Maximum entropy methods

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

We briefly repeat now the essential parts of the maximum entropy method for details we refer to the literature [167-169]. We seek to obtain information on the dynamics of the internal degree of freedom of the model from PIMC simulations. The solution of this problem is not... 

The well-known maximum entropy method (MEM) can be implemented thanks to a non-quadratic regularization term which is the so-called negen-tropy ... 

Table 1 Spectral analysis for Nino 3.4 index, streamflow, and AF using the Maximum Entropy Method. Only results for the low frequency band are showed ...

We started by analyzing the dominant modes of oscillation showed by oxygen content, streamflow, and ENSO using the Maximum Entropy Method (MEM) (Table 1). It is remarkable that both the streamflow to the reservoir and the AF showed common oscillations with ENSO at fi-equencies between 0.016 and 0.035 cycles month These frequencies are very close to the two main periods of ENSO (the quasi-biennial and quasi-quadrennial periods) [56]. Although we do not have a mechanistic explanation for this teleconnection (in fact the extratropical influence of ENSO is a hot topic in climate research, Merkel and Latif [57]), it is certainly difficult to propose an alternative explanation for the oscillations in AF and stream-flow observed at these frequencies. 

The maximum entropy method thus consists of maximizing the entropy under the constraint. An algorithm to maximize entropy is the so-called Cambridge algorithm [16]. 

Papoular, R.J., Vekhter, Y. and Coppens, P. (1996) The two-channel maximum-entropy method applied to the charge density of a molecular crystal a-glycine, Acta Cryst., A52, 397 107. 

Takata, M. and Sakata, M. (1996) The influence of completeness of the data set on the charge density obtained with the maximum-entropy method. A re-examination of the electron-density distribution in Si, Acta Cryst., A52, 287-290. 

Jauch, W. and Palmer, A. (1993) The maximum-entropy method in charge-density studies aspects of reliability, Acta Cryst., A49, 590-591. 

Iversen, B.B., Larsen, F.K., Souhassou, M. and Takata, M. (1995) Experimental evidence forthe existence of non-nuclear maxima in the electron density distribution of metallic beryllium. A comparative study ofthe maximum entropy method and the multipole method, Acta Cryst., B51, 580-591. 

Sakata, M., Uno, T., Takata, M. and Howard, C. (1993) Maximum-entropy-method analysis of neutron diffraction data, J. Appl. Cryst., 26, 159-165. 

Reliability of charge density distributions derived by the maximum entropy method... 

Dobrzynski, L., Papoular, R.J. and Sakata, M. (1995) Internal magnetization density distributions of iron and nickel by the maximum entropy method, J. Phys. Soc. Japan, 65, 255-263. 

Y. Kubota, M.Takata, M. Sakata, T. Ohba, K. Kifune, and T.Takati, A Charge Density Study of the Intermetallic Compound MgCu2 by the Maximum Entropy Method, J. Phys. Condensed Matter, 12,1259 (2000). 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

To answer the question as to whether the fluorescence decay consists of a few distinct exponentials or should be interpreted in terms of a continuous distribution, it is advantageous to use an approach without a priori assumption of the shape of the distribution. In particular, the maximum entropy method (MEM) is capable of handling both continuous and discrete lifetime distributions in a single analysis of data obtained from pulse fluorometry or phase-modulation fluorometry (Brochon, 1994) (see Box 6.1). 

The maximum entropy method has been successfully applied to pulse fluorometry and phase-modulation fluorometry3- . Let us first consider pulse fluorometry. For a multi-exponential decay with n components whose fractional amplitudes are a , the d-pulse response is... 

Brochon J. C. (1994) Maximum Entropy Method of Data Analysis in Time-Resolved Spectroscopy, Methods in Enzymology, 240, 262-311. 

A. Seimiarczuk and W. R. Ware, Temperature dependence of fluorescence lifetime distributions in l,3-di(l-pyrenyl)propane with maximum entropy method, J. Phys. Chem, 93, 7609-7618 (1989). 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

A. K. Livesey and J. C. Brouchon, Analyzing the distribution of decay constants in pulse-fluorimetry using the maximum entropy method, Biophys. J. 52, 693-706 (1987). 

THE MAXIMUM ENTROPY METHOD OF SOLVING CRYSTAL STRUCTURES FROM ELECTRON DIFFRACTION DATA... 

The Maximum Entropy Method of Solving Crystal Structures from Electron Diffraction Data... 

Gilmore, C.J., Shatikland, K. and Fryer, J.R. (1993), Phase extension in electron crystallography using the maximum entropy method and its application to two-dimensional purple membrane data fromHalobacterium halobium. Ultramicroscopy, 49, 147-178. 

Giknore, C.J. Shankland, K. and Bricogne, G. Application of the maximum entropy method to powder diffraction and electron crystallography. Proc. Roy. Soc. (London) A442 97-lll (1993). 

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