The Boundary and Resampling Problem

As sketched in Fig. 3, registered images are influenced by fractions of the original signal from outside the domain of measured data. We distinguish three characteristic regions:

Inner Region

Points in this region are covered by the complete point spread function. Therefore we can use the maximum available information on these data and should be able to obtain a reasonable restoration.

Outer Region

Points in this region are not covered by the point spread function at all. Therefore we do not have any information on these data and they cannot be restored.

Intermediate Region

Points in this region are covered by only a fraction of the point spread function. If we move from the inner region towards the outer region, the available information on these data decreases more and more. Therefore, the quality of the restoration decreases in the same manner.

If the region of the restoration covers only the inner region, it is assumed implicitly that the original signal has a region of support which is equal to the support of the restored data. But in reality the underlying original signal (the whole sky in astronomy) has an unlimited region of support. Therefore the assumption of a limitation can lead to inconsistencies and to oscillations at the boundaries of the region of the restoration.

A reduction of the boundary problem is possible by suitable pre-processing of the measured data, e.g., by even continuation (Yaroslavsky 1985). A rigorous solution which completely avoids the boundary problem is the extension of the region of restoration by the extent of the point spread function. If the extent of the measured data is 512512 and the extent of the point spread function is 256256, the extent of the region of the restoration should be (512 + 256)(512 + 256) = 768768. In this way the measured data are not influenced at all by original data from outside the region of restoration and no inconsistencies can occur. Basically this means that the measured data are padded with zeroes up to the extent of the real measured data plus twice the extent of the point spread function. The extent of the region of virtual measured data should be (512 + 256 + 256)(512 + 256 + 256) = 10241024. The concentration on the real measured data is obtained by means of the weighting function outside the region of real measured data. We would obtain the same effect if all the padded data were available with infinite variance of the noise outside the region of real measured data.

A fact which first appears to be peculiar is that the domain of restored data must be larger than the domain of available measured data!

The weighting function can also be used to incorporate resampling on a finer grid directly into the restoration process. The measured data are again padded with zeroes, but now the zeroes are positioned between the real data points (see Fig. 3). Again we assume that the variance of the noise tends to infinity where we do not have measured data. Of course this kind of resampling is only possible if the point spread function is known completely on the finer grid. Basically it should even be possible to incorporate an interpolation directly into the restoration process if the measured data are given on a non-uniform and/or a non-rectangular grid.