Iterative deconvolution

In image processing, blind deconvolution is a technique used to correct blurred images when the system Point Spread Function (PSF) is unknown or poorly known. The PSF is then estimated from the image. This technique has been used for decades (Ayers and Dainty 1988 Levin et al. 2009) and in this section a blind deconvolution iterative algorithm is applied to spectro-spatial data to study its applicability. 

When 5 blind deconvolution iterations are applied to the dirty data cube, the beam size does not correspond to the one expected from theory anymore, from which one can infer that the recovered datacube is unrealistic. 

We, now, illustrate result given by Papoulis deconvolution, reduced to its first iterate. 

According to the RG calculations, vahd for relatively low functionalities, the mean contribution of EV in the numerator and denominator of Eq. (20) should cancel for any branched structure in a good solvent. Therefore, ratio g for a star in a good solvent should be very close to gg, Eq. (21). Different experimental data included in [49] seem to support this conclusion. Croxton [50] carried out iterative deconvolution theoretical calculations for uniform stars with up to six arms of model lengths that yielded, g=f a result that is not in agreement with Eq. (21) for the considered range of low functionalities. On the other hand, Eq. (14) shows that the Daoud and Cotton theory gives or, approxi-... 

In addition, the time-dependence of these concentrations also contains (albeit in encoded form) the homogeneous parameters of the particular mechanism being considered. These latter techniques are termed convolutions. Convolution (and its reverse, i.e. deconvolution) are ideal for the electroanalyst because the theoretical calculation of current, and direct comparison with experimental data, is often not viable. This alternative of testing experimental currents via convolutions results in expressions for concentrations at the electrode which arise directly from the data rather than requiring iterations(s). The electrode concentrations thus estimated for a particular mechanism then allow for correlations to be drawn between potential and time, thereby assessing the fit between the data and the model. 

Natural minerals may contain simultaneously up to 20-25 luminescence centers, which are characterized by strongly different emission intensities. Usually one or two centers dominate, while others are not detectable by steady-state spectroscopy. In certain cases deconvolution of the liuninescence spectra may be useful, especially in the case of broad emission bands. It was demonstrated that for deconvolution of luminescence bands into individual components, spectra have to be plotted as a function of energy. This conversion needs the transposition of the y-axis by a factor A /hc (Townsend and Rawlands 2000). The intensity is then expressed in arbitrary imits. Deconvolution is made with a least squares fitting algorithm that minimizes the difference between the experimental spectrum and the sum of the Gaussian curves. Based on the presumed band numbers and wavelengths, iterative calculations give the band positions that correspond to the best fit between the spectrum and the sum of calculated bands. The usual procedure is to start with one or... 

Iterative Target Testing is another approach. The preliminary approximations of the real factors are chosen, based on the first (VARIMAX) rotation of the abstract PCA solution. With iterative target testing the factors are transformed to the best approximations. It can be considered as LSO, where PCA and VARIMAX are supplying the model. Clusters with an arbitrary number of peaks can be deconvoluted. Six component systems are tested (Vandeginste et al. 

It has been noted that deconvolution methods, most of which were linear, had a propensity to produce solutions that did not make good physical sense. Prominent examples were found when negative values were obtained for light intensity or particle flux. As noted previously, the need to eliminate these negative components was generally accepted. Accordingly, Gold (1964) developed a method of iteration similar to Van Cittert s but used multiplicative corrections instead of additive ones. 

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... 

Thomas (1981) and Schafer et al (1981) have also discussed the reblurring method. In addition, Schafer et al. have studied a generalized class of iterative deconvolution algorithms. They examined the convergence properties of the iteration... 

The actual deconvolution of a data set is formally straightforward. Let dik)(x) be the kth iterative estimate of the actual spectrum o(x), where x is nominally time viewed as a sequence-ordering variable. Further, let i(x) be the actual observed spectrum that has been instrumentally convolved with the observing system response function s(x). The observed data set i(x) is assumed to be related to o(x) by the convolution integral equation... 

For all the tests detailed in this chapter, deconvolution was carried out for 100 iterations using a relaxation function of the form... 

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. 13 Deconvolution results for a dynamic weighting test (a) a = 2, (b) a = VRMSEq/RMSE , (c) a = RMSE0/RMSEfc 1, (d) a = 2 RMSE /RMSE x, (e) a = 2RMSE0/RMSEfc 1, all for 100 iterations (f) a = 2RMSE0/RMSEk 1 for 40 iterations.

Fig. 19 Deconvolving s(x) = sinc(jc) and s(x) = sinc2(x) convolved data. Trace (a) is the original spectrum o(x), trace (b) the result of convolving with an eight-point sine function, trace (c) the result of unconstrained deconvolution using the same sine function for just 10 iterations. Trace (d) is the result of 100 iterations using zero clipping. Trace (e) is the original spectrum convolved with an 8 /T-point sine-squared function, trace (f) the result deconvolving trace (e) with the same sine-squared function for 100 iterations using the Jansson-type relaxation function r k)[oik X)(x)].

With Deconvolution 1 you have access to a fully automatic and interactive mode. In the automatic mode only the region used for deconvolution and a few optional parameters (type of lineshape. number of peaks,. ..) may be set. Whilst the interactive mode allows you to set the initial values for the parameters controlling the iterative fitting process and to create, edit and delete peaks. 

Another option is to use a multiwave technique, whereby the fundamental frequency and several other frequencies (harmonics) are added together into a single complex wave. Each one of these multiwave iterations can be deconvoluted into its components after the test is complete. Thus, each complex data point can... 

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