Absorption spectrophotometry signal de-noising using invariant wavelets

Rubiel Vargas Cañas¹, Humberto Loaiza Correa²

¹ Systems Engineer, Universidad Industrial de Santander. Specialist in Industrial Electronics, Universidad del Cauca. Magíster Master in Engineering with emphasis in electronics, Universidad del Valle. PhD candidate in Biomedical Engineering, City University, Reino Unido. Current affiliation, Universidad del Cauca. rubiel@unicauca.edu.co

² Electrical Engineer, Master in Automation, Universidad del Valle. PhD in Robotics, Universite D'evry Val D'essonne, Francia. Humberto.loaiza@correounivalle.edu.co

ABSTRACT

Diseases such as cancer, hepatitis and AIDS cause body fluid concentration and amount to become modified; their measurement can thus be useful as a diagnostic technique. Spectroscopy is one of the most widely used techniques for biological substance detection and quantification. The presence of unwanted signals is the main limiting factor for sensitivity and quality; this is called noise. Noise has different backgrounds which range from physical assumptions to environmental influence. Eliminating or reducing noise in spectroscopy has thus been studied for many years and the applicability of wavelet transform has been demonstrated in recent decades. This paper presents invariant wavelet transform for increasing signal to noise ratio in spectrophotometer signals and thus improve the quality of spectrophotometric analysis and biological substance quantification. The proposed technique was applied to artificially-generated signals and signals from two spectrometers, one having a continuum source and another with a laser radiation source. The results obtained with this technique were compared to those obtained from traditional filters: Gaussian, Wiener and orthogonal wavelets. This technique's main advantages are a substantial increase in signal to noise ratio and preservation of spectral peak location and width. These advantages increase biological substance detection and quantification quality and accuracy and allow automatic analysis of the spectrum. They can also lead to better understanding of experimental limitations and allow a quantitative study of the influence of changes in substance concentration in related diseases.

Keywords: absorption spectrophotometry, signal filtering, invariant wavelet, biological substance detection.

Received: May 20th 2010 Accepted: November 24th 2011

Introduction

There are different ways for detecting and quantifying biological and chemical species; spectroscopy is particularly attractive since it is highly sensitive, very fast, robust, extensible and its implementation involves medium complexity. The Lambert-Beer-Bourguer law or general law of spectrophotometry provide the physical-mathematical fundamentals for quantifying substance concentration from measured absorption in absorption spectro-photometry (Valea and Girón, 1998). This involves knowing the average power affecting a sample and transmitted power; the amount of light absorbed by a sample can be calculated by using:

where A is absorbance, a is absorption coefficient (or extinction coefficient) given in litres per gram-centimetre, b is the length of the optic path given in centimetres and c is the concentration. Equation (1) is defined for a monochromatic and parallel radiation beam. It is often also written as:

where T is transmittance, showing the ratio between incident and transmitted radiation (Valea Pérez and Giron, 1998) according to:

There is a degree of uncertainty here due to some deviation produced by the noise or randomness. Such deviations can be classified as deviations typical of the so-called law of spectropho-tometry, deviations produced by instrumentation and deviations produced by chemical reactions (Portal Unal virtual, 2007; Valea and Girón, 1998). Deviations produced by physical fundamentals are related to discrepancies between actual conditions and ideal conditions assumed by definition, i.e. constant wavelength, constant optical path, monochromatic radiation beam and linearity. Chemical deviations are produced by reactions occurring during measurement and instrumental deviations are due to radiation or parasitic noise due to instrumentation, appearing during signal acquisition and digitisation and/or its transmission (Oppemheim and Willsky, 1998; Van and Haykin, 2002).

