Property Analysis of Noise Suppressing L-Filters for Speckle Image Processing Vladimir Lukin a , Vladimir Melnik a,b , Victor Chemerovsky a , Alexander Zelensky a , Jaakko Astola b , Pauli Kuosmanen b , Heikki Huttunen b a Dept. 507, State Aerospace University, Kharkov, 310070, Ukraine b Signal Processing Lab., Tampere Univ. of Technology, Tampere, P.O. Box 553, FIN-33101, Finland ABSTRACT The images formed by coherent imaging systems are characterized by presence of multiplicative noise with non- symmetrical p.d.f.s. The examples are the Rayleigh and one-side (negative) exponential distributions. For these cases the optimal L-filters are derived for different coefficient censoring by minimizing MSE of residual fluctuations. Some sub- optimal L-filters are considered as well. They are the Lpq-filters that use only two order statistics and the trimmed filters with symmetrical and nonsymmetrical coefficient censoring. Those filters are parametrically optimized according to the same criterion. The robust features of the considered filters are analyzed both theoretically using empirical influence functions and numerically with application of contamination noise model. As contaminating factor we exploited the salt- and-pepper noise with different probabilities and amplitudes of positive and negative spikes. Output estimate bias and variance were calculated and examined. It is shown that the use of sub-optimal filters is well motivated from different points of view. Keywords: L-filters, speckle, optimization, spikes, robustness, efficiency comparison 1. INTRODUCTION The coherent imaging systems are widely used in many practical applications: remote sensing (synthetic aperture radars), biomedical imaging, acoustic holography, etc. A high level of speckles is typical for obtained images, especially if they are formed using one-look systems. One more peculiarity consists of the fact that the multiplicative noise distribution occurs to be non-gaussian and, moreover, it is non-symmetrical with respect to the mean value. The most typical examples of such distributions are the Rayleigh and negative exponential p.d.f.s 1 . Considered images can be simultaneously corrupted by spikes and speckles. The reasons of spike occurrence are various, for example, bit errors. The problem is that the probability of spikes and their characteristics are often unknown (unpredictable). That is why it is desirable to apply robust filters for image processing. The corresponding algorithms should satisfy a set of contradictory requirements: to provide appropriate noise suppression efficiency, reliable spike removal and good edge/detail preservation simultaneously. In our papers 2,3,4 we have proposed the robust locally adaptive filtering algorithms ensuring rather good trade-off of required properties. The proposed approach assumes that for processing of edge/detail neighborhoods (i.e. locally active areas) the detail preserving filters (DPFs) are applied and for image homogeneous regions (locally passive areas) the noise suppressing filters (NSFs) are used. The decision on filter selection is based on local activity indicator analysis (in the simplest case its comparison to some predetermined threshold). This approach permits to formulate different requirements to DPFs and NSFs. The main items for the latter ones are to ensure an effective reduction of speckle noise combined with appropriate robustness with respect to spikes. In this paper, we analyze the properties of L-filters when they are used as NSFs for considered noise models. A standard median filter causes bias in the output signal for non-symmetrical p.d.f.s. The same effect takes place for some other traditional nonlinear filters, such as Hodges-Lehmann, Wilcoxon and α-trimmed filters with equal number of rejected elements in sorted data sample. The L-filters are known to be a wide class of nonlinear filters with flexible (adjustable) parameters and properties. By means of weighting factor optimization one can get a proper solution for wide variety of distributions and models. The theory of L-filter optimization and the corresponding tools are developed well 5,6,7 . For given p.d.f. of noise it is rather easy to get the coefficients of an L-filter providing the unbiased mean estimate and ensuring its minimal mean square error (MSE). The robust properties of L-filters can be controlled by means of coefficient censoring 5 , i.e., by imposing some preliminary constraint. In this case it is also possible to optimize the weights of L-filters. However, for image processing