Deconvolution, removing
instrumental broadening, need not be an untamed operation with many different
answers. Clearly the desired result is not the largest
spectrum, nor the smallest, nor the prettiest, but the most likely
spectrum that a better instrument with a narrower bandpass would produce. So
we construct a function which maximizes that probability. We ask for the most
likely output that a hypothetical "better instrument" would produce.
The resulting algorithm depends upon the noise statistics, which may be either
Gaussian, signalindependent (additive), or Poisson, signaldependent
(counting event). It also contains other explicit constraints (for example,
positivity) which the nature of the data requires. The most common of these is
the informational entropy of the output spectrum, which should be maximized if
the result is to be a maximum entropy spectrum. The constraint of the
broadening function of the instrument that took the data is obviously also
needed. The better this is known the better the results.
We offer four deconvolution functions, DEC,
LUC, ASH and LME. Two of them, DEC and LME, are
Maximum Entropy deconvolution functions specifically tailored for data with
gaussian noise statistics, such as infrared. Two are Maximum Likelihood
deconvolution methods for data with Poisson (counting) noise
statistics, such
as encountered in xray spectroscopy. One of the Maximum Entropy functions accepts a Bayesian prior.
Fig 1 shows a DEC (Maximum Entropy) deconvolution
of a UV spectrum (benzene) overlaying the data. The operator used a
measured peakshape of an experimental peak known to be much narrower
than the instrument function. This is the best way to obtain an
instrument function. Measure it. This is a remarkable result; rarely
will you do better! The instrument function was very well known and the
noise was low. 

Fig 2 shows the same DEC algorithm applied to
noisy data. This is as far as it went because this is as far as it should
go.



Fig 3 shows a
LUC deconvolution of a Poisson spectrum of xray data. LUC is a
Maximum Likelihood algorithm due to Lucy. This time we used a peakshape
derived from the data itself. (Sometimes that's all one has.) A factor of
two to three is about the best LUC will do; but the Lucy algorithm is
very robust and it is rare to come up empty handed because of
instabilities even if the spread function is not quite right. It's a good
choice even for Gaussian statistics if the instrument function is not well
known. 
One can get higher resolution than LUC will
produce in an acceptable number of iterations, by using a preprocessor
starting function. ASH works this way; the starting function is
produced by a linearized version of the Maximum Entropy algorithm. Fig 4
shows the result of such a marriage. The spectrum is again a Raman
spectrum, this time of sulfur. A characteristic multiplet, not resolved
even at 1 /cm, is shown here taken at 2/cm, with the result of ASH
superimposed. You certainly can't count the peaks by eye in the original
data!


Linear deconvolution is sometimes useful because it is
fast, and does not enforce positivity. Not every spectroscopy is positive!
(Think of first derivative NMR data). LME is the linear form of the Maximum
Entropy function DEC. If you want a linear algorithm that is not iterative,
and permits negativity (all linear algorithms do), this is the way you should
go. Noise is properly accounted for.
Fig 5 shows the same data as Fig 1, with the output
of LME overlaid. 

Many other deconvolution algorithms are known. One
class allows you to fiddle with the constraining parameters until you get
something you like. Fourier SelfDeconvolution is an example. Avoid these, all
of them. Most of them take no account of noise. They permit you to get about
any spectrum you want out of your data, so you have to know the answer before
you ask the question, or you may get garbage. But if you know what the
spectrum should look like, why even take data? The purpose of data processing
of any kind is to improve your state of knowledge.
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