Deconvolution problem

In the remainder of this paper, we exhibit the solution of the deconvolution problem in the frequency domain, but it is possible to establish an analogy with tlie temporal solution exposed by G. Demoment [5,6]. 

Solving such a myopic deconvolution problem is much more difficult because its solution is highly non-linear with respect to the data. In effect, whatever are the expressions of the regularization terms, the criterion to minimize is no longer quadratic with respect to the parameters (due to the first likelihood term). Nevertheless, a much more important point to care of is that unless enough constraints are set by the regularization terms, the problem may not have a unique solution. 

Stated like this, conventional and blind deconvolution appear to be just two extreme cases of the more general myopic deconvolution problem. We however have seen that conventional deconvolution is easier to perform than myopic deconvolution and we can anticipate that blind deconvolution must be far more difficult. 

Also, application of these methods is not limited to one-dimensional problems. Most of the concepts may readily be extended to multiple dimensions. Digital image processing, which usually deals with two independent variables, has been the object of considerable investigation and the source of many advances in our understanding of the deconvolution problem. 

Some techniques for solving deconvolution problems also adapt readily to the more-general Fredholm case. The relaxation methods of, for example, Van Cittert (1931) and Jansson can be so adapted (Jansson, 1968, 1970 Jansson, Hunt and Plyler, 1968, 1970). 

Difficulties in solving the deconvolution problem are revealed if we examine it in the continuous representation given by Eq. (86). Suppose that the solution for o(x) is to some degree uncertain, and that it may be written as the sum of a desired solution o(x ) and a spurious part 0(x ) ... 

In deconvolution problems, such dominance is rarely the case. It is a very modest resolution correction indeed when the response function is so narrow that its central element dominates all the others. Fortunately, the necessary conditions are far less severe, and convergence is usually not a serious problem. Equation (25) seems to tell us, however, that each new row in the matrix brings more independent information when [s]ww/Sn9tw [s]ww is as large as possible. 

A large number of linear methods have been developed with particular characteristics that tend to suit them to specific deconvolution problems. None of these adaptations shows beneficial results nearly so profound as those resulting from the imposition of the physical-realizability constraints discussed in the next chapter. Furthermore, the present work is not intended... 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

We first mentioned the applicability of optimization (minimization) methods in Section V.C of Chapter 1. Constraints pose no particular problem to many of these methods. It would seem that the deconvolution problem with object amplitude bounds should be a straightforward application. The most general case, however, deals with each sampled element om of the estimate as a parameter of the objective function and hence the solution. Excessive computation is then required. The likelihood is great that only local minima of the objective function O will be found. Nevertheless, the optimization idea may be teamed with a Monte Carlo technique and a decision rule to yield a method having some promise. 

We propose here to apply the technique to the bounded deconvolution problem. Trial grain positions may be selected at random or by some other process, such as bit-reversed sampling (Allebach, 1981 Deutsch, 1965). The trial solution after t grains have been placed is d. The corresponding objective function may be, but is not limited to, a sum of squares ... 

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. 

Comment. All the less- or more-disordered packing modes introduced above are frequently encountered for arrays of helical molecules. The problem of disorder results in an additional (compared with most single crystal analyses) deconvolution problem when X-ray diffraction patterns of such systems are being interpreted. Although complications from disorder effects are not unique to fibrous systems, they are more frequently encountered there. I suspect that this has... 

Lessons learned from natural products dereplication and prioritization in high-throughput assays can also be applied to the deconvolution problems, faced by combinatorial chemists when assaying pools of compounds. With limited resynthesis capabilities, not every active pool can be followed up, and the use of prioritization methods will be required, similar to those outlined above. 

A deconvolution problem in general does not have a unique solution. Instead there are an infinite number of possible solutions that can fit the same set of cascade impactor measurements. It is well recognized that, for most engineering applications, actual particle size distributions can be reasonably represented by a set of log-normal distributions. With this concern, in the following, a deconvolution method (chi-squared method) to extract particle size distributions from cascade impactor data is introduced, which is based on multimodal log-normal size distributions (Dzubay and Hasan, 1990). 

The main deconvolution problem, problem 2, requires problem 1 to be solved first. 

Problem 2 Several deconvolution methods exist for determining the inpnt to solve deconvolution problem 2. Basically these can all be classified in two categories, namely the direct methods and the prescribed inpnt fnnction methods. 

To solve a univariate deconvolution problem, approaches such as evolving factor analysis (EFA) (Maeder, 1987) or multivariate curve resolution (MCR) (Tauler Barcelo, 1993), among others (Vivo-Truyols et al., 2002 Sarkar et al., 1998 Kong et al. 2005) can be used. When these approaches are used with univariate data, the variables to be solved for are the number, positions, and abundances of each of the peaks that make up the signal. 

Klementev K. V., Deconvolution problem in x-ray absorption fine structure spectroscopy,/ Phys. D Appi. Phys., 34,2241-2247 (2001). Jovari R, Saksl K., Pryds N., Lebech B., Bailey N. R, Mellergard A., Delaplane R. G., and Franz H., Atomic structure of glassy MggoCu3oY4o investigated with EXAFS, x-ray and neutron diffraction, and reverse Monte Carlo simulations, Phys. Rev. B, 76, 054208 (8 pages) (2007). 

From Eq. 21, it is clear that the determination of the distribution function G x) requires the deconvolution of the integral equation knowing both the current response /(/) or 7(thermal wave technique) and the time derivative of the temperature profile 7 x, t). Various approaches to solve the deconvolution problem have been discussed (Sessler 1997). 

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