blind poissonian reconstruction algorithm via curvelet

Blind Poissonian reconstruction algorithm via curvelet regularization for FTIR spectrometer

LIU, Hai; LI, Youfu; ZHANG, Zhaoli; LIU, Sanya; LIU, Tingting

Published in:Optics Express

Published: 03/09/2018

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Published version (DOI):10.1364/OE.26.022837

Publication details:LIU, H., LI, Y., ZHANG, Z., LIU, S., & LIU, T. (2018). Blind Poissonian reconstruction algorithm via curveletregularization for FTIR spectrometer. Optics Express, 26(18), 22837-22856.https://doi.org/10.1364/OE.26.022837

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https://doi.org/10.1364/OE.26.022837

Blind Poissonian reconstruction algorithm via curvelet regularization for FTIR spectrometer

HAI LIU,1,2,3 YOUFU LI,2,* ZHAOLI ZHANG,1,3 SANYA LIU,1 AND TINGTING

LIU1

1National Engineering Research Center for E-Learning, Central China Normal University, Wuhan 430079, China 2Department of Mechanical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China 3These authors contributed equally to this work *[email protected]

Abstract: FTIR spectrometer often suffers from common problems of band overlap and Poisson noises. In this paper, we show that the issue of infrared (IR) spectrum degradation can be considered as a maximum a posterior (MAP) problem and solved by minimized a cost function that includes a likelihood term and two prior terms. In the MAP framework, the likelihood probability density function (PDF) is constructed based on the observed Poisson noise model. A fitted distribution of curvelet transform coefficient is used as spectral prior PDF, and the instrument response function (IRF) prior is described based on a Gauss-Markov function. Moreover, the split Bregman iteration method is employed to solve the resulting minimization problem, which highly reduces the computational load. As a result, the Poisson noises are perfectly removed, while the spectral structure information is well preserved. The novelty of the proposed method lies in its ability to estimate the IRF and latent spectrum in a joint framework, thus eliminating the degradation effects to a large extent. The reconstructed IR spectrum is more convenient for extracting the spectral feature and interpreting the unknown chemical or biological materials. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

OCIS codes: (300.6340) Spectroscopy, infrared; (200.4560) Optical data processing; (040.2235) Far infrared or terahertz; (100.1455) Blind deconvolution; (100.3190) Inverse problems.

References and links

1. A. L. Serrano, A. Ghosh, J. S. Ostrander, and M. T. Zanni, “Wide-field FTIR microscopy using mid-IR pulse shaping,” Opt. Express 23(14), 17815–17827 (2015).

2. G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).

3. M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J.Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopyto analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).

4. N. P. Ayerden and R. F. Wolffenbuttel, “The Miniaturization of an Optical Absorption Spectrometer for Smart Sensing of Natural Gas,” IEEE Trans. Ind. Electron. 64(12), 9666–9674 (2017), doi:10.1109/TIE.2017.2719600.

5. W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).

6. K. Kondo, Y. Ichioka, and T. Suzuki, “Image restoration by Wiener filtering in the presence of signal-dependent noise,” Appl. Opt. 16(9), 2554–2558 (1977).

7. M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).

8. F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “IndustrialApplications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).

9. Y. Senga, K. Minami, S. Kawata, and S. Minami, “Estimation of spectral slit width and blind deconvolution of spectroscopic data by homomorphic filtering,” Appl. Opt. 23(10), 1601–1608 (1984).

10. 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(3), 271–276 (1981).

11. V. A. Lórenz-Fonfría and E. Padrós, “The Role and Selection of the Filter Function in Fourier Self-Deconvolution Revisited,” Appl. Spectrosc. 63(7), 791–799 (2009).

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12. P. A. Jansson, Deconvolution: with applications in spectroscopy (Academic Press, New York, USA, 1984). 13. P. B. Crilly, “Increased throughput for process chromatography using constrained deconvolution,” IEEE Trans.

Ind. Electron. 39(1), 20–24 (1992).14. S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic

peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).15. J. Katrašnik, F. Pernuš, and B. Likar, “Deconvolution in Acousto-Optical Tunable Filter Spectrometry,” Appl.

Spectrosc. 64(11), 1265–1273 (2010).16. S. Chen, G. Wang, X. Cui, and Q. Liu, “Stepwise method based on Wiener estimation for spectral reconstruction

in spectroscopic Raman imaging,” Opt. Express 25(2), 1005–1018 (2017).17. C. W. Helstrom, “Image Restoration by the Method of Least Squares,” J. Opt. Soc. Am. 57(3), 297–303 (1967).18. J. Yuan, Z. Hu, and J. Sun, “High-order cumulant-based blind deconvolution of Raman spectra,” Appl. Opt.

44(35), 7595–7601 (2005).19. T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial

Infrared Spectrometers," IEEE Trans. Ind. Inf. DOI: 10.1109/TII.2018.2794449 (2018). 20. L. Yan, H. Liu, S. Zhong, and H. Fang, “Semi-blind spectral deconvolution with adaptive Tikhonov

regularization,” Appl. Spectrosc. 66(11), 1334–1346 (2012).21. K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components

characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015). 22. H. Liu, Z. Zhang, S. Liu, T. Liu, L. Yan, and T. Zhang, “Richardson-Lucy blind deconvolution of spectroscopic

data with wavelet regularization,” Appl. Opt. 54(7), 1770–1775 (2015).23. H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind

deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).24. H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified

φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).25. A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast

Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).26. L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl.

Spectrosc. Rev. 37(4), 429–450 (2002).27. S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by

sparse deconvolution with an interframe penalty,” J. Chemom. 31, e2847 (2017).28. Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse

representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).29. C. J. Manning and P. R. Griffiths, “Noise Sources in Step-Scan FT-IR Spectrometry,” Appl. Spectrosc. 51(8),

1092–1101 (1997). 30. E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model.

