AN ADAPTIVE LENGTH RECURSIVE WEIGHTED MEDIAN FILTER WITH IMPROVED PERFORMANCE IN IMPULSIVE NOISY ENVIRONMENT

S.Manikandan, O.Uma MaHeswari, D.Ebenezer

Department of Electronics and Communication Engineering

College of Engineering, Guindy.,Anna University

Chennai-25, India

Abstract : - In this paper, an algorithm employing adaptive-length recursive weighted median filters for improving impulse noise removal performance for image processing is proposed. This algorithm can achieve significantly better image quality than fixed length median filters when the images are corrupted by impulsive noise. The weights for the Recursive weighted median filter (RWM) are selected by threshold decomposition or by optimization techniques. The performance of the proposed algorithm is compared with fixed length median filters, weighted median filters, and Recursive weighted median filters.

Key-Words: - Weighted Median Filter, Adaptive length, Optimization technique, Impulsive Noise,

Less MSE, Lin’s Algorithm.

1 Introduction

Generally, the median filters are well known for removing impulse noise and preserving edges. Over the last two decades, there is a significant improvement in the development of median filters. Weighted median filter(WMF) [1](Arce.G), RWM filter are examples. Recently it has been proved that RWM[2] filter produces better result when compared to other median type filter. But results are available only for the fixed window length. The median type filters exhibit blurring for fixed window sizes and insufficient noise suppression for small window sizes. In this paper an adaptive window size RWM filter algorithm is proposed, which achieve a high degree of noise suppression and preserve image sharpness.

In case of general RWM filter [2], the weights are chosen in accordance with window length. In some windows the signal may be noise free. However attenuation of the amplitude of the signal cause blurring. The adaptive window size RWM algorithm is used to overcome blurring. Window lengths are selected based on the amount of noises present in the original signal. The window length is

calculated by Lin’s algorithm [5]. After calculating the window length, the RWM operation is performed. The weights are selected by using the different algorithms .It is observed that optimization technique by steepest descent algorithm produces better result. And also it has been proved that use of negative weights [2] in RWM filter gives good mean square error when compared to positive weights and zero valued weights.

2 Recursive Weighted Median Filter

In linear signal processing, the reason why recursive filters are used is simple. With same number of operations, they achieve more than non-recursive filters. The goal may be, for example, steepest possible transition between the pass band and stop band. The traditional stability concept is achieved by applying the negative feedback in the recursive part.

2.1 Selection of weights

The success of the median filters in the image processing is based on two intrinsic

properties: edge preservation and efficient attenuation of the impulsive noise properties not shared by traditional filters. However, noise removal capability of WMF is lower than that of RWM filter The general weighted median filter structure [1] admitting positive and negative weights as

B=MEDIAN(|W1|sgn(W1)a1,|W2|sgn(W2)a2,…………..…,…..|Wn|sgn(Wn)an), (1)

with Wi  R for I=1,2,3,4……………….n, and

where  is the replication operator defined as

Wi  ai =( ai, ai …….. ai)Wi times.

Admitting only positive weights, weighted median and nomalised weighted mean filters are, essentially filters having ‘low pass’ filtering characteristics [2],[4].

A large number of engineering applications require ‘band pass’ or ‘high pass’. Frequency filtering characteristics equalization, deconvolution, prediction, beam forming and system identification.

To overcome these limitations, a weighted median filters admitting negative weights are introduced. The properties on interpretation of the negative weights [4] are:

1.The output sample is strictly one of the input samples.

  1. A negative indicates that the corresponding sample cannot appear in the output, and other samples of the same magnitude have a reduced sample selection probability. Again, other equally valued samples may appear in the output, provided that the sum of the relevant weights is positive.
  1. A negative weight can also be said to favour samples with extreme magnitudes. That is, the minimum or maximum of the sample set, if their weights are positive.

2.2 Optimization technique

In general, the design of the RWM filters is done by adaptive optimization algorithm. The main objective of the optimization is to find the best filter coefficients such that a performance cost criterion is minimized. A criteria widely used in the design of the median based filters is the mean absolute error (MAE) between the filter’s output and the desired signal

The optimal WM filtering with structural constraints is formulated as

N

Minimize  Wi 2

i=1

subject to (2)

N

 Wi = 1

i=1

3 Structure of the Filter

The general structure of the recursive weighted median filter[1] is given as

Y(n) = MEDIAN ( |A1|sgn(A1)Y(n-l)|N +

|Bk|sgn(X(n-k)|M2) (3)

Algorithm

Lin’s technique Recursive Structure

Fig. 2. Stages of Algorithm

Let us consider the algorithm as stages.

Stage 1:Determination of the window size:

First the amount of the impulsive noise in the signal is determined and then length of the window is determined by using the Lin’s algorithm.

