Deconvolution methods constrained

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. 

Verotta, D., Two constrained deconvolution methods using spline functions, Journal of Pharmacokinetics and Biopharmaceutics, Vol. 21, No. 5, 1993, pp. 609-636. 

A simple and reliable way of constraining the input function is to apply the prescribed input function deconvolution method becanse this method allows the input function to be directly constrained. For example, a simple linear spline may be used as an input function. The non-negativity constraint is introdnced by a simple parameterization of the spline with parameters defined as the function valnes at the so-called knots where the linear line segments are joined. The inpnt fnnction will be non-negative by ensuring that all parameters, i.e., the function valnes at the knots are non-negative. 

Vardya and Hester used their OEX model in a constrained linear optimi2ation procedure, based on the Box-Complex method which is essentially a constrained simplex minimization technique. The method does not require derivatives of the object function and is not subject to scaling problems. As an example. Fig. 8 shows a de-convolution of a chromatogram of Dextran T-2000 with water as the mobile phase in a controlled porous glass column. The badly fused peaks are successfully deconvoluted. 

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. 

In Section II.C we will present novel tricyclic xanthene derived amino acid templates, which allow the construction of libraries of cyclic conformationally constrained peptide loop mimetics using the split-and-mix method without having to use tagging and deconvolution strategies. In Section III we will focus on parallel and combinatorial approaches devoted to the synthesis of small molecule, non-peptidic compound collections, which in addition offer the possibility to incorporate structural features derived from protein epitope mapping into conformationally constrained peptide mimetics. 

The final step before structure determination by traditional methods is extraction of integrated intensities from the powder diffraction pattern (see below). Intensities can nearly always be determined only after deconvolution of partially overlapping Bragg reflections. This can be done for individual peaks or for groups of peaks, but usually full decomposition methods are used. There are basically two techniques the iterative Le Bail method [6] and the constrained linear least-squares approach developed by Pawley [7]. 

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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