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CSCE 641 Computer Graphics: Image Sampling and Reconstruction

CSCE 641 Computer Graphics: Image Sampling and Reconstruction. Jinxiang Chai. Review: 1D Fourier Transform. A function f(x) can be represented as a weighted combination of phase-shifted sine waves How to compute F(u) ?. Inverse Fourier Transform. Fourier Transform. Review: Box Function.

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CSCE 641 Computer Graphics: Image Sampling and Reconstruction

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  1. CSCE 641 Computer Graphics:Image Sampling and Reconstruction Jinxiang Chai

  2. Review: 1D Fourier Transform A function f(x) can be represented as a weighted combination of phase-shifted sine waves How to compute F(u)? Inverse Fourier Transform Fourier Transform

  3. Review: Box Function f(x) x |F(u)| u If f(x) is bounded, F(u) is unbounded

  4. Review: Cosine  -1 1 If f(x) is even, so is F(u)

  5. Review: Gaussian If f(x) is gaussian, F(u) is also guassian.

  6. Review: Properties Linearity: Time shift: Derivative: Integration: Convolution:

  7. Outline 2D Fourier Transform Nyquist sampling theory Antialiasing Gaussian pyramid

  8. Extension to 2D Fourier Transform: Inverse Fourier transform:

  9. Building Block for 2D Transform Building block: Frequency: Orientation: Oriented wave fields

  10. Building Block for 2D Transform Building block: Frequency: Orientation: Oriented wave fields Higher frequency

  11. Some 2D Transforms From Lehar

  12. Some 2D Transforms Why we have a DC component? From Lehar

  13. Some 2D Transforms Why we have a DC component? From Lehar

  14. Some 2D Transforms Why we have a DC component? From Lehar

  15. Some 2D Transforms Why we have a DC component? From Lehar

  16. Some 2D Transforms Why we have a DC component? - the sum of all pixel values From Lehar

  17. Some 2D Transforms Why we have a DC component? - the sum of all pixel values Oriented stripe in spatial domain = an oriented line in spatial domain From Lehar

  18. 2D Fourier Transform Why? - Any relationship between two slopes?

  19. 2D Fourier Transform Why? - Any relationship between two slopes? Linearity

  20. 2D Fourier Transform Why? - Any relationship between two slopes? Linearity Why is the spectrum bounded?

  21. Online Java Applet http://www.brainflux.org/java/classes/FFT2DApplet.html

  22. 2D Fourier Transform Pairs Gaussian Gaussian

  23. 2D Image Filtering Fourier transform Inverse transform From Lehar

  24. 2D Image Filtering Fourier transform Inverse transform Low-pass filter From Lehar

  25. 2D Image Filtering Fourier transform Inverse transform Low-pass filter high-pass filter From Lehar

  26. 2D Image Filtering Fourier transform Inverse transform Low-pass filter high-pass filter band-pass filter From Lehar

  27. Aliasing Why does this happen?

  28. Aliasing How to reduce it?

  29. f(x) x fs(x) x … … -2T -T 0 T 2T Sampling Analysis Sampling

  30. f(x) x fs(x) x … … -2T -T 0 T 2T Sampling Analysis Sampling Reconstruction

  31. f(x) x fs(x) x … … -2T -T 0 T 2T Sampling Analysis What sampling rate (T) is sufficient to reconstruct the continuous version of the sampled signal? Sampling Reconstruction

  32. Sampling Theory • How many samples are required to represent a given signal without loss of information? • What signals can be reconstructed without loss for a given sampling rate?

  33. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x ?

  34. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x ? What happens in Frequency domain?

  35. Fourier Transform of Dirac Comb T

  36. Review: Dirac Delta and its Transform f(x) x |F(u)| 1 u Fourier transform and inverse Fourier transform are qualitatively the same, so knowing one direction gives you the other

  37. Review: Fourier Transform Properties Linearity: Time shift: Derivative: Integration: Convolution:

  38. Fourier Transform of Dirac Comb T

  39. Fourier Transform of Dirac Comb

  40. Fourier Transform of Dirac Comb T 1/T Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!

  41. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x ? What happens in Frequency domain?

  42. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u

  43. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u How does the convolution result look like?

  44. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u

  45. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u

  46. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u G(0)? G(fmax)? G(u)?

  47. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u G(0) = F(0) G(fmax) = F(fmax) G(u) = F(u)

  48. F(u) fmax u -fmax Sampling Analysis: Freq. Domain How about … -1/T 0 1/T … u Fs(u) fmax u -fmax -1/T 1/T

  49. F(u) fmax u -fmax Sampling Analysis: Freq. Domain How about … -1/T 0 1/T … u Fs(u) fmax u -fmax -1/T 1/T

  50. F(u) fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u Fs(u) fmax u -fmax -1/T 1/T

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