Statistical analysis of backscatter coefficients in ERS-1 SAR images

Inhaltsverzeichnis (Table of Contents)

• Summary in Danish
• Acknowledgement
• Chapter 1
  • Abstract and introductory remarks
• Chapter 2
  • Historical background
  • Some initial/practical problems
  • SAR and bathymetry
  • Start of statistical analysis
  • Contents of the thesis
• Chapter 3
  • Contributions to the theory of K-distribution and generalized gamma distribution
  • Review
• Chapter 4
  • A statistical model for the scattering mechanism
  • The Rayleigh model
  • The K-distribution model
  • Problems with the K-distribution model
• Chapter 5
  • Digamma function
  • Basic properties of the digamma function
• Chapter 6
  • Generalized gamma distribution
  • Basic properties of the generalized gamma distribution
  • Maximum likelihood estimation
  • The ordinary gamma distribution
• Chapter 7
  • Exponential gamma distribution
  • Basic properties of the exponential gamma distribution
• Chapter 8
  • K-distribution
  • Basic properties of the K-distribution
  • Mode of the K-distribution
  • Maximum likelihood estimation in the K-distribution
• Chapter 9
  • Moment method
  • Brief introduction to the moment method
• Chapter 10
  • Method of moments
  • Moments and uniqueness of distributions
  • Moments and cumulants
• Chapter 11
  • Raghavan's method
  • General introduction to the method
  • The generalized gamma distribution
  • The K-distribution
• Chapter 12
  • Chi-square test for goodness-of-fit
  • Introduction to the chi-square test
  • Construction of categories
  • Number of observations and categories
  • Concluding remarks on the chi-square test
• Chapter 13
  • The runs test and the U test
  • The runs test
  • The U test
• Chapter 14
  • Kolmogorov's Dn test
  • Description of the Dn test
• Chapter 15
  • Which test should be chosen?
  • General problems with the category construction
  • The Dn test and the runs test
  • The Dn test and the chi-square test
  • Concluding remarks on the test comparisons
• Chapter 16
  • Data analysis, Parameter analysis
  • Parameter values in six homogeneous ROls
  • Parameter values in two inhomogeneous ROls
  • First consider 16014_1215, ROI
  • Second consider 16588_1215, ROI
  • Analysis of individual cells in 16014_1215, ROI
  • Analysis of individual cells in 16588_1215, ROI
  • Summary of analyses of the inhomogeneous ROls
• Chapter 17
  • Data analysis, Statistical analysis
  • Test of statistical distributions
  • Tests of different test methods
  • Statistical analysis of 16481_2385, ROl2
  • Statistical analysis of 16014_1215,ROI and 16588_1215,ROI
  • Relationship between the K-distribution and the gamma distribution
• Chapter 18
  • Conclusions
• Chapter 19
  • Future recommendations
• References
• Chapter 20
  • Appendix, Text material
  • Chapter 20.1
    • Appendix, A statistical model for the scattering mechanism
  • Chapter 20.2
    • Appendix, Calibration of ERS-1 SAR images
  • Chapter 20.3
    • Appendix, Exponential gamma distribution
  • Chapter 20.4
    • Appendix, K-distribution
  • Chapter 20.5
    • Appendix, Saddlepoint method for approximating the K-distribution
  • Chapter 20.6
    • Appendix, Raghavan's method
  • Chapter 20.7
    • Appendix, Chi-square test for goodness-of-fit
  • Chapter 20.8
    • Appendix, The runs test and the U test
  • Chapter 20.9
    • Appendix, Kolmogorov's Dn test
  • Chapter 20.10
    • Appendix, Numerical and iterative calculations
  • Chapter 20.11
    • Appendix, Probability theory
  • Chapter 20.12
    • Appendix, Statistical definitions and properties
  • Chapter 21
    • Appendix, Data material
    • 21.1 Data results for image: 16481_2385, ROl1
    • 21.2 Data results for image: 16481_2385, ROl2
    • 21.3 Data results for image: 17055_2385, ROl1
    • 21.4 Data results for image: 17055_2385, ROl2
    • 21.5 Data results for image: 17055_2385, ROl3
    • 21.6 Data results for image: 17299_2385, ROI
    • 21.7 Data results for image: 16014_1215, ROI
    • 21.8 Data results for image: 16588_1215, ROI
  • Chapter 22
    • Appendix, Images

  Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

  The main objective of this thesis is to develop statistical methods that can be used to analyze SAR (Synthetic Aperture Radar) images for bathymetric purposes, particularly at Greenland.

