1“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL...

1“Principles & Applications of SAR”Instructor: Franz Meyer

© 2009, University of AlaskaALL RIGHTS RESERVED

Dr. Franz J MeyerEarth & Planetary Remote Sensing

University of Alaska [email protected]

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Mathematical Basics



Complex Numbers and Oscillating Signals

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Complex Numbers

Motivation: Polynomials of order n should have n roots.

1;101

0;00

1;101101

212

212

212

zzz

zzz

zzzzz



Complex Numbers

Motivation: Polynomials of order n should have n roots.

Def.: 1j imaginary unit (in mathematics mostly: „ i “)

real part

1;101

0;00

1;101101

212

212

212

zzz

zzz

zzzzz

yxyjxz ,

zx Re

zy Im imaginary part

complex number



Complex Numbers as Vectors

zjzzjzz expImRe

z

Im

Re

zz argCheck for quadrants!

22 ImRe zzz

z

zzz Re

Imarctanarg

0



Corresponds to vector sum:

212121 ImImReRe zzjzzzz

1z

2z

21 zz

Im

Re

Summation of Complex Numbers



Complex Numbers & Harmonic Oscillations

Phase (t)

A = |z|



Complex Numbers & Harmonic Oscillations

t

tuRe

tuIm

tsin

tcos

t

Phase (t)

z

A = |z|



Spectral Analysis Through

Fourier Transformation

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Spectral Analysis

• Fourier analysis of signals:

Representation of a signal by sine and cosine oscillations of varying amplitude and frequency

=> Transformation from time domain to frequency domain & analysis of signal energy in different frequencies



Spectral Analysis

• Fourier analysis of signals:

Representation of a signal by sine and cosine oscillations of varying amplitude and frequency

=> Transformation from time domain to frequency domain & analysis of signal energy in different frequencies

• Example of spectral analysis in nature:– Optical Prisma

– Human ear: Transformation of pressure waves to sounds of different tone levels (frequencies)

White light Blue componentRed component

Spectral Analysis

Example:

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Spectral Analysis

• Benefit of Fourier Analysis:– Transformation of difficult signal into a set of very simple signals eases signal

analysis– Removal of frequencies with low amplitude allows signal compression without loss of

signal quality (e.g. mpeg, jpeg, ...)

• Analysis of image content and image manipulation:– E.g. Design of low-pass and high-pass filters– Texture analysis– ...

• Some mathematical operations much easier in frequency domain– Correlation in time domain: many thousand operations (depending on signal length)– Correlation in frequency domain: only a single multiplication

nc

tu

1,0,1n nc

Approximation of a Signal by Finite Fourier Series

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“Zero frequency” corresponds to the signal mean

Increasing pos. frequenciesIncreasing neg. frequencies

nc

tu


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nc4...,,4n

nc

tu


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20...,,20n nc



Fourier TransformationProblem: Discontinuities

Discontinuities require infinite base functions for approximation


Discontinuities require infinite base functions for approximation

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t f

T

2T 2T

1

T1 T1

Fourier domain Time domain

Discontinuities in spectrum Sidelobes in time domain



Width of Spectrum and Resolution of Image

Full resolution and smoothed

image (smoothing filter sigma = 1)



Width of Spectrum and Resolution of Image

Full resolution and smoothed

image (smoothing filter sigma = 5)



Auto and Cross-Correlation

Functions

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Correlation

• Correlation between two signals is a measure of signal similarity– Auto-correlation: Correlating signal s1 with itself (measure of self-similarity)– Cross-correlation: Correlating signal s1 with signal s2

• Calculating correlation properties of a signal (time domain procedure):

* ∑

Past positions Future positions

Sharp peak: favorable correlation properties

Wide peak: unfavorable correlation properties

Sharp peak: favorable correlation properties

Wide peak: unfavorable correlation properties

s1

s2

t

Low bandwidth signal (only few frequencies)

tu

Correlation and Signal BandwidthExample: Autocorrelation

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Wide correlation peak

t

tu


Medium bandwidth signal

narrower correlation peak

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t

tu


High bandwidth signal

Narrow correlation peak

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Correlation and SAR

• We learned that high bandwidth signals have a narrow peak in the auto-correlation function

• In SAR, we will correlate the observed signal with a synthetic filter for image focusing

• As high-bandwidth signals result in a narrower correlation peak → high bandwidth SARs produce higher resolution (narrower peak → closer objects can be discriminated → higher resolution)

1“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL...

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