Weighted Median Filters for Complex Array Signal

Processing

Yinbo Li - Gonzalo R. Arce

Department of Electrical and Computer EngineeringUniversity of Delaware

May 20th, 2005

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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.

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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.

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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.

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Limitations of WVM

Original image

Corrupted image

WVM filtered image

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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.

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X

X

X

X

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12X

11X

21X

Independent & IdenticalIndependent & Identical

Vector Median

Vector median emerges from the ML location estimate of i.i.d. vector-valued samples.

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12

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X

X

X

X

11X

12X

11X

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Independent & not Identical

Independent & Identical

Weighted Vector Median

WVM extends VM to the case of independent but not identically distributed vector-valued samples.

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Exploiting Correlations

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

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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

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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)

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+ =

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1

Y

Y

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1

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Y

Y

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112

1

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WW

WW=

Weight matrix for time 1Sample at time 1

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X

2

2

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Y

Y

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2122

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WW

WW=

Weight matrix for time 2Sample at time 2

Multivariate Filtering Structure (cont’d)

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Frequently correlation matrices differ only by scale factors:

Then, the ML estimate can be rewritten as:

Reducing Complexity

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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

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+ =

2

1

Y

Y

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X

X

2

1

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Y

Y=1V

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12

X

X

2

2

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Y

Y=2V

Cross-channel weightmatrix for all samples

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2111

WW

WW

Time-dependent weightsfor times 1 & 2

Reducing Complexity (cont’d)

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

where

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22

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21

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X

X

X

X

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12X

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Independent & not Identical

Correlated & not Identical

Multivariate filtering structure This new multivariate filtering structure

deals with spectrum correlation intrinsically.

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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:

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The Complex MWM Filter is defined as:

where

and

Complex MWM Filter for Array Processing

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))()(sgn()())(()()1( nYnPnenVnVnV tR

tR

tR

tiv

ti

ti i

))()(sgn()( nYnPne tI

tI

tI i

))()(()()(2 nYnPnPnej tR

tR

tI

tR ii

))()(()()(2 nYnPnPnej tI

tI

tR

tI ii

Filter Optimization

The update for time dependent weights:

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The update for cross-channel weights:

))(()()1( nWnWnW stw

stst

)())(( * t

RtR

i

sti

ti

sti

ti

tR YPQBVQAVe

i

)())(( * t

ItI

i

sti

ti

sti

ti

tI YPQBVQAVje

i

Filter Optimization (cont’d)

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Performance Results

Simulation for MWMII

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Performance Results (cont’d)

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Performance Results (cont’d)

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Nonlinear Signal Processing

Nonlinear Signal Processing : A

Statistical Approachby Gonzalo R. Arce

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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