MERIDIAN ESTIMATOR PERFORMANCE FOR SAMPLES OF GENERALIZED GAUSSIAN DISTRIBUTION

1. MERIDIAN ESTIMATOR PERFORMANCE FOR SAMPLES OF GENERALIZED GAUSSIAN DISTRIBUTION D.A. Kurkin1,V.V. Lukin1, I. Djurovic2, S. Stankovic2 1 National Aerospace University, 61070, Kharkov, Ukraine 2 University of Montenegro, 81000, Podgorica, Montenegro

2. Outline of presentation • Considered applications • Blind estimation of noise variance • Blind determination of noise spatial correlation properties • Generalized Gaussian distributions • Meridian estimator and its properties • Accuracy analysis • Conclusions and future research

3. Considered applications • Blind determination of noise spatial correlation properties • Blind estimation of noise variance • Non-Gaussian noise filtering in communication channels Common properties: • A considered distribution has maximum of bell or peaky shape; • Distribution tail is heavy; • One needs to find (estimate) distribution mode with appropriate accuracy. • Estimator is to be robust with respect to outliers (heavy tail)

4. Blind estimation of noise variance • Is needed in hyperspectral imaging for determination of sub-band SNR, for image filtering and lossy compression; • Can be used in radar imaging for determination of multicative (relative) noise variance (efficient number of looks in SAR imaging). 3-step procedure of noise variance blind estimation: To obtain local estimates of noise variance in blocks of size from 5x5 to 9x9 pixels; To form a histogram of obtained estimates; To find the distribution mode.

5. Blind estimation of noise variance A sample histogram of local estimates obtained for test image corrupted by additive noise with

6. Blind determination of noise spatial correlation properties The main purpose is to determine whether noise spatially correlated or not, thus estimation of both parameters and is required. Where Where denotes q-th order statistic of DCT coefficients for an l-th block.

7. Blind determination of noise spatial correlation properties a) b) Histogram of the parameter Rl for the test image Barbara (a) and Histogram of the parameter Elfor the test image Lena (b) both corrupted with i.i.d. Gaussian noise with

8. Generalized Gaussian distributions 8 where ; σ – scale parameter; μ – location parameter (expectation); k – shape parameter.

9. Meridian estimator 9 • xi –sample element; • N – sample size; • δ – medianity parameter of meridian estimator. Cost functions for δ = 0.1; δ = 1; δ = 10000; Output of the meridian estimator is always one element of initial data sample to which the estimator is applied. Probability that sample meridian and median coincide

10. Meridian estimator main properties 10 • Meridian estimator belongs to class of M-estimates (maximum-likelihood estimates) • Cost function of meridian estimator has global minimum and its argument is just the estimate, it is always placed between the minimal and maximal values of the data processed • Outside this limits, the cost function monotonically decreases or increases • If δ is large enough, meridian estimator properties are the same as for the median estimator x = {1,1; 0,3; 7,2; 4,5; 3,8; 2,9; 6,0; 8,2; 1,9}

11. Statistical characteristics of the meridian estimator: criteria and properties 11 • standard deviation (SD) of meridian estimates; • median of absolute deviations (MAD): Properties: For symmetric distributions, sample meridian is an unbiased estimate; Distribution of obtained estimates can be non-Gaussian (might have heavy tail; thus, it is reasonable to analyze not only standard deviations of the estimates but also their MAD values

12. Accuracy analysis 12 For GGDs with p larger than 1, the meridian estimator is not able to provide better (more accurate) estimation than the standard median; For GGDs with p smaller than 1 there can be accuracy improvement (compared to the median estimator) due to applying the meridian estimator under condition that δ is set correctly (optimally) When p decreases, ratio of estimate SD to MAD becomes larger, this shows that the distribution of estimates is no more Gaussian and it has heavy tail.

13. Comparison of the meridian and myriad estimators 13 Comparison has been done for data samples of different size. If p decreases, optimal δ also reduces; Myriad and meridian estimators (under condition of optimally set parameters) produce approximately the same accuracy for p smaller than 1; For larger N, provided SD and MAD of estimates are smaller

14. Conclusions 14 • The performed studies have shown that the meridian estimator can be used as an effective estimator for a generalized Gaussian distribution family. • Maximum effectiveness could be reached by choosing an optimal value of the tunable parameter δopt. • In this case, the meridian estimator with optimal δ performs either as the sample median or better. The latter is observed if p<1. • The location of δopt is affected by tail heaviness of an estimated sample distribution and by scale of this distribution (MADX). • For GGD with p<1, the meridian estimator performs similarly to the sample myriad estimator.

15. Future research Performance analysis of the meridian estimator for one-side heavy tail distributions; Determination of potential accuracy (lower bound of location estimation) for ML and optimal L-estimators for GG distributions; comparison of accuracy; Design of methods for adaptive tuning the parameter δ for meridian estimator; Analysis of accuracy for adaptive myriad estimators designed earlier for the GG distribution; Design of practical algorithms of mode estimation for several distributions with aforementioned properties.