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

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

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

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

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

  12. Building Block for 2D Transform Building block: Frequency: Orientation: Oriented wave fields A function f(x,y) can be represented as a weighted combination of phase-shifted oriented wave fields. Higher frequency

  13. Some 2D Transforms From Lehar

  14. Some 2D Transforms From Lehar

  15. Some 2D Transforms From Lehar

  16. Some 2D Transforms From Lehar

  17. Some 2D Transforms From Lehar

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

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

  20. 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

  21. 2D Fourier Transform Why? - Any relationship between two slopes?

  22. 2D Fourier Transform Why? - Any relationship between two slopes? Linearity

  23. 2D Fourier Transform Why? - Any relationship between two slopes? Linearity The spectrum is bounded by two slopes.

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

  25. 2D Fourier Transform Pairs Gaussian Gaussian

  26. 2D Image Filtering Fourier transform Inverse transform From Lehar

  27. 2D Image Filtering Fourier transform Inverse transform Low-pass filter From Lehar

  28. 2D Image Filtering Fourier transform Inverse transform Low-pass filter high-pass filter From Lehar

  29. 2D Image Filtering Fourier transform Inverse transform Low-pass filter high-pass filter band-pass filter From Lehar

  30. Aliasing Why does this happen?

  31. Aliasing How to reduce it?

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

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

  34. 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

  35. 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?

  36. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? discrete signal (samples)

  37. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? What happens in Frequency domain? discrete signal (samples)

  38. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? What happens in Frequency domain? discrete signal (samples)

  39. Fourier Transform of Dirac Comb T

  40. 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

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

  42. Fourier Transform of Dirac Comb T

  43. Fourier Transform of Dirac Comb

  44. Fourier Transform of Dirac Comb T 1/T - Fourier transform of a Dirac comb is a Dirac comb as well. - Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!

  45. fs(x) x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function What happens in Frequency domain? ? discrete signal (samples)

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

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

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

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

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

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