1. CPSC 641 Computer Graphics: 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 Throw away every other row and column to create a 1/2 size image - called image sub-sampling 1/4 1/8

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

5. Difference between Lines

6. Even Worse for Synthetic Images

7. Really Bad in Video Click herehere

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

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

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

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

16. A Sum of Sine Waves 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?

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)? Amplitude:Phase angle: Inverse Fourier Transform Fourier Transform

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

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)| How much of the sine wave with the frequency u appear in the original signal f(x)? 5 ?

26. 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)? 5

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

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

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

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

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

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

33. Box Function and Its Transform x u f(x) |F(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 x f(x)

37. Dirac Delta and its Transform x 1 u f(x) |F(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  If f(x) is even, so is F(u)

39. Sine and Its Transform 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. 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

49. Filtering  =   = = Low-pass High-pass Band-pass FGH

50. Filtering Identify filtered images from low-pass filter, high-pass filter, and band-pass filter?

51. Convolution Compute the integral of the product between f and a reversed and translated version of g

52. Convolution Compute the integral of the product between f and a reversed and translated version of g

53. Convolution Compute the integral of the product between f and a reversed and translated version of g Reversed and translated function g

54. Convolution Compute the integral of the product between f and a reversed and translated version of g Reversed and translated function g

55. Convolution Compute the integral of the product between f and a reversed and translated version of g t=2.5

56. Convolution Compute the integral of the product between f and a reversed and translated version of g t=4.6

57. Examples Convolution of the function f(t) and a delta function σ(t)

58. Examples Convolution of the function f(t) and a delta function σ(t)

59. Convolution Theory

64. Correlation Correlation between f and g Similarly, we have its fourier transform

65. Auto-correlation Self-correlation for f Similarly, we have its fourier transform

66. t t   Auto-correlation in Action

67. Qualitative Property The spectrum of a functions tells us the relative amounts of high and low frequencies –Sharp edges give high frequencies –Smooth variations give low frequencies A function is band-limited if its spectrum has no frequencies above a maximum limit –sine, cosine are bandlimited –Box, Gaussian, etc are not

68. Summary Convert a function from the space and frequency domain and vice versa. Interpret a function in either domain, e.g., filtering and correlation. Build your intuition about functions and their spectra.

69. Next lecture Fourier transform for 2D signals (images) Sampling and reconstruction How to avoid aliasing Fourier transform