Introduction to wavelet applications in surface spectroscopies

In experimental sciences, recorded data are often modelled as the noisy convolution product of an instrumental response with the ‘true’ signal. High‐resolution electron energy‐loss spectroscopy (HREELS) and x‐ray photoelectron spectroscopy (XPS) constitute two examples of this. A series of three papers is proposed about an estimation method of this ‘true’ signal in the particular cases of HREELS and XPS. This method uses wavelets that, as functions well localized in time and frequency, are properly adapted to signal analysis. In this first article, the wavelet theory is introduced and its rapid expansion is justified by a comparison of the wavelet transform with the Fourier transform. Afterwards, in order to illustrate the efficiency of the wavelet approach, some wavelet‐based signal analysis tools are presented. These tools include: filtering of a noisy signal, localization of irregular signal structures such as singularities or peaks and deconvolution itself. Copyright © 2004 John Wiley & Sons, Ltd.     

HREELS signal processing via wavelets

In spectroscopy, the recorded spectra can often be modelled as the noisy convolution product of an instrumental function with the ‘true’ signal to be estimated. Such models have often been used for high‐resolution electron energy‐loss spectroscopy (HREELS). In this article, a new method is suggested to estimate the ‘true’ HREELS signal, i.e. the original electronic diffusion function with ‘true’ peak intensities. Our method relies upon the use of wavelets that, because they exhibit simultaneous time and frequency localization, are well‐suited for signal analysis. Firstly, a wavelet shrinkage algorithm is used to filter the noise. This is achieved by decomposing the noisy signal into an appropriate wavelet basis and then thresholding the wavelet coefficients that contain noise. This algorithm has a particular threshold related to frequency and time. Secondly, the broadening due to the instrumental response is eliminated through a deconvolution process. This step mainly rests on the existing relation between the Lipschitz regularity of the signal and the decay with scale of its wavelet coefficients and on least squares. The efficiency of this technique is highlighted by comparing the results obtained with those provided by other published methods. This work is the second in a series of three papers in this issue. The first one presents background knowledge on the wavelets required to understand the estimation methods. The third paper explores the application of wavelet filtering and deconvolution techniques to x‐ray photoelectron spectroscopy. Copyright © 2004 John Wiley & Sons, Ltd.     

Wavelet transforms in separation science for denoising and peak overlap detection

Wavelet transform is a versatile time‐frequency analysis technique, which allows localization of useful signals in time or space and separates them from noise. The detector output from any analytical instrument is mathematically equivalent to a digital image. Signals obtained in chemical separations that vary in time (e.g., high‐performance liquid chromatography) or space (e.g., planar chromatography) are amenable to wavelet analysis. This article gives an overview of wavelet analysis, and graphically explains all the relevant concepts. Continuous wavelet transform and discrete wavelet transform concepts are pictorially explained along with their chromatographic applications. An example is shown for qualitative peak overlap detection in a noisy chromatogram using continuous wavelet transform. The concept of signal decomposition, denoising, and then signal reconstruction is graphically discussed for discrete wavelet transform. All the digital filters in chromatographic instruments used today potentially broaden and distort narrow peaks. Finally, a low signal‐to‐noise ratio chromatogram is denoised using the procedure. Significant gains (>tenfold) in signal‐to‐noise ratio are shown with wavelet analysis. Peaks that were not initially visible were recovered with good accuracy. Since discrete wavelet transform denoising analysis applies to any detector used in separation science, researchers should strongly consider using wavelets for their research.     

Combined Wavelet Transform with Curve‐fitting for Objective Optimization of the Parameters in Fourier Self‐deconvolution

Fourier self‐deconvolution was the most effective technique in resolving overlapping bands, in which deconvolution function results in deconvolution and apodization smoothes the magnified noise. Yet, the choice of the original half‐width of each component and breaking point for truncation is often very subjective. In this paper, the method of combined wavelet transform with curve fitting was described with the advantages of an enhancement of signal to noise ratio as well as the improved fitting condition, and was applied to objective optimization of the original half‐widths of components in unresolved bands for Fourier self‐deconvolution. Again, a noise was separated from a noisy signal by wavelet transform, therefore, the breaking point of apodization function can be determined directly in frequency domain. Accordingly, some artifacts in Fourier self‐deconvolution were minimized significantly.     

