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Sampling

Sampling. Outline  FT of comb function  Sampling • Nyquist Condition  sinc interpolation • Truncation • Aliasing. Sampling.

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Sampling

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  1. Sampling Outline FT of comb function  Sampling • Nyquist Condition  sinc interpolation • Truncation • Aliasing

  2. Sampling As we saw with the CCD camera and the pinhole imager, the detector plane is not a continuous mapping, but a discrete set of sampled points. This of course limits the resolution that can be observed.

  3. Finite # of data points Finite field of view High spatial frequency features can be missed or recorded incorrectly Limits of Sampling

  4. Formalism of Sampling recall that

  5. Formalism of Sampling

  6. Formalism of Sampling f(x) Comb(x/x) TopHat(2x/fov)

  7. Formalism of Sampling f(x) Comb(x/x) TopHat(2x/fov)

  8. Formalism of Sampling {fn}

  9. Frequency of Sampling x FT k

  10. Frequency of Sampling every point x FT every second point k

  11. Frequency of Sampling every point x FT every third point k

  12. Frequency of Sampling every point x FT every fourth point k

  13. Sampling

  14. Sampling

  15. Sampling

  16. Sampling Nyquist theorem: to correctly identify a frequency you must sample twice a period. So, if Dx is the sampling, then π/Dx is the maximum spatial frequency.

  17. Sampling A Simple Cosine Function Consider a simple cosine function, We know the Fourier Transform of this What happens when we sample this at a rate of

  18. Sampling A Simple Cosine Function (cont …) So that, Note, I’ve taken a short cut and left off the integrals that are needed to sample with the delta function.

  19. Define A New Frequency: Case 1 Define a new frequency Case 1: 2f0 -fs 0 fs In this case, there is no overlap and regardless of the complexity of this spectrum (think of having a number or continuum of cosine functions), the frequency spectrum correctly portrays the time evolution of the signal.

  20. Case 2 2f0 -fs 0 fs Now the frequency spectrum overlaps and look what happens to our picture, 2f’0 -fs 0 fs So it appears as though we are looking at a frequency of - this is an aliased signal.

  21. Nyquist Theorem Consider what happens when there is a complex spectrum. Then the entire spectrum overlaps. Either way, the frequency spectrum does not correspond to a correct picture of the dynamics of the original time domain signal. Nyquist Theorem: In order to correctly determine the frequency spectrum of a signal, the signal must be measured at least twice per period.

  22. Return to the sampled cosine function So Now let

  23. (cont…) Therefore we can see that it is not the Fourier Transform that fails to correctly portray the signal, but by our own sampling process we mis-represented the signal.

  24. A Fourier Picture of Sampling

  25. A Fourier Picture of Sampling

  26. Bandwidth Limited time domain

  27. Limitations of Sampling {fn}

  28. Limitations of Sampling perfect sampling average 3 data points

  29. Limitations of Sampling perfect sampling average 5 data points

  30. Limitations of Sampling perfect sampling average 7 data points

  31. Reciprocal Space real space reciprocal space

  32. Sampling real space under sampled

  33. Sampling real space zero filled

  34. Sinc interpolation

  35. Filtering We can change the information content in the image by manipulating the information in reciprocal space. Weighting function in k-space.

  36. Filtering We can also emphasis the high frequency components. Weighting function in k-space.

  37. Fourier Convolution

  38. Deconvolution Recall in an ideal world Try deconvolution with

  39. Deconvolution (cont…) This appears to be a well-balanced function, but look what happens in a Fourier space however: Recall that the Fourier Transform is linear where PSF(k) << n(k), then noise is blown up. The inverse filter is ill-conditioned and greatly increases the noise particularly the high-frequency noise since This is avoided by employing a “Wiener” filter.

  40. Original Nyquist Problem A black & white TV has 500 lines with 650 elements per line. These are scanned through Electron beam Deflection magnets Electron beam is scanned over the screen by deflection magnets and the intensity of the beam is modulated to give the intensity at each point. The refresh rate is 30 frames/second. By the sampling theorem, if the frequency is , independent information is available once every seconds. Need information

  41. Fourier Transform of a Comb Function The Fourier Transform we wish to evaluate is, One trick to this is to express the comb function as a Fourier series expansion, not the transform. Normalization to keep the series unitary Limits chosen since this is a periodic function

  42. Fourier Transform of a Comb Function Over the interval to the comb function contains only a single delta function. A series representation of the comb function is

  43. Fourier Transform of a Comb Function Now, …1 … 0 x 0 k Note: Scaling laws hold

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