# J.S. Lee refined speckle filter

### Mathematical Description

Unlike optical remote sensing images, characterized by very neat and uniform features, SAR images are affected by speckle. Even if speckle confers a random aspect to SAR images, it may not be considered as a simple noise. It is, in fact, tightly related to SAR measurement principle.

Speckle phenomenon corrupts polarimetric observables (phase and intensity) in an important way. Specific procedures have to be used to retrieve relevant polarimetric information and to reduce the randomness of the acquired signals.

Synthesized SAR data may be considered as the result of the integration of a scene coherent response within each resolution cell, resulting from the convolution of the SAR impulse response with the coherent contribution of each elementary scatterer. As the number of contributing scatterers, within a resolution cell, tends to be large (it is the case for common resolution SAR measurements), the resulting integrated response is random in phase and amplitude and is shown to follow, over homogeneous areas, a Normal distribution.

A speckled response is usually represented under the form of a simple product model: where represents a complex speckled scattering coefficient, the original unspeckled scattering coefficient and the multiplicative speckle contribution.

The speckle term, is composed of independent real and imaginary parts, following both real centered Normal distribution . The corresponding speckled intensity, , is given by : .

The principle of speckle filtering consist in reducing the variance of in order to improve the estimate of its mean. The sample mean, , is defined as the empirical average of L independent realizations of a speckled intensity as follows:

The J. S. Lee speckle filter determines the unspeckled intensity estimate that minimizes a mean squared error following . This MMSE filter is based on a linearized speckle model leading to the following estimate expression where k is an adaptive filtering coefficient, based on local statistics, given by with the a priori speckle variance.

Over homogeneous areas, and , whereas over point targets and highly heterogeneous areas,and the pixel intensity remains unaffected by the filtering procedure.

In order to reduce the sensitivity of the adaptive filtering coefficient, , to isolated heterogeneities, this filter uses directional masks to determine the most homogeneous part of the sliding window where local statistics have to be estimated. This modification permits to preserve relatively sharp edges. This filter is named the J.S. Lee refined speckle filter.

Speckle filtering is based on incoherent averaging and requires to handle second order representations. The intensity information used in the scalar case has to be extended to the vector case when dealing with two or more polarization channels in order to take into account the different intensities as well as the cross-correlation related information.

A simple way to build an incoherent polarimetric representation consists in vectorizing a scattering matrix to create a target vector and computing the corresponding (3×3) covariance matrix or the (3×3) coherency matrix .

A polarimetric speckle filter should be developed based on the following principle:

● To preserve polarimetric properties, each term of the covariance / coherency matrix should be filtered in a manner similar to multi-look processing by averaging the covariance / coherency matrices of the same neighboring pixels. Like that, all terms of the covariance / coherency matrix should be filtered by the same amount.

● To avoid cross-talk between polarization channels, each element of the covariance / coherency matrix has to be filtered independently in the spatial domain.

● To preserve scattering characteristics, edge sharpness and point targets, the filtering has to be adaptive, and the filtering should select neighboring pixels for averaging.

J. S. Lee proposed to estimate the unspeckled covariance matrix and/or coherency matrix according to the following corresponding expressions: and .

Where k remains a scalar coefficient computed from the span statistics with: .

This approximation is allows to filter polarimetric data in a fast and simple way and avoids additional coupling (or cross-talk) between the polarimetric channels.

Once the edge-aligned window is selected based on the span, pixels in the edge-aligned window are then used to compute the mean for each element of the covariance / coherency matrix and the same filtering weights computed for the span image are then applied to each element equally and independently. The computation of the local variance is not required for each element of the covariance / coherency matrix, because the filtering weights are determined by the span. Only the local variance of the span image is required for the computation of the filtering weight. The use of the same weights makes this algorithm computationally efficient. Additionally, the polarimetric information is preserved in homogeneous areas, and cross-talk between channels is avoided. This is because, for each pixel, each element of the covariance matrix is filtered independently to avoid cross-talk, and the same edge-aligned window and the same filtering weight are applied to filter all elements of the covariance / coherency matrix to preserve polarimetric information. Furthermore, the image sharpness is maintained, because of the use of edge-aligned windows.

The following figure proposes a flowchart of the J.S. Lee refined speckle filter when applied on a 3x3 coherency matrix .

## 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

Journals:

● S. Goze, A. Lopes, « A MMSE Speckle Filter for Full Resolution SAR Polarimetric Data », J.E.W.A., vol 7, n°5, pp 717-737, May 1993.

● J.S. Lee, « Digital Image Enhancement and Noise Filtering by Use of Local Statistics », IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol PAMI-2, n°2, pp 165-168, March 1980.

● J.S. Lee, « Refined Filtering of Image Noise Using Local Statistics », Computer Graphics and Image Processing, 15, pp 380-389, 1981.

● J.S. Lee, « Speckle Analysis and Smoothing of Synthetic Aperture Radar Images », Computer Graphics and Image Processing, 17, pp 24-32, 1981.

● J.S. Lee, « A Simple Speckle Smoothing Algorithm for Synthetic Aperture Radar Images », IEEE Transactions on Systems, Man and Cybernetics, Vol SMC 13, n°1, pp 85-89, January/february 1983.

● J.S. Lee, « Speckle Suppression and Analysis for Synthetic Aperture Radar Images », Optical Engineering 25(81), pp 636-643, May 1986.

● J.S. Lee, M.R. Grunes, « Speckle Reduction in Multipolarization, Multifrequency SAR Imagery », IEEE Transactions on Geoscience and Remote Sensing, vol 29, n°4, pp 535-544, July 1991.

● J.S. Lee, K.W. Hoppel, S.A. Mango, A.Miller, “Intensity and Phase Statistics of Multi-Look Polarimetric and Interferometric SAR Imagery”, IEEE Trans GE-32, pp. 1017-1028, 1994.

● J.S. Lee, I. Jurkevich, P. Dewaele, P. Wambacq, A. Oosterlinck. « Speckle Filtering of Synthetic Aperture Radar Images: A Review », Remote Sensing Review, 1994, Vol n°8, pp 313-340.

● Lopes, R. Touzi, E. Nezry, « Adaptative Speckle Filters and Scene Heterogeneity », IEEE Transactions on Geoscience and Remote Sensing, vol 28, n°6, pp 992-1000, November 1990.

● Lopes, E. Nezry, R. Touzi, H. Laur, « Structure Detection and Statistical Adaptive Speckle Filtering in SAR Images », International Journal of Remote Sensing, 1993, Vol 14, n°9, pp 1735-1758.

● R. Touzi, A. Lopes, « The Principle of Speckle Filtering in Polarimetric SAR Imagery », IEEE Transactions on Geoscience and Remote Sensing, vol 32, n°5, pp 1110-1114, September 1994.