Methods in Image Analysis

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
Eulers Identity
5
Phasor Complex Number
6
multiplication rotate and scale
7
Spinning phasor
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
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 gt 2 samples per cycle
20
Sampling lt 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 xn 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 Rayleighs 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.