Problems of Photometry on Restored Images

Photometry of restored images has two problems. First, many restorations are photometrically unfaithful, and you get the wrong answer. Second, in a restoration the pixels are no longer statistically independent, and this makes a mess of processes like least-squares fitting. It is important to note this characteristic. You don't want to use DAOPHOT on a restored image, because DAOPHOT chooses the weights of the pixels on the assumption that they are statistically independent. (Note also that the variances are no longer Poisson, which is what DAOPHOT assumes.)

What is needed instead is a least-squares method that works on correlated data points. I have actually worked out how to do least squares correctly on a restoration, but I haven't carried it all the way through to a programmable algorithm, because I felt I didn't have any real use for the result. Briefly, the normal equations look, in a formal way, exactly the same as the normal equations for least squares on independent data points, except for one horrible complication: instead of weights of individual pixels you have the covariance matrix of the pixels. One certainly doesn't want, for a image, to generate a covariance matrix whose size is , or 68 billion elements. Undoubtedly there is a way of dealing with small parts of the image - say, the surroundings of a single star; but as I say, I haven't wanted to pursue this further.

I have taken a dim view of photometry on restored images, but I do hope that by the end of this meeting I will have heard about restoration methods that don't distort the photometry, and even methods that do better photometry on the restored image than you can do on the original. In that case I think that it will be valuable to resuscitate the study that I have just mentioned and to bring it to a practical fruition.

And of course there are situations where it is absolutely necessary to restore the image. There is no way of doing correct photometry on an extended object without restoration. But again, we have to be very sure that the restoration process doesn't distort the photometric values. And that often happens. My former postdoc, Adam Stanford, made up a test image that is closely similar to the center of M31, with its double peak. He then convolved it with a PSF, and added noise. We ran the lucy task that is in STSDAS, for the usual 80 iterations, and it gave the wrong answer; the ratio of the heights of the two peaks was wrong by about 10 or 20%. We sent the test to one of the practitioners of maximum entropy, and he failed it. On the other hand, Tod Lauer has told me that he has tested ``lucy'' and found it to reproduce a known answer correctly. Maybe Tod's test wasn't severe enough; maybe we did something wrong in our test. I would certainly like to know what the answer is.

Also, we have run ``lucy'' on a star field and then done photometry. The faint stars were systematically wrong. I have been told by at least one expert that what you need to do is to run ``lucy'' to complete convergence, but we haven't been able to do that - at least with the lucy task in STSDAS; the field begins to fill up with false stars that come from noise peaks. But note the distinction: I am not talking about the Richardson-Lucy algorithm in general; I am only telling what happened when I used the version in STSDAS. Maybe others can do it better. As I said, I'm not an expert.

Nor do I wish to detract from all the excellent restoration methods that are available, but just to say, ``buyer beware.'' If you just turn a crank, there is a real danger that all you'll get is sausage.