[crntaylor]

This is interesting. I am generally very skeptical about taking derivatives of functions computed from real-world data for exactly this reason. What I normally end up doing is applying some form of kernel smoothing (e.g. nearest K points with a Gaussian kernel) to approximate the function, and then compute derivatives of that instead.

Would you recommend another way?

[backprojection]

This is pretty much exactly what I'm saying. So you're convolving with a gaussian kernel. That's the exact same thing as multiplying, in the frequency domain, by another gaussian. (convolution theorem, and the FT of a gaussian is a gaussian). The Gamma function filter goes one step further, it throws in a polynomial to make that gaussian 'flatter', which is more suitable for a low-pass filter.

Try the script at the end of this comment. Here's the output http://imgur.com/5948cW6

Notice that the 0-th one is just a gaussian. If you're interested, these things have traditionally been called HDAFs, I have a library to do this kind of thing https://bitbucket.org/nmaxwell/hdaf it may need more documentation. I don't mind if anyone provides constructive feedback on it.

The functionality you'd be interested in is hdaf.hdaf_periodic_samples, - you can convolve with that to low-pass filter, and use hertz_to_sigma to get the sigma parameter (from cutoff frequency).

  import numpy, scipy.special
  import matplotlib.pyplot as plt

  x=numpy.linspace(-2,5,1000)
  for n in [0,1,4,10]:
      s=1 if n==0 else n
      h=scipy.special.gammaincc(n+1,s*x**2)
      plt.plot(x, h)
  plt.savefig("filters")

EDIT: formatting.