Deconvolution method

It is still possible to enhance the resolution also when the point-spread function is unknown. For instance, the resolution is improved by subtracting the second-derivative g x) from the measured signal g x). Thus the signal is restored by ag x) - (7 - a)g Xx) with 0 < a < 1. This llgorithm is called pseudo-deconvolution. Because the second-derivative of any bell-shaped peak is negative between the two inflection points (second-derivative is zero) and positive elsewhere, the subtraction makes the top higher and narrows the wings, which results in a better resolution (see Fig. 40.30). Pseudo-deconvolution methods can correct for sym-... 

Nelson, T. J., Deconvolution method for accurate determination of overlapping peak areas in chromatograms, /. Chromatogr., 587, 129, 1991. 

Both methods are also limited in accuracy of secondary structure determinations because spectral peaks must be deconvolved estimates are made of the overlapping contributions of different structural regions. These estimates may introduce error based on the reference spectra used and because deconvolution methods equate crystallographic secondary structure with the secondary structure of the protein in solution (Pelton and McLean, 2000). As amyloid fibrils are neither crystalline nor soluble, there may be even greater error in estimates of secondary structure. To compound the problem, estimates of /f-sheet content are less reliable than those of a-helix, because of the flexibility and variable twist of / -structure (Pelton and McLean, 2000). In addition, / -sheet and turn bands overlap in FTIR spectroscopy (Jackson and Mantsch, 1995 Pelton and McLean, 2000). Side chains also contribute to spectral peaks in both methods, and they can skew estimates of secondary structure if not properly accounted for. In FTIR spectra, up to 10-15% of the amide I band may arise from side chain contributions (Jackson and Mantsch, 1995). 

Gillespie, W.R. and Veng-Pedersen, P., A polyexponential deconvolution method evaluation of the gastrointestinal bioavailability and mean in vivo dissolution time of some ibuprofen dosage forms, /. Pharmacokinet. Biopharm., 13, 289-307, 1985. 

Polarimetric detection of enantiomers eluted from liquid chromatographic columns employing chiral stationary phases has been described 57 58 and interesting applications have been report-ed 59-60, for example, the study of enantiomerizations during chromatography and the evaluation of optica] purity despite incomplete chromatographic enantiomer separation. By this deconvolution method, based on Beer s and Biot s expressions, optical purities rather than enantiomeric purities are determined60. 

Deconvolution methods may be arranged in different ways for instance by final object, the nature of the pre-information, or the applied basic mathematical theory. Objects are . Peak purity test. 

A linear deconvolution method is one whose output elements (the restoration) can be expressed as linear combinations of the input elements. Until recently, the only seriously considered methods of deconvolution were linear. These methods can be developed and analyzed in detail by use of long-standing mathematical tools. Analysis of linear methods tends to be simpler than that of nonlinear methods, and computations are shorter. This point is especially important, because deconvolution is inherently computation intensive. It is not surprising that linear methods have historically dominated deconvolution research and applications. 

We have presented two deconvolution methods from an intuitive point of view. The approach that suits the reader s intuition best depends, of course, on the reader s background. For those versed in linear algebra, methods that stem from a basic matrix formulation of the problem may lend particular insight. In this section we demonstrate a matrix approach that can be related to Van Cittert s method. In Section IV.D, both approaches will be shown to be equivalent to Fourier inverse filtering. Similar connections can be made for all linear methods, and many limitations of a given linear method are common to all. 

Any linear deconvolution method that is to achieve even modest success... 

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... 

In preceding chapters we laid a foundation for the study of deconvolution. We presented several linear methods that exemplify the groundwork available before recent developments revolutionized the deconvolution field. Why, in their simplicity and elegance, did the linear methods fail to stimulate the wide adoption of deconvolution methods in spectroscopy After all, available instrumental resolution is limiting in many applications, and the simplicity of the microcomputer makes numerical processing attractive. 

It has been noted that deconvolution methods, most of which were linear, had a propensity to produce solutions that did not make good physical sense. Prominent examples were found when negative values were obtained for light intensity or particle flux. As noted previously, the need to eliminate these negative components was generally accepted. Accordingly, Gold (1964) developed a method of iteration similar to Van Cittert s but used multiplicative corrections instead of additive ones. 

The matrix notation serves to stress that the technique is applicable to shift-variant spread functions, that is, where sjk slm for j — k = l — m. Many of the deconvolution methods described here with the convolution notation may thus be generalized. In the convolution notation, the present method may be expressed by the equation... 

Although we have not proved that 6(a)) approaches 0(co) for co > Q, experimentation tends to bear this out. This is entirely consistent with current knowledge of the general properties of constraints applied to various deconvolution methods (Biraud, 1969 Frieden, 1972 Howard, 1981a, 1981b). 

Next, we ask whether anything is lost by use of the bounded methods. Although the answer is yes, the loss is almost always vastly outweighed by the benefits. In using bounded methods, we do, in fact, lose the ability to observe a certain kind of unexpected result. Consider, for example, a fluorescing sample placed between a source and an optical absorption spectrometer. A deconvolution method imposing an upper transmittance bound of 100% will be confused by the data, to say the least. The moral Be sure of the validity of constraints before you apply them. 

Finally, we suggest that a backlog of data on objects (spectra) of known properties be analyzed until the characteristics of the deconvolution method used are completely familiar. Only then should results yielded by unknown objects be judged. As in any other experimental work, the experiment should be repeated to verify reproducibility and develop confidence in the result. In this sense, the deconvolution process may be treated just like any piece of laboratory apparatus. Indeed, it takes on that identity when packaged in a laboratory microcomputer. 

Linear deconvolution methods have served to educate us as to the pitfalls of the deconvolution problem. Their occasional successful applications both tantalized and discouraged us. Now, there are fewer and fewer circumstances in which use of linear methods is justified. The more-generally useful nonlinear methods have teamed with the powerful hardware that they demand to enhance future prospects for wide application of deconvolution methods. 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

Fig. 19, an unapodized spectrum [response function (sin nx)/nx = sinc(x)] is shown in trace (b). For such a spectrum there will be sidelobes and negative absorption if the natural linewidths are narrower than the full width of the sine-shaped response function. These are seen in Fig. 19, where the linewidth is three points and the response function width eight points. Here the phrase instrument response function may have a slightly different definition, but the meaning is clear. For such a response function, the direct deconvolution methods fall short. 

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