Simul. 5(3), 861–899 (2006).31. J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image

Process. 11(6), 670–684 (2002).32. T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375

(1998).33. T. Goldstein and S. Osher, “The Split Bregman Method for L1-Regularized Problems,” SIAM J. Imaging Sci.

2(2), 323–343 (2009).34. A. Mukherjee and A. Sengupta, “Estimating the Probability Density Function of a Nonstationary Non-Gaussian

Noise,” IEEE Trans. Ind. Electron. 57(4), 1429–1435 (2010).35. H. G. Schulze, M. M. L. Yu, C. J. Addison, M. W. Blades, and R. F. B. Turner, “Automated Estimation of White

Gaussian Noise Level in a Spectrum With or Without Spike Noise Using a Spectral Shifting Technique,” Appl. Spectrosc. 60(7), 820–825 (2006).

36. S. B. Engelson, Infrared Spectral of Butyl propionate http://www.models.life.ku.dk/specarb (2018). 37. S. B. Engelson, Infrared Spectral of D (+) -Glucopyranose http://www.models.life.ku.dk/specarb (2018).

1. Introduction

Fourier transform infrared (FTIR) spectrometer has been widely used in the industrial fields of rapid identification of chemicals [1], analytical techniques and biomedicine [2, 3], remote sensing detection [4]. Unfortunately, the Fourier transform infrared spectrometer has intrinsic physical limitations, which in combination with blurring effect, resolution decimation and Poisson noises (shown in Fig. 1), tending to reduce the overall quality of spectral resolution. Also, the resolution is affected by the ambient temperature and pressure. As a result of those degradations, the measured spectral data are prone to misinterpretation, affecting the further processing accuracy. Thus, it is crucial to suppress the noise in IR spectrum before the succeeding spectral interpretation processes. The purpose of spectral reconstruction algorithm is to seek the best estimations of latent spectrum and instrument response function from the measured IR spectrum.


Fig. 1. Spectral reconstruction for FTIR spectrometer. Original spectrum is contaminated by the band overlap and Poisson noises, which are caused by blurring effect and resolution decimation. The degraded spectrum can be recovered by the spectral reconstruction algorithm.

1.1 Related reconstruction algorithms

In recent decades, many IR spectral reconstruction algorithms have been proposed [5–9]. The current arsenal of spectral reconstruction techniques can broadly be classified into two classes: filter-based reconstruction method (FBR) and regularization-based reconstruction method (RBR). For the FBR method, Wiener filtering [6] is one of the most popular methods. In [7], Kalman-filter-based algorithms were proposed for improving the resolution of IR spectrum, which are widely used for industrial applications [8]. In [9], Senga developed a homomorphic filter to reconstruct the IR molecular absorption spectra successfully. Kauppinen et al. proposed the Fourier-self deconvolution method (FSD), which is one of the most famous method used in IRS [10]. However, the filtering function is fixed for all the reconstruction tasks. To solve the problem of filter function selection, Lórenz-Fonfría [11] discussed nine kinds of filter functions for different cases in FSD method. Their conclusion is that the Bessel filter and Gaussian filter are the most suitable filter functions for the FSD method. The constrained reconstruction algorithm with a relaxation weighting function proposed by Jansson had been proved to be practically effective [12, 13]. In [14], a blind reconstruction method was proposed which allows the reconstruction of broad-band signals passing through a low-pass filter. But it only works well on some specific spectrum cases. To address this problem, KatraSnik [15] developed an acousto-optical tunable filter to enhance the spectral resolution, which gave good results for line spectra and highly dynamic spectra. However, it failed to work with low signal-to-noise ratio (SNR) IR spectrum. Recently, Wiener estimation method [16] has been proposed for spectra with heavy noise, which has achieved the impressive reconstructed results.

For the RBR method, least squares method [17] was first proposed without any prior knowledge. However, this method amplifies noise after a few iterations, which can be avoided by incorporating prior knowledge about the solution. The prior knowledge include high-order statistic [18], Tikhonov regularization (TR) [19, 20] and its variants [21], Shannon-entropy regularization, and total variation (TV) [22] regularization. In the existing regularization methods, the TV regularization-based method [22] is commonly used for IR spectral data processing. However, the TV often yields the so-called staircasing effect, a phenomenon which stems from the fact that TV regularization has a tendency to produces piecewise constant solution. To overcome this problem, the authors proposed an adaptive Huber-Markov function [23] to process the spectral random noise successfully. Recently, Zhu et al. [24] proposed a novelty reconstruction method to recover the spectrum with the Hebert-Seahy (φHS) regularization. The reconstructed results seemed rather unnatural, especially when IR spectrum suffered from significant noises and heavy band overlap.


1.2 Proposed algorithm

Generally, the IR spectrum can be divided into three different regions. Flat region is where the region has no absorption peak. The spectral points have the equational intensity value with their neighborhood points. Shoulder region includes the points at the shoulder of absorbance peak. The spectral points have the dramatic different their surrounding points. Noises region includes the Poisson noise and overlap band points, whose gradient value is between the flat region and the shoulder region. In Figs. 2(a) and 2(b), it shows the three different types regions in IR spectrum.

It is worth noting that most of the aforementioned IR spectral reconstruction methods remove the degradation effects according to the gradient value of noise points. Their gradient values are between the flat points and the shoulder points. The drawback of processing the spectral degradation according the gradient information is that it is hard to set a precise value to distinguish the noise from the degradation spectrum. Many IR spectral denoising methods are developed using the frequency domain information to improve the IR spectral quality, such as fast Hartley transform [25], and wavelet transform [26], in which the coefficients are simply set to 0 when each wavelet coefficient is smaller than a predetermined value. Thus, how to utilize the frequency domain information in IR spectrum reconstruction is a noteworthy point. Recently, two IR spectrum reconstruction algorithms using sparse representation to exploit the spectral characteristics have been developed [27, 28].

