Direct deconvolution method

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. 

Any chemically encoded library requires a double orthogonal chemical strategy, or the construction of elaborate tags/linkers on the solid support prior to the synthesis. Even when these are easily prepared and inert, the synthetic scheme becomes more complicated than direct deconvolution methods, where only the library synthesis is required. Sometimes the tag chemistry and the... 

Encoding methods provide generally a larger set of more detailed data but also require additional complexity to be added to the library structure. Direct deconvolution methods also have appealing features such as being often independent from analytical techniques, easy to perform, and often reliable. An absolute choice among the two main structure determination method classes, and among different methods in the same class, does not exist. The skilled chemist,... 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

In the specific case of a linear disposition, the above formula simply becomes equal to the direct deconvolution method previously described. 

An IV reference is available. The absorption rate may be determined by deconvolution using a direct deconvolution method or a prescribed input function method as described above. 

Numerical reconvolution has been used to retrieve the kinetic, enthalpic and volumetric parameters. Direct deconvolution methods have been developed as well. The reconvolution methods assume a sum of single-exponential decay functions for the time evolution of the pressure ... 

The deconvolution method we propose here is also parametric and is based on direct integral parameter estimation (ref. 27). We consider a "hypothetical" linear system S with input u = h, where h is the known weighting function of the real system S, and the output of S is assumed to be = , the known response function. Then by (5.66) we have... 

It is much easier, though, to compare encoding techniques with direct deconvolution (see elsewhere in this book) [1, 6], The two main classes of structure determination methods for pool libraries are significantly different, and clear distinctions about their usefulness can be made. The example of a 240-member (without considering diastereoisomers) mercaptoacyl pyrrolidine library, which was prepared as an encoded (secondary amine tags) [40] or as a nonencoded library and then submitted to iterative deconvolution [88], will be used for this comparison. The following considerations were either reported by MacLean et al. [40] or derived from the critical analysis of the results obtained. 

In conclusion, let me underscore once more the need for combinatorial chemists to be aware both of the current status of encoding/direct deconvolution techniques, of their main features and requirements, and of the new trends/methodologies/ideas emerging in literature. If the above is true, chemists will pick the best method to be applied to the combinatorial problem they are facing. It is also important to consider not only the synthesis of the library but also the screenings which will be used for testing it, the analytical and automation facilities available, and most of all the timelines of the project, so as to choose the best compromise which will give the best overall results for the project needs. 

Seneci P, Direct deconvolution techniques for pool libraries of small organic molecules, Combinatorial Chemistry and Combinatorial Technologies Methods and Applications (Eds. Miertus S, Fassina G), 153-192, 2005, Chapter 8, this volume. 

The first section will be devoted to the synthesis of these libraries using the so-called mix-and-split or divide-and-recombine approach (2, 3) and to their analytical characterization. The following sections will focus on different methods to determine the structure of an active component from an SP pool library direct structure determination (Section 7.2) and indirect structure determination, via deconvolutive methods (Section 7.3) or encoding methods (Section 7.4), will be covered. Finally, a section will be devoted to new trends in SP pool libraries, paying particular attention to innovative methods for the fast and reliable discovery of new active structures through miniaturization (bead-based techniques). 

Direct structure determination methods, where positives are characterized directly via off-bead or on-bead identification of their chemical structure, will be described in detail in this section. Indirect methods that determine the structure of positives from the library architecture will be covered later they use either deconvolutive methods (Section 7.3), where the iterative synthesis of library pools with decreasing complexity via sequential determination of the best monomers leads to the identification of a positive structure, or encoding methods (Section 7.4), where, during the library synthesis, the structure of each component is coupled to a tag that can be read from a single bead after the library screening. 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

It is, finally, worth mentioning that in addition to deconvolution methods it is also possible to use a convolution approach. The advantage of the latter is that LPA can be performed in a one-step procedure, directly on the measured data, provided that the procedure can make use of a parametric description of the IP preliminarily determined. Software packages are available for this type of analysis. ... 

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. 

The above analytically exact direct deconvolution algorithm does not apply when UIR(O) = 0. However, the method is readily extended to include UIR(0) = 0 as follows. Differentiation of R with respect to time in this case gives... 

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. 

The direct demodulation algorithm provides a general approach to handle a large variety of image restoration or reconstruction problems. Computer simulations and analysis results for COS-B and CGRO 7-ray data show that in comparison with traditional techniques, e.g. maximum entropy method, cross-correlation deconvolution or likelihood approach, the direct demodulation method has high sensitivity, high resolution ability and capability to effectively reduce the effect of statistical fluctuations and noise in data and to simultaneously restore both the extended and discrete features in the object. 

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