Methods in Image Analysis  Lecture 3 Fourier

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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 3Fourier

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.
• We will review the classic temporal description first, and then move to 2D and 3D space.
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.
Continuous Fourier Series

is the Fundamental Frequency

Synthesis

Analysis

Special Transform Pairs
• Impulse has all frequences
• Average value is at frequency = 0
• Aperture produces sync function
Discrete signals introduce aliasing

Frequency is no longer the rate of phase change in time, but rather the amount of phase change per sample.

Discrete Time Fourier Series

Sampling frequency is 1 cycle per second, and fundamental frequency is some multiple of that.

Synthesis

Analysis

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

Sampling frequency is still 1 cycle per second, but now any frequency are allowed because x[n] is not periodic.

Synthesis

Analysis

2D Fourier Transform

Analysis

or separating dimensions,

Synthesis

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)
Projection

Combine with rotation, have arbitrary projection.

Gaussian

seperable

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

Hankel Transform

Elliptical Fourier Series for 2D Shape

Parametric function, usually with constant velocity.

Truncate harmonics to smooth.

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

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

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.