Removing or deducing noise levels is a very important task in spectroscopy for obtaining highly accurate data, allowing automatic analysis of the spectrum and understanding some experimental limitations (Werner and Kjell-Arild, 2005). An alternative for reducing instrumental drift and increasing quality measurement is to digitally process signals delivered by the measuring equipment. Radiant energy is converted into an electrical signal (digitised in some cases) which is filtered, amplified, mathematically compensated for, manipulated and presented as absorbance or transmittance on a digital measurement scale on a screen or meter. Traditional de-noising methods are based on differential absorption, digital filters (Gaussian or Wiener) and orthogonal wavelets (Schuck et al., 2010).

This paper presents a novel approach to eliminating or minimising spurious signals contained in a light signal from a spectrophotometer which is used for quantifying biological substances' concentrations. The ultimate goal was to improve signal to noise ratio, to increase instrument sensitivity, improve measurement time and increase measurement accuracy; translation invariant wavelet transform was thus used.

To facilitate evaluating the technique, it was applied to artificially created signals and signals from two spectrophotometers, one in which the light source was continuous and the other in which the light source was a laser. The signal to noise ratio was quantified and compared against traditional approaches in both cases.

Filtering spectroscopic signals

Motivation

The main limiting factor regarding any instrumental analytical method's accuracy and sensitivity is the presence of unwanted signals superimposed on the signal being generated by the substance to be analysed (Werner and Kjell-Arild, 2005). Such signals are inevitable in most cases and they reduce the quality of analysis, leading to errors if useless information cannot be effectively removed. Spectrum analysis is based on signal variation, specifically peak location, amplitude and width. Hence, such analysis is sensitive to noise, even in small quantities. The need for a methodology removing additional signals whilst keeping the characteristics of the original signal thus becomes evident, since even small variations can be related to biophysical attributes of the substance being studied. This methodology is known as filtering and involves the removal or attenuation of noise by electronic or digital arrays.

Signals are usually affected by two types of noise: random and periodic noise (Oppemheim and Willsky, 1998; Van V and Haykin, 2002). Random noise presents a pattern which can be characterised from the probability density function, whilst periodic noise parameters are typically estimated by inspecting signals' spectra due to them being appreciated as peaks in the frequency domain. Periodic noise parameters can also be estimated using prior knowledge regarding the location of the interference frequency components or sensor specifications (Oppemheim and Willsky, 1998). Some traditional techniques such as Gaussian and Wiener filters exploit prior knowledge of parameters characterising the noise, specifically the mean and variance.

The main techniques

Differential absorption is the most common technique for suppressing noise in absorption spectrophotometry (Barnard, 2009); it involves splitting the light signal into a test signal and a reference signal. This technique assumes that the noise equally affects both signals; subtracting or dividing the two signals then removes excess noise. Such assumption is not always correct in practice and therefore using a receiver or divider does not provide the expected results. Moreover, the most common method for increasing the signal to noise ratio in laser absorption spectrophotometry is to chop (cut) and amplify the laser and current in the detector with a look-in amplifier. This technique has some issues due to "chopper" frequency or when the background signal is low frequency. Another approach is to modulate the signal frequency and average a set of spectral scans (Hong et al., 2009). This technique provides very good results but its implementation requires a very sophisticated electro-optical design.

Other approaches take advantage of prior knowledge regarding the characteristics of noise, particularly its variance (Hong et al., 2009). Gaussian filters reduce random and periodic noise, whilst demonstrating good performance signals affected by noise with Erlang distributions (Gonzalez and Eddins, 2004; Van V and Haykin, 2002) which are common in laser systems (virtual Portal Unal, 2007). These filters assume constant variance throughout the signal and sometimes (Barnard, 2009) divide the signal into different segments and estimate the variance for each of them.

Each aforementioned technique has advantages and disadvantages and the price to pay for eliminating noise is the distortion of the original signal. For example, Notch filters do not shift peaks in time because they have a zero phase response; however, the impulse response is not sufficiently short, which becomes degenerated into a broadening spectral peak (Werner and Kjell-Arild, 2005). According to the pertinent literature, there is a tendency to use wavelet filters (Arizmendi et al., 2010; Garcia and Ramirez, 2007; Posú and Landrove, 2008; Werner and Kjell-Arild, 2005) to filtrate spectroscopic signals as they preserve the characteristics of the peaks. A summary of some relevant work in this field is given below.

