Theory

General Radar Theory

This page will eventually contain some basic radar theory. For now this is illustrated below using the Matlab demo scripts from the toolbox.

In addition to the references below, for a general introduction to radar, the texts by M.Skolnik are standard references. For a good introduction to imaging radar leading to advanced techniques, Understanding Synthetic Aperture Radar Images by C. Oliver and S. Quegan is very easy to follow. For more advanced polarimetric theory, papers by J-S.Lee in IEEE Transactions on Geoscience and Remote Sensing are standard references.

Matlab Demonstrations

The following demo files are available: (Please close all windows and clear all variables before running each demo)

polarisation_demo.m
statistics_demo.m
segmentation_demo.m
separation_demo.m
k_demo.m
weibull_demo
correlation_demo.m
change_demo.m
k_detection_demo.m
cfar_theory_demo.m
cfar_sim_demo.m
k_noise_demo.m

Click on the thumbnails for a full size image.

polarisation_demo.m

This simulates multivariate polarimetric samples and compares to theory. Figures are plotted from the papers below.

[1] `Intensity and Phase Statistics of Multilook Polarimetric and Interferometric SAR Imagery', J-S. Lee, K.W. Hoppel, S.A. Mango and A.R. Miller. IEEE TGRS (32)5 Sep 1994 pp 1017-1028
[2] `Statistics of the Stokes Parameters and of the Complex Coherence Parameters in One-Look and Multilook Speckle Fields', R. Touzi and A. Lopes. IEEE TGRS (34)2 Mar1996 pp 519-531

statistics_demo.m

This simulates univariate speckle distributions from exponential, gamma, Weibull and K distributions. Sample results are compared to theoretical PDF and CDFs.

segmentation_demo.m

Shows how to load actual AirSAR data from a rice growing area in Japan. Decompresses from the Compressed Stokes form and corrects the scene to have positive-definite covariance matrices due to the slightly lossy compression. Maximum likelihood segmentation is performed by merging single pixels. Expectation maximisation is used to automatically classify the scene into 5 classes. The sample covariance matrices are generated. The theoretical separation distance (related to classification error) between Urban and Vegetation classes is calculated using components of the covariance matrix. .

See `Polarimetric Classification using Expectation Methods', G Davidson et al. for a detailed description of this theory

separation_demo.m

This script verifies the class separation theory in `Polarimetric Classification using Expectation Methods', G Davidson et al. that, as far as I know, is an unpublished result.
It gives an expression for the Maximum Likelihood polarimetric classification accuracy between two classes based on the number of looks and the eigenvalues of the ratio of covariance matrices. This assumes that the polarimetric samples are completely described by their covariance matrices - i.e. no texture or environmental variation.
There is a small chance of it failing, if so run it again (I don't how to generate example class covariance matrices that are guaranteed positive definite).
The output is text only and basically tests the accuracy of multiple_polsep.m , if the code is changed for looks=1 then single_polsep.m can be tested as well. Due to Matlab's inability to recognise certain types of matrices, operations like the determinant of a positive definite covariance matrix are O(1 + j*eps). Floating point accuracy is repeatedly checked to remove the erroneous complex component.

k_demo.m

Independent Identically Distributed samples from the K distribution are generated for various shape parameter nu. The normalised log estimate is used to obtain the sample estimate of nu. See correlation_demo.m for a method of generating correlated K.

[1] `Estimation of texture parameters in K-distributed clutter', P. Lombardo and C.J. Oliver, IEE Proc. Radar, Sonar and Navigation, 1994, 141(4), pp196-204
[2] `Estimation of parameter estimators for K-distribution', D. Blacknell, IEE Proc. Radar, Sonar and Navigation, 1994, 141(1), pp 45-58
[3] `Parameter estimation for the K-distribution based on [z log(z)]', D. Blacknell and R.J.A. Tough, IEE Proc. Radar, Sonar and Navigation, 2001, 148(6), pp 309-312

weibull_demo.m

Independent Identically Distributed samples from the Weibull distribution are generated for various shape parameter a. A numerical MLE search is used to obtain the sample estimate of a and the mean mu. See [1] for a thorough treatment of Weibull clutter.

[1] `Weibull Radar Clutter' by M. Sekine and Y. Mao, IEE Publishing.

correlation_demo.m

Generation of correlated compound K is demonstrated using the algorithm from [1]. Correlated gamma is generated with a desired autocorrelation function by modifying the correlation of the source Gaussian. The compound formulation is used to generate the appropriate K. Great effort and a bit of luck has helped get this numerically stable.

