Maximum Entropy deconvolution

Fuchs GM, Prohaska T, Friedbacher G, H utter H, Grasserbauer M (1995) Maximum entropy deconvolution ofAFM and STM images. Fresenius J Anal Chem 351 143... 

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... 

Ferrige, A. G., Seddon, M. J., Jarvis, S. (1991). Maximum-entropy deconvolution in electrospray mass-spectrometry. Rapid Commun Mass Sp. 5(8), 374-377. 

A recent application of the maximum-entropy deconvolution is the interpretation of ESl-MS spectra from the giant, extracellular hexagonal bilayer hemoglobin from the leech Nephelopsis oscura, consisting of three monomer globin chains, five non-globin linker chains, and two subunits, ranging between 16 and 33 kDa [18],... 

A.G. Ferrige, M.J. Seddon, S. Jarvis, Maximum entropy deconvolution in ESI-MS, Rapid Commun. Mass Spectrom., 5 (1991) 374. 

VanderNoottried to extract the distribution function G(x) from the CPE model. He found that the Fourier inversion method is not suitable, but that the maximum entropy deconvolution works relatively well. However, because this is an ill-posed problem, the results obtained are very sensitive to the experimental errors (noise). 

Maximum entropy deconvolution, however, is normally iteratively combined with other methods. For example, least squares curve fitting attempts to create structure in a spectrum. Maximum entropy picks solutions with least structure the reason is that in the absence of external evidence, there is no reason to assume structure in the reconstructed spectrum the flat map is the most likely answer. Most algorithms try to decrease mean square errors iteratively. A target residual is selected, according to the desired speed of computational convergence, and the solution with maximum entropy is chosen. Then, the residual is decreased and, again, a maximum entropy solution is found, until convergence. 

The central engine of this data workflow is the process of spectral deconvolution. During spectral deconvolution, sets of multiply charged ions associated with particular proteins are reduced to a simplified spectrum representing the neutral mass forms of those proteins. Our laboratory makes use of a maximum entropy-based approach to spectral deconvolution (Ferrige et al., 1992a and b) that attempts to identify the most likely distribution of neutral masses that accounts for all data within the m/z mass spectrum. With this approach, quantitative peak intensity information is retained from the source spectrum, and meaningful intensity differences can be obtained by comparison of LC/MS runs acquired and processed under similar conditions. 

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) ... 

JJ Hu, FH Li. Maximum entropy image deconvolution in high resolution electron microscopy. Ultramicroscopy 35 339-350, 1991. 

CRYSTAL STRUCTURE DETERMINATION BY MAXIMUM ENTROPY IMAGE DECONVOLUTION IN COMBINATION WITH IMAGE SIMULATION... 

Key words HRTEM, image deconvolution, maximum entropy method... 

In the present paper, the crystal structure of Ndi ssCcq i5Cu04. has been studied by maximum entropy image deconvolution in combination with image simulation. This work is focused on demonstrating the effectiveness of image deconvolution technique rather than determining the structure itself... 

Fig.n.9 Deconvolved mass spectra [deconvolution by a maximum entropy algorithm (MaxEnt) supplied by instrument manufacturer] for S-peptide and S-protein, showing differences upon oxidation with 15 mM H2O2 using 15 mM Gin as scavenger. 

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. 

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

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. 

KAL 95] KALCEFF W., ARMSTRONG N., CLINE J.P., An evaluation of deconvolution techniques in x-ray profile broadening analysis and the application of the maximum entropy method to alumine data , [Pg.332]

Fig. 3.18 Example of the improvement in resolution obtainable by the use of the maximum entropy method on FDS (LANSCE) data. INS spectrum of 1,3,5-triamino-2,4,6 trinitrobenzene before (a) and after (b) deconvolution.

D.S. Sivia, P. Vorderwisch R.N. Silver (1990). Nucl. Insts. Meth. Phys. Res., A290, 492—498. Deconvolution of data from the filter difference spectrometer -from hardware to maximum-entropy. 

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