Noisy data

The importance of distinct a priori knowledge account becomes more perceptible if noisy data are under restoration. The noise / ( shifts the solution of (1) from the Maximum Likelihood (ML) to the so called Default Model for which the function of the image constraint becomes more significant. 

To implement the reconstruction of the initial image, using denoised and/or noisy data given by simulated projections The algorithm (1) and the Gibbs functional in the form (12) were used for the reconstruction. The coefficients a and P were optimized every time. 

Finally, it is shown in terms of the presented example that the proposed adaptive reconstruction algorithm is valuable for image reconstruction from projections without any prior information even in the case of noisy data. The number of required projections can be determined by investigating the convergence properties of the reconstruction algorithm. 

Next, we find the first eigenvector of the noisy data set and plot it in Figures 41 and 42. We see that it is nearly identical to the first eigenvector of the noise... 

Figure 41. The noisy data from Figure 39 plotted together with the first eigenvector (factor) for the data.

Continuing, we find the second eigenvector for the noisy data. Figures 43 and 44 contain plots of the first two eigenvectors for the noisy data. Again, the second eigenvector for the noisy data is nearly identical to that of the noise-free data. 

Completing the cycle, we calculate the third eigenvector for the noisy data. Figures 45 and 46 contain the plots of all three eigenvectors for the noisy data. 

Referring to Table 7, we see that the eigenvalue for the third eigenvector o the noisy data is no longer equal to zero. Of course, this makes perfect sens because the noisy data no longer lie exactly in a plane and so the thir< eigenvector is now able to capture some variance from the data. 

Figure 77. Projections of the concentration data onto each concentration factor vs. the corresponding projections of the spectral data onto each spectral factor for the noisy data using eigenvectors as factors.

Maximum likelihood methods are commonly used to estimate parameters from noisy data. Such methods can be applied to image restoration, possibly with additional constraints (e.g., positivity). Maximum likelihood methods are however not appropriate for solving ill-conditioned inverse problems as will be shown in this section. 

As seen, there is significant variability in the estimates. This is the reason why we should avoid using this technique if possible (unless we wish to generate initial guesses for the Gauss-Newton method for ODE systems). As it was mentioned earlier, the numerical computation of derivatives from noisy data is a risky business ... 

In the last section of the paper, we discuss a Bayesian approach to the treatment of experimental error variances, and its first limited implementation to obtain MaxEnt distributions from a fit to noisy data. 

Finally, recent work of Iversen et al. has carefully examined the bias associated to the accumulation of the error on low-order reflexions, and attempted a correction of the MaxEnt density [39]. The study, based on a number of noisy data sets generated with Monte Carlo simulations, has produced less non-uniform distribution of residuals, and has given quantitative estimate of the bias introduced by the uniform prior prejudice. For more details on this work, we refer the reader to the chapter by Iversen that appears in this same book. 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

Figure 5. L-Alanine. Fit to noisy data. Calculation A. Distribution of residual structure factor amplitudes at the end of the MaxEnt calculation on 2532 noisy data up to 0.463A. Residuals plotted ... 

We briefly discuss in this section the results of the valence MaxEnt calculation on the noisy data set for L-alanine at 23 K we will denote this calculation with the letter A. The distribution of residuals at the end of the calculation is shown in Figure 5. It is apparent that no gross outliers are present, the calculated structure factor amplitudes being within 5 esd s from the observed values at all resolution ranges. 

Figure 6. l-Alanine. Fit to noisy data. Calculation A. MaxEnt deformation density and error map in the COO- plane Map size, orientation and contouring levels as in Figure 2. (a) MaxEnt dynamic deformation density A uP. (b) Error map qME - Model. 

The MaxEnt method will always deflate deformation features by the (<80 ) ,1S corresponding to measurements error [39]. To obtain an empirical estimate of this intrinsic spread allowed by the noise, twenty noisy data sets were generated as in formula (31), and fitted with BUSTER using the fragment and NUP already described in the previous paragraph. 

To check for the dependence of this bias on the noise level, a number of 20 noisy data sets were generated with variances lowered to 10% of their experimental values, and MaxEnt calculations run against these low-noise data. 

Figures, l-Alanine.Fits to noisy data Calculations A (experimental noise) and B (10% experimental noise). MaxEnt, deformation and error density profiles along the Cl-01 bond. Solid line Model valence density. Dashed line MaxEnt density A. Dot-dashed line MaxEnt density B. Dotted line valence-shells non-uniform prior.

Doerffel K, Kuchler L, Meyer N (1990) Treatment of noisy data from distribution analysis using models from time-series analysis. Fresenius J Anal Chem 337 802... 

If the slit length is finite and the scattering intensity shall be desmeared, the profile W (si, S3) of the primary beam must be known. In order to carry out the desmear-ing numerically, different algorithms have been proposed, but few of these methods are able to manage the derivative problem from Eq. (4.6) properly for noisy data. One of them is the method developed by Glatter [68],... 

We are processing noisy data. Thus, intensities may have become negative by accident. In order to mark such spots as invalid data, they should be set to zero. This is accomplished by the masking formalism of image processing... 

In any case, even if all the above statements are 100% true, it does not affect our discussion because they are beside the point. The behavior of calibration algorithms in the face of noisy data is an important topic and perhaps should be studied in depth, but it was not at issue in the Linearity in Calibration column. 

We also have noted before that adding or subtracting noisy data causes the variance to increase as the number of data points added together [2], The noise of the first derivative, therefore, will be larger than that of the underlying absorbance band by a factor of the square root of two. 

These schemes require the calculations of the second and mixed derivatives, which normally result in poor accuracy when the computations are performed on discrete data. For noisy data, computed values of H and K depend on the finite element scheme used to calculate the first, second, and mixed derivatives. 

For several years application of MRE to biological tissues has been an active area of research. Strategies either turns towards excised tissues (ex vivo) or towards living animals or humans (in vivo). The former allows more detailed studies of strain-stress relationships and thus are important for the characterization of heterogeneous biological systems. From a biological point-of-view in vivo results might appear more relevant. However they are difficult to obtain and one must keep in mind biases that could arise from incomplete or noisy data. 

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