Optimal Combination

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.