In this work, from a frequency transform domain smoothness perspective, we develop a blind reconstruction method to recover the latent spectrum and estimate the IRF from the measured IR spectrum. In this method, the IR spectrum is sparsely represented in the curvelet transform. The main ideas and novelties of the proposed method can be summarized as three aspects.

1) Our probabilistic model converts the IR spectrum reconstruction problem into themaximum a posteriori (MAP) formulation.

2) The curvelet coefficient distribution of high-resolution IR spectrum is fitted with anexponential function, which is utilized to constrain the degraded IR spectrum. In thiswork, we develop a blind spectral reconstruction model with the curvelet transform regularization.

3) To make the proposed method simple and fast to implement, the split Bregmanmethod is employed to optimize the curvelet transform-based minimization problem,where the optimization of the reconstruction model is split into some easier sub-problems.

1.3 Organization of this paper

The remainder of this paper is organized as follows. In the next section, the IR spectrum observation model and spectral characteristics are introduced. The MAP-based reconstruction functional is proposed in Section 3. Section 4 provides the numerical details of the minimization procedure. The experimental results and discussions are illustrated in Section 5. Finally, the conclusions are given in Section 6.


Fig. 2. Curvelet transform coefficient comparison between the high-resolution IR spectrum and the degraded one. (a) Observed IR spectrum of methyl (C15H29NO7), which suffers from the band overlap and random noises. This degradation often appears in applications of IR spectroscopy. (b) Measured by a high-resolution spectrometer. IR spectrum can be classified into three different regions, such as flat region, shoulder region and noises region. (c)-(d) Histogram statistics for curvelet transform coefficient in Figs. 2(a) and 2(b).

2. Analysis of IR spectral characteristics

2.1 Problem formulation

IR spectra degradation process can be modelled by convolutional instrument response function (IRF) with contamination by Poisson noises [29]. Generally, the observed noisy spectrum o∈RN (N denotes the IR spectrum length) can be illustrated in two equivalent matrix-vector forms,

( ) = { { ( )}} { { ( )}}v P v P v=o F g G f (1)

where v denotes spectral wavenumber, F∈RN × N represents the matrix notation of the IRF f∈RN, and G∈RN × N denotes the matrix form of the ground-truth spectrum g∈RN. { }P • is

the process of Poisson noises in IR spectrum.

2.2 Curvelet transform

The curvelet transform, developed by Candes [30], is a higher dimensional generalization of the wavelet transform designed to represent signals at different scales and different angles. Rather than tracking the shape of the discontinuity set as the wavelet transform does, the curvelet provides a stable, efficient, and near-optimal representation of otherwise smooth objects having discontinuities along smooth curves. An important ingredient of the curvelet transform is to restore sparsity by reducing redundancy across scales. The curvelet provides a sparse representation of both smooth functions and perfectly straight functions combined with a spatial bandpass filtering operation to isolate different scales.

Curvelet transform [30] is designed to handle curves using only a small number of coefficients. Hence the curvelet handles curve discontinuities well. In this work, we analyze the coefficients distribution of the IR spectrum based on the famous curvelet transform.


Fig. 3. Probability density function of the curvelet coefficients of the IR spectra (in Fig. 2). (a) Natural logarithm of the probability density function in Fig. 2(c) and 2(d), as shown by the black and blue curves. (b) Two curves are fitted by the constructed functions in Fig. 3(a). The red, yellow, and blue regions denote the low, middle, and high frequency, respectively.

2.3 IR spectrum curvelet coefficient distribution

Figure 2(a) is the observed IR spectrum of methyl (C15H29NO7). Figure 2(b) is measured by a high-resolution spectrometer, without the problems of band overlap and random noises. It is observed that the peaks in Fig. 2(b) are sharper and narrower than those in Fig. 2(a).

First, Figs. 2(a) and 2(b) are transformed by the curvelet transform. Then, their curvelet transform coefficients are calculated by the histograms, which are plotted in Figs. 2(c) and 2(d), respectively. The close-ups (from 0.02 to 0.04) are shown at the upper-right corner. It can be observed that most curvelet transform coefficients approach zero value. Moreover, the degradation effect of the spectrometer changes the coefficients distribution of the ground-truth IR spectrum. It is considered that the high-frequency components (such as shoulder region, structural details) of Fig. 2(b) are richer than those of Fig. 2(a). It motives us that the degraded spectrum can be reconstructed by adjusting the ratios among high-frequency components (shoulder region, peaks), mid-frequency components (random noises and overlap band), and low-frequency (flat region).

Figure 3 shows the probabilistic distribution of curvelet transform coefficient. This

distribution is fitted with an exponential function ( )expp

y a b= − −Cg , where a>0 is a

scale factor and 0<p<2, b is a constant (b can be ignored as it does not impact the optimization). In Fig. 3(a), the black and blue curves denote the natural logarithm of the probabilistic distribution. The ground-truth distribution can be fitted as

( )1exp 18 0.96y = − × −Cg , namely,

( ) ( )1

1 1exp .p λ∝ −Cg Cg (2)

3. Proposed model

3.1 Spectrum reconstruction for FTIR spectrometer

In this paper, the MAP framework is introduced for infrared spectrum reconstruction. Given the overlap and noisy IR spectrum, the latent spectrum and IRF can be estimated by the following MAP model:

arg max ( , )MAP p=g g f o (3)

By using the Bayes theorem, the posteriori probability is ( ) ( ) ) ( ).p p p p∝g,f o o g,f (g f .

