INTRODUCTION

The recent discussion of magnitude biases in 2MASS point source photometry has highlighted a rather fundamental problem in PSF generation that was first identified and documented in January 2000, namely undersampling. The details are as follows:

In order to properly represent the spatial information in the PSF, the sampling grid must provide a representation of all spatial frequencies incident on the focal plane, which, in the case of J-band would require a sampling interval of 0.1 arcsec based on the Nyquist criterion (lambda/(2D)). This, of course, means that we would need to subsample the observed pixels by a factor of 20.

Due to reasons of computational economy, our current PSF generation falls far short of that requirement. What we do is first to generate a PSF on a coarse grid (interpolation factor of 2 with respect to a focal-plane pixel) using a most-probable linear estimator, and then as a subsequent step, interpolate that image onto a Nyquist-sampled grid using sinc functions. The price we pay for the undersampling of the intermediate coarse grid is that the sinc-interpolated image suffers from Helmholtz ringing -- a set of concentric rings of alternately positive and negative sidelobes. These rings introduce biases into the photometry.

If we were applying uniform weighting to the PSF during point-source photometry, the ringing would not matter, because all stars would suffer the same effects. However, a maximum likelihood estimate requires that we weight the data pixels in inverse proportion to their variance, which results in a nonuniform weighting which varies with the source magnitude. In the case of faint sources, the noise is background dominated, and hence the PSF is uniformly weighted in the solution. By contrast, the bright sources are seeing-error dominated, whereby the error is largest near the center of the PSF; the photometric solution is then weighted towards the wings of the PSF, and can therefore suffer a significant bias due to the Helmholtz lobes. The result is a magnitude-dependent flux bias, as illustrated by a plot of the (fitted - aperture) magnitude difference as a function of magnitude, an example of which is presented in Figure 1 of:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/delmag.html

More recent examples are shown in:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/psfmake_devel#May10.html

The PSF distortion also manifests itself in another way: anomalously low values of the reduced chi squared for the brightest sources. The distortion actually induces a reciprocity relationship between trends in magnitude bias and chi squared as a function of magnitude. The slopes of the magnitude bias and chi squared trends behave like non-commuting variables in the sense that we can eliminate either one individually, but not both at the same time.


THE OPTIMAL SOLUTION TO THE PROBLEM

The optimal solution is to generate the PSF estimate directly onto a Nyquist-sampled grid without the intermediate sinc-interpolation step. Unfortunately, this would be very consuming computationally. In the current implementation, the processing time goes up as the fourth power of the interpolation interval, i: The number of points in the rectangular grid goes up as the square of i, while the number of matrix elements goes up as the square of that.

Currently, on a 400 MHz workstation, 1 PSF (an average of the images of 40-50 stars) takes 10 minutes to generate, using an interpolation interval of i = 2. Based on the above scaling law, even increasing i to 8 would require a processing time of about 40 hours per PSF. Further, since each hardware period involves about 700 PSFs, and since there are 5 hardware periods, the computational burden of i=8 (not to mention full Nyquist sampling) would seem to be beyond our present resources. On the other hand, it may be possible to reduce the burden by the use of more efficient numerical techniques such as FFT.

It may be that we do not have to go all the way to Nyquist sampling, since high spatial frequencies are attenuated due to seeing and also due to the pixel response function. Because of the daunting nature of the computational burden, however, we have so far not gone higher than an intepolation factor of 4. An example of the improvement gained in going from i=2 to i=4, is presented in:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/delmag.html#undersampling .


POSSIBLE SUBOPTIMAL (BUT COMPUTATIONALLY FEASIBLE) FIXES

(1) Generate PSFs on the coarsely-sampled i=2 grid, but truncate them just inside the first Helmholtz ring. That is, in fact, what we did for the current generation of Version 3 PSFs. The results initially appeared to be satisfactory, in that the magnitude bias was greatly reduced, and the chi squared values were distributed around unity, as described in:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/delmag.html#Nov8

Subsequently, it was found that the magnitude bias exceeded the Version 2 results, and also, the sigmas for the PSF-fitted magnitudes were anomalously high. The latter was due to the fact that the truncation produces a sharp gradient, and hence contaminates the derivatives used in the error calculations.

(2) Desensitize the photometric solution to the problem parts of the PSF using an ad hoc weighting function. This was tried with some success, and is described in:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/delmag.html#May10

(3) Apply a correction to the sinc-interpolated image using the residuals with respect to the data values. This is described in:

http://spider.ipac.caltech.edu/staff/kam/2mass/psf/delmag.html#May20

A drawback of this technique is that high spatial frequencies in the correction are lost as a result of a smoothing operation. Recent tests of the technique on southern PSFs were unsuccessful; we believe that this was due to the choice of too large a smoothing window.


COMPARISON BETWEEN VERSION 2 AND VERSION 3 RESULTS

A compilation of the results of Version 3 RTB runs (with and without fix #2 above) with their Version 2 counterparts is given in:

http://spider.ipac.caltech.edu/staff/roc/2mass/2mapps/v30/photom/photperf.html

These results include plots of the magnitude bias and chi squared as a function of magnitude. The Version 3 results are presented for processing runs made on 010504 (without fix #2) and 010512 (with fix #2).

In comparing Version 2 and Version 3 results it must be emphasized that although the Version 3 PSFs can be expected to be superior in some respects (such as better signal to noise, since they are averages of typically 10 individual PSFs), they cannot be expected to be any better in terms of magnitude bias, since they were generated on the same undersampled grid.


OPTIONS

We need to verify that the finer sampling interval will indeed solve the problem. Specific issues are:

(1) What is the minimum subsampling factor that will produce satisfactory results?

(2) Can we improve the computation efficiency to make it feasible?

In parallel with this, we need to have a contingency plan ready to ensure that we allow enough time for final processing. Options that we might consider include:

(1) Use Version 3 PSFs, and correct the sigma problem by removing the discontinuties associated with truncation. Tradeoff involved: we would get good chi squared distributions, but we would need to do some empirical corrections after the fact to remove magnitude biases.

(2) Regenerate PSFs in a similar fashion to Version 2, but with the improvement gained from combining sets of PSFs. Actually, the chief difference between Version 2 and Version 3 PSFs (other than the truncation) is the variance maps. In version 2, they were suboptimal from the point of view of chi squared, but fairly good with respect to magnitude biases. So in reverting to the Version 2 techniques, we would take a hit in the chi squared distributions.

(3) Regenerate PSFs using improvements developed recently, which include the "fix 3" correction (with a better-optimized smoothing window), and also the use of prior knowledge in the form of a positivity constraint in the sinc interpolation. We could minimize the magnitude bias (to a somewhat better degree than Version 2), but again would take a hit in the chi squared distributions. An example of the results of a run in which we incorporated these improvements is plotbias_j083.ps.