1. 1“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Dr. Franz J Meyer Earth & Planetary Remote Sensing University of Alaska Fairbanks fmeyer@gi.alaska.edu Mathematical Basics

2. 2“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers and Oscillating Signals

3. 3“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers Motivation: Polynomials of order n should have n roots.

4. 4“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers Motivation: Polynomials of order n should have n roots. Def.: imaginary unit (in mathematics mostly: „ i “) real part imaginary part complex number

5. 5“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers as Vectors Check for quadrants! 0

6. 6“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Corresponds to vector sum: Summation of Complex Numbers

7. 7“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers & Harmonic Oscillations Phase  (t)   A = |z|

8. 8“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Complex Numbers & Harmonic Oscillations Phase  (t)        A = |z|

9. 9“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Spectral Analysis Through Fourier Transformation

10. 10“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED 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

11. 11“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED 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 component Red component

12. Spectral Analysis Example: 12

13. 13“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED 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

14. Approximation of a Signal by Finite Fourier Series 14 “Zero frequency” corresponds to the signal mean Increasing pos. frequenciesIncreasing neg. frequencies

15. Approximation of a Signal by Finite Fourier Series 15 “Zero frequency” corresponds to the signal mean Increasing pos. frequenciesIncreasing neg. frequencies

16. Approximation of a Signal by Finite Fourier Series 16 “Zero frequency” corresponds to the signal mean Increasing pos. frequenciesIncreasing neg. frequencies

17. 17“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Fourier Transformation Problem: Discontinuities Discontinuities require infinite base functions for approximation

18. Fourier Transformation Problem: Discontinuities Discontinuities require infinite base functions for approximation 18

19. 19“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Fourier Transformation Problem: Discontinuities Fourier domainTime domain Discontinuities in spectrumSidelobes in time domain

20. 20“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Width of Spectrum and Resolution of Image Full resolution and smoothed image (smoothing filter sigma = 1)

21. 21“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Width of Spectrum and Resolution of Image Full resolution and smoothed image (smoothing filter sigma = 5)

22. 22“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Auto and Cross-Correlation Functions

23. 23“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED 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 positionsFuture positions Sharp peak: favorable correlation properties Wide peak: unfavorable correlation properties Sharp peak: favorable correlation properties Wide peak: unfavorable correlation properties s1 s2

24. Low bandwidth signal (only few frequencies) Correlation and Signal Bandwidth Example: Autocorrelation 24 Wide correlation peak

25. Correlation and Signal Bandwidth Example: Autocorrelation Medium bandwidth signal narrower correlation peak 25

26. Correlation and Signal Bandwidth Example: Autocorrelation High bandwidth signal Narrow correlation peak 26

27. 27“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED 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)