / May 25 / Applied Optics 4343 B. Deconvolution of Real Experimental Spectra In order to demonstrate the validity of the SBD-AHL and SBD-HL methods, the real Raman spectrum of a-lactose (C 2 H 22 O -H 2 O)[24] is taken as an example. The spectrum from 6 to cm at cm resolution is shown in Fig. 7, and the recovery results deconvolved by the SBD-HL and SBD-AHL methods are shown as Figs. 7 and 7, respectively. Obviously, the deconvolved spectra are far more resolved than the measured experimental. It is worth to mention that, here we do not explain the spectrum in terms of its physical interpretation, but observe the width and height of existed bands, and the inflection points which would be the spectral wavenumber(cm ) wavenumber(cm ) wavenumber(cm ) Fig. 7. Deconvolution results of real Raman spectrum. Raman spectrum of a-lactose (C 2 H 22 O -H 2 O) from 6 to cm. Deconvolved by SBD-HL method. Deconvolved by SBD-AHL method. bands, such as the inflection points at band 379, 263, and 4 cm. Compared with the measured experimental spectrum, the existed bands such as 47, 348, 335, 327, 86, 53, and 2 cm are narrowed obviously in the deconvolved spectra. Band 379 cm is split into bands 38 and 369 cm ; band 263 cm is split into bands 285, 273, 263, and 246 cm ; band 4 cm is split into bands 4 and 3 cm. In addition, the bands that are just split slightly in the original experimental spectrum are separated and narrowed clearly, such as bands 358, 42, and 34 cm. 6. CONCLUSIONS Two SBD methods based on φ HL and φ AHL (SBD-HL and SBD-AHL) are proposed for spectral recovery. First, we explain the necessity of adding the regularization term in the spectral deconvolution model and analyze the conditions on regularization for smoothing noise while preserving details. Based on these conditions, φ HL is introduced in the spectral deconvolution model. Then in view of the deficiency of φ HL under noisy condition, φ AHL regularization is proposed. Second, the two SBD methods (SBD-HL and SBD-AHL) based on φ HL and φ AHL regularization are proposed. Lastly, the recovery performance of the SBD-HL and SBD-AHL methods are verified by deconvolving simulated IR spectra and a real experimental Roman spectrum. The recovery results indicate that the proposed SBD-HL and SBD-AHL methods not only can recover the spectra with better details-preserving performance, but also can estimate the blur kernel adaptively. Moreover, SBD-AHL has better performance than SBD-HL, especially on the noisy condition. However, the proposed SBD-AHL and SBD-HL methods also have some limitations. First, the ability of the proposed models on suppressing noise is still inferior to MaxEntD (it can be seen from Fig. 5), so it should be improved in further research. Second, although the blur kernel can be approximated as the form of Gaussian or Lorentzian in general, it would not be well applied in recovering sophisticated real spectra whose shape could be the synthetic of Gaussian, Lorentzian, and so on. Third, the model is solved by alternate iterative optimization, so it makes sense to further study a fast and robust iterative algorithm for reducing the time cost. Finally, during the numerical solution procedure, the regularization parameters are set manually according to the test spectra and the experience value, so a strategy for setting regularization parameters adaptively should be researched in further study. Natural Science Foundation of Jiangsu Province (BK24874); School Foundation of Nanjing University of Posts and Telecommunications (NY2339, NY244); Startup Funds for Distinguished Scholars (NY2366). REFERENCES. L. Xu, H. Yang, K. Chen, Q. Tan, Q. He, and G. Jin, Resolution enhancement by combination of subpixel and deconvolution in miniature spectrometers, Appl. Opt. 46, (27). 2. K. Chen, H. Zhang, H. Wei, and Y. Li, Improved Savitzky Golay-method-based fluorescence subtraction algorithm for rapid recovery of Raman spectra, Appl. Opt. 53, (24).

8 4344 Vol. 54, No. 4 / May 25 / Applied Optics Research Article 3. P. A. Jansson, Deconvolution: With Applications in Spectroscopy (Academic, 984). 4. J. K. Kauppinen, D. J. Moffatt, H. H. Mantsch, and D. G. Cameron, Fourier self-deconvolution: a method for resolving intrinsically overlapped bands, Appl. Spectrosc. 35, (98). 5. J. K. Kauppinen, D. J. Moffatt, M. R. Hollberg, and H. H. Mantsch, New line-narrowing procedure based on Fourier self-deconvolution, maximum entropy, and linear prediction, Appl. Spectrosc. 45, 4 46 (99). 6. P. E. Saarinen, Spectral line narrowing by use of the theoretical impulse response, maximum entropy, and linear prediction, Appl. Spectrosc. 5, 88 2 (997). 7. W. I. Friesen and K. H. Michaelian, Deconvolution and curvefitting in the analysis of complex spectra: The CH stretching region in infrared spectra of coal, Appl. Spectrosc. 45, 5 56 (99). 8. A. Mohammad-Djafari, J. F. Giovannelli, G. Demoment, and J. Idier, Regularization, maximum entropy and probabilistic methods in mass spectrometry data processing problems, Int. J. Mass Spectrom. 25, (22). 9. Z. Xu and E. Lam, Maximum a posteriori blind image deconvolution with Huber Markov random-field regularization, Opt. Lett. 34, (29).. S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, and F. E. Abd El-Samie, Regularized super-resolution reconstruction of images using wavelet fusion, Opt. Eng. 44, 972 (25).. M.-Y. Zou and R. Unbehauen, A deconvolution method for spectroscopy, Meas. Sci. Technol. 6, (995). 2. J. Yuan and Z. Hu, High-order statistical blind deconvolution of spectroscopic data with a Gauss Newton algorithm, Appl. Spectrosc. 6, (26). 3. V. A. Lorenz-Fonfra and E. Padros, Maximum entropy deconvolution of infrared spectra: use of a novel entropy expression without sign restriction, Appl. Spectrosc. 59, (25). 4. L. Yan, H. Liu, S. Zhong, and H. Fang, Semi-blind spectral deconvolution with adaptive Tikhonov regularization, Appl. Spectrosc. 66, (22). 5. T. Hebert and R. Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag. 8, (989). 6. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Trans. Image Process. 6, (997). 7. S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, Variational approach for edge-preserving regularization using coupled PDE s, IEEE Trans. Image Process. 7, (998). 8. J. Yuan, Z. Hu, and J. Sun, High-order cumulant-based blind deconvolution of Raman spectra, Appl. Opt. 44, (25). 9.Q.Wang,D.D.Allred,and L.V.Knight, Deconvolution of the Raman spectrum of amorphous carbon, J. Raman Spectrosc. 26, (995). 2. T. F. Chan and C.-K. Wong, Total variation blind deconvolution, IEEE Trans. Image Process. 7, (998). 2. P. C. Hansen, The blurring function, in Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 26), pp C. B. Morler, Numerical Computing with MATLAB (SIAM, 24). 23. P. R. Griffiths and L. Shao, Self-weighted correlation coefficients and their application to measure spectral similarity, Appl. Spectrosc. 63, (29). 24. S. B. Engelson,