Equation (3) can be written

, arg max{ ( , ) ( ), ( )}p p p< >=g f o g f g f (4)


where p(o|g, f), p(g) and p(f) denote the likelihood probability, the prior probability of g and f, respectively. In the following, the likelihood probability and two prior probabilities will be formulated. Using the negative monotonic logarithm function, (4) can be expressed as

, arg max{ log ( , ) log ( ) log ( )}.p p p< >= − − −g f o g f g f (5)

It can be seen that three probability functions need to be formulated. First, we define the likelihood probability function. For the independent and identically distributed Poisson noises, the likelihood probability as

( )o( ) exp( )

, .!

vNv v

v v

p−

= ∏ Fg Fgo g f

o(6)

The first term ( )log ,p− o g f in (5) can be computed as

( ) ( ) exp( )log , log

!

[ log( ) ( ) log !]

[( ) log( ) ] log ! !.

vNv v

v v

N

v v v vv

N N

v v v v vv v

p−

− = −

= − − −

= − +

oFg Fgo g f

o

o Fg Fg o

Fg o Fg o o

(7)

Dropping the constant log !N

vv o term, the (7) can be formulated as

( )log , [( ) log( ) ]N

v v vv

p− ∝o g f Fg - o Fg (8)

3.2 Prior for latent spectrum p(g)

The second probability function in (5) is the spectrum prior, which imposed the spatial constraints on the spectrum. According to the analysis of the curvelet transform coefficients,

it finds that the coefficients obey the fitted distribution (2), and ( ) ( )p p∝g Cg , namely,

( ) ( )1 1expp λ∝ −g Cg (9)

where C is the curvelet transform [31]. The criticism of the use of the fitted distribution is that some middle-frequency components in the curvelet coefficients tend to be adjusted. For the flat region (red region in Fig. 3(b)), the low-frequency components should be increased. For the noise region (green region), the mid-frequency components should be decreased. For the shoulder region, the high-frequency components should be decreased, since the noises exist in the observed spectrum. The structure information in the IR spectra can be preserved effectively by the curvelet transform prior.

3.3 Prior for IRF p(f) The third probability function in (5) is the IRF prior. Without any a prior knowledge about the IRF, the IRF estimated from the observed spectrum often approximates a Gaussian-like line shape [17, 18]. To describe the smoothness of the IRF, the difference matrix is employed. Since the length of spectrum is N, the first-order difference matrix will be of size of (N-1) × N. For the Gaussian-shape IRF, Gauss-Markov prior is selected to constrain the smoothnessof the IRF,


( ) ( )2

2exp ,p λ∝ −f Df (10)

where Df stands for the first derivative of f.

3.4 Proposed model

Substituting the definitions (9) and (10) to (5), it is straightforward to show that the MAP estimation problem is equivalent to a cost function minimization problem. After some manipulation, the constant terms can be dropped, we get the proposed model:

2

1 2 011

, arg min [( ) log( ) ( )],N

v

λ λ ≥=

< >= − + + + Θ gg,h

g f Fg o Fg Cg Df g (11)

where λ1 and λ2 are the regularization parameters, and the 0 ( )≥Θg g means that the latent

spectrum intensity is a set of positive values. We explain each term of the model as follows:

1) The first term is the reconstruction constraint, i.e. the reconstructed result should beconsistent with the observation according to the observed model.

2) The second term penalizes the L1-norm of the curvelet transform coefficients of thespectrum g, so as to suppress the Poisson noises. In other words, it favors a solutiong that its curvelet coefficients obey the groundtruth distribution fitted in Fig. 3(b).

3) The third term enforces the L2-norm (Gauss-Markov) constraint on the gradient of theIRF, aiming to achieve a solution f, whose derivative is rather smooth.

4) The fourth term imposes the non-negative constraint on the latent spectrum g. It isvery reasonable since the transmittances of the IR spectrum are positive values.

In summary, the basic idea of the model is to penalize the curvelet coefficients distribution of the latent spectrum while enforcing the L2-norm constraint on the IRF to regularize its smoothness. And the latent spectrum should keep the positive intensity value at the same. We call the proposed method as fast blind Poissonian spectral reconstruction method (FBPSR) via curvelet transform regularization.

4. Optimization algorithm

The alternating minimization (AM) algorithm [32] is applied to find g and f. This iterative method is initialized g0 with the observed spectrum o, and the cost function is minimized with respect to f, thereby a latent IRF is estimated. Then, a new g is obtained for a fixed f. To be specific, the following two steps are implemented iteratively: estimating the IRF f, assuming the latent spectrum g is fixed, and estimating the g, assuming f is fixed. Before f is fixed, the latent spectrum is first initiated as g0 = o. This procedure is repeated until convergence is reached.

4.1 Updating the IRF f In the f-step, we fix g and optimize f. The function (11) can be simplified:

2

21

min [( ) log( )]N

v

λ=

− +f

Fg o Fg Df (12)

The update of the IRF f can be written as

1 *

2

( )1

k kk k

T kλ

+ = −

f of G

D Df G f

(13)


11

1

1

( )

kk

N k

vv

++

+

=

=

ff

f

(14)

where the superscript * is the adjoint operation.

4.2 Updating the latent spectrum g

In this step, we fix f and compute the optimal g. The (11) is simplified to

1 011

arg min [( ) log( )] ( ),N

N

R v

λ ≥∈ =

− + + Θ gg

Fg o Fg Cg g (15)

the difficulty in minimizing this sub-problem lies in optimizing the non-smooth and non-separable term ||Cg||1. The split Bregman iteration method [33] is utilized to optimize the function. The basic idea of split Bregman iteration is to convert the unconstrained minimization problem on spectrum g in (15) into a constrained one by introducing auxiliary variables 1d = Fg , 2d = Cg , and 3d = g . Then, the problem is transformed into an

equivalent problem:

3

1 2 3

1 2 0 31, , , 1

2 3

arg min [( log ] ( )

,

N

N

R v

such that

λ ≥∈ =

− + + Θ

= =

1 1 dg d d d

d o d d d

d = Fg d Cg, d g.

