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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.

Methods in Image Analysis – Lecture 3 Fourier

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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

for real

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

Perceived as

### 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.

### 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.