How to implement Discrete Integration of the Gaussian Function on a Grid?

Hi!

I want to implement a physical problem, where the photons are incident on a 2D sensor as a Gaussian Function. I need to find out the number of photons hitting each pixel.

I know the total number of photons incident on the sensor as well as the spread of the Gaussian Function.

I can't use the PDF of a Gaussian Function as its a point value.

Any ideas on how to implement this on Matlab would be extremely helpful.

Answer 1

[Bjorn Gustavsson]

In that case you better integrate your Gaussian (point-spread-function?) over the pixel-areas. You could use something like this:

ph_cnt = @(x,y,x0,y0,sx,sy) integral2(@(u,v) exp(-((u-x0).^2/sx^2)+(v-y0).^2/sy^2^2),x,x+1,y,y+1);

Where x and y are the pixel-indices, x0 and y0 are the centroid of your photon-beam, sx and sy are the horizontal and vertical widths of the beam. You will still have to manage the normalization of the integral to give you the correct total photon-count.

HTH

[Neilabh Banzal]

Hey!

Thanks for the answer.

Is there any other way than integral? This is the last option that I have. Any other approach would be immensely helpful.

- N

[Bjorn Gustavsson]

You might get away with simply assuming that you can approximate the integral with: you might get a good enough fit just ignoring the integration, it might get you a slightly different set of parameters () than if you do it "properly". If you do some experimentation you might get some correction-factors. The integration is not too slow, I get 5-10 ms, so might not be too much of a problem unless you need to do very many of them.

HTH

[Neilabh Banzal]

Bjorn,

I have to perform the integration on a grid of 1024 x 1280 pixels, for some 20 beams of photons. So, Intnegration would take a lot of time. And I can't vectorise the integration (As far as I know)

I am currently using the PDF value at the midpoint of the pixel. The problem is that most of my sigma spreads are too small for the PDF approximation to be valid.

Thanks for all your inputs. :)

- N

[Bjorn Gustavsson]

But if your beams are narrow, you certainly don't need to integrate over the entire 1024x1280 area. If you restrict the integration to an aera around each centre-point your relative error should be on the order of 1-erf(5)^2 or ~3e-12. To detect that small differences would require "very good" accuray for your photon-count.