Data Constrained Deconvolution

The actual deconvolution of a data set is formally straightforward. Let dik)(x) be the kth iterative estimate of the actual spectrum o(x), where x is nominally time viewed as a sequence-ordering variable. Further, let i(x) be the actual observed spectrum that has been instrumentally convolved with the observing system response function s(x). The observed data set i(x) is assumed to be related to o(x) by the convolution integral equation 

Step by step, the constrained deconvolution algorithm may be stated as follows. 

Convergence may be monitored by forming the root-mean-square error of the current estimate as 

The key to the success of constrained deconvolution is the relaxation function r[x]. For absorption spectra the data are scaled to lie in the range from 0 to 1. For example, Jansson (1970) used 

In Chapter 7 several tests are presented using a Gaussian relaxation function such that 

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VI. Deconvolving the Data Constrained Deconvolution A Deconvolution Algorithm... 

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. 

The constrained deconvolution algorithm produces estimates that cannot be obtained from the data by simple linear inverse filtering. This is most readily seen using the Blass-Halsey weight function as an example. 

Two conclusions follow readily. First, the constrained deconvolution algorithm is decidedly nonlinear in the observed data. Second, no easy analytical interpretation of the effects of a particular relaxation function may be obtained by considering the Fourier transform of o(k) cast in terms of i(x) and r[i(x)]. 

Constrained deconvolution is an experimental tool. Happily, it is a useful tool testable at each step in its development. Do not, however, assume that it can do no wrong. Utilization of state-of-the-art scientific instrumentation is a demanding task and is not for the faint of heart. Constrained deconvolution is a probable addition to many state-of-the-art instruments. Constrained deconvolution is rapidly becoming a part of data acquisition rather than a part of data analysis. 

Figures 4b and 4c show that neither unconstrained nor non-negative maximum likelihood approaches are able to recover a usable image. Deconvolution by unconstrained/constrained maximum likelihood yields noise amplification — in otfier words, the maximum likelihood solution remains iU-conditioned (i.e. a small change in the data due to noise can produce arbitrarily large changes in the solution) regularization is needed.

If at the outset the data are very noisy and if the noise predominates in the Fourier frequency range needed to effect a restoration, constraints provide the only hope for improvement. The reason is that many of the noise values in the data would restore to physically unrealizable values by linear deconvolution. The constrained methods are inherently more robust because they must find a solution that is consistent with both data and physical reality. 

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. 

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. 

FIGURE 11.7 (a) Affinity distributions obtained at different ionic strengths from experimental data for purified peat humic acid (PPHA) (b) deconvolution of the distribution obtained for PPHA with Gaussian functions. (Data from Milne. C.J. et al.. Environ. Sci. TechnoL, 37, 958-971, 2003 Reprinted from J. Colloid Interface Sci., 336, Orsetti, S. et al., Application of a constrained regularization method to extraction of affinity distributions Proton and metal binding to humic substances, 377-387. Copyright 2009, with permission from Elsevier.)... 

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