Application of empirical mode decomposition in the field of polymer physics

Noisy data has always been a problem to the experimental community. Effective removal of noise from data is important for better understanding and interpretation of experimental results. Over the years, several methods have evolved for filtering the noise present in the data. Fast Fourier transform (FFT) based filters are widely used because they provide precise information about the frequency content of the experimental data, which is used for filtering of noise. However, FFT assumes that the experimental data is stationary. This means that: (i) the deterministic part of the experimental data obtained from a system is at steady state without any transients and has frequency components which do not vary with respect to time and (ii) noise corrupting the experimental data is wide sense stationary, that is, mean and variance of the noise does not statistically vary with respect to time. Several approaches, for example, short time Fourier transform (STFT) and wavelet transform‐based filters, have been developed to handle transient data corrupted with nonstationary noise (mean and variance of noise varies with respect to time) data. Both these approaches provide time and frequency information about the data (time at which a particular frequency is present in the signal). However, these filtering approaches have the following drawbacks: (i) STFT requires identification of an optimal window length within which the data is stationary, which is difficult and (ii) there are theoretical limits on simultaneous time and frequency resolution. Hence, filtering of noise is compromised. Recently, empirical mode decomposition (EMD) has been used in several applications to decompose a given nonstationary data segment into several characteristic oscillatory components called intrinsic mode functions (IMFs). Fourier transform of these IMFs identifies the frequency content in the signal, which can be used for removal of noisy IMFs and reconstruction of the filtered signal. In this work, we propose an algorithm for effective filtering of noise using an EMD‐based FFT approach for applications in polymer physics. The advantages of the proposed approach are: (i) it uses the precise frequency information provided by the FFT and, therefore, efficiently filters a wide variety of noise and (ii) the EMD approach can effectively obtain IMFs from both nonstationary as well as nonlinear experimental data. The utility of the proposed approach is illustrated using an analytical model and also through two typical laboratory experiments in polymer physics wherein the material response is nonstationary; standard filtering approaches are often inappropriate in such cases. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Signals ratio method combined with wavelet transform: application to resolution of overlapped electrochemical signals

A signals ratio method combined with wavelet transform was proposed for the resolution of a weak voltammetric signal overlapped by other components. The signals ratio method usually suffers from interference from noise and baseline contained in the original signals because these factors cause distortion of the signals ratio. The multiresolution capability of the wavelet transform method was exploited here to simultaneously remove or reduce the noise and background. As a result, a deformation-free signals ratio with good signal-to-noise ratio (SNR) was obtained even for very noisy signals. The properties of the proposed method were compared to other resolution methods. It was demonstrated that the combined signals ratio wavelet transform method was particularly applicable to resolve a minor component in the presence of large amount of other components, suggesting that it can provide improved detection limits and quantified results for minor components. The method was employed for the voltammetric determination of residual chlorine in the presence of N,N-diethyl-p-phenylenediamine (DPD).     

Sub-optimal wavelet denoising of coaveraged spectra employing statistics from individual scans

This paper proposes a novel wavelet denoising method, which exploits the statistics of individual scans acquired in the course of a coaveraging process. The proposed method consists of shrinking the wavelet coefficients of the noisy signal by a factor that minimizes the expected square error with respect to the true signal. Since the true signal is not known, a sub-optimal estimate of the shrinking factor is calculated by using the sample statistics of the acquired scans. It is shown that such an estimate can be generated as the limit value of a recursive formulation. In a simulated example, the performance of the proposed method is seen to be equivalent to the best choice between hard and soft thresholding for different signal-to-noise ratios. Such a conclusion is also supported by an experimental investigation involving near-infrared (NIR) scans of a diesel sample. It is worth emphasizing that this experimental example concerns the removal of actual instrumental noise, in contrast to other case studies in the denoising literature, which usually present simulations with artificial noise. The simulated and experimental cases indicate that, in classic denoising based on wavelet coefficient thresholding, choosing between the hard and soft options is not straightforward and may lead to considerably different outcomes. By resorting to the proposed method, the analyst is not required to make such a critical decision in order to achieve appropriate results.     

化学信号的近似高阶导数计算

提出了化学信号近似四阶导数计算的新方法——小波卷积法。该法通过信号与二阶样条小波函数的卷积运算对信号求导,能用于高噪音信号的直接求导,避免了普通导数运算将噪音放大的缺陷,即使对信噪比低至0．5的信号也能得到光滑的导数信号。详细讨论了尺度值、噪音、信号类型对求导的影响并建立了参数确定规则。将该法用于含噪音重叠分析化学信号的求导,能同时提高信号的分辨率和信噪比,结果满意。     

Product or sum: comparative tests of Voigt,and product or sum of Gaussian and Lorentzian functions in the fitting of synthetic Voigt‐based X‐ray photoelectron spectra

A comparative study for the fitting of X‐ray photoelectron spectra (XPS) using different model functions is presented. Synthetically generated test spectra using Gaussian/Lorentzian convolution and a real measured spectrum are fitted with the three commonly used models: product, sum and Gaussian/Lorentzian convolution functions. In these limited tests, it was found that the sum function is superior to the product function, particularly for low‐noise spectra. Copyright © 2007 John Wiley & Sons, Ltd.     

