Treatment of Undersampled Image Frames

In principle, both WF/PC cameras undersample their PSF. In the case of WF/PC-1 the undersampling problem is usually obscured by the combined effects of spherical aberration and noise, rendering many WF/PC frames sufficiently sampled in effect. For WF/PC-2 it is expected that, in view of its sharp PSF, non-ideal pixel response function, and reduced detector noise (Burrows 1994), the undersampling problem will be much more prominent.

The anticipated problems associated with undersampled image frames have been addressed in a number of articles (Adorf 1989a, 1989b, 1990). Fosbury (1993) discusses several image sampling strategies for the proposed future Advanced Camera for HST, and lists recent references concerning undersampled multi-frames. The major difficulty with any undersampled data frame is that it does not represent a continuous image. Thus generally, it cannot be shifted, stretched or rotated.

In order to overcome the undersampling problem, in the context of HST, it is generally recommended to obtain two or more data frames of the same field (i.e., multi-frames) with different (e.g., shifted or rotated) sampling patterns, and to subsequently combine them. The generic procedure for reconstructing an image from a set of undersampled multi-frames consists of carrying out a joint restoration following the standard Lucy-Hook recipe, which internally requires algorithms for shifting, stretching, or rotating the model (= best restoration) in order to map it onto the sampling patterns of the different observations. It is important that these mappings are carried out faithfully. Of these operators the rotation operator is the most difficult one to realize in a faithful manner.

A Novel Faithful Rotation Algorithm

The rotation algorithm described here strictly preserves the spatial Fourier-spectrum and is based on the fact that a rotation matrix , which rotates an image by an angle , can be decomposed in several ways, including one consisting of three consecutive shear matrices

Here and designate a shear by an angle in the - and -direction, respectively.

Since a sufficiently sampled image can be faithfully sheared using efficient 1D sinc interpolations (based on fast Fourier tranforms, FFTs), the whole rotation operation can also be carried out faithfully. The only problem to watch out for are potential Nyquist frequency components, which can only appear in even-sized frames. Since the three shears, out of which the rotation is composed, are carried out independently on individual rows (or columns), the faithful rotation algorithm is particularly suited to a vector-parallel computer architecture.

The rotation algorithm can be incorporated into the generalized Lucy-Hook ``co-addition'' scheme, in order to faithfully rotate the current best restoration from model to observational space, and then to rotate the correction factor back

Note that neither the WF/PC PSF nor the pixel response function are isotropic.

Restoration to a Finer ``Subsampled'' Grid

The standard implementations of the RL-algorithm include a subsampling option (Lucy 1991b; White 1993), so the problem of restoring to a finer grid is widely considered to be ``solved.'' However, it has not been proven so far that the RL-algorithm with subsampling is unique or has desirable properties when unaliasing or reconstructing missing high spatial frequency components.

When the RL-method with subsampling option is used to restore an undersampled ``image'' frame, it will create a restoration that, while consistent with the data, is not smooth, but rather maximally pixelated (see the Appendix). So the question emerges which method to use in the case of an undersampled optical PSF, and a detector with non-ideal pixel response, as expected for WF/PC-2.

The following observations show that there is indeed some opportunity for algorithmic development:

1.

The sampling theorem states: It is possible to exactly reconstruct a continuous bandlimited signal from discrete samples, if the signal is sufficiently densely sampled (and noise-free). The sampling theorem, however, does not state: an undersampled continuous signal cannot be reconstructed from discrete samples.

2.

An undersampled signal is data with missing values (e.g. every other data row and column is missing).

3.

One possesses some prior knowledge, to help in filling in the missing values: - the continuous signal is non-negative (apart from noise); - the observed signal is strictly bandlimited (apart from noise); - the PSF is known, and since its Fourier-transform (the optical transfer function, OTF), has a finite support, it always stretches beyond more than one pixel (it is in fact infinitely large).

4.

The interplay between the (linear) band-limitation and the (non-linear) non-negativity constraint is not yet completely understood. So there is latitude for theoretical developments and experimentation.

1.

Create a static mask which is 1 where data values have been observed and 0 elsewhere.

2.

Replace the predicted values by the observed ones wherever the latter are known, i.e., where the mask and apply other constraints such as non-negativity (projection onto convex set, POCS); a dynamically relaxing band-limit constraint should presumably be included here.

3.

Proceed with a standard RL-restoration.

Detection and Treatment of Cosmic Ray Hits

The cosmic ray (CR) problem involves two separate parts: detecting CR-hits and somehow ``filling in'' the missing data.

Several stand-alone algorithms have previously been published to detect CR-hits (Murtagh 1992; Murtagh &Adorf 1992; Adorf et al. 1993). However, it is clear that if a restoration is carried out anyway, the CR-contamination problem is best treated within that restoration process (Freudling 1993; White 1993). For sufficiently sampled images these algorithms work well. However, for undersampled images they are affected by the pixelation problem described above.

Let us, for a moment, consider how the human visual system would find cosmic ray hits: suspicious pixels are compared with their respective neighbourhoods. More precisely, the brain, using PSF-knowledge, predicts how the observation ought to look, and the eye compares this prediction with the actual data.

The essence of this detection method can be captured in mathematical terms, using Lucy's (1974) original notation, as: The prediction , computed from the current best restoration via convolution with the PSF, represents our current understanding of the observation. Thus any faults in the observed data are best detected by comparing the data with the prediction . We are led to the following spatially adaptive, dynamic - clipping algorithm for detecting CR-hits (and potentially other faults):

1.

Estimate a map of spatially variable standard deviations . If the Poisson-noise assumption holds, one can simply take the square root of the signal counts. Otherwise one may estimate from the spatial variance within a small neighborhood around location .

2.

Flag all those pixels whose data values are outside thereby producing a dynamic mask , which is where the data is ``good'' judged via , and elsewhere. Here denotes a threshold value.

Several methods are available to treat CR-hits once detected. One might replace the predicted values by the observed ones wherever the latter are known (projection onto convex set) and proceed with a quasi standard RL-restoration. In mathematical notation this algorithm reads:

This method of filling in missing data values might also be beneficial in situations when the information is not completely destroyed, e.g., in the case of saturated pixels, where the recorded value can kept as a lower bound for a number of iterations, but later on would be left free to be inferred from the data (White, pers. comm.).