Nonlinear deconvolution methods

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. 

Like the other nonlinear constrained methods, the maximum-entropy method has proved its capacity to restore the frequency content of 6 that has not survived convolution by s and is entirely absent from the data (Frieden, 1972 Frieden and Burke, 1972). Its importance to the development of deconvolution arises from the statistical concept that it introduced. It was the first of the nonlinear methods explicitly to address the problem of selecting a preferred solution from the multiplicity of possible solutions on the basis of sound statistical arguments. 

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. 

There are numerous books on digital signal processing (DSP) and Fourier transforms. Unfortunately, many of the chemically based books are fairly technical in nature and oriented towards specific techniques such as NMR however, books written primarily by and for engineers and statisticians are often quite understandable. A recommended reference to DSP contains many of the main principles [29], but there are several similar books available. For nonlinear deconvolution, Jansson s book is well known [30]. Methods for time series analysis are described in more depth in an outstanding and much reprinted book written by Chatfield [31]. 

It is important to realise that least squares and maximum entropy solutions often provide different best answers and move the solution in opposite directions, hence a balance is required. Maximum entropy algorithms are often regarded as a form of nonlinear deconvolution. For linear methods the new (improved) data set can be expressed as linear functions of the original data as discussed in Section 3.3, whereas nonlinear solutions cannot. Chemical knowledge often favours nonlinear answers for example, we know that most underlying spectra are all positive, yet solutions involving sums of coefficients may often produce negative answers. 

There are several other nonlinear methods applied in different spectroscopies, the book edited by Jansson provides a broad review of nonlinear deconvolution. 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

In applying the technique of deconvolution, we take as known the spectrometer response function. It seems reasonable that the more accurately we know this function, the more accurate will be the deconvolved result. Although the nonlinear methods described in Chapter 4 are more tolerant of error, they too require a knowledge of the response function. 

In spite of performance advantages in the use of nonlinear methods, it is instructive to start our deconvolution study by examining the linear methods they will give us insight into the process. The ensuing development will also define the applicability domain of linear methods and reveal their limitations. We shall see that in some circumstances a linear method is the method of choice. 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

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