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Methods in Image Analysis – Lecture 3 Fourier PowerPoint PPT Presentation

Methods in Image Analysis – Lecture 3 Fourier George Stetten, M.D., Ph.D. U. Pitt Bioengineering 2630 CMU Robotics Institute 16-725 Spring Term, 2006 Frequency in time vs. space Classical “signals and systems” usually temporal signals. Image processing uses “spatial” frequency.

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Methods in Image Analysis – Lecture 3 Fourier

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Methods in image analysis lecture 3 fourier l.jpg

Methods in Image Analysis – Lecture 3Fourier

George Stetten, M.D., Ph.D.

U. Pitt Bioengineering 2630

CMU Robotics Institute 16-725

Spring Term, 2006


Frequency in time vs space l.jpg

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.


Phase vs frequency l.jpg

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.


Euler s identity l.jpg

Euler’s Identity


Phasor complex number l.jpg

Phasor = Complex Number


Multiplication rotate and scale l.jpg

multiplication = rotate and scale


Spinning phasor l.jpg

Spinning phasor


Continuous fourier series l.jpg

Continuous Fourier Series

is the Fundamental Frequency

Synthesis

Analysis


Selected properties of fourier series l.jpg

Selected properties of Fourier Series

for real


Differentiation boosts high frequencies l.jpg

Differentiation boosts high frequencies


Integration boosts low frequencies l.jpg

Integration boosts low frequencies


Continuous fourier transform l.jpg

Continuous Fourier Transform

Synthesis

Analysis


Selected properties of fourier transform l.jpg

Selected properties of Fourier Transform


Special transform pairs l.jpg

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.


Sampling 2 samples per cycle l.jpg

Sampling > 2 samples per cycle


Sampling 2 samples per cycle20 l.jpg

Sampling < 2 samples per cycle


Under sampled sine l.jpg

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


Matrix representation l.jpg

Matrix representation


Fast fourier transform l.jpg

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)


Discrete time fourier transform l.jpg

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


The periodic spectrum l.jpg

The Periodic Spectrum


Aliasing outside the base band l.jpg

Aliasing Outside the Base Band

Perceived as


2d fourier transform l.jpg

2D Fourier Transform

Analysis

or separating dimensions,

Synthesis


Properties l.jpg

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)


Separability l.jpg

Separability


Rotation invariance l.jpg

Rotation Invariance


Projection l.jpg

Projection

Combine with rotation, have arbitrary projection.


Gaussian l.jpg

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


Elliptical fourier series for 2d shape l.jpg

Elliptical Fourier Series for 2D Shape

Parametric function, usually with constant velocity.

Truncate harmonics to smooth.


Fourier shape in 3d l.jpg

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


Quaternions 3d phasors l.jpg

Quaternions – 3D phasors

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


Summary l.jpg

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.


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