# Wishart - H / A / Alpha unsupervised 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 objective of an unsupervised classification process is to gather the complementary information contained in polarimetric data in order to deliver highly descriptive clusters as well as an interpretation of their characteristics.

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.

An optimal segmentation necessitates maximizing a global ML function over the entire polarimetric data set and requires an unreasonable amount of time. A sub-optimal solution consists in iteratively optimising this function using a k-mean clustering algorithm. It is well known that such an algorithm may get stuck in local minima and is then highly sensitive to the initialisation conditions. It was found that an initialisation of the different clusters using the results of the H-Alpha classification procedures led to satisfying and stable results (J.S. Lee et al.).

A similar ML segmentation scheme explicitly including Anisotropy related information may be built from the Wishart statistics and led to improved segmentation results (E. Pottier et al.).

● 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 joint likelihood optimization for all the sample matrices cannot be performed in an easy way. Indeed, in the frame of an unsupervised segmentation, the global coherency is built from the sample matrices belonging to the class . An optimal solution would consist in testing all the possible segmentations of a given number of sample matrices into N classes. This optimal solution cannot be applied due to the unrealistic computational load it requires.

In 1994, J.S. Lee et al. propose an alternative method, based on a k-mean iterative clustering algorithm. At each iteration of the algorithm, 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:

The following figure depicts the unsupervised segmentation process based on a maximum likelihood and using a k-mean clustering algorithm.

It is known that the initialization of the pixel distribution into N classes is a critical stage of the k-mean clustering algorithm. An adequate initialization permits a fast convergence and provides correctly segmented clusters.

The convergence of the algorithm is evaluated by testing a condition of termination. Such a criterion may be defined from the estimation of the classification quality, or consist in a maximum number of iterations or in a sufficiently low number of pixels that are differently classified from one iteration to the other.

● The combined Wishart – H / alpha segmentation

In 1998, J.S. Lee et al. proposed an unsupervised classification method that uses the two-dimensional H / alpha classification plane to initially classify the polarimetric SAR image.

The initial classification map defines training sets for classification based on the Wishart distribution. This initialization provides 8 classes relating to the underlying physical scattering mechanism and giving a stable initial approximation of the segmentation.

The classified results are then used as training sets for the next iteration using the Wishart method. Significant improvement in each iteration has been observed, and the analysis of the final class centers in the two-dimensional H / alpha classification plane are useful for interpretation of terrain types.

The polarimetric H / alpha segmented image is used as training sets for the initialization of the supervised Wishart classifier. The cluster centers of the coherency matrices, , is computed for each zone, with :

where Nm is the number of pixels in the a priori class Each pixel in the whole image is then reclassified by applying the distance measure procedure. The reclassified image is then used to update the , and the image is then again classified by applying the same distance measure procedure.

To classify similar objects in the same image, which can have different orientation angles, the orientation dependence is removed from the coherency matrix during the Wishart classification. The classification procedure stops when a termination criterion, defined by the user, is met. The termination criterion we used is the number of iterations and is here equal to 10. In this case, the ratio of pixels switching class with respect to the total pixel number is smaller than 10%.

The following figure depicts the Wishart H / alpha unsupervised maximum likelihood segmentation.

● The combined Wishart – H / A / alpha segmentation

In order to improve the capability to distinguish between different classes whose cluster centers end in the same zone, the combined Wishart classifier is extended and complemented with the introduction of the anisotropy (A) information which indicates the relative importance of secondary mechanisms obtained from the expansion of a coherency matrix.

This polarimetric indicator is particularly useful to discriminate scattering mechanisms with different eigenvalue distributions but with similar intermediate entropy values. In such cases, a high anisotropy value indicates two dominant scattering mechanisms with equal probability and a less significant third mechanism, while a low anisotropy value corresponds to a dominant first scattering mechanism and two non-negligible secondary mechanisms with equal importance.

This original method consists in comparing the anisotropy value of all the pixels to ½. This comparison procedure leads to the definition of an « equivalent » projection of the three-dimensional H / A / alpha space in two complemented H / alpha planes,

Among the different approaches tested, the best way to introduce the anisotropy information in the classification procedure consists in implementing two successive combined Wishart classifiers. The first one is identical to the previous one. Once the first classification procedure has met its termination criterion, the anisotropy comparison for all the pixels, is then introduced, which leads to the definition of 16 new training sets used for the initialization of the second Wishart classifier.

The entire unsupervised Wishart H / A / alpha classification procedure is as follows :

1 : Apply target decomposition to compute the entropy H and alpha angle.

2 : First initial classification of the image into 8 classes by zone in the two-

dimensional H / alpha plane.

3 : For each class, compute the initial cluster center [Tm](0) (k=iteration number

and m=1..8)

4 : Classify the whole image using the distance measure procedure

5 : Compute [Tm](k+1) for each class using the classified pixels of step 4

6 : Return to step 4, until a termination criterion defined by the user is met.

7 : Apply target decomposition to compute the anisotropy A.

8 : Second initial classification of the image into 16 classes by zone in the

projected three-dimensional H / A /  space, with :

9 : For each class, compute the new initial cluster center [Tm](0) (k=iteration

number and m=1..16)

10 : Classify the whole image using the distance measure procedure

11 : Compute [Tm](k+1) for each class using the classified pixels of step 10

12 : Return to step 10, until a termination criterion defined by the user is met.

Improvements in classification and details are observed. Some classes, indistinguishable in the classification based on entropy (H) and alpha angle (alpha) are now clearly visible with the introduction of the anisotropy information. It is also possible to discriminate different areas, belonging to the same scattering type (same entropy H and alpha angle) but differentiated with the associated anisotropy information which is there significative of the presence of several scattering mechanism types.

The introduction of the anisotropy in the clustering process permits to split large segments into smaller clusters discriminating small disparities in a refined way.

## 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:

● L. Ferro-Famil, E. Pottier, J. S. Lee, "Unsupervised classification of multifrequency and fully polarimetric SAR images based on the H/A/Alpha-Wishart classifier", IEEE Transactions on Geoscience and Remote Sensing, vol. 39, n°11, pp 2332-2342, November 2001.

● L. Ferro-Famil, E. Pottier, J.S. Lee, "Classification and Interpretation of Polarimetric SAR data", IEEE International Geoscience and Remote Sensing Symposium, June 2002, Toronto, Canada.

● 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.

● J.S. Lee, M.R. Grunes, T.L. Ainsworth, L.J. Du, D.L. Schuler, S.R. Cloude, “Unsupervised Classification using Polarimetric Decomposition and the Complex Wishart Distribution”, IEEE Transactions Geoscience and Remote Sensing, Vol 37/1, No. 5, p 2249-2259, September 1999.

● E.Pottier, J.S. Lee "Unsupervised Classification Scheme of POLSAR Images Based on the Complex Wishart Distribution and the H/A/ Polarimetric Decomposition Theorem" 3th European Conference on Synthetic Aperture Radar, EUSAR 2000, Munich, 23-25 May 2000.