Noise filtering and deconvolution of XPS data by wavelets and Fourier transform

In experimental sciences, the recorded data are often modelled as the noisy convolution product of an instrumental response with the ‘true’ signal to find. Different models have been used for interpreting x‐ray photoelectron spectroscopy (XPS) spectra. This article suggests a method of estimate the ‘true’ XPS signal that relies upon the use of wavelets, which, because they exhibit simultaneous time and frequency localization, are well suited to signal analysis. First, 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 similar to that developed in the previous paper in this series for the analysis of HREELS data. This step mainly rests on least‐squares and on the existing relation between the Fourier transform, the wavelet transform and the convolution product. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

An economic prediction of refinement coefficients in wavelet‐based adaptive methods for electron structure calculations

The wave function of a many electron system contains inhomogeneously distributed spatial details, which allows to reduce the number of fine detail wavelets in multiresolution analysis approximations. Finding a method for decimating the unnecessary basis functions plays an essential role in avoiding an exponential increase of computational demand in wavelet‐based calculations. We describe an effective prediction algorithm for the next resolution level wavelet coefficients, based on the approximate wave function expanded up to a given level. The prediction results in a reasonable approximation of the wave function and allows to sort out the unnecessary wavelets with a great reliability. © 2012 Wiley Periodicals, Inc.     

一种新的小波滤波方法在化学谱图信号滤噪中的应用

仪器分析测定中，噪声的存在往往影响分析的准确度和仪器的检出限。小波变换多分辨分析的特性使得它成为一种很好的滤噪方法。基于小波分解后信号与噪声的小波系数随尺度变化规律不同的特性，提出了一种新的滤波滤方法-空域相关法，即通过不同尺度上相关系数模值与小波系数模模值的比较，达到滤波滤的目的。本文提出的方法具有无需人为选定无需人为选定滤噪阈值和小波函数、方法简单、失真度小等优点，可以大在提高信号的信噪比。模拟数据和ICP-AES实验数据证明了该方法的有效性。     

A wavelet‐PCA method saves high mass resolution information in data treatment of SIMS molecular depth profiles

A detailed depth characterization of multilayered polymeric systems is a very attractive topic. Currently, the use of cluster primary ion beams in time‐of‐flight secondary ion mass spectrometry allows molecular depth profiling of organic and polymeric materials. Because typical raw data may contain thousands of peaks, the amount of information to manage grows rapidly and widely, so that data reduction techniques become indispensable in order to extract the most significant information from the given dataset. Here, we show how the wavelet‐based signal processing technique can be applied to the compression of the giant raw data acquired during time‐of‐flight secondary ion mass spectrometry molecular depth‐profiling experiments. We tested the approach on data acquired by analyzing a model sample consisting of polyelectrolyte‐based multilayers spin‐cast on silicon. Numerous wavelet mother functions and several compression levels were investigated. We propose some estimators of the filtering quality in order to find the highest ‘safe’ approximation value in terms of peaks area modification, signal to noise ratio, and mass resolution retention. The compression procedure allowed to obtain a dataset straightforwardly ‘manageable’ without any peak‐picking procedure or detailed peak integration. Moreover, we show that multivariate analysis, namely, principal component analysis, can be successfully combined to the results of the wavelet‐filtering, providing a simple and reliable method for extracting the relevant information from raw datasets. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Application of wavelet transforms and an approximate deconvolution method for the resolution of noisy overlapped peaks in DNA capillary electrophoresis

A new procedure for resolving noisy overlapped peaks in DNA separations by capillary electrophoresis (CE) is developed. The procedure combines both a wavelet-based denoising method that effectively denoises the signal and a novel approximate deconvolution technique that resolves the fragment peaks and improves the ability to separate highly overlapped peaks early in the electrophoresis process. Different kinds of overlapped peaks with and without noise simulated by computer as well as some DNA experimental electropherograms were submitted to the new procedure. A second order differential operator with variable coefficients is applied to the entire electrophoresis signal at any given time and approximate deconvolutions of the individual Gaussian peaks are performed. The operator incorporates the effect of the superposition and gives exact annihilation in the neighborhood of each peak. Overlapped peaks with a resolution higher than 0.46 can be resolved directly. Also, the method can determine the peak components of signals with a signal to noise ratio higher than 1.4     

去卷积伏安法中离散数据处理的方法研究(Ⅰ)──正交小波变换平滑

一系列的离散数据处理方法已成为化学计量学的重要组成部分[1],去卷和伏安法就是结合计算机技术的新一代电分析方法,其激励信号与输出信号均为计算机发生和采集的数字信号,对采集到的信号一般采用移动平均法[2]和Fourier变换处理法[3]进行平滑处理．但是,Fourier变换在电分析化学领域的难度较大,运算复杂,为此,Aubarel等[4]提出了不用FFT的Fourier变换平滑算法,但是该法要先对信号进行预处理,并且对Fourier的和式要反复进行折叠,计算量较大．80年代末发展起来的小波变换引起了人们广泛的关注[5],被称为数学“显微镜”,具有…     

