Deconvolution smoothing

Equation 9 represents the IR spectmm (intensity versus wavenumber), which can be derived from expression (8) using a mathematical technique known as Fourier transformation. Needless to say, this requires spectrometer-interfaced computing power, which additionally provides the capacity for spectral manipulation such as deconvolution, smoothing, and subtraction. 

The data-analysis part of LabVTEW includes such lools as curve litling. signal generation, peak analysis, Fourier analy.sis. deconvolution, smoothing, and various mathematical operations. The program also inte grates with standard mathematics software such as Mathcad and M.ATLAB. 

Instrumental transmission (convolution by the PSF) is always a smoothing process whereas noise is usually non-negligible at high frequencies, the noise amplification problem therefore always arises in deconvolution. This is termed as ill-conditioning in inverse problem theory. 

The situation is illustrated in Fig. 2.15 where a signal is shown which has been obtained in the analytical reality, distorted and disfigured by noise and broadening. All of these effects can be returned to a certain degree by techniques of signal treatment like deconvolution, signal accumulation and smoothing, etc. 

The demand that the solution 6 be consistent with the data i results in the improved resolution that we expect from a deconvolution method. As we have explained, however, it also results in the amplification of high-frequency noise. The smoothing of this noise to some extent defeats the purpose of deconvolution. The tradeoff between smoothness and consistency is explicit in the formulation of a method first described by Phillips (1962) and further developed by Twomey (1965). In this method, we minimize the quantity... 

For this work, the spectrometer function s(x) was determined by the method outlined in Section II.G.3 of Chapter 2. In digitizing the data, a sample density was chosen to accommodate about 70 samples taken across the full width at half maximum of s(x). A 25-point cubic polynomial smoothing filter was used in the deconvolution procedure to control high-frequency noise. Instead of the convolution in Eq. (13), the point-successive modification described in Section III.C.2 of Chapter 3 was employed. In Eq. (24) of Chapter 3, we replaced k with the expression... 

Execution time can be significantly reduced without loss of effectiveness by data sample density reduction and corresponding alteration of the smoothing polynomial. Further improvements can be realized by using modern highspeed microcomputers. Constrained deconvolution times of well under one minute for equivalent spectra should be possible with compact and inexpensive equipment. 

Because the most serious problem arising in the deconvolution of spectra is that of noise, detailed attention to smoothing in a fashion consistent with the uniform attenuation of high-frequency noise will result in the best possible deconvolution results. 

In Section II, the deconvolution examples used noise-free simulated spectra. Any real spectrum will be corrupted by noise. The noise can be reduced by smoothing, but smoothing generally attenuates high spectral frequencies in data. There is an operational conflict, however, because it is these same high spectral frequencies that we wish to enhance by deconvolution. In this section, the effects of noise on deconvolution are demonstrated and several smoothing techniques are evaluated. 

It can be concluded from the results shown in Fig. 3 that without some form of electronic or digital smoothing the minimum required signal-to-noise ratio for successful deconvolution might be as high as 200 1. In Fig. 4,... 

In testing a smoothing technique, two criteria should be considered. Obviously, a smoothing technique should reduce the magnitude of the noise and the impact of noise on the deconvolved spectrum. Second, a smoothing technique should not seriously affect the deconvolution process or the deconvolved spectrum. That is, if the smoothing is too severe, it will further... 

Fig. 7 Deconvolution results for simulated data with a signal-to-noise ratio of 60 1. Trace (a) is the original spectrum o(jc), trace (b) the convolved spectrum i(x). Traces (c)-(f) are the results of deconvolution with no smoothing, 5-point quadratic smoothing, 13-point quartic smoothing, and multismoothing, respectively.

Prior to deconvolution, the background was subtracted and the data were smoothed with a 15-point quadratic least-squares polynomial followed by a 19-point quartic least-squares polynomial. The data were then scaled from 0 to 1. The S3 profile was deconvolved using a weight constraint of the form... 

P.J.M. Smulders, A deconvolution technique with smooth non-negative results, Nucl. Instr. Meth. B14 (1986) 234-239. 

Deconvolution Algorithms. The correlation function for broad distributions is a sum of single exponentials. This ill-conditioned mathematical problem is not subject to the usual criteria for goodness-of-fit. Size resolution is ultimately limited by the noise on the measured correlation function, and measurements for several hours (13) are required to obtain accurate widths. Peaks closer than about 2 1 are unlikely to be resolved unless a-priori assumptions are invoked to constrain the possible solutions. Such constraints should be stated explicitly otherwise, the results are misleading. Constraints that work well with one type of distribution and one set of data often fail with others. Thus, artifacts including nonexistent bi-, tri-, and quadramodals abound. Many particle size distributions are inherently nonsmooth, and attempts to smooth the data prior to deconvolution have not been particularly successful. 

There is a large body of literature on FTIR spectroscopy including, for example, Bracewell (1965), Horlick (1968), Bell (1972), Griffiths (1975, 1983), Ferraro and Basile (1978), Koenig (1981), Griffiths and de Haseth (1986), Perkins (1986, 1987), Mackenzie (1988). Cameron and Moffatt (1984), and Gillette et al. (1985) have explained the basic mathematical concepts of deconvolution, derivation and smoothing in FTIR spectroscopy. 

Deconvolution is manipulation of the interferogram with an exponential weighting function. It may be viewed as the inverse of smoothing. In deconvoluted... 

Cameron DG, Moffatt DJ (1984) Deconvolution, derivation, and smoothing of spectra using Fourier transforms Test Eval 76 83... 

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