Deconvolution methods linear

A linear deconvolution method is one whose output elements (the restoration) can be expressed as linear combinations of the input elements. Until recently, the only seriously considered methods of deconvolution were linear. These methods can be developed and analyzed in detail by use of long-standing mathematical tools. Analysis of linear methods tends to be simpler than that of nonlinear methods, and computations are shorter. This point is especially important, because deconvolution is inherently computation intensive. It is not surprising that linear methods have historically dominated deconvolution research and applications. 

We have presented two deconvolution methods from an intuitive point of view. The approach that suits the reader s intuition best depends, of course, on the reader s background. For those versed in linear algebra, methods that stem from a basic matrix formulation of the problem may lend particular insight. In this section we demonstrate a matrix approach that can be related to Van Cittert s method. In Section IV.D, both approaches will be shown to be equivalent to Fourier inverse filtering. Similar connections can be made for all linear methods, and many limitations of a given linear method are common to all. 

Any linear deconvolution method that is to achieve even modest success... 

In preceding chapters we laid a foundation for the study of deconvolution. We presented several linear methods that exemplify the groundwork available before recent developments revolutionized the deconvolution field. Why, in their simplicity and elegance, did the linear methods fail to stimulate the wide adoption of deconvolution methods in spectroscopy After all, available instrumental resolution is limiting in many applications, and the simplicity of the microcomputer makes numerical processing attractive. 

It has been noted that deconvolution methods, most of which were linear, had a propensity to produce solutions that did not make good physical sense. Prominent examples were found when negative values were obtained for light intensity or particle flux. As noted previously, the need to eliminate these negative components was generally accepted. Accordingly, Gold (1964) developed a method of iteration similar to Van Cittert s but used multiplicative corrections instead of additive ones. 

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. 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

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... 

P. Veng-Pedersen, Novel deconvolution method for linear pharmacokinetic systems with polyexponential impulse response, J. Pharm. Sci.,... 

It is important to realise that least squares and maximum entropy solutions often provide different best answers and move the solution in opposite directions, hence a balance is required. Maximum entropy algorithms are often regarded as a form of nonlinear deconvolution. For linear methods the new (improved) data set can be expressed as linear functions of the original data as discussed in Section 3.3, whereas nonlinear solutions cannot. Chemical knowledge often favours nonlinear answers for example, we know that most underlying spectra are all positive, yet solutions involving sums of coefficients may often produce negative answers. 

Kiwada, H. Morita, K. Hayashi, M. Awazu, S. Hanano, M. A new numerical calculation method for deconvolution in linear compartmental analysis of pharmacokinetics. Chem. Pharm. Bull. 1977, 25, 1312-1318. 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

UV examination has been proved to be a relevant method for the study of water and wastewater quality using deconvolution methods of UV spectra. The absorbency spectrum of water can be decomposed from a few number of characteristic spectra (reference spectra). Therefore, a given spectrum can be reconstructed with a linear combination of reference spectra and all additive parameters can be computed with the same linear combination. Qualitative and quantitative results in terms of classical parameters such as TOC, COD, BOD5, TSS, nitrate,... can be provided. 

The deconvolution methods are multi-wavelength procedures which can be classified with regard to the selection procedure of reference spectra. These spectra can be chosen from specific compounds (Maier, 1981), from independent spectra of real samples (Thomas et al., 1993), statistically selected (Gallot and Thomas, 1993) or from a mixed choice of spectra of specific compounds and of real samples. Reference spectra are not universal recently, according to the complexity of the composition of wastewater, SECOMAM has developed UVPro software based on advanced UV spectral deconvolution (Patent 00402038-4, 17 July 2000) which allows creating dedicated models and determination of reference spectra from a set of studied water and wastewater UV spectra an automatic calibration step is carried using parameters values obtained by standard or reference method. Deconvolution is used in order to find a linear relation between measured and UV estimated values for any parameter. 

Zhong Y, Liu Z (2012) Gene expression deconvolution in linear space. Nat Methods 9 8-9, author reply 9... 

Veng PP. Novel Deconvolution Method for Linear Pharmacokinetic Systems with Polyexponential Impulse Response.]Pharm Sci 1980 69 312-318. 

Microscopic dissociation constants of 3-hydroxy-a-(methylamino) methylbenzene-methanol have been calculated from the titration spectrophotometer data (c = 3.8 x 10" M. Ionic strengdi = 0.16 buffer system H3BO3/KOH) by application of a spectral deconvolution method. The results found (pKa = 9.48 pJCb = 9.71 pJCc = 10.12 pi d = 9.88) are in good concordance with those obtained from the conventional regression linear method (pJCa = 9.45 pXb = 9.77 pK, = 10.14 p d = 9.81)."... 

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. 

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. 

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

The in vivo release rate or input function is first determined. For drugs with a linear disposition kinetics this may be done using the LSA-based deconvolution methods previously discussed. Step 1 requires a suitable reference administration for the deconvolution. Dual-step methods are also denoted as reference-based methods. Three scenarios exist depending on the references given below. 

Deconvolution is briefly treated in books on the Fourier transform, however, as the literature devoted to the subject shows, it is a much broader subject. One example is the 1984 monograph by Jansson. The theoretical sections are of general interest but the applications discussed focus on optical spectroscopy and, in a more recent edition, image processing and relaxation. The first chapter is a very pedagogical and intuitive introduction, while the third - and fourth - chapters provide an introduction to linear and non-linear deconvolution methods... 

The peak area errors for the two most studied deconvolution methods (i.e., perpendicular drop and linear tangential skim) are dependent on a complex combination of resolution, relative peak width, relative peak height, and asymmetry ratio.Exponential skimming assumes that the tailing of the first peak can be described by an exponential decay and that the peaks are sufficiently resolved to... 

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