Hybrid Vector Filters based on Marginal Ordering for Impulsive Noise Suppression in Color Images

Abstract

To attenuate impulsive noise in color images, a hybrid basic vector filter and its switching extensions are introduced in this paper. By utilizing reliable components provided by the marginal filter and retaining the inherent correlation between multi-channels, the new method selects the vector, which has minimal distance to the output of the marginal median filter. Based on this scheme, some well-known switching filters are easily modified to improve their noise suppression capability. The experiments demonstrate that the proposed filtering approach is more effective to suppress multichannel impulsive noise in color images, and its computation is more efficient than state-of-the-art basic vector filters. Moreover, extended experiments indicate that the noise suppression capability of several well-known switching vector filters can also be improved by adopting this basic scheme as an alternative approach at the replacement stage.

Appendix:

Theorem 1

Let {z ₁ ,..,z _m ,...,z _N } represents an ordered set with z _i ∈R and N is an odd integer(>1), such that z _m (m=(N+1)/2) is the median of the set. The following inequation holds for any z _l from the set :

$$ \sum\limits_{j=1}^{N}{\mid z_{m}-z_{j}\mid} \leq \sum\limits_{j=1}^{N}{ \mid z_{l}-z_{j} \mid} $$

(15)

Proof

When l = m, two sides of (15) is obviously identical.

When l<m, the left side of (15) becomes

$$\begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N}{\mid z_{m}-z_{j}\mid} &=& \sum\limits_{j=1}^{l}{\mid z_{m}-z_{j}\mid} +\sum\limits_{j=l+1}^{m}{\mid z_{m}-z_{j}\mid} \\&&+\sum\limits_{j=m+1}^{N}{\mid z_{m}-z_{j}\mid} \end{array} $$

(16)

The right side of (15) becomes

$$\begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N}{ \mid z_{l}-z_{j} \mid} &=& \sum\limits_{j=1}^{l}{\mid z_{l}-z_{j}\mid} +\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid}\\&& +\sum\limits_{j=m+1}^{N}{\mid z_{l}-z_{j}\mid} \\ &=& \sum\limits_{j=1}^{l}{\mid z_{l}-z_{j}\mid} +\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid}\\&& +\sum\limits_{j=m+1}^{N}{\mid z_{l}-z_{m}\mid} +\sum\limits_{j=m+1}^{N}{\mid z_{m}-z_{j}\mid} \\ &= & \sum\limits_{j=1}^{l}{\mid z_{l}-z_{j}\mid} +\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid}\\&&+ (N-m)\mid z_{l}-z_{m}\mid\ +\sum\limits_{j=m+1}^{N}{\mid z_{m}-z_{j}\mid} \\ &=& \sum\limits_{j=1}^{l}{\{ \mid z_{l}-z_{j}\mid+\mid z_{l}-z_{m}\mid\}} \\&&+(N-m-l)\mid z_{l}-z_{m}\mid\\&& +\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid}+ \sum\limits_{j=m+1}^{N}{\mid z_{m}-z_{j}\mid} \end{array} $$

$$\begin{array}{@{}rcl@{}} {\kern55pt}&=& \sum\limits_{j=1}^{l}{\mid z_{m}-z_{j}\mid}+(N-m-l)\mid z_{l}-z_{m}\mid\\ &&+\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid}+ \sum\limits_{j=m+1}^{N}{\mid z_{m}-z_{j}\mid} \end{array} $$

(17)

Using these substitution (16) and (17), some identical items of two sides of (15) are canceled

$$ \sum\limits_{j=l+1}^{m}{\mid z_{m}\!-z_{j}\mid} \leq (N-m-l)\mid z_{l}-z_{m}\mid\ \!+\sum\limits_{j=l+1}^{m}{\mid z_{l}-z_{j}\mid} $$

(18)

The right side of (18) becomes

$$\begin{array}{@{}rcl@{}} \sum\limits_{j=l+1}^{m-1}{\mid z_{l}\,-\,z_{m}\mid} \,+\,\sum\limits_{j=l+1}^{m}{\mid z_{l}\,-\,z_{j}\mid} &\!\geq\!& \sum\limits_{j=l+1}^{m}{\mid z_{m}\,-\,z_{j}\mid}\\ &\!\geq\!& \sum\limits_{j=l+1}^{m}{\mid z_{m}\,-\,z_{j}\mid} \end{array} $$

(19)

Hence, the inequality (15) is proved on the condition of $l\!{\kern -.5pt}<\!m$.

When l>m , set y _i =−z _i , we can get another ordered set {−y ₁,..,−y _m ,..,−y _N } and have

$$ \sum\limits_{j=1}^{N}{\mid z_{m}-z_{j}\mid} = \sum\limits_{j=1}^{N}{\mid y_{N-l+1}-y_{j}\mid} $$

(20)

According to the proof on the condition of l<m, we know that (N−l+1)<m on the condition l>m. We have

$$ \sum\limits_{j=1}^{N}{\mid y_{N-l+1}-y_{j}\mid} \geq \sum\limits_{j=1}^{N}{\mid y_{m}-y_{j}\mid} =\sum\limits_{j=1}^{N}{\mid z_{m}-z_{j}\mid} $$

(21)

So we have ${\sum }_{j=1}^{N}{\mid z_{m}-z_{j}\mid } \leq {\sum }_{j=1}^{N}{\mid z_{m}-z_{j}\mid }$ when l>m. This completes the proof. □