# Find lone peak with high sampling cost

(I'm not sure if this is the correct stackexchange place to ask; please redirect me if it doesn't belong here.)

I have a 2D function f(x,y) which is (near-)zero everywhere, except for a single region where there is a gaussian peak of "width" 0.15mm (created by a normal gaussian + noise). The image below shows this, all red dots are background noise, the peak is centered.

The problem is, there's a high cost for sampling f(x,y). If there was no noise, no problem, estimate the gradient and off you go. But with noise this technique will just get me to random locations.

I can manually narrow the region of interest down to an area of 4x4mm, but that'll still take me around 274 samples (based on circle covering problem here) to catch at least a glimpse of the edge of the gaussian.

Are there any techniques for this kind of problem? I'm thinking towards optimal sampling location/sampling sequence? Ie spiralling out from the center; hexagonal grid; start coarse, then finetune...

• I may be wrong, but I think either our site or the statistics site may be able to help you. I don't know this stuff, but taking averages will tend to cancel (additive) noise, so there are probably known methods for detecting a positive gradient if you estimate it using more points. Of course, the gradient of a gaussian drops off very fast, so the signal-to-noise ratio becomes unmanageably low fairly soon. Commented Jan 28, 2015 at 6:09