The Lin’s algorithm for Filter length decision is given by

Pixels inside window buffer:

Pj =x(n+j-1)-x(n+j), for j=1,2,3;

Pj =x(n+j+1)-x(n+j), for j=-1,-2,-3; (4)

The conditions for the threshold values are as given as [5]

If (p2>T3 and p-2>T3 ) or (p1>T3 and p-3> T3 ) or ( p3 > T3 and p-1 > T3 ),

choose L = 7;

If (p1 > T2 and p-2 > T2 ) or ( p2 > T2 and p-1 > T2 )

choose L = 5;

If ( p1 > T1 and p-1 > T1 )

Choose L=3;

Otherwise, choose L=1.

Stage 2: Filtering operation:

After the determination of the window length, the recursive-weighted median filtering operation is done. The algorithm for the RWM filtering is given as

  1. Calculate the threshold

N M

T0 = (1/2) (   |A1| +  |Bk| ) . (4)

i=1 k=0

  1. Jointly sort the ‘signed’ past output samples sgn(Al)Y(n-l) and the ‘signed’ input observations sgn(Bk)X(n+k).
  1. Sum the magnitudes of the weights corresponding to the sorted ‘signed’ samples beginning with the maximum and continuing down in the order.
  1. If 2T0 is an even number, the output I is the average between the signed sample whose weight magnitude causes the sum to become  T 0 and the next smaller signed sample; otherwise, the output is the signed sample whose weight causes the sum to become  T 0.

4 Results

By using the above algorithm, the filtering operation is done for various median techniques. After calculating the window length by lin’s algorithm, the RWM operation is performed by using the negative weights. RWM proves that the weights selected by using the optimization technique gives better result when compared to the other median types algorithms. The mean square error for the various median types are calculated and tabulated, From the table adaptive length RWMF gives better result than any other median type filter.

5 CONCLUSION REMARKS

Generally, the RWM filters are designed only for the fixed window length. This causes blurring in the output samples. Because in the fixed window length, the noise may absent in some windows in that condition, the filtering operation is done for the original samples which

causes the blurring in the output. To overcome this filter is designed where the window length is determined by the width of the impulsive noise presented in the input sample. So that there is no chance of filtering the original sample without noise which reduces the blurring in the output sample.

References

[1] G. Arce, "A General Weighted Median Filter Structure Admitting Negative Weights", IEEE Tr. on Signal Proc., vol.46, Dec. 1998.

[2] G. Arce and J. Paredes, "Recursive Weighted Median Filters Admitting Negative Weights and Their Optimization", IEEE Tr. on Signal Proc., vol.48, nr. 3, March 2000.

[3] O. Yli-Harja, J. Astola and Y. Neuvo, "Analysis of the Properties of Median and Weighted Median Filters Using Threshold Logic and Stack Decomposition", IEEE Tr. Signal Proc., vol. 39,no. 2, pp. 395-410, Feb. 1991.

[4] O.Yli-Harja,Heikki Huttunen,Antti and Karen”Design of Recursive weighted median filters with negative weights”signal processing lab,tampere university of tech.,Finland

[5] Ho-Ming Lin and Alan”Median filters with Adaptive Length”, IEEE transactions of the circuits and systems,vol..35,no.6,june 1988.

[ [6]I.Pitas and .N.Venetsanopoulos,

”Nonlinear digital filters Principles and applications”.

[7] Jakko Astola and Pauli,”Fundamentals of nonlinear digital filtering”.

Noise density / 0.25 / 0.40
Mean filter (fixed window) / 693.00 / 1148.8
Median filter (fixed window) / 225.92 / 931.44
Weighted median filter
(fixed window) / 198.32 / 850.12
Recursive weighted median
Filter (fixed window) / 153.14 / 543.67
Recursive weighted median
Filter (adaptive window) / 123.99 / 325.95

Table I

Results for impulsive noise removal (MSE)

(a) (b)

© (d)

(e) (f)

Fig 1. Image denoising (a) Original (b) Image with salt and pepper noise with 0.25 (c) Median Filtered image (fixed window length (d) Recursive weighted median filtered image (fixed window length) (e) Recursive Weighted median filter with (fixed window length) (f) Recursive weighted median filter (adaptive window length)

(a) (b)

© (d)

(e) (f)

Fig 2 : Image denoising (a) Original (b) Image with salt and pepper noise with 0.4 (c) Median Filtered image (fixed window length (d) Recursive weighted median filtered image (fixed window length) (e) Recursive Weighted median filter with (fixed window length)(f) Recursive weighted median filter (adaptive window length)

Dr.D.Ebenezer., is working as a Assistant Professor in College of Engineering, Anna University. Guindy ,Chennai. His area of interest is Digital Communication system, Digital Signal Processing. He is currently working in Nonlinear signal processing.

O.Uma Maheswari is currently working as a Lecturer in College of engg., Anna University Chennai. She is doing in Ph.D in Nonlinear signal Processing

S.Manikandan is currently doing his M.E degree in Applied Electronics at Anna University. His Area of interest is Digital signal processing, VLSI signal processing.