Minimisation of instrumental noise in the acquisition of FT-NIR spectra of bread wheat using experimental design and signal processing techniques

Spectral resolution (R) and number of repeated scans (S) have a significant effect on the S/N ratio of Fourier transform-near infrared (FT-NIR) spectra, but the optimal values of these two parameters have to be determined empirically for a specific problem, considering separately both the nature of the analysed matrix and the specific instrumental setup. To achieve this aim, the instrumental noise of replicated FT-NIR spectra of wheat samples was modelled as a function of R and S by means of the Doehlert design. The noise amounts in correspondence to different experimental conditions were estimated by analysing the variance signals derived from replicate measurements with two different signal processing tools, Savitzky–Golay (SG) filtering and fast wavelet transform (FWT), in order to separate the “pure” instrumental noise from other variability sources, which are essentially connected to sample inhomogeneity. Results confirmed that R and S values leading to minimum instrumental noise can vary considerably depending on the type of analysed food matrix and on the different instrumental setups, and helped in the selection of the optimal measuring conditions for the subsequent acquisition of a wide spectral dataset.     

A novel method to calculate the approximate derivative photoacoustic spectrum using continuous wavelet transform

A novel method based on continuous wavelet transform (CWT) using Haar wavelet function for approximate derivative calculation of analytical signals is proposed and successfully used in processing the photoacoustic signal. An approximate nth derivative of an analytical signal can be obtained by applying n times of the wavelet transform to the signal. The results obtained from four other different methods – the conventional numerical differentiation, the Fourier transform method, the Savitzky-Golay method, and the discrete wavelet transform (DWT) method – were compared with the proposed CWT method; it was demonstrated that all the results are almost the same for signals without noise, but the proposed CWT method is superior to the former four methods for noisy signals. The approximate first and second derivative of the photoacoustic spectrum of Pr(Gly)₃Cl₃· 3 H₂O and PrCl₃· 6 H₂O were obtained using the proposed CWT method; the results are satisfactory. Received: 21 December 1999 / Revised: 28 February 2000 / Accepted: 7 March 2000     

A novel method to calculate the approximate derivative photoacoustic spectrum using continuous wavelet transform

A novel method based on continuous wavelet transform (CWT) using Haar wavelet function for approximate derivative calculation of analytical signals is proposed and successfully used in processing the photoacoustic signal. An approximate nth derivative of an analytical signal can be obtained by applying n times of the wavelet transform to the signal. The results obtained from four other different methods--the conventional numerical differentiation, the Fourier transform method, the Savitzky-Golay method, and the discrete wavelet transform (DWT) method--were compared with the proposed CWT method; it was demonstrated that all the results are almost the same for signals without noise, but the proposed CWT method is superior to the former four methods for noisy signals. The approximate first and second derivative of the photoacoustic spectrum of Pr(Gly)3Cl3.3H2O and PrCl3.6H2O were obtained using the proposed CWT method; the results are satisfactory.     

小波变换用于二次微分交流示波计时电位信号中有用信息的提取

小波分析是８０年代发展起来的一种新的数学分支。由于小波变换具有许多其它的信号处理手段所不具备的优良特性，如正交性，可变的时－频分辨率和可调节的局部支持等，使它成为信号处理的一种强有力的工具。     

Wavelet transform for Shah convolution velocity measurements of single particles and solutes in a microfluidic chip

Wavelet transform analysis is applied to determine the speed of fluorescent polystyrene microspheres and fluorescent solutes in a microchip. The data analysed consist of the periodical signal (Shah convolution) obtained when fluorescent particles or solute plugs move in a channel that is covered with a chromium grid pattern. This setup converts velocity into a (fluorescence emission) frequency, and previous analyses therefore used Fourier transform to extract the frequency information. In this paper it is shown that wavelet transform has some advantages over Fourier transform. With wavelet transform, time information can be obtained in addition to frequency information. Thus the speed of individual particles was determined together with their moments of appearance and disappearance in the system. With solutes small changes of velocity during the analysis were detected, and an improvement in peak frequency resolution was obtained.     