Classification of Rat FTIR Colon Cancer Data Using Waveletsand BPNN

本文提出了一种新的基于水平衰减全反射-傅里叶变换红外光谱（HATR-FTIR）的小波特征提取与反向传播人工神经网络模式分类方法以提高FTIR对早期大鼠结肠癌的诊断准确率．对60只DMH诱导的SD大鼠,44只诱导鼠的第二代鼠,36只正常SD大鼠的结肠正常组织、异常增生、早癌及进展期癌组织所获得的的HATR-FTIR,利用连续小波多尺度分析法提取12个特征量,采用反向传播人工神经网络进行分类,识别准确率分别为100%、94%、97.5%及100%．实验结果表明此方法对早期结肠癌具有较高的诊断率．     

Experimental shift allowance in the deconvolution of SIMS depth profiles

We study the deconvolution of the secondary ion mass spectrometry (SIMS) depth profiles of silicon and gallium arsenide structures with doped thin layers. Special attention is paid to allowance for the instrumental shift of experimental SIMS depth profiles. This effect is taken into account by using Hofmann's mixing‐roughness‐information depth model to determine the depth resolution function. The ill‐posed inverse problem is solved in the Fourier space using the Tikhonov regularization method. The proposed deconvolution algorithm has been tested on various simulated and real structures. It is shown that the algorithm can improve the SIMS depth profiling relevancy and depth resolution. The implemented shift allowance method avoids significant systematic errors of determination of the near‐surface delta‐doped layer position. Copyright © 2013 John Wiley & Sons, Ltd.     

Improvement of prediction ability of PLS models employing the wavelet packet transform: A case study concerning FT-IR determination of gasoline parameters

The wavelet packet transform (WPT) is a variant of the standard wavelet transform that offers greater flexibility in the decomposition of instrumental signals. Although encouraging results have been published concerning the use of WPT for signal compression and denoising, its application in multivariate calibration problems has received comparatively little attention, with very few contributions reported in the literature. This paper presents an investigation concerning the use of WPT as a feature extraction tool to improve the prediction ability of PLS models. The optimization of the wavelet packet tree is accomplished by using the classic dynamic programming algorithm and an entropy cost function modified to take into account the variance explained by the WPT coefficients. The selection of WPT coefficients for inclusion in the PLS model is carried out on the basis of correlation with the dependent variable, in order to exploit the joint statistics of the instrumental response and the parameter of interest. This WPT-PLS strategy is applied in a case study involving FT-IR spectrometric determination of four gasoline parameters, namely specific mass (SM) and the distillation temperatures at which 10%, 50%, 90% of the sample has evaporated. The dataset comprises 103 gasoline samples collected from gas stations and 6144 wavelengths in the range 2500-15000 nm. By applying WPT to the FT-IR spectra, considerable compression with respect to the original wavelength domain is achieved. The effect of varying the wavelet and the threshold level on the prediction ability of the resulting models is investigated. The results show that WPT-PLS outperforms standard PLS in most wavelet-threshold combinations for all determined parameters.     

Enhancement of the Nuclear Spin Noise Signal Using Wavelet Transform

Spin noise spectroscopy has attracted considerable attention recently owing partly to intrinsic interest in the phenomenon and partly to its significant application potential. Here, we address the inherent problem of low sensitivity of nuclear spin noise and examine the utility of wavelet transform to mitigate this problem by distinguishing real peaks from the noise contaminated data. Suppression of the random circuit noise and the consequent enhancement of the correlated nuclear spin noise signal have been demonstrated with discrete wavelet transform. Spectra of both ¹H and ¹³C nuclear spins have been considered and significant signal enhancements in both the cases have been observed. A detailed analysis of several possible wavelet, thresholding and decomposition solutions have been made to obtain the optimum condition for signal enhancement. It is observed that the application of wavelet transform leaves the spin noise signal line shape essentially unchanged, which is an advantage for several applications involving spin noise spectra.     

Performance evaluation of spectral deconvolution analysis tool (SDAT) software used for nuclear explosion radionuclide measurements

The Spectral Deconvolution Analysis Tool (SDAT) software was developed to improve counting statistics and detection limits for nuclear explosion radionuclide measurements. SDAT utilizes spectral deconvolution spectroscopy techniques and can analyze both β-γ coincidence spectra for radioxenon isotopes and high-resolution HPGe spectra from aerosol monitors. Spectral deconvolution spectroscopy is an analysis method that utilizes the entire signal deposited in a gamma-ray detector rather than the small portion of the signal that is present in one gamma-ray peak. This method shows promise to improve detection limits over classical gamma-ray spectroscopy analytical techniques; however, this hypothesis has not been tested. To address this issue, we performed three tests to compare the detection ability and variance of SDAT results to those of commercial-off-the-shelf (COTS) software which utilizes a standard peak search algorithm.     

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。