Weighted Median Filters for Complex Array Signal Processing Yinbo Li - Gonzalo R. Arce Department of Electrical and Computer Engineering University of Delaware May 20 th, 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 Independent & Identical Vector Median Vector median emerges from the ML location estimate of i.i.d. vector-valued samples.

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

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 M x M cross channel correlation matrix. Multivariate Filtering Structure

11 The ML estimate of location is: Inspiring the following filtering structure: NM 2 weights. For 3 color image with 5x5 window, 25*32=225 Multivariate Filtering Structure (cont’d)

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

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 = [V 1,…,V N ] T is the time/spatial weight vector W = (W jl ) MxM is the cross-channel weight matrix (N+M 2 ) weights. For 3 color images with 5x5 window, =34

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

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

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

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