Weighted Median Filters for Complex Array Signal Processing

1. Weighted Median Filters for Complex Array Signal Processing Yinbo Li - Gonzalo R. Arce Department of Electrical and Computer Engineering University of Delaware May 20th, 2005

2. Weighted Median Filters for Complex Array Signal Processing • Array processing: sonar, radar, seismology, etc. • Problem: impulsive noise and interference is expected. • We present a new multi-channel WMF that captures general correlation structure in array signals.

3. Nonlinear Signal Processing in Arrays • Median filtering, the optimal solution in impulsive-noise environments. • Extension of median filtering for use in multidimensional signals present high computational complexities. • Vector median [Astola, 1990] arises as a basic (very limited) solution.

4. Vector Median and Weighted Vector Median • Vector median is defined as: • VM is extensively used in color imaging and vector signal processing. • Problems: • Weights confined to be non-negative. • WVM does not fully utilize the cross-channel correlation from data.

5. Limitations of WVM Original image Corrupted image WVM filtered image

6. Multivariate Weighted Median (MWM) • Our solution: a filtering structure capable of capturing and exploiting both spatial and cross-channel correlations embedded in the data. • Exploit multiple frequency and phase shifts in array processing: complex processing domain.

7. Vector Median Vector median emerges from the ML location estimate of i.i.d. vector-valued samples. Independent & Identical Independent & Identical

8. Weighted Vector Median WVM extends VM to the case of independent but not identically distributed vector-valued samples. Independent & not Identical Independent & Identical

9. Exploiting Correlations Very often the multi-channel components of the samples are not independent at all.

10. Consider a set of independent, not identically distributed samples obeying : where and are M-variate vectors, and is the inverse of the MxM cross channel correlation matrix. Multivariate Filtering Structure

11. The ML estimate of location is: Inspiring the following filtering structure: NM2 weights. For 3 color image with 5x5 window, 25*32=225 Multivariate Filtering Structure (cont’d)

12. Multivariate Filtering Structure (cont’d) Weight matrix for time 1 Sample at time 1 = + = = Sample at time 2 Weight matrix for time 2

13. Frequently correlation matrices differ only by scale factors: Then, the ML estimate can be rewritten as: Reducing Complexity

14. Reducing Complexity (cont’d) Leading to the following filtering structure: V = [V1,…,VN]T is the time/spatial weight vector W = (Wjl)MxM is the cross-channel weight matrix (N+M2) weights. For 3 color images with 5x5 window, 25+32=34

15. Reducing Complexity (cont’d) Cross-channel weight matrix for all samples = + = = Time-dependent weights for times 1 & 2

16. Multi-channel Weighted Median Structure • The nonlinear multi-channel filter: where

17. Multivariate filtering structure This new multivariate filtering structure deals with spectrum correlation intrinsically. Correlated & not Identical Independent & not Identical

18. Extending to the Complex Domain • MWM must be extended to allow complex weighting when the filter input vector is complex. • Complex Weighted Medians are defined as: where:

19. The Complex MWM Filter is defined as: where and Complex MWM Filter for Array Processing

20. Filter Optimization The update for time dependent weights:

21. The update for cross-channel weights: Filter Optimization (cont’d)

22. Performance Results Simulation for MWMII

23. Performance Results (cont’d)

24. Performance Results (cont’d)

25. Nonlinear Signal Processing Nonlinear Signal Processing : A Statistical Approach by Gonzalo R. Arce

26. Introduced multi-channel median filter for complex array processing Derived its optimal filter Simulations show the gain in performance when multi-channel signals are correlated Can be used on more applications Need to analyze implementation complexity Conclusions