The Problem of Deconvolution

In Section II we developed the concept of the convolution integral and its discrete approximation. No conceptual difficulty is therefore encountered in computing A from B and G  

Even considering our brief treatment of this subject thus far, we may have enough information to foresee that difficulties could arise when we attempt to compute values of G from given values of A and B. Such difficulties do indeed occur. The problem of deconvolution has therefore been the subject of a vast literature spanning the numerous special fields of science. 

Let us begin our study of the problem by reformulating it in the continuous representation, with definitions of variables that have enjoyed some popularity  

If we are observing a spectrum o(x ) with the aid of an instrument having a characteristic response function s(x — x ), then i represents the data acquired. If we have a perfectly resolving instrument, then s(x — x ) is a Dirac function, and our data i(x) directly represent the true spectrum, that is, o(x). In this case we have no need for deconvolution. 

Conversely, we may observe an exceedingly narrow spectral line, so that o(x ) is approximated by 5(x ). Now the data i(x) represent the response function. This principle can, in fact, be used to determine the response function of a spectrometer. The laser, for example, is a tempting source of monochromatic radiation for measuring the response function of an optical spectrometer. Coherence effects, however, complicate the issue. We present further detail in Section II of Chapter 2. 

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Before we can confront the problem of undoing the damage inflicted by spreading phenomena, we need to develop background material on the mathematics of convolution (the function of this chapter) and on the nature of spreading in a typical instrument, the optical spectrometer (see Chapter 2). In this chapter we introduce the fundamental concepts of convolution and review the properties of Fourier transforms, with emphasis on elements that should help the reader to develop an understanding of deconvolution basics. We go on to state the problem of deconvolution and its difficulties. 

Wing, G., A Primer on Integral Equations of the First Kind The Problem of Deconvolution and Unfolding, Society for Industrial and Applied Mathematics, Philadelphia, 1991. 

In addition to natural product sources, chemical compound libraries are used frequently in the drug discovery process. These libraries can range from small, focused libraries specifically synthesized with a particular target in mind to massive, randomly generated libraries. While there is still the problem of deconvolution of an active library, synthetically generated libraries do have a few advantages over natural product extract mixtures. There are typically equal... 

The definition of the convolution product is quite clear like the one of the Fourier transforms, it has a given mathematical expression. An important property of convolution is that the product of two functions corresponds to the Fourier transform of the convolution product of their Fourier transforms. In the context of high-resolution FT-NMR, a typical example is the signal of a given spin coupled to a spin one half. In the time domain, the relaxation gives rise to an exponential decay multiplied by a cosine function under the influence of the coupling. In the frequency domain, the first corresponds to a Lorentzian lineshape while the second corresponds to a doublet of delta functions. The spectrum of such a spin has a lineshape which is the result of the convolution product of the Lorentzian with the doublet of delta functions. In contrast, the word deconvolution is not always used with equal clarity. Sometimes it is meant as the strict reverse process of convolution, in which case it corresponds to a division in the reciprocal domain, but it is often used more loosely to mean simplification. This lack of clarity is due to the diversity of solutions offered to the problem of deconvolution, depending on the function to be deconvoluted, the quality one wishes to obtain, and other parameters. 

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) ... 

There are many facets of this study which we feel merit further investigation. In particular it is necessary to consider am extension of the proposed model, which in its present form is confined to the performance of a simple column, to cover the behaviour of any set of columns since it is column sets which are normally used. In addition, it is important to consider the input to the model which should be truly representative of polymers with a molecular weight distribution and not merely a concentration pulse of perfectly monodisperse polymer. In relation to this latter suggestion it would be significant if it were possible to link this model to the very real problem of deconvolution, i.e. the removal of instrumental and column broadening from the observed chromatogram to produce the true molecular weight distri-... 

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. 

Consider now the problem of identifying a linear system in the form of its weighting function h(t), using the relationship (5.66). This problem is called deconvolution. Discrete Fourier transformation offers a standard technique performing numerical deconvolution as mentioned in Section 4.3.3. It... 

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... 

Only a high-resolution structural method that overcomes the difficulties of deconvoluting multiple, spatially highly overlapping structures (Fig. 4) is likely to be able to accurately answer this question. The spectral overlap problem is a principal one since even under ideal conditions it is not possible to achieve occupancies of the K or L intermediate higher than about 50% and 70%, respectively, because of the extensive spectral overlap between the ground state and both the K and the L intermediate (Amax differences of +22 nm and —28 nm, respectively) (Balashov and Ebrey, 2001). 

From the point of view of synthetic effort, preparation of combinatorial mixtures is by far the most economical approach. It can be done with ordinary laboratory equipment and does not take more time than the synthesis of any one of the individual components of the library. This simplicity, however, has its price firstly, the more components a mixture contains the more difficult it becomes to follow the reaction analytically and to determine the actual composition of the reaction product. Secondly, if hits are found in a biological assay, deconvolution is required. In most cases this is done via resynthesis either of the individual components or of subsets of the mixture. If the composition of the initial mixtures was carefully planned it may be possible to identify the active component(s) by simply comparing the composition of the active mixtures with those of the inactive ones. Corresponding procedures have been reported in the literature (e.g., the techniques of indexed [1,2] and orthogonal [3] chemical libraries have been used in solution-phase synthesis). However, the biological effect of a mixture may also be due to a combined action of several weakly active members, with the result that deconvolution does not identify a significantly active compound. Finally, the problem of impurities multiplies with the complexity of the mixtures. 

An important advantage of the autocorrelation function lies in the possibility of deconvolution which can lead to a unique solution, if the electron density profile is centrosymmetric Despite stringent requirements on the precision of the data to be analyzed this method in principle overcomes the phase problem. Special methods for this deconvolution have been developed by several authors however most of... 

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