(16)

Further, the (16) can be converted into an unconstrained problem via the Bregman iteration:

}||||||||||{||2

1

)(||||]log[min

2233

2222

2211

3031211

113,2,1,

dgbdCgbdFgb

dddod ddddg

−++−++−++

Θ++ − ≥=∈

α

λN

vNR (17)

where α is the Bregman penalty parameter. There are certain advantages to formulate the problem since the unconstrainted problem (17) can be solved by a simple alternating minimization scheme. Thus, the minimization of (17) can be performed alternately with the following four subproblems:

−+=−+=−+=

−++=

−++=

−++−=

−++

−++−+=

+++

+++

+++

+≥

+

++

+

=

+

∈

+

)(

)(

)(

||||2

1)(Θminarg

||||2

1||||minarg

||||2

1]log[minarg

},||||

||||||{||2

1minarg

13

13

13

12

12

12

11

11

11

223

1330

13

222

12121

12

221

11

111

11

2233

2222

2211

1

32

1

1

kkkk

kkkk

kkkk

kkk

kkk

kkN

v

k

kk

kkkk

R

k

α

αλ

α

αN

dgbb

dCgbb

dFgbb

dgbdd

dCgbdd

dFgbdodd

dgb

dCgbdFgbg

dd

d

d

g

(18)

where k is the iteration number. In (18), each iteration of the g subproblem is obtained by solving the following linear system:

T 1 T T1 1 2 2 3 3( 2 ) ( ) ( ) ),k k k k k k k++ = − + − + −F F I g F d b C d b d b (19)

which can be solved with a closed-form solution by the fast Fourier transform (FFT), (18) as shown in the following,


T T1 1 1 1 2 2 3

T

( ( ) ( ) )

( ) 2

k k k k kk F

FF

+ − − + − − += +

F d b C d b bg

F F I(20)

where I is the identity matrix, ( )F • denotes the fast Fourier transform and 1( )F − • the

inverse transform. The superscript T denotes the transpose operation. The second subproblem in (18) can be solved easily as follows:

1 22

1( ( ) 4 )

2k k kη η α+ = + +d o (21)

where 11

k k kη α+= + −b Fg . Similarly, 12k +d and 1

3k +d can be also derived simultaneously:

11 1 1 2

2 2 1 12 2

1 1 13 3

max{ ,0}.

max{ ,0}

k kk k k

k k

k k k

λ α+

+ + ++

+ + +

+= + − +

= +

Cg bd Cg b

Cg b

d g b

(22)

The split Bregman method can split the difficult optimization problem in Eq. (16) into four sub-problems, which can be easily optimized. The g-related sub-problem is accelerated by FFT efficiently. Moreover, d1, d2 and d3–related sub-problems are independent and can be efficiently calculated in parallel. Thus, b1, b2 and b3 can be updated parallelly.

4.3 Optimization details and parameters

In (19), the auxiliary variables are set as 0 0 0 0 0 01 2 3 1 2 3 0= = = = = =d d d d d d , matrix F is the

block-circulant with circulant-block matrix. Discrete curvelet transform is utilized to transform the original IR spectrum into the curvelet coefficients. Then, the curvelet coefficients are adjusted by the fitted distribution, and the adjusted curvelet coefficients can be achieved. In the following, those coefficients are transformed back by the inverse discrete curvelet transform (IDCT).

Two regularization parameters λ1, and λ2 in proposed method are adjustable, which are shown in Eqs. (9) and (11). They correspond to the probability parameters in (12) and (16) for the IRF and latent spectrum. And their values are adapted from their initial values with the number of iterations. We set λ1 = λ1/t1, λ2 = λ2/t2. Then, after each iteration of optimization, the values of λ1 and λ2 are divided by t1 and t2, respectively, where we usually set t1 = 1.02, t2 = 1.01 to reduce the influence of the spectrum smoothness and increase that of the spectrum likelihood.

To estimate the noises level, the median absolute differences are used to estimate δ defined in the following equation [34]:

1

1.4826{ , 2...., }.

2v vMedian v Nδ −= − =o o (23)

It is worth noting that the method proposed by Turner [35] using second-order difference

spectra to compute the noise level. Initialize the regularization parameters, we set 01λ = 20δ,

02λ ∈ [150, 350] and α = 20/λ1. The intensity of IR spectra is normalized to the range [0, 1].

To measure the convergence, the “normalized step difference energy” is computed for each iteration: ||fk+1-fk||2/||fk||2<η1 and ||gk+1-gk||2/||gk||2<η2, where η1 and η2 are the small constants. The algorithm is implemented as a visual program in Matlab 2016a, and the source codes are available upon request. Taking into account the above equations, the complete iteration can be achieved for blind Poissonian spectral reconstruction as Algorithm 1.


Algorithm 1. Numerical algorithm for fast blind Poissonian spectral reconstruction via curvelet transform regularization

Input: g0 = o, k = 0, and set the parameters λ1, λ2 and α.

While ||fk+1-fk||2/||fk||2<η1, ||g k+1-g k||2/||g k||2<η2 and k< MaxItern do

(i) Solve f sub-problem using (13) and (14);

(ii) Solve g sub-problem using (15)

Update gk + 1 by (20),

Solve (21) and (22) for 1

1k +d ,

12k +d ,

13k +d ,

Update 1

1k +b ,

12k +b , and

13k +b by (18);

(iii) k: = k + 1

End while Output: IRF = fk+1, Latent spectrum = gk+1

5. Experimental results and discussions

To demonstrate the effectiveness of the developed method, both simulated and real data experiments are demonstrated. and the experimental results are evaluated quantitatively and visually. The FBPSR method is compared with three state-of-the-art methods, which are Richardson-Lucy (RL) method [15], shift-excitation blind super-resolution [21] (SE-BSR) method, and Winer estimation for spectral reconstruction (WE-SR) method [16], respectively. In order to give an overall evaluation, several qualitative and quantitative metrics are introduced.

