# Can a point source be located more accurately out-of-focus or in-focus?

Let's say I am taking a picture, and I know a priori that the image is of a single ideal point source of light at infinity.

With a perfect imaging system in focus, the image shows an Airy disk. I already knew to expect an Airy disk, so I can do some mathematical fitting to guess the center of the Airy disk. That's my best guess for the source location.

Alternatively, the imaging system might be somewhat out of focus. The image would show a much bigger blurry circle. Again, I can do some mathematical fitting to guess the center of the blurry circle. Again, that's my best guess for the source location.

My question is: Which approach would give me more accurate information about the location of the point source?

I can't see an obvious answer. In the out-of-focus image, you are drawing useful information from a much greater number of pixels, which is usually helpful in mathematical fitting. (Particularly if a large pixel size limits the resolution.) On the other hand, we normally say that defocusing an image simply erases its small-scale information, and I've never heard of anyone defocusing an imaging system on purpose like that!

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## 2 Answers

If the Airy disk is smaller than a pixel (rather common), then you want to defocus. Star trackers on satellites do this in order to get sub-pixel pointing accuracy. If the Airy disk is much larger than a pixel, then you probably don't want to defocus. In the latter case the situation is complicated by aberrations and the problem of modeling the shape of the spot on the focal plane, which in general is no longer circular. That modeling problem might be more accurate for the in-focus case. I suppose as a practical matter one might do the best one can in the design stage, but then actually measure point spread functions for various angles. That is, calibrate the actual device. (That's speculation.)

But if you calculate the size of the Airy disk for typical devices you will find that it's generally smaller than a pixel, so defocussing usually wins.

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To add to that: if you can shift focal length to both sides of best focus, you probably can calculate the best-focus position for a sub-pixel target. – Carl Witthoft Jul 3 '14 at 16:01

In the absence of noise they can both work the same (assuming you know the exact amount of defocusing, and you over-sample the Airy disk)

In the presence of realistic noise, you are better off focusing the object due to the details of the noise. For intensity images, you are (likely) dealing with Riciean distributed data, and you are better off using a smaller number of samples with larger real intensity than a large number of samples all with low intensty.

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