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Homework Help: Optimization problem (imaging science)

  1. Nov 14, 2013 #1
    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
  2. jcsd
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