# Optimization problem (imaging science)

1. Nov 14, 2013

### richard7893

1. The problem statement, all variables and given/known data

Assume a transmission imaging system with the following parameters: focal spot PSF
equal to a Gaussian with FWHM = 0.5 mm, a detector PSF equal to a Gaussian with
FWHM = 0.1 mm, source-to-detector distance = 100 cm. Compute:
1. the optimal source-to-object distance
2. the FWHM (in mm) of the total PSF at the object plane at this optimal source-to object distance.

2. Relevant equations

a= S1/(S1+S2); b=S2/(S1+S2); where S1 is the distance from focal spot to object, S2 is the distance from the object to the detector, & S1+S2 is the distance from the focal spot to detector (100cm).

a+b=1 (S1 + S2 have to add to 100cm, the detector-source distance)

FWHM_object plane = ( (an 'a' dependent factor *FWHM_focal spot)^2 + (an 'a' dependent factor *FWHM_detector)^2)^(1/2) ==> (pythagorean, in case the text looks confusing. Hard doing this without pencil & paper).

3. The attempt at a solution
I know I have to take the derivative wrt 'a' of the FWHM_obj plane equation, set it equal to 0 and solve for 'a'.

I know at this distance,'a', the PSF of the object is smallest because it's FWHM is at a minimum, making that location the optimum place to put the object.

Ultimately I'm trying to solve for S1 which I'll know once I find the value of 'a'.

The answer is supposed to come out to: 96.1cm, this is how far the object needs to be from the focal spot. And the FWHM of the object PSF is 0.98mm. I just don't know how to arrive at these numbers.

My teacher gave me the above equations, but I just don't know what the 'a' dependent factor needs to be.

And my prof. also said I didn't need to use any exponentials, even though the problem mentions gaussian functions in it.

Any help will be appreciated.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution