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Tuesday, March 22, 2011

Talked to Dr. Jaggard

I talked to Dr. Jaggard yesterday about Fourier optics.  He gave me some suggestions, including looking at Classical Electrodynamics by J.D. Jackson, and the journals Radio Science and IEEE Transactions on Antennas and Propagation.  So this morning I checked out Classical Electrodynamics, and quickly read the chapter on propagation.  Unfortunately, I don't think that that chapter will be of much use; it breaks up the regions of radio propagation into very near, intermediate, and far, and then makes approximations for each of the regions so that you can solve them by hand.  I'm not solving anything by hand, so I don't need those approximations.  So at this point, I need to re-derive the equations.  This isn't particularly difficult as I'm going to be using many different books as guides, but it does increase my workload.

I'm also trying to figure out what translation and rotation in Fourier space really means.  Ideally, I'd find a general solution for an object, and then do an affine transform on its Fourier transform to get any translated, rotated, or scaled version of it that I want.  I don't know if this is mathematically possible, or it is better to do a pure FFT each time the environment changes.

I'm also trying to see if it is possible to perform limited Fourier transforms; that is, since I know the wavelengths I'm interested in (which is always some range), is it possible to reduce the amount of calculation necessary, so instead of integrating over all of space, I only integrate over those portions that are near the wavelength.  This may not be possible simply because of resonance; if I have a cavity (like the inside of a hallway), which happens to be a multiple of the wavelength, then there will be bright and dark spots.  If I don't include the higher spatial frequencies,  I risk making the hallways look 'blobby' rather than rectilinear, which means that resonance won't occur.

Most importantly, I still don't know if I can take one 3D spatial Fourier transform, and then convolve my cameras/lights whenever and wherever I want.  I think I can, but I can't state that for certain.

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