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Computational Photography: Image Sampling and Reconstruction

Dive into the world of 2D Fourier transforms, image sampling, and reconstruction. Learn about properties, algorithms, and applications to enhance your computational photography skills.

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Computational Photography: Image Sampling and Reconstruction

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  1. Computational Photography: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? Linearity Why is the spectrum bounded?

  21. Online Java Applet Click here.

  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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