  • Identifying and discussing relevant statistical distributions for intensities in SAR images
  • Developing methods for statistical analysis of intensities in SAR images over homogeneous open sea areas
  • Examining different types of statistical test methods to assess the goodness-of-fit of the proposed distributions
  • Analyzing the statistical behavior of intensities in SAR images using the developed methods
  • Investigating the impact of noise and inhomogeneity on the statistical analysis of SAR images

  Zusammenfassung der Kapitel (Chapter Summaries)

  • Chapter 1 provides a general overview of the thesis, outlining its objectives and the focus on analyzing homogeneous sea areas.
  • Chapter 2 delves into the historical context of the research, highlighting initial challenges and the shift from a bathymetric analysis to a focus on statistical behavior of backscatter coefficients in homogeneous areas.
  • Chapter 3 reviews contributions from the literature regarding the theory of K-distribution and generalized gamma distribution, emphasizing the importance of statistical testing.
  • Chapter 4 explains the derivation of the K-distribution model for radar images and discusses situations where it may not be applicable.
  • Chapter 5 introduces the digamma function, which is crucial for understanding statistical distributions and estimation methods.
  • Chapter 6 discusses the generalized gamma distribution and its properties, including special cases and maximum likelihood estimation.
  • Chapter 7 explores the exponential gamma distribution and its importance in decibel calculations and logarithmic transformations.
  • Chapter 8 describes the K-distribution, its properties, and maximum likelihood estimation.
  • Chapter 9 examines the moment method for parameter estimation, highlighting its limitations and its applicability for different distribution families.
  • Chapter 10 introduces the method of moments, a more useful alternative to the moment method, and discusses its relationship to cumulants and the uniqueness of distributions.
  • Chapter 11 presents Raghavan's method, a parameter estimation technique that is particularly useful for cases where maximum likelihood estimation is difficult.
  • Chapter 12 covers the chi-square test for goodness-of-fit, its application in hypothesis testing, and the importance of constructing appropriate categories.
  • Chapter 13 describes the runs test, which is used to determine whether deviations between a hypothetical distribution and the true distribution are random or systematic. It also discusses the U test, a combination of the chi-square test and the runs test.
  • Chapter 14 focuses on Kolmogorov's Dn test, an alternative goodness-of-fit test that does not involve categories but relies on the distribution function instead of the category probabilities.
  • Chapter 15 analyzes the strengths and weaknesses of the four test methods, emphasizing the importance of category construction and highlighting the potential problems with the Dn test and the runs test in the case of composite hypotheses.
  • Chapter 16 presents the parameter analysis, including results from six homogeneous and two inhomogeneous regions of interest (ROIs). It discusses the different parameter values obtained using various estimation methods and examines their correlations.
  • Chapter 17 focuses on the statistical analysis, presenting results from testing whether backscatter coefficients follow different distributions. It also evaluates the performance of the four test methods under the assumption that data follow a generalized gamma distribution.
  • Chapter 18 summarizes the conclusions of the thesis, highlighting the applicability of the generalized gamma distribution and the K-distribution for describing backscatter coefficients in homogeneous sea areas. It also discusses the limitations of the statistical methods in detecting inhomogeneity and the need for supplementary methods.
  • Chapter 19 provides future recommendations for research, focusing on refining the K-distribution model, using airborne SAR, and investigating the significance of wind and other satellite measurements for bathymetric detection.

  Schlüsselwörter (Keywords)

  The main keywords and focus topics of the text include statistical analysis of SAR images, backscatter coefficients, K-distribution, generalized gamma distribution, parameter estimation, goodness-of-fit tests, chi-square test, runs test, U test, Kolmogorov's Dn test, homogeneous sea areas, inhomogeneity, noise, bathymetry, Greenland, ERS-1 SAR images.

  Frequently Asked Questions

  What is the main goal of this thesis on SAR images?

  To find relevant statistical distributions for backscatter coefficients in SAR images, particularly for detecting bathymetry or objects like ships in homogeneous sea areas.

  Why is the generalized gamma distribution recommended?

  It describes homogeneous sea areas as well as the K-distribution, but its parameters are easier and more robust to estimate.

  What is the K-distribution used for in radar imagery?

  The K-distribution is a standard model for the scattering mechanism in radar images, though it can be difficult to estimate parameters for.

  What statistical tests are examined for goodness-of-fit?

  The thesis examines the chi-square test, the runs test, the U test, and Kolmogorov's Dn test.

  What is the role of the digamma function in this analysis?

  The digamma function is used in the maximum likelihood estimation of parameters for the statistical distributions discussed.