Computational Photography: Fourier Transform

1. Computational Photography:Fourier Transform Jinxiang Chai

2. Image Scaling This image is too big to fit on the screen. How can we reduce it? How to generate a half- sized version?

3. Image Sub-sampling 1/8 1/4 • Throw away every other row and column to create a 1/2 size image • - called image sub-sampling

4. Image Sub-sampling 1/2 1/4 (2x zoom) 1/8 (4x zoom) Why does this look so crufty?

5. Difference between Lines

6. Even Worse for Synthetic Images

7. Really Bad in Video Click here

8. Aliasing occurs when your sampling rate is not high enough to capture the amount of detail in your image Can give you the wrong signal/image—an alias Where can it happen in computer graphics and vision? During image synthesis: • sampling continuous signal into discrete signal • e.g. ray tracing, line drawing, function plotting, etc. During image processing: • resampling discrete signal at a different rate • e.g. Image warping, zooming in, zooming out, etc. To do sampling right, need to understand the structure of your signal/image– signal processing

9. Signal Processing Analysis, interpretation, and manipulation of signals - images, videos, geometric and motion data - sampling and reconstruction of the signals. - minimal sampling rate for avoiding aliasing artifacts - how to use filtering to remove the aliasing artifacts?

10. Periodic Functions • A periodic function is a function defined in an interval that repeats itself outside the interval • What’s the interval for sinx? • What’s the interval for sin2πfx?

11. Jean Baptiste Fourier (1768-1830) had crazy idea (1807): Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies. Don’t believe it? • Neither did Lagrange, Laplace, Poisson and other big wigs • Not translated into English until 1878! But it’s true! • called Fourier Series

12. Our building block: Add enough of them to get any signal f(x) you want! A Sum of Sine Waves

13. Our building block: Add enough of them to get any signal f(x) you want! A Sum of Sine Waves

14. Our building block: Add enough of them to get any signal f(x) you want! A Sum of Sine Waves

15. Our building block: Add enough of them to get any signal f(x) you want! A Sum of Sine Waves

16. Our building block: Add enough of them to get any signal f(x) you want! How many degrees of freedom? What does each control? Which one encodes the coarse vs. fine structure of the signal? A Sum of Sine Waves

17. How about Non-peoriodic Function? • A non-periodic function can also be represented as a sum of sin’s and cos’s • But we must use all frequencies, not just multiples of the period • The sum is replaced by an integral.

18. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves

19. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves How to compute F(u)?

20. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves How to compute F(u)?

21. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves How to compute F(u)? Amplitude: Phase angle:

22. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves How to compute F(u)? Inverse Fourier Transform Fourier Transform Amplitude: Phase angle:

23. Fourier Transform A function f(x) can be represented as a sum of phase-shifted sine waves How to compute F(u)? Inverse Fourier Transform Fourier Transform Dual property for Fourier transform and its inverse transform Amplitude: Phase angle:

24. Fourier Transform Magnitude against frequency: f(x) |F(u)| How much of the sine wave with the frequency u appear in the original signal f(x)?

25. Fourier Transform Magnitude against frequency: f(x) |F(u)| ? 5 How much of the sine wave with the frequency u appear in the original signal f(x)?

26. Fourier Transform Magnitude against frequency: f(x) |F(u)| 5 How much of the sine wave with the frequency u appear in the original signal f(x)?

27. Fourier Transform f(x) |F(u)| |F(u)| f(x)

28. Fourier Transform f(x) |F(u)| |F(u)| f(x)

29. Fourier Transform f(x) |F(u)| |F(u)| f(x)

30. Fourier Transform f(x) |F(u)| |F(u)| f(x)

31. Box Function and Its Transform f(x) x

32. Box Function and Its Transform f(x) x

33. Box Function and Its Transform f(x) x |F(u)| u If f(x) is bounded, F(u) is unbounded

34. Another Example If the fourier transform of a function f(x) is F(u), what is the fourier transform of f(-x)?

35. Another Example If the fourier transform of a function f(x) is F(u), what is the fourier transform of f(-x)?

36. Dirac Delta and its Transform f(x) x

37. 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

38. Cosine and Its Transform  -1 1 If f(x) is even, so is F(u)

39. Sine and Its Transform  -1 1 - If f(x) is odd, so is F(u)

40. Gaussian and Its Transform If f(x) is gaussian, F(u) is also guassian.

41. Gaussian and Its Transform If f(x) is gaussian, F(u) is also guassian. what’s the relationship of their variances?

42. Gaussian and Its Transform If f(x) is gaussian, F(u) is also guassian. what’s the relationship of their variances?

43. Properties Linearity:

44. Properties Linearity: Time-shift:

45. Properties Linearity: Time-shift:

46. Properties Linearity: Time-shift:

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

48. 1D Signal Filtering

49. 2D Image Filtering

50. Signal Filtering A filter is something that attenuates or enhances particular frequencies Easiest to visualize in the frequency domain, where filtering is defined as multiplication: Here, F is the spectrum of the function, G is the spectrum of the filter, and H is the filtered function. Multiplication is point-wise