Methods in Image Analysis – Lecture 3 Fourier

1. Methods in Image Analysis – Lecture 3Fourier George Stetten, M.D., Ph.D. U. Pitt Bioengineering 2630 CMU Robotics Institute 16-725 Spring Term, 2006

2. Frequency in time vs. space • Classical “signals and systems” usually temporal signals. • Image processing uses “spatial” frequency. • We will review the classic temporal description first, and then move to 2D and 3D space.

3. Phase vs. Frequency • Phase, , is angle, usually represented in radians. • (circumference of unit circle) • Frequency, , is the rate of change for phase. • In a discrete system, the sampling frequency, , is the amount of phase-change per sample.

4. Euler’s Identity

5. Phasor = Complex Number

6. multiplication = rotate and scale

7. Spinning phasor

11. Continuous Fourier Series is the Fundamental Frequency Synthesis Analysis

12. Selected properties of Fourier Series for real

13. Differentiation boosts high frequencies

14. Integration boosts low frequencies

15. Continuous Fourier Transform Synthesis Analysis

16. Selected properties of Fourier Transform

17. Special Transform Pairs • Impulse has all frequences • Average value is at frequency = 0 • Aperture produces sync function

18. Discrete signals introduce aliasing Frequency is no longer the rate of phase change in time, but rather the amount of phase change per sample.

19. Sampling > 2 samples per cycle

20. Sampling < 2 samples per cycle

21. Under-sampled sine

22. Discrete Time Fourier Series Sampling frequency is 1 cycle per second, and fundamental frequency is some multiple of that. Synthesis Analysis

23. Matrix representation

24. Fast Fourier Transform • N must be a power of 2 • Makes use of the tremendous symmetry within the F-1 matrix • O(N log N) rather than O(N2)

25. Discrete Time Fourier Transform Sampling frequency is still 1 cycle per second, but now any frequency are allowed because x[n] is not periodic. Synthesis Analysis

26. The Periodic Spectrum

27. Aliasing Outside the Base Band Perceived as

28. 2D Fourier Transform Analysis or separating dimensions, Synthesis

29. Properties • Most of the usual properties, such as linearity, etc. • Shift-invariant, rather than Time-invariant • Parsevals relation becomes Rayleigh’s Theorem • Also, Separability, Rotational Invariance, and Projection (see below)

30. Separability

31. Rotation Invariance

32. Projection Combine with rotation, have arbitrary projection.

33. Gaussian seperable Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.

34. Hankel Transform For radially symmetrical functions

35. Elliptical Fourier Series for 2D Shape Parametric function, usually with constant velocity. Truncate harmonics to smooth.

36. Fourier shape in 3D • Fourier surface of 3D shapes (parameterized on surface). • Spherical Harmonics (parameterized in spherical coordinates). • Both require coordinate system relative to the object. How to choose? Moments? • Problem of poles: singularities cannot be avoided

37. Quaternions – 3D phasors Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.

38. Summary • Fourier useful for image “processing”, convolution becomes multiplication. • Fourier less useful for shape. • Fourier is global, while shape is local. • Fourier requires object-specific coordinate system.