SIMILAFUTY BASED IMPULSIVE NOISE REMOVAL IN COLOR IMAGES

B. Sinoiku’ K.N. Plataniotis’ R. Lukuc’ A.N. Venet~anopoitlos~

Department of Automatic Control, Silesian University of Technology, Poland

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering,
University of Toronto, 10 King’s College Road, Toronto ON, M5S 3G4, Canada

ABSTRACT 2. PROPOSED ALGORITHM

In this paper a novel approach to the problem of impulsive 2.1. Gray-scale Images
noise reduction in color images based on the nonparametric Let us assume a filtering window W containing nf 1 image
density estimation is presented. The basic idea behind the pixels, {Fo, F l , . . . ,F,,}, where n is the number of neigh-
new image filtering technique is the maximization of the
similarities between pixels in apredefined filtering window.
The new method is faster than the standard vector median
-
bors of the central pixel Fo, (see Fig. 2a) and let us de-
fine the similarity function p : 10;cu) R which is non-
ascending in [O;03) , convex in 10;00) and satisfies p ( 0 ) =
filter (VMF) and preserves better edges and fine image de- 1, p ( m ) = 0 . The similarity between two pixels of the
tails. Simulation results show that the proposed method out- same intensity should be 1, and the similarity between pix-
performs standard algorithms of the reduction of impulsive els with far distant gray scale values should he very close to
noise in color images. 0. The function p(F;, F j ) defined as p(F?,F j ) = ~ ~ ( l f -
i;
Fjl) satisfies the three above conditions.
1. NOISE REMOVAL IN COLOR lMAGES Let us additionally define the cumulated sum M of sim-
ilarities between the pixel Fk and all its neighbors. For the
A number of nonlinear, multichannel filters, which utilize central pixel and its neighhors we define 11.1,and hfk as
correlation among multivariate vectors using various dis-
tance measures, have been proposed [ 1-71. The most popu- ”
lar nonlinear, multichannel filters are based on the ordering fifo=Clr(~OII.;),
AJ~= C
~ . ( F ~ , P ~()i ,j
of vectors in a predefined moving window. The output of j=1 j=1 .j#b
these filters is defined as the lowest ranked vector according
vector ordering technique. which mcans that for Fk which are neighbors of FO we
Let F(z) represents a multichannel image and let W do not take into account the similarity between Fk and Fo,
+
be a window of finite size 11 1, (filter length). The noisy which is the main idea hehind the new algorithm. The omis-
image vectors inside the filtering window W are denoted sion of the similarity p ( F k , Fo) privileges the central pixel,
as Fj, ,f = 0,1, ...,n . If the distance between two vec- as in the calculation of MOwe have TI similarities p(F0, Fk),
tors F i , Fj is denoted as p ( F i , Fj) then the scalar quantity k = 1,2, . . . 71, and for A4,, I; > 0 we have only n - 1 sim-
~

Ri = E,”=, p ( F i , Fj) is the distance associated with the ilarity values, as the central pixel FOis excluded from Af,.
noisy vector F i . The ordering of the Ri ’s: R p ) 5 R ( , ) 5 In the construction of the new filter the reference pixel
... 5 implies thc same ordering to the corresponding Fo in the window H’ is replaced by one of its neighbors
vectors Fi : F(0)5 F(,) 5 ... 5 F(n).Nonlinear ranked if hfO < A{,, k = 1 , . . . , n. If this is the case, then FO
type multichannel estimators define the vector F(”) as the is replaced hy that Fk for which I; = arg tnax Mi, i =
filter output. However, the concept of input ordering, ini- 1 , . . . , T I . . In other words FO is detected as being cormpted
tially applicd to scalar quantities is not easily extended lo if A40 < A t k , I; = 1, . . . , ? I and is replaced by its neighhors
multichannel data, since there is no universal way to define Fi which maximizes the sum of similarities A t between all
ordering in vector spaces. To overcome this problem, dis- its neighbors excluding the central pixel. This is illustrated
tance functions are often utilized to order vectors, [1,2]. in Figs. 2 and 5.
The majority of standard filters detect and replace well The basic assumption is that a new pixel must he taken
noisy pixels, hut their propeny of preserving pixels which from the window W ,(introducing pixels which do not occur
were not corrupted hy the noise process is far from the ideal. in the image is prohibited like in VMF). For this purpose p
In this paper we show the construction of a simple, efficient must be convex, which means that in order to find a maxi-
and fast filter which removes noisy pixels, but ha? the ahility mum of the sum of similarity functions A4 it is sufficient to
of preserving original image pixel values. calculate the values of A l only in points PO:.. . F,?, [7].
~
2.2. Color Images Applying the linear similarity fuuction / L , we obtain
'The presented approach can be applied in a straightforward
way to color images. We usc the similarity function defincd 1 - p ( F i l F k ) / h for p ( F i , F k<
) /I,
/L(FI,Fk) =
by b{Fi, Fj} = / I ( llFi -F,j)ll where 1 ) . 11 denotes the spe- 0 otherwise.
cific vector norm. Now, in exactly the same way we maxi-
mize the total similarity function A4 for the vector case. Then we have from (2), A40 =n - p(F0, F 3 ) ,and
We have checked several convex functions in order to ?=I
compare our approach with the standard filters uscd in color
image processing presented in Tab. 1 and we have obtained
very good results (Tab. 21, when applying