1 Methods in Image Analysis – Lecture 3 Fourier CMU Robotics Institute U. Pitt Bioengineering 2630 Spring Term, 2004 George Stetten, M.D., Ph.D.

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

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11 Continuous Fourier Series SynthesisAnalysis is the Fundamental Frequency

12 Selected properties of Fourier Series for real

13 Differentiation boosts high frequencies

14 Integration boosts low frequencies

15 Continuous Fourier Transform SynthesisAnalysis

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. SynthesisAnalysis

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(N 2 )

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

26 The Periodic Spectrum

27 Aliasing Outside the Base Band Perceived as

28 2D Fourier Transform Analysis Synthesis or separating dimensions,

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 Variable Conductance Diffusion (VCD) Attempt to get around the global nature of Fourier. Smoothing with a Gaussian in the spatial domain yields multiplication by a Gaussian in the frequency domain, i.e., a low pass filter. This lowers noise, but also blurs boundaries. Gaussian smoothing simulates uniform heat diffusion. VCD makes conductance an inverse function of gradient, so that “heat” does not flow well across boundaries. This homogenizes already homogenious regions while preserving boundaries.

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

37 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

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

39 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.