Nuclear magnetic resonance (NMR) spectroscopy: Cancino-De-Greiff et al., 2002, used simulation for analysing potential wavelet shrinkage in filtering noise in magnetic resonance spectros-copy; combining Cadzow's algorithm in their work. Posú and Landrove, 2008, presented a methodology for processing data obtained by NMR spectroscopy for metabolic mapping. Once a power spectrum has been obtained it is filtered to remove noise using Haar wavelets and its baseline is approximated by a third order polynomial. Trbovica et al., 2005, applied a series of filters to experimentally obtained NMR spectroscopy data to optimise computational efficiency and minimise the weight of small chemical translations. The best results were obtained when applying principal component analysis to the wavelet coefficients.

Absorption spectroscopy: Garcia and Ramirez, 2007, used wavelet neural networks to reduce noise in stellar spectra. To validate the technique, they simulated stellar spectra and found that the result obtained with the wavelet neural network was almost equal to the desired output, i.e. the stellar spectrum without noise. Xiaoguan and Dahai, 2009, described a method for simultaneous spectrophotometric determination of copper, zinc, nickel and cobalt in water. The absorbance spectrum was processed with wavelet transform to remove interference.

Infrared Spectroscopy: Zhu et al., 2007, investigated the influence of the shape of the wavelet, the number of levels used and the type of thresholding in filtering signals from near-infrared spectroscopy. The best results were obtained with a wavelet bior3.3, using two levels of decomposition and soft thresholding. Berry and Ozaki, 2002, compared noise filtering by using infrared spectroscopy, diffusion eigenvectors and reconstruction wavelets, demonstrating that using wavelets was better than the aforementioned techniques. Zhan et al., 2004, applied wavelet analysis and noise filter correlation infrared spectroscopy to traditional Chinese medicine. This technique allowed them to separate spectral peaks which were superimposed prior to analysis.

Terahertz and Raman spectroscopy: Gao et al., 2004, discussed noise filtering in Raman spectroscopy using a stationary Haar wavelet. Noise variance was estimated for each wavelet level and fixed threshold, being equal to the square root of twice the natural logarithm of the number of samples present in the signal. Li-Na Yinglang and Cai, 2009, presented Mexican hat-shaped wavelets which were applied to the analysis of terahertz spectro-scopic data in the time domain. The results of the simulations showed their ability to remove noise in low and high frequency signals.

Procedimiento

Wavelets

The wavelet theory is a branch of mathematics whose research focuses on building a model of systems or processes using a special type of signal known as a wavelet. A wavelet is a wave having limited effective duration which has a zero average value and may represent a signal in terms of translated and dilated versions of the wave (Rioul O. and Vetterli M., 1991; Cuesta FD et al., 2000). Wavelets are families having the following function:

where a and b are the expansion and translation parameters, respectively. The wavelet transform of a signal is then given by:

This mathematical technique has become very important recently in applications of all kinds related to non-stationary signal processing. Wavelet analysis consists of decomposing a signal into scaled and shifted versions of the mother (original) wavelet. Such decomposition allows observing a correspondence between wavelet scale and frequency; therefore, wavelets are related to a signal's harmonic analysis. The graphs of wavelet transform coefficients are a much more natural time-scale representation of the signal. This representation reveals patterns which were not previously visible, preserving the signal's temporal aspect. The major advantage lies in performing local analysis, in other words to analyse a specific area of a large non-stationary and fast transience signal.

The process involves analysis and reconstruction. Wavelet analysis involves filtering and down sampling while reconstruction involves oversampling (up sampling) and filtering. The expansion parameter is set to ^{a=2 j}with jÎZ (called dyadic wavelet) (Cuesta FD et al., 2000) to make the calculation quickly. A fast algorithm similar to the FFT is used for automatic computation.

