Deconvolution errors

In fact a sensor measures a flow and proceeds an integration of across a surface, which operates as a spacial lowpass filter. To avoid a critical deconvolution, the error due to this integration must be kept negligible. 

Typical approaches for measuring diffusivities in immobilised cell systems include bead methods, diffusion chambers and holographic laser interferometry. These methods can be applied to various support materials, but they are time consuming, making it onerous to measure effective dififusivity (Deff) over a wide range of cell fractions. Owing to the mathematical models involved, the deconvolution of diffusivities can be very sensitive to errors in concentration measurements. There are mathematical correlations developed to predict DeS as... 

Figure 4b. Typical size distribution of 218Po, full line BOM, dashed line EML, for growth regime. CN level was in general around 4 X 105 per cm , humidity 3-5%, SO2 introduced concentration 10-20 ppm. The presence of a peak at size. 075y to. lly is not fully understood and may be an artifact of the deconvolution program. Error bars are standard deviations for the measurements.

Both methods are also limited in accuracy of secondary structure determinations because spectral peaks must be deconvolved estimates are made of the overlapping contributions of different structural regions. These estimates may introduce error based on the reference spectra used and because deconvolution methods equate crystallographic secondary structure with the secondary structure of the protein in solution (Pelton and McLean, 2000). As amyloid fibrils are neither crystalline nor soluble, there may be even greater error in estimates of secondary structure. To compound the problem, estimates of /f-sheet content are less reliable than those of a-helix, because of the flexibility and variable twist of / -structure (Pelton and McLean, 2000). In addition, / -sheet and turn bands overlap in FTIR spectroscopy (Jackson and Mantsch, 1995 Pelton and McLean, 2000). Side chains also contribute to spectral peaks in both methods, and they can skew estimates of secondary structure if not properly accounted for. In FTIR spectra, up to 10-15% of the amide I band may arise from side chain contributions (Jackson and Mantsch, 1995). 

O Connor D. V., Ware W. R. and Andre J. C. Pouget J., Mugnier J. and Valeur B. (1989) (1979) Deconvolution of Fluorescence Decay Correction of Timing Errors in Multi-Curves. A Critical Comparison of Tech- frequency Phase/Modulation Fluoro-... 

Table 5 Prediction Errors Associated with ISMN GEOMATRIX IVIVC Developed Using the Study-Specific Reference Data for Deconvolution or the Reference Data from Study 194.573 for Deconvolution of All Study Data. Prediction Errors Outside of the FDA Acceptance Criteria Are Indicated in Bold...

A complication is introduced for systems that readily lose protons, or have two different coincident species overlapping in the same m/e region. In this case, deconvolution of the overlapping spectra becomes necessary. Once begun, however, deconvolution must be carried to completion. As shown in Table V, for B3N3H6 serious errors can occur if... 

In applying the technique of deconvolution, we take as known the spectrometer response function. It seems reasonable that the more accurately we know this function, the more accurate will be the deconvolved result. Although the nonlinear methods described in Chapter 4 are more tolerant of error, they too require a knowledge of the response function. 

Usual noise levels produce considerable error in such measurements. Slit openings wide enough to yield the high signal-to-noise ratio needed would probably be unsuitable for data acquisition runs of the spectra to be deconvolved. Whether or not we can tolerate various errors in the response-function measurement depends on the way in which it is applied. Deconvolution with an artificially wide response function yields artificially narrow deconvolved lines and probably some artifacts as well. Experimentation may be the best guide in deciding this issue. 

Have we solved the problem Is this all there is to deconvolution Let us look a little closer at the three-point example by examining Eq. (5). Suppose that the first spread-function value s x is small relative to the other values of s, as is typical. The value i0 would also then be small. We are dealing with data acquired from the real world, and no observation can be without error. Suppose that the observation of im is subject to the error nm. We may then write our object estimate... 

Just as others who have used linear methods, this author was disappointed to note the appearance of spurious nonphysical components when he applied the linear relaxation methods (Chapter 3) to the deconvolution of infrared spectra. Infrared absorption spectra, and other types of spectra as well, must lie in the transmittance range of zero to one. Spurious peaks appeared to nucleate on specific noise fluctuations in the data and grow with successive iterations, even though the mean-square error... 

First let us deal with deconvolution in general. We have a few admonitions to the reader of a literature report on a new method. They should ask, does the writer deal fairly with noise Even the most volatile of the linear methods can produce a reasonable restoration when noise is limited to roundoff error in the seventh significant figure of the data. A method s capability of yielding acceptable restorations in the presence of realistic noise is critical to its practicality. 

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. 

Fig. 1 Deconvolution of simulated noiseless data using the Jansson weighting scheme. Trace (a) is the original spectrum o x trace (b) the convolved spectrum i x). Traces (c) and (d) are the power and phase spectra of o(x), traces (e) and (f) the power and phase spectra of i(x), traces (g) and (h) the power and phase spectra of the error spectrum E(jc). Traces (i)-(m) are the deconvolution result, the power and phase spectra of the deconvolution result, and the power and phase spectra of the error spectrum, respectively, after 10 iterations with r(jjjax = 1.0. Traces (n)-(r) are the same results after 20 additional iterations with r ax= 2.0. Traces (s)-(w) are the same results after 20 additional iterations with r(3.5. Traces (x)-(bb) are the same results after 20 additional iterations with r( Jax= 5.0.

Fig. 10 Root-mean-square-error (RMSE) plots for the deconvolutions of Fig. 9 (a) no relaxation, (b) Jansson-type relaxation, (c) clipping, (d) Gaussian weighting.

The deviations due to some of these destructive influences are reversible. These are usually described as systematic errors. Many of the degradation processes that affect images and most recorded data are classified as systematic errors. For many of these cases the error may be expressed as a function known as the impulse response function. Much mathematical theory has been devoted to its description and correction of the degradation due to its influence. This has been discussed in some detail by Jansson in Chapter 1 of this volume. In that correction of this type of error usually involves increasing the higher frequencies of the Fourier spectrum relative to the lower frequencies, this operation (deconvolution) may also be classified as an example of form alteration. ... 

Addressing first the limitations of a periodic representation, such as with the DFT or Fourier series, we see that it is evident that these forms are adequate only to represent either periodic functions or data over a finite interval. Because data can be taken only over a finite interval, this is not in itself a serious drawback. However, under convolution, because the function represented over the interval repeats indefinitely, serious overlapping with the adjacent periods could occur. This is generally true for deconvolution also, because it is simply convolution with the inverse filter 1 1/t(w). If the data go to zero at the end points, one way of minimizing this type of error is simply to pad more zeros beyond one or both end points to minimize overlapping. Making the separation across the end points between the respective functions equal to the effective width of the impulse response function is usually sufficient for most practical purposes. See Stockham (1966) for further discussion of endpoint extension of the data in cyclic convolution. 

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