小波变换用于示波信号中有用信息提取的研究

示波分析是近年来在我国发展起来的一个新的电化学分析研究领域［１～４］．它根据阴极射线示波器荧光屏上示波图及其变化进行分析测试,从原理上可以将其分为示波电位法和示波计时电位法;从测定方式上可以将其分为示波滴定和示波测定．关于示波电位法的一些理论问题（如...     

End-point detection in potentiometric titration by continuous wavelet transform

The aim of this work was construction of the new wavelet function and verification that a continuous wavelet transform with a specially defined dedicated mother wavelet is a useful tool for precise detection of end-point in a potentiometric titration. The proposed algorithm does not require any initial information about the nature or the type of analyte and/or the shape of the titration curve. The signal imperfection, as well as random noise or spikes has no influence on the operation of the procedure.The optimization of the new algorithm was done using simulated curves and next experimental data were considered. In the case of well-shaped and noise-free titration data, the proposed method gives the same accuracy and precision as commonly used algorithms. But, in the case of noisy or badly shaped curves, the presented approach works good (relative error mainly below 2% and coefficients of variability below 5%) while traditional procedures fail. Therefore, the proposed algorithm may be useful in interpretation of the experimental data and also in automation of the typical titration analysis, specially in the case when random noise interfere with analytical signal.     

Accurate grading of brain gliomas by soft independent modeling of class analogy based on non‐negative matrix factorization of proton magnetic resonance spectra

Hydrogen magnetic resonance spectroscopy (¹H‐MRS) is a non‐invasive technique which provides a ‘frequency‐signal intensity’ spectrum of biochemical compounds of tissues in the body. Although this method is currently used in human brain studies, accurate classification of in‐vivo ¹H‐MRS is a challenging task in the diagnosis of brain tumors. Problems such as overlapping metabolite peaks, incomplete information on background component and low signal‐to‐noise ratio disturb classification results of this spectroscopic method. This study presents an alternative approach to the soft independent modeling of class analogy (SIMCA) technique, using non‐negative matrix factorization (NMF) for dimensionality reduction. In the adopted strategy, the performance of SIMCA was improved by application of a robust algorithm for classification in the presence of noisy measurements. Total of 219 spectra from two databases were taken by water‐suppressed short echo‐time ¹H‐MRS, acquired from different subjects with different stages of glial brain tumors (Grade II (26 cases), grade III (24 cases), grade IV (41 cases), as well as 25 healthy cases). The SIMCA was performed using two approaches: (i) principal component analysis (PCA) and (ii) non‐negative matrix factorization (NMF), as a modified approach. Square prediction error was considered to assess the class membership of the external validation set. Finally, several figures of merit such as the correct classification rate (CCR), sensitivity and specificity were calculated. Results of SIMCA based on NMF showed significant improvement in percentage of correctly classified samples, 91.4% versus 83.5% for PCA‐based model in an independent test set. Copyright © 2015 John Wiley & Sons, Ltd.     

基于小波的恒电量瞬态响应信号的滤波处理

利用小波变换的阈值法对恒电量响应信号进行滤波处理, 同时与传统的时域和频域的滤波方法进行分析比较, 并且讨论了小波变换的分解层数对恒电量响应信号滤波效果的影响. 结果表明, 利用小波变换可以在时域和频域同时对恒电量响应信号取得良好的去噪效果. 这不仅能提高时域曲线拟合的精度, 还大大地提高了恒电量频谱解析的可靠性. 在实际应用中, 小波变换的分解层数取5～7层可以收到满意的效果.     

Comparing different means of signal treatment for improving the detection power in HPLC-ICP-MS

The purpose of detecting trace concentrations of analytes often is hindered by occurring noise in the signal curves of analytical methods. This is also a problem when different arsenic species (inorganic As(III) and As(V) as well as organic dimethylarsinic acid and arsenobetaine) are to be determined in food and feeding stuff by HPLC-ICP-MS, which is the basis of this work. In order to improve the detection power, methods of signal treatment may be applied. We show a comparison of convolution with Gaussian distribution curves, Fourier transform, and wavelet transform. It is illustrated how to estimate decisive parameters for these techniques. All methods result in improved limits of detection. Furthermore, applying baselines and evaluating peaks thoroughly is facilitated. However, there are differences. Convolution with Gaussian distribution curves may be applied, but Fourier transform shows better results of improvement. The best of the three is wavelet transform, whereby the detection power is improved by factors of about 6.