Filtering signals using wavelets

The effect of wavelet transform is to filter the signal through high pass filtering which preserves fine details and low pass filtering which provides a rough approximation of the signal. The choice of filter determines the shape of the wavelet to be used to make a better reconstruction. The advantage of this procedure for filtering by frequency bands is that it results in an almost noise-free signal with small changes in its characteristics (presence of high frequency peaks, etc.). These results are different from those obtained with traditional smoothing methods which get rid of the noise at the expense of smoothing or distorting some of the signal components (Rioul O. and Vetterli M., 1991).

To reduce white Gaussian noise, acquired signal was considered as:

where x_n was a free-noise signal,   represented white Gaussian noise with zero mean and unitary variance, n represented the level and S_nwas the base-line approximated by low frequency components.

The procedure for filtering noise was similar to that carried out in the frequency domain by means of Fourier transform, i.e. the wavelet transform was calculated to change the signal domain and a series of operations was carried out on the coefficients in this new domain. These operations are often nonlinear and usually consist of a thresholding. The basic idea is to remove components below a certain threshold (fixed threshold), or multiplied by a weighting factor (flexible threshold). The filtered signal was returned to the time domain after performing the thresholding to calculate the inverse transform.

The most significant differences between most methods proposed in work related to this application lie in the type of threshold used or given to the weighted coefficients (Cuesta FD et al., 2000). Threshold specification depends on data application and quality. Ling and Ren, 2008, combined principal component analysis and wavelets in analysing overlapping spectrophotometric signals and proposed a method based on optimisation to select the thresholding level. Gao et al,. 2004b, discussed a method based on the principle of entropy maximisation; this method had good selectivity on simulated signals and its performance was less sensitive to changes in the signal to noise ratio. Gao et al., 2004, used a fixed threshold equal to the square root of twice the natural logarithm of the number of samples present in the signal.

Invariance to translation

The translational invariance of operator L meant that if the input was shifted by   then the output would also be shifted by the same factor:

Continuous wavelet:   was a replica of   shifted by  :

where was invariant to tranlations:

Discrete wavelet: rewriting (5) as,

translation ^u was uniformly sampled at time intervals that proportional to   as,

This sample procedure destroyed translation invariance.

Shifting   for   gave,

If the sampling interval were relatively large, then coefficients   and   could take on different values that were not shifted relative to one another. It is well known that orthogonal wavelet transforms are variant to translation (Garcia and Ramirez, 2007; Cuesta et a/., 2000), meaning that the wavelet transform of a shifted signal is different from the wavelet transform shifted from the original signal.

Wavelet transform maintains translational invariance only by sampling the scale of a continuous wavelet transform. However, other strategies maintain translation invariance; one of them is to keep the sampling interval small enough and then samples of   will be shited when   is also shited. Another approach the signal by cyclic translations (Rioul and Vetterli, 1991), averaging the results; i.e. shifting the signali samples, and then i samples at the end of the signal are taken to the beginning, which is why this procedure is known as cyclical. The idea is to obtain a better estimate of the signal averaging estimates given for each movement made; total N-1 can travel, for which it would take O (N2) operations.

Another way is to implement the transformation without sub-sampling, this is equivalent to decomposing the signal into a family of N * (j +1) redundant coefficients, where N is signal length and j the number of scales. Let ^(t) be a continuous signal characterised by N samples sampled at N~¹distance. To simplify notation, original signal sampling interval was normalised to 1. Dyadic wavelet transform of normalised discrete signal   may be calculated only at scales   or the same as  .The samples   were written as:

for any ^j≥0

with

The coefficients for the dyadic wavelet transform for ^j>0 were given by,

then

where   were the basis or filters used to compute the wavelet transform. An interested reader can refer to Mallat, 2009, and Holshnecder et al., 1989, for a more detailed derivation and the proof of this.

A representation of   can be calculated with a convolution cascade whose computational complexity is equivalent to the complexity of FFT, i.e.  . The method described is thus known as fast discrete wavelet transform (Mallat, 2009).

