Methods in Image Analysis

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Methods in Image Analysis Lecture 3Fourier
George Stetten, M.D., Ph.D.

• CMU Robotics Institute 16-725
• U. Pitt Bioengineering 2630
• Spring Term, 2004

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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.

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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.

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Eulers Identity
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Phasor Complex Number
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multiplication rotate and scale
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Spinning phasor
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Continuous Fourier Series
is the Fundamental Frequency
Synthesis
Analysis
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Selected properties of Fourier Series
for real
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Differentiation boosts high frequencies
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Integration boosts low frequencies
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Continuous Fourier Transform
Synthesis
Analysis
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Selected properties of Fourier Transform
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Special Transform Pairs

• Impulse has all frequences
• Average value is at frequency 0
• Aperture produces sync function

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Discrete signals introduce aliasing
Frequency is no longer the rate of phase change
in time, but rather the amount of phase change
per sample.
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Sampling gt 2 samples per cycle
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Sampling lt 2 samples per cycle
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Under-sampled sine
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Discrete Time Fourier Series
Sampling frequency is 1 cycle per second, and
fundamental frequency is some multiple of that.
Synthesis
Analysis
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Matrix representation
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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)

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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
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The Periodic Spectrum
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Aliasing Outside the Base Band
Perceived as
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2D Fourier Transform
Analysis
or separating dimensions,
Synthesis
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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)

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Separability
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Rotation Invariance
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Projection
Combine with rotation, have arbitrary projection.
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Gaussian
seperable
Since the Fourier Transform is also separable,
the spectra of the 1D Gaussians are, themselves,
separable.
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Hankel Transform
For radially symmetrical functions
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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.

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

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