# Wishart supervised segmentation

## Description

There is a great deal of interest in the use of polarimetry for radar remote sensing. In this context, polarimetric SAR data classification has been widely addressed in the 1990’s.

The tight relation between natural media physical properties and their polarimetric features leads to highly descriptive classifications results that can be interpreted by analyzing underlying scattering mechanisms.

The Wishart polarimetric classification scheme performs a Maximum Likelihood (ML) statistical segmentation of a polarimetric data sets based on the multivariate complex Wishart probability density function of second order matrix representations.

After having learnt the Wishart statistics of user-defined training areas, the whole data set is then classified by assigning each pixel to the closest class using a Maximum Likelihood decision rule.

● Polarimetric SAR data statistics

It has been verified that when the radar illuminates an area of random surface of many elementary scatterers, a target vector k can be modeled as having a multivariate complex gaussian probability density function of the form:, where q stands for the number of elements of k, equal to three in the monostatic case, represents the determinant, and is the global 3x3 coherency matrix of the target vector .

It has been shown that assuming that target vectors have a distribution, a sample L-look coherency matrix follows a complex Wishart distribution with L degrees of freedom, , given by:

with and where is the gamma function, and the trace of .

● Maximum likelihood (ML) segmentation based on the Wishart distribution

A Maximum Likelihood (ML) segmentation process assigns sample coherency matrices to the class represented by the coherency matrix of its cluster center , maximizing its likelihood function over N possible classes. This decision may be expressed under the following form:

The ML assignment of a sample coherency matrix following a Wishart distribution becomes:

with:

where corresponds to the global coherency matrix of the cluster center evaluated over the class .

● The Wishart supervised segmentation

In the case of supervised classification, training areas are required to estimate for each class.

Training areas may be defined by the way of a graphic interface which permits to delimitate areas by defining regions of interest on a visual representation of the data to be classified.

Once training areas are defined, the training process consists in collecting the coordinates of each training area and computes each class centre matrix .

In 1994, J.S. Lee et al. proposed the following procedure: a sample coherency is assigned to the class according to the following decision rule:

The statistical distance between the sample matrix and the class , , derives from the Log-likelihood function and is given by:

This relation shows that if the number of look (L) increases, the a priori probability of the class does not play a significant role for the classification. It is generally assumed that without a priori knowledge, the different are equal, in which case the distance measure is not a function of the number of look (L).

Thus, for each pixel, represented by its 3x3 coherency matrix , the distance is computed for each class, and the class associated to the minimum distance is assigned to the pixel and, after simplification, is given by:

This procedure based on a distance measure is simple and easy to apply. In addition, this algorithm based on the Wishart distribution uses the full polarimetric information.

## References

Books:

● Jong-Sen LEE – Eric POTTIER, Polarimetric Radar Imaging: From basics to applications, CRC Press; 1st ed., February 2009, pp 422, ISBN: 978-1420054972

● Shane R. CLOUDE, Polarisation: Applications in Remote Sensing, Oxford University Press, October 2009, pp 352, ISBN: 978-0199569731

● Charles ELACHI – Jakob J. VAN ZYL, Introduction To The Physics and Techniques of Remote Sensing, Wiley-Interscience; 2nd edition (July 31, 2007), ISBN-10 0-471-47569-6, ISBN-13 978-0471475699

● Harold MOTT, Remote Sensing with Polarimetric Radar, Wiley-IEEE Press; 1st edition (January 2, 2007), ISBN-10 0-470-07476-0, ISBN-13 978-0470074763

● Jakob J. VAN ZYL – Yunjin KIM, Synthetic Aperture Radar Polarimetry, Wiley; 1st edition (October 14, 2011), ISBN-10 1-118-11511-2, ISBN-13 978-1118115114

● Yoshio Yamaguchi, Polarimetric SAR Imaging : Theory and Applications, CRC Press; 1st ed., August 2020, pp 350, ISBN: 978-1003049753

● Irena HAJNSEK – Yves-Louis DESNOS (editors), Polarimetric Synthetic Aperture Radar : Principles and applications, Springer; 1st edition (Marsh 30, 2021), ISBN 978-3-030-56502-2

● L. Ferro-Famil, E. Pottier, J.S Lee, Unsupervised Classification of Natural Scenes from Polarimetric Interferometric SAR Data in "Frontiers of Remote Sensing Information Processing". C.H. CHEN. Chief Editor, Ed. World Scientific Publishing, July 2003

ISBN 981-238-344-1

● L. Ferro-Famil, E. Pottier, Radar Polarimetry Basics and Selected Earth Remote Sensing Applications In “Academic Press's Library in Signal Processing” collection. Volume 2 “Communications and radar Signal Processing”, S. Theodoridis and R. Chelappa (Directors), N. Sidiropoulos and F. Gini (Eds.), 4 October 2013, ISBN: 978-0-124-16616-5, Academic Press.

Journals:

● J.S. Lee, M.R. Grunes, R. Kwok « Classification of multi-look polarimetric SAR imagery based on the complex Wishart distribution» International Journal of Remote Sensing, vol. 15, No. 11, pp 2299-2311. 1994.