Results and Discussion

An artificial signal having different time and frequency components was generated to evaluate the proposed technique's performance. This signal was distorted by adding Gaussian noise with zero mean and unit variance, as shown in Figure 1.

This signal was filtered using Gaussian, Wiener and orthogonal wavelets (Haar) using a fixed threshold. The results obtained with these filters are shown in Figure 2.

As shown in Figure 2, the best signal to noise ratio was obtained with the Wiener filter. The results obtained with the wavelet were better than those obtained with the Gaussian filter which was implemented using the Fourier transform with a window width equal to 4, approximating unit variance. A more detailed analysis of this figure revealed that sharp peaks were slightly shifted to the left in the results achieved with the Gaussian filter, while Wiener and fixed threshold wavelet preserved their location. Regarding signal appearance, the signal recovered from the wavelet was smoother while most variation was obtained with the Wiener filter.

The signal in Figure 1 was also filtered with a translation invariant cyclic wavelet. Fixed threshold was used and 4 scales.

Figure 3 shows the result of filtering the signal presented in Figure 1 with an invariant wavelet by applying the described method. The resulting signal to noise ratio superseded that obtained with the Wiener filter and the appearance of the signal was better than that obtained with the orthogonal wavelet. It could also be observed that peaks retained their original location. The fixed threshold used for filtering was equivalent to 3σ which provided excellent results in filtering Gaussian noise (Mallat, 2009).

Table 1 summarises the results obtained from simulated signal and filtering the different alternatives. The method used is shown in the left-hand column, the SNR of the filtered signal in the middle and the right-hand column describes the filtered signal's appearance.

The cyclic wavelet (i.e. the proposed technique) was tested in two light signals coming from a spectrophotometer which used a continuous light source and another that used a laser radiation source; the test substance consisted of blood samples pre-processed with markers attached to a protein, thereby enhancing radiation absorption (results are illustrated in Figures 5 and 6, respectively). As can be seen, the filtered signal preserved original signal characteristics to a large extent. The location of the peaks can be seen best in the signal from the spectrophotometer having continuous radiation source, while the low frequency components were best appreciated in the signal from the spectrophotometer having a laser light source.

Conclusions

This paper has shown the applicability of translation invariant wavelet transform in filtering light signals from spectrophotometers. This technique was shown to maximise the signal to noise ratio compared to Gaussian, Wiener and orthogonal wavelet filters using a fixed threshold. These improvements lead to better quality results regarding measurements made with a spectrophotometer.

A substance's spectrum is like its signature; therefore, preserving spectral peak location, height and width is of the utmost importance. Spectra are inevitably corrupted by noise in spectrophotometric analysis. Filtering techniques have advantages and disadvantages regarding the removal of such noise, but a closer look at the results obtained with wavelets revealed superior performance in filtering signals from spectrophotometers as they preserve a signal's original spectral peak characteristics.

Although the signal to noise ratio did not differ much between the results obtained with the Gaussian filter and orthogonal wavelet, the latter was preferred for its higher signal to noise ratio compared to the Gaussian filter, especially because it did not modify the location of a signal's original singularities thus making wavelets attractive in the field of spectrophotometry.

Translation invariant wavelet filtering had the highest signal to noise ratio, and its implementation through fast discrete wavelet transform required O(Nlog2 (N)) operations, making it more attractive than the cyclically turning wavelet which required O (N2) operations. Invariant wavelet transform's computational complexity was comparable to fast Fourier transform.

The benefits of applying this technique to spectrophotometry lead to an automated analysis of the spectrum and better understanding of the technique's experimental limitations. They may also lead to a quantitative study of variation in the concentration of biological substances related diseases. Thus, future work will focus on analysing the signal to noise ratio using a wider spectrum range and quantifying the concentration of biologic substances present in a sample.

Acknowledgements

Rubiel Vargas would like to thank the Universidad del Cauca for supporting this work.

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