5.1 Evaluation indexes

The performance of the reconstruction methods on the simulated and real spectra are evaluated on the basis of the following quantitative merits among the observed spectrum o,

original spectrum g, and the reconstructed spectrum g .

(i) Normalized mean-squared error (NMSE):

2

2F

F

NMSE−

=g g

g(24)

where ||•||F is the Frobenius matrix norm (i.e.) F|| || , = < >g g g . NMSE represents the average

difference between the two spectra, with a small NMSE corresponding to a good match.

(ii) Noises suppression ratio (NSR) [15]

/ ,NSR = Do Dg (25)

which is defined as the ratio of the total variation of the observed spectrum o to total variation

of the latent spectrum g .

(iii) Ratio of full width at half-maximum (RFWHM) [15]

( ) ( )1

/N

v vFWHM

v

R FWHM FWHMN

= o g(26)

which ( )vFWHM

g and ( )vFWHMo denote the bands width in the latent spectrum g and

observed spectrum o, respectively. The NMSE index can only be utilized in the simulation


experiments, since it requires the existence of a reference spectrum g. Moreover, the NSR and RFWHM can also be utilized in the real IR spectral data experiments because they can be computed without the ground-truth spectrum. In [15], both indexes have been verified to represent the noises suppression and width reduction. The larger the values of NSR and RFWHM are, the higher the IR spectral quality achieves.

5.2 Simulated experiments for noise-free

To verify the performance of the developed approach, three IR spectra are executed simulations under Poisson noises process (Methyl propionate (C4H8O2), Cumene (C9H12), D( + )- Cellobiose (C12H22O11)). Each spectrum is convoluted by the IRF of Gaussian functionwith σ = 8 cm−1 (in Fig. 4(b)). Then, the overlap spectrum is further contaminated by Poissonnoises.

Fig. 4. Simulated experiments for IR spectra. (a) methyl propionate (C4H8O2) from 3300 to 2100 cm−1. (b) IRF Gaussian function with σ = 8 cm−1. (c) Overlap spectrum, convolute with the IRF. (d) Corrupted by Poisson noises (SNR = 200).

We consider the IR spectrum of methyl propionate (C4H8O2) from 3300 to 2100 cm−1 (Fig. 4(a)). The degraded IR spectrum is shown in Fig. 4(c). It can be seen that the spectral line becomes much smoother and less resolved, with the bands becoming wider and lower. For example, it is difficult to distinguish the peaks at 2464 cm−1, 2435 cm−1 and 2339 cm−1. Then, the degraded spectrum is contaminated with Poisson noises, which is shown in Fig. 4(d). The pure Poisson noises are shown at the bottom of the Fig. 4(d), which equals to the noisy spectrum (Fig. 4(d)) subtracting the overlap spectrum (Fig. 4(c)).


Fig. 5. Simulated experiments for the noise-free case (Fig. 4(c)). (a) RL method [15]. (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR method.

First, we executed four methods on the noise-free spectrum (Fig. 4(c)). For the RL method, the accurate Gaussian function was utilized as the IRF (Fig. 4(b)). Using the (23) to Fig. 4(c), we calculate the δ = 0.00131, and set λ1 = 0.026, λ2 = 250, α = 20/λ1, and 50 iterations. Figures 5(a)-5(d) show reconstruction results by RL, SE-BSR, WE-SR, and the proposed method. All the reconstructed spectra are clearly resolved. Visual analysis of the results clearly shows that the FBPSR method splits the overlap peaks very well. In the Fig. 5(d), the peaks at 2460 cm−1 is split into three peaks 2466, 2432 and 2378 cm−1, respectively. While in Fig. 5(b) (SE-BSR method) and Fig. 5(c) (WE-SR method), the overlap bands are separated slightly. Thus, for the noise-free spectrum, it seems clear that the FBPSR method produces a much narrower spectrum with more spectral structures than the compared methods.

Table 1. Peak Distortions in Reconstructed Spectra by RL, SE-BSR, WE-SR and FBPSR in Fig. 5 (methyl propionate (C4H8O2)).

Peak position a 2856 2464 2435 2379 RMSE

Height b

RL −0.124 + 0.098 −0.172 −0.145 0.1245

SE-BSR −0.109 + 0.085 −0.124 −0.159 0.1097

WE-SR −0.097 −0.086 + 0.099 −0.125 0.1019

FBPSR −0.094 −0.034 0.085 −0.092 0.0851

Position

RL + 1 −2 + 1 −2 1.7320

SE-BSR 0 −3 −1 −1 1.2584

WE-SR −1 −1 + 1 −1 1.0251

FBPSR 0 0 + 1 −1 0.8164 a In cm−1, obtained from the band maximum. b plus or minus symbol indicates larger or smaller than the original spectrum, respectively.

For the reconstructed IR spectra by four compared methods, the band distortions are investigated. In the original spectrum (Fig. 4(a)), four absorbance bands at 2856, 2464, 2435, 2379 cm−1 are taken as references. Table 1 shows their bands distortions in height and position between the reconstructed and original spectrum. Roots of mean square error (RMSE) of their distortions are computed. In terms of RMSE, the distortion of the positions is


smaller than those by the compared methods. It means that the result reconstructed by FBPSR method is the most similar one to the original IR spectrum.

