Deconvolution algorithm

Low and High frequency can be restored by use of a deconvolution algorithm that enhances the resolution. We operate an improvement of the spectral bandwidth by Papoulis deconvolution based essentially on a non-linear adaptive extrapolation of the Fourier domain. 

The characteristic peaks must be deconvolved to eliminate peak interference powerful deconvolution algorithms exist for EDS and WDS. 

Nevertheless a number of blind deconvolution algorithms have been devised which are able to notably improve the quality of real (i.e. noisy) astronomical images (e.g. Ayers and Dainty, 1988 Lane, 1992 Thiebaut and Conan, 1995). 

Thiebaut, E., Conan, J.-M., 1995, Strict a priori constraints for maximum likelihood blind deconvolution, JOSA.A, 12, 485 Thiebaut, E., 2002, Optimization issues in blind deconvolution algorithms, SPIE 4847, 174... 

Because online separations provide such a wealth of information about target proteins, interpretation becomes of critical importance in order to make full use of the data. The first step in any analysis of LC-MS data involves integration and deconvolution of sample spectra to determine protein mass and intensity. In manual analysis (Hamler et al., 2004), users identify protein umbrellas, create a total ion chromatogram (TIC), integrate the protein peak, and deconvolute the resulting spectrum. Deconvolution of ESI spectra employs a maximum entropy deconvolution algorithm often referred to as MaxEnt (Ferrige et al., 1991). MaxEnt calculates... 

The deconvolution algorithms were developed by Rudzki et al. [284,300] and are now commercially available as computer software [301]. For instance, in the case of two sequential reactions (equation 13.27) producing two heat decays,... 

The relative molecular mass determination of an unknown protein is generally performed automatically using various deconvolution algorithms, but the procedure is limited to relatively simple mixtures. 

These properties carry back to the discrete formulation. We shall use both discrete and continuous formulations in this volume, changing back and forth as needs require. The continuous regime allows us to avoid consideration of sampling effects when such consideration is not of immediate concern. Deconvolution algorithms, on the other hand, are numerically implemented on sampled data, and we find the discrete representation indispensable in such cases. 

Although a number of effective deconvolution algorithms do not use Fourier methods, these methods shed considerable light on the performance of the algorithms. For this reason, we introduce the Fourier transform and outline some of its most-useful properties. Only a brief treatment is given here. For additional detail, we again refer the reader to the excellent practical text on this subject by Bracewell (1978). 

There is another difficulty, however. Additive instrumental noise is also logarithmically converted by the operation. This in turn may affect the validity of the noise assumptions employed in developing certain deconvolution algorithms. When absorptions are very small, we may approximate... 

Thomas (1981) and Schafer et al (1981) have also discussed the reblurring method. In addition, Schafer et al. have studied a generalized class of iterative deconvolution algorithms. They examined the convergence properties of the iteration... 

VI. Deconvolving the Data Constrained Deconvolution A Deconvolution Algorithm... 

Step by step, the constrained deconvolution algorithm may be stated as follows. 

The constrained deconvolution algorithm produces estimates that cannot be obtained from the data by simple linear inverse filtering. This is most readily seen using the Blass-Halsey weight function as an example. 

Two conclusions follow readily. First, the constrained deconvolution algorithm is decidedly nonlinear in the observed data. Second, no easy analytical interpretation of the effects of a particular relaxation function may be obtained by considering the Fourier transform of o(k) cast in terms of i(x) and r[i(x)]. 

There are many reasons why deconvolution algorithms produce unsatisfactory results. In the deconvolution of actual spectral data, the presence of noise is usually the limiting factor. For the purpose of examining the deconvolution process, we begin with noiseless data, which, of course, can be realized only in a simulation process. When other aspects of deconvolution, such as errors in the system response function or errors in base-line removal, are examined, noiseless data are used. The presence of noise together with base-line or system transfer function errors will, of course, produce less valuable results. 

To date, the prior knowledge built into deconvolution algorithms has mainly been deterministic in nature, such as by use of positivity or boundedness. But a unified approach to estimating spectra should accommodate all possible physical and statistical prior knowledge about formation of the spectra. In particular, the approach should be physical, based on the Bose-Einstein nature of photons. Such an approach will be presented here. One general restoring principle will be derived, from which particular estimators... 

Deconvolution Algorithms. The correlation function for broad distributions is a sum of single exponentials. This ill-conditioned mathematical problem is not subject to the usual criteria for goodness-of-fit. Size resolution is ultimately limited by the noise on the measured correlation function, and measurements for several hours (13) are required to obtain accurate widths. Peaks closer than about 2 1 are unlikely to be resolved unless a-priori assumptions are invoked to constrain the possible solutions. Such constraints should be stated explicitly otherwise, the results are misleading. Constraints that work well with one type of distribution and one set of data often fail with others. Thus, artifacts including nonexistent bi-, tri-, and quadramodals abound. Many particle size distributions are inherently nonsmooth, and attempts to smooth the data prior to deconvolution have not been particularly successful. 

Several deconvolution algorithms are now available including EST, NNLS, and CONTIN (3). 

Samples with higher polydispersity do not generally yield symmetrical elution profiles. Moreover, a detailed MWD [either m(M) or n(M)] is often desired. As mentioned earlier, band broadening does not significantly affect the elution profile when low flow rates are used. Therefore, the elution profile can be directly converted into a MWD by the procedures outlined. If, on the other hand, fast flow rates are used to shorten the analysis time, the observed elution profile must be adjusted to account for the effects of band broadening. A deconvolution algorithm that filters out the band-broadening contribution to the elution... 

The embedded microcomputer board applies the ASD (Advanced Spectral Deconvolution) algorithm to the Diode Array signal, stores the spectra, the results together with the traceability data (time, user, sampling site...). All the results are displayed on a bright graphic screen with the traceability data and the multi-parameters concentrations. PASTEL UV can be used for a wide range of applications such as ... 

Where the multicharged series is due to cationization, the mass of H should be replaced by that of the cation (e.g., Na+ or K+). Fortunately, most modern ESI-MS data systems have computer-based deconvolution algorithms to automate this process (Figure 4(c)). 

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