[1] `The correlation properties of gamma and other non-Gaussian processes generated by memoryless nonlinear transformation', R. J. A. Tough and K. D. Ward, J. Physics D: Applied Physics, vol. 32, pp. 3075-3084, Dec. 1999.
[2] `Cell-averaging CFAR gain in spatially correlated K-distributed clutter', S. Watts, IEE Radar, Sonar and Navigation, 1996, 143(5), pp 321-327
[3] `Estimating the correlation properties of K-distributed SAR clutter', P. Lombardo and C.J. Oliver, IEE Proc. Radar, Sonar and Navigation, 1995, 142(4), pp167-178

change_demo.m

Illustrates the general theory of change detection for homogenous clutter. It illustrates Section 12.4 of [1] . The maximum likelihood measure for a change in the underlying intensity between two speckled areas is the ratio of their mean (e.g. [1] Eqn 12.49). Unfortunately, very few references recognise that this measure is distributed according to an F distribution. By using this, the maths in [1] can be considerably simplified.

[1] `Understanding Synthetic Aperture Radar Images' by C. Oliver and S. Quegan, 1998

k_detection_demo.m

This demonstrates the generation of K-distributed clutter in the presence of thermal noise. It shows that for a reasonable Probability of Detection, and typical Probability of False Alarm, the existing Swerling Theory can be used for detection prediction of the Swerling target models. It also shows the significant increase in False Alarm Rate if a threshold is based on theory which assumes Gaussian clutter (i.e. exponential in intensity).
See RSNlap for a more detailed description of this theory (and a method of determining the correct threshold for multilook K distribution in thermal noise).
The generated figure shows the performance curve for K distributed clutter (shape parameter nu=1) with a Clutter to Noise Ratio of 0dB. Targets fluctuating according to the Swerling models are introduced for various Signal to Interference Ratios. The approximate theory is in agreement with Monte Carlo results.

The following references give some background to this problem:

[1] `Radar detection prediction in K-distributed sea clutter and thermal noise', S. Watts, IEEE Trans., 1987, AES-23(1), pp 40-45
[2] `Maritime surveillance radar, Part 1: Radar scattering from the ocean surface', K.D. Ward, C.J. Baker and S. Watts, IEE Proc. F., 1990, 137(2), pp 51-62
[3] `Maritime surveillance radar, Part 2: Detection performance prediction in sea clutter', S. Watts, C.J. Baker and K.D. Ward, IEE Proc. F., 1990, 137(2), pp 63-72

cfar_theory_demo.m

This demo reproduces the results from [1] which calculated the expected detection losses associated with various Constant False Alarm Rate (CFAR) processors. These include Cell Averaging (CA), Cell Averaging Greatest Of (CAGO), Cell Averaging Smallest Of (CASO), Order Statistics (OS) and Trimmed Mean (TM). Comparison can be made to the Optimum Processor as assumed by the Swerling Detection theory. All plots assume a Swerling 1 target in Gaussian clutter (i.e. exponential in intensity).

[1] `Analysis of CFAR Processors in Nonhomogeneous Background', P.P.Gandhi and S.A.Kassam, IEEE Trans. AES, 24(4), July 1988, pp 427-445

cfar_sim_demo.m

Various CFAR processor architectures have been implemented in C and compiled as a mex file for speed. This demonstrates these functions upon simulated speckle. The False Alarm rate is measured from CA, CAGO, CASO, OS and TM processors using similar window sizes and parameters to that in cfar_theory_demo.m
The output is text only, and shows that the desired Pfa of 1e-4 is in agreement with theory, within statistical uncertainty, for all considered processors.

k_noise_demo.m

This simulates clutter drawn from the K distribution in the presence of thermal noise. Sample results are compared to theoretical PDF and CDFs. These are non-trivial expressions that require time-consuming numerical integration. The integration is quite sensitive; the code demonstrates how a singularity was removed and how to integrate to an effective infinity. Note the results plot both the PDF and the Complementary CDF (equal to 1-CDF) on semilogy paper.

In addition to the references below k_detection_demo.m , some of the best papers on detection theory within K-distributed clutter and thermal noise are by Simon Watts. A reasonable seach should find conference papers such as:

http://www.aspc.qinetiq.com/Events/July99/swatts.pdf

Note that parameter estimation of K-distributed clutter is very much complicated by the presence of thermal noise - inevitable in real systems. In particular the normalised log estimate U, while being close to optimum without noise, can be inaccurate even at high clutter to noise ratios [1].

[1] `Effect of noise on order parameter estimation for K-distributed clutter', P. Lombardo, C.J. Oliver and R.J.A. Tough, IEE Proc. Radar, Sonar and Navigation, 1995, 142(1) pp 33-40