Fig. 6. NMSE versus the iteration number of the three methods for the IR spectrum of methyl propionate (C4H8O2).

Table 2. NMSE, NSR and RFWHM of Degraded Spectrum and the Best Reconstructed Spectrum. The larger the NSR and RFWHM values, the higher the spectral quality.

Metrics IR spectra Spectral reconstruction by

RL [15] SE-BSR [21] WE-SR [16] FBPSR

NMSE

Methyl propionate 0.0372 0.0351 0.0298 0.0257

Cumene 0.0351 0.0336 0.0279 0.0213

D( + )-Cellobiose 0.0398 0.0373 0.0369 0.0302

NSR


Cumene 1.43 1.76 1.86 2.27

D( + )-Cellobiose 1.36 1.58 1.84 2.03

RFWHM


Cumene 1.45 2.13 2.65 3.23

D( + )-Cellobiose 1.38 2.25 2.63 3.01

Furthermore, the NMSE, NSR and RFWHM of three degraded spectra and the best reconstruction spectra by four methods are compared in Table 2. It shows that the FBPSR has achieved the lowest NMSE in comparison with other methods. Also, the NMSE values of the three methods are plotted in Fig. 6, where the convergence of FBPSR achieves the lowest value. The NSR and RFWHM indices have the similar trend to the NMSE index, which validates the good performance of the non-reference indices. Thus, they can be utilized to evaluate the reconstructed results of the real IR spectra.


Fig. 7. Experimental results for the noisy degraded spectrum (Fig. 4(d), SNR = 200). (a) RL method [15]. (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR method.

5.3 Effect of the noises

Moreover, we execute the four methods on the noisy degraded spectrum case (Fig. 4(d)). For the SE-BSR and WE-SR methods, we carefully adjust the parameters to achieve the best results. For the FBPSR method, it computes the noise level δ = 0.0178 according to (23), and set the parameters as λ1 = 0.035, λ2 = 250, α = 20/λ1, and 50 iterations.

Figure 7(a) shows the reconstructed result by RL method, and noticeable residual noises are meanwhile showed in the result. In Fig. 7(b), the SE-BSR approach recovers more structure details than the comparative method (RL), but it also produces some ringing artifacts (shown by red arrow). This phenomenon is caused by Poisson noises. In Fig. 7(d), FBPSR noticeably reduces the adverse effects. The close-ups of residual noises (from 2350 to 2100 cm−1) are shown in the top-left corner in Fig. 7. Thus, it seems clear that the FBPSR method not only estimates the IRF reliably, but also produces a much narrower spectrum with more details than the compared methods.

Fig. 8. Comparison of the NMSE values of RL, WE-SR, SE-BSR and the proposed method in all SNR conditions. The lower NMSE values imply improved performance.

The proposed method has been validated by direct comparisons with other methods at the index of NMSE. The NMSE values of the all-level SNR spectra are shown in Fig. 8.

As the SNR→0, the noises become dominant, resulting in an increase of the NMSE and invalidation of all the comparative approaches. The smaller the SNR value, the lower quality


the reconstructed IR spectrum. If the SNR→∞, the values of NMSE decrease, in other words, the reconstructed spectra approximate to the ground-truth one. When the SNR is larger than 400, the order of comparative approaches of NMSE values is RL, SE-BSR, WE-SR and FBPSR method. And when the SNR is less than 400, that is WE-SR, SE-BSR, RL and FBPSR. Thus, it concludes that the FBPSR method outperforms the RL, WE-SR, and SE-BSR methods in preserving the spectral details and suppressing Poissonian noises.

Table 3. Comparison of the speed between the proposed method and RL, SE-BSR, and WE-SR (Unit: Second).

Real IR Spectra Spectral reconstruction by

RL [15] SE-BSR [21] WE-SR [16] FBPSR


Cumene 18.803 9.863 14.254 0.914

D( + )-Cellobiose 20.723 11.294 15.931 1.122

The average run time of the proposed method on those IR spectra is about 1.0 seconds on a 3.4-GHz Pentium CPU and 8-GB memory with Matlab code. Table 3 shows the calculating times of the proposed method and comparative methods. From this table, it can be seen that the Matlab implementation of the FBPSR method is very fast. The reason is that the optimization is accelerated by the split Bregman iteration method.

Fig. 9. Sensitivity analysis of regularization parameters λ1, and λ2. Change of the NMSE value versus the parameters (a) λ1 value, and (b) λ2 value.

5.4 Regularization parameters determination

In the proposed model (11), two regularization parameters λ1 and λ2 need to be determined. Parameter λ1 depends on the noise level, and larger values should be chosen for heavy noise. Parameter λ2 controls the smoothness of IRF.

To show their effects on the reconstruction performance, using simulated experiment Fig. 7 as an example, we give a sensitivity discussion for the two regularization parameters. The change of the NMSE with the change of the parameters λ1 and λ2 are shown in Figs. 9(a)-9(b). In Fig. 9(a), the lowest NMSE value is achieved with parameter λ1 between [0.03, 0.05]. From Fig. 9(b), it is shown that the reconstruction results exhibit robustness with the change of parameter λ2. In all our implementations, the parameters are set with the range of λ1∈[0.03, 0.05], λ2∈[150, 350].


Fig. 10. Real IR spectrum experiment. (a) Butyl propionate (C7H14O2) [36] from 3100 to 2150 cm−1, reconstructed by (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR. (The estimated IRF is plotted at up-top corner).

5.5 Real data experiments

Furthermore, we test the effectiveness of the FBPSR method on real IR spectra. Figure 10(a) shows the IR absorbance spectrum of butyl propionate (C7H14O2) [36] from 3100 to 2150 cm−1. The IRF is plotted in Fig. 10(a), which is experimentally measured. Here, we compute δ = 0.0015 and set λ1 = 0.03, λ2 = 200, α = 20/λ1, and 50 iterations and start with Gaussian function (σ = 1 cm−1) for the initial IRF.

The restored result by SE-BSR method is illustrated in Fig. 10(b). The residual noises can be observed in Fig. 10(b). In Fig. 10(c), WE-SR method reconstructs more band structures than SE-BSR, but it also produces noticeable ringing artifacts, which are introduced by Poisson noises. Notably, FBPSR method observably reduces the adverse effects in Fig. 10(d). In Fig. 10(a), the overlap peak 2941 cm−1 is split into 2941 and 2925 cm−1. The reconstruction result seems sharp and its band structures are preserved well. The estimated IRF is plotted at up-top corner in Fig. 10(d). It is very similar to the ground-truth one measured by experiment (Fig. 10(a)).

The FBPSR method is also applicable to Raman spectra. Figure 11(a) is the Raman spectrum of D( + )-Cellobiose (C6H12O6) [37] from 900 to 350 nm. Here, we compute δ = 0.0016 and set λ1 = 0.032, λ2 = 230, α = 20/λ1 and 50 iterations. From Fig. 11(c), it can be seen that the overlap bands (661 and 631 nm in Fig. 11(a)) have been split into four bands by FBPSR, such as 663, 631, 605, and 579 nm. The WE-SR result can be found in Fig. 11(b), but only three peaks 663, 631, and 606 nm. Residual noises can be remarkably observed in Fig. 11(b). The recovered IR spectrum is convenient for extracting the spectral features, such as the band position, height, and number.


Fig. 11. Reconstruction experiment for real Raman spectra. (a) IR spectrum of D( + )-Cellobiose [37] from 900 to 350 cm−1, (b) WE-SR [16] method. (c) FBPSR result. (d) IRF estimated by FBPSR method.

Table 4 shows the NSR and RFWHM values of all the seven IR spectra. It is obvious that all the reconstructed algorithms raise the RFWHM and NSR values, but the FBPSR method achieves the highest values. It suppresses noises successfully while preserving significantly more details. The main reason lies in that the numbers of curvelet coefficients of the noise region and overlap band are large, which are markedly penalized by the regularization term ||Cg||1. Thus, the noises are suppressed substantially. The curvelet transform regularization can adjust the ratio of the flat, shoulder and noises regions.

Table 4. RFWHM and NSR (in Bracket) values of different reconstruction methods on the real IR spectra.

Real spectra RL [15] SE-BSR [21] WE-SR [16] FBPSR

Figure 10 1.89 (1.21)

2.43 (1.43)

2.93 (2.02)

3.23 (2.76)

Figure 11 1.42 (1.89)

1.68 (2.11)

3.11 (2.13)

3.58 (2.86)

IR 3 1.67 (1.41)

2.21 (1.75)

3.06 (1.98)

3.54 (2.40)

IR 4 1.78 (1.28)

1.98 (1.52)

3.12 (2.13)

3.61 (2.91)

IR 5 1.56 (1.22)

2.16 (1.67)

3.03 (2.08)

3.59 (2.56)

IR 6 1.84 (1.31)

2.51 (1.59)

3.16 (2.15)

3.52 (2.51)

IR 7 1.85 (1.67)

2.25 (1.51)

2.95 (2.08)

3.46 (2.55)


Fig. 12. Clustering comparison between experimental before-after results based on PCA. (a) Original spectra. Experimental results by (b) WE-SR [16]. (c) FBPSR method.

5.6 Application for cluster analysis

In our real experiments, the real spectra are a class of hydrocarbon molecule, which have the similar chemical structure, such as fingerprint region and functional group. If discrepancy between data sets achieved from the same industrial infrared spectrometer is primarily due to band overlap and noises, the effective reconstruction of them will increase the similarity for analytical chemometrics, which will be reflected in a tighter cluster. Thus, an approach based on principle component analysis has been used to evaluate the effectiveness of the spectral reconstruction approach. Figure 12 shows the comparative results after spectral reconstruction processing. It shows that the reconstruction results have a tighter PCA cluster.

6 Conclusion

In this paper, we show how the issue of the IR spectrum reconstruction can be considered as MAP-based problem and solved by minimizing a cost function that includes a likelihood term and two prior terms. The curvelet transform is employed for analyzing the spectral characteristics. The fitted distribution of the curvelet transform coefficients is utilized to describe the characteristic of three types of regions (such as flat, shoulder and noise regions) in IR spectra data. Moreover, the Gauss-Markov is introduced to constrain the smoothness the instrumental response functions characteristic. The split Bregman algorithm is also introduced to solve the proposed method. The quantitative assessment demonstrates that the proposed method outperforms those traditional reconstruction methods, which can reconstruct the spectral structural details as well as suppress noise effectively. The novelty of the proposed method is that it can estimate the IRF and the latent spectrum simultaneously. The recovered IR spectrum is of much more effectiveness and efficiency when extracting the spectral features and interpreting the chemical or biological materials.

Funding

Research Grants Council of Hong Kong (No. CityU 11205015 and CityU 11255716), National Natural Science Foundation of China under Grant (No. 61505064, 61673329), Hong Kong Scholars Programs under Grant (No. XJ2016063), the National Natural Science


Foundation of Hubei Province under Grant (No. 2016CFB497), the National key Research and Development Program (No. 2017YFB1401301, 2017YFB1401303, 2017YFB1401305), the Specific Funding for Education Science Research by Self-determined Research Funds of CCNU (No. CCNU18ZDPY10, CCNU16JYKX031), and the Cultivating Excellent Doctoral Dissertations Program of CCNU (No. 2017YBZZ009).


blind poissonian reconstruction algorithm via curvelet

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