The Richardson-Lucy restoration technique can be used to optimally combine sub-stepped GHRS data to produce a spectrum with two or four times finer pixels, with consequent increase in spectral resolution. The technique is a modification of the restoration with increased sampling (Lucy &Baade 1989). The equation of image formation for a pixelated spectrum can be written
where
relates the object to the pixelated image
.
acts as a modified Line Spread Function (LSF) which
gives the pixelated image of a point source.
is the
pixel function, viz., 1/
for
within a pixel of width
containing
, and zero elsewhere. The estimate of the
desired solution
is produced on a fine grid with the
integration performed for each sub-stepped spectrum (which can be
weighted other than equally if the sub-stepped spectra have
different exposure times). The LSF
needs to be
evaluated on the fine grid. This might require fitting of the LSF
by analytical means in order to form the LSF on the sub-diode
grid.
In restoration with non-linear algorithms such as the Richardson-Lucy method, overfitting of the data occurs as the maximum likelihood solution is approached. The result shows a fit to noise features and is clearly undesirable. By incorporating a regularization procedure into the iterative restoration, the approach to the maximum likelihood solution can be controlled by penalizing departures from smoothness. A wide choice of regularization functions is possible but two are:
which is entropy, or
where is some default image.
may be derived from
such as by convolution with a Gaussian or other line spread
function. The latter regularization term gives more flexibility in
the extent of the smoothing and can be matched to the extent of
spectral lines. Moreover this regularization greatly diminishes
the bias resulting from regularization with the classical form of
entropy. A multiplicative factor,
, controls the level of
regularization and since this is not known a priori some
experiment is required to determine the best value.
A Kolmogorov-Smirnov test can be applied after each complete
restoration with a given regularization constant (e.g., Skilling,
Strong, and Bennett 1979). The Kolmogorov-Smirnov test determines
the greatest distance between two cumulative distributions; the
distance is called the Kolmogorov-Smirnov statistic. This
statistic can be expressed in terms of the significance level of
an observed value of the statistic, giving the probability for the
null hypothesis that both data sets are drawn from the same
distribution (see Press et al. 1987, p. 472ff for details). Values
of the Kolmogorov-Smirnov probability for the null hypothesis of
0.5 provide a basis on which to choose the value of
such that the fit is neither bad nor too close to the input data.