Deconvolution of Images and Spectra: Second Edition (Dover Books on Engineering) by Engineering

Author:Engineering [Engineering]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2014-05-04T16:00:00+00:00

Figure 9 Deconvolving s(x) = sinc(x) 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, and trace c the result of unconstrained deconvoltuion 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√2-point sinc-squared function, trace f the result of deconvolving trace e with the same sinc-squared function for 100 iterations using the Jansson-type relaxation function r(k) [o(k-1) (x)].

The negative sidelobes of the unapodized spectrum render the constraints defined earlier useless. Trace c of Fig. 9 shows the result of unconstrained deconvolution. After only five iterations the algorithm begins to diverge and, as seen in Fig. 9 , the spectrum is worse for our trouble. Trace d shows the result of zero clipping. Although there is some improvement and convergence does occur, the resulting spectrum appears real but is not a faithful representation of the original spectrum shown in trace a, particularly where several lines overlap in the convolved spectrum. Trace e represents the same spectrum with triangular apodization used to generate a response function s(x) = (sin 2 πx)/π 2 x 2 = sinc 2 (x). Although some resolution is lost, the negative lobes are eliminated and the positive sidelobes greatly reduced. Such a spectrum can be deconvolved with the Jansson weighting technique even in the presence of the sidelobes (which are now all nonnegative). The result is shown in trace f. This spectrum shows resolution equal to or slightly better than the unapodized spectrum shown in trace b and the sidelobes have been almost completely eliminated. For a very crowded spectrum this would appear to be a useful technique, particularly if there is interest in the weaker lines of the spectrum.

B. PRESSURE-BROADENED INFRARED AND RAMAN SPECTRA

In all the previous simulations we have assumed a narrow Gaussian line profile that results from Doppler broadening. This is often not the case, owing to the presence of pressure broadening that results in a Lorentzian line profile at sufficiently high pressures. This often means an unavoidable loss of resolution. To observe a spectrum when the transition intensities are low, it is often necessary to increase the sample gas pressure. In this case the pressure broadening can be sufficient to mask important spectral features. Conversely, when the pressure is low enough to allow these important spectral features to be resolved, they may be too weak to observe at achievable signal-to-noise ratios. To some extent this problem can be alleviated by using deconvolution to remove pressure-broadening effects (as well as instrumental effects). This is demonstrated for two cases: a simulated absorption spectrum and a simulated Raman spectrum.

For an absorption spectrum the effects of pressure broadening can be approximated by convolving the spectrum with a Lorentzian of the proper width. (An accurate representation would require us to carry out the convolution on the absorption coefficient spectrum.) Thus a

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