Wiener distribution

To summarize, Wiener inverse-filter is the linear filter which insures that the result is as close as possible, on average and in the least squares sense, to the true object brightness distribution. 

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

Rada, R.G., D.E. Powell, and J.G. Wiener. 1993. Whole-lake burdens and spatial distribution of mercury in surficial sediments in Wisconsin seepage lakes. Canad. Jour. Fish. Aquat. Sci. 50 865-873. 

The interpretation of the Langevin equation presents conceptual difficulties that are not present in the Ito and Stratonovich interpretation. These difficulties are the result of the fact that the probability distribution for the random force rip(f) cannot be fully specihed a priori when the diffusivity and friction tensors are functions of the system coordinates. The resulting dependence of the statistical properties of the random forces on the system s trajectories is not present in the Ito and Stratonovich interpretations, in which the randomness is generated by standard Wiener processes Wm(f) whose complete probability distribution is known a priori. 

The evenness measurement, calculated from the Shannon-Wiener formula, suggests that trees which have an uneven distribution of terpenes are more resistant to the budworm. It is likely that this Imbalance In the terpene distribution Is represented by the specific terpenes (acetate fraction, myrcene, and the unidentified terpene) that were found to be important In the analysis. The analysis also Indicated that the polyphenol and protein complexlng capacity of the extracts from the foliage... 

Wiener Powders (Baked Powders). Introduced in Russ in 1873, they were prepd by compression of the usual ingredients of BlkPdr, preheated to 120°. This was done in order to melt the sulfur, and thus achieve its better distribution thruout the mass... 

Exercise. It has been remarked in 1 that a Markov process with time reversal is again a Markov process. Construct the hierarchy of distribution functions for the reversed Wiener process and verify that its transition probability obeys the Chapman-Kolmogorov equation. 

Figure 5.5 Bond distribution of Wiener index (W) for 2,3-dimethylhexane.

In a pioneer work, Marcus established the link between some usual time-varying forms of h ( ) and / (a) in a single compartment [300]. For instance in h(t) = (f +/ ), a = 1 leads to A Gam(A,/3) and 1 < a < 2 defines the standard extreme stable-law density with exponent a. In the case of a = 1.5, the obtained distribution is known as the retention-time distribution of a Wiener process with drift. 

Here, X. is the stochastic state vector, B(r,X.j) is a vector describing the contribution of the diffusion to the stochastic process and W. is a vector with the same dimensions as X. and B(t,X.j). After Eqs. (4.94) and (4.95), the W,. vector is a Wiener process (we recall that this process is stochastic with a mean value equal to zero and a gaussian probability distribution) with the same dimensions as D(t,X,) ... 

The Wiener process. We consider a particle governed by the transition probabilities of the simple random walk. The steps of the particle are Z(l), Z(2),. .. each having for n = 1, 2,... the distribution ... 

Normalized distance distribution moments were used to define —> molecular profiles and the second moment Dx takes part in defining the —> hyper-Wiener index) moreover, the index D2 was demonstrated to be equal to half the trace of the distance matrix D raised to the second power [Diudea, 1996a Diudea and Gutman, 1998]. 

Of course, for A, = 1, Wj is the Wiener index, for A, = — 1, the Harary index (or RDS UM index), and for A, = —2 the Harary number. Other X values, namely, 2, = 1/3 and 2,= —1/3, were used to model boiling points of C8 alkane isomers [Lucic, Milicevic et al., 2003], Note that for positive integer X values, W). indices coincide with the distance distribution moments Dx [Klein and Gutman, 1999]. 

Models are constructed which suggest that these optical measurements can be used to determine the effective particle size distribution parameters, mean diameter and sigma. Assumptions include multilayer particle deposit, the lognormal distribution of the diameters of the spherical, opaque particles, and no sorting of size classes during particle deposition. The optical measurement include edge trace analysis to derive the contrast transfer function, and density fluctuation measurements to derive the Wiener spectrum. Algorithms to perform these derivations are outlined. 

The corresponding functions which do contain information over a range of dimensions are the contrast transfer function, CTF, and the Wiener spectrum, or noise content of the deposit as a function of spatial frequency (8). These functions also have been found to contain information which correlates to the particle size distribution, as will now be discussed. 

The statistical fluctuation or noise level of a toner image is also by these postulates dependent on the particle size distribution. The function which relates the statistical fluctuations to spatial frequency is the Wiener spectrum, which is the Fourier transform of the optical density autocorrelation function. In terms of toner images, it is a measure of the dimensional extent over which the presence or absence of a particular toner particle will contribute to density. The density fluctuations can be measured as a function of position, normally with a slit aperture. This is schematically represented in Figure 6 where the left-hand sketch is related to large particles and the right-hand one to small particles. The density data can be used to calculate noise power or Wiener spectrum (8). Formally, the Wiener spectrum is ... 

A theoretical model to relate the Wiener spectrum to the toner deposit parameters is difficult to construct because the mathematical difficulties of dealing with projections of transforms of probability distributions quickly "hide" any simple relationships. Models have been constructed however for a crowded monolayer photographic emulsion (11), and for multilayers of emulsion (12). Although the analysis was done for one-dimensional geometry, extension to two dimensions was outlined. A different approach will be used here, which relies on the linearity property of the Fourier transform, and assumes that the location of the toner particles is independent of neighbors. 

For a first approximation to a model for the Wiener spectrum constructed from the toner particle distribution data, consider a fraction of a deposit consisting of n-j toner particles of circular projection area and diameter d-j. The expected circular aperture Wiener spectrum for a deposit of these particles is (10,2),... 

Two estimates of how well the particle size distribution shows through the computed Wiener spectrum were made. First, the extrapolated value of (0) was determined. This should be proportional to the granularity of the deposit (2, 6), and according to the relation of equation (3), should in turn correlate with dm. Figure 9 shows this comparison, yielding a correlation coefficient of 0.985. Secondly, the spatial frequency at which (v) falls to the extrapolated maximum /w(0) should be correlated with the average particle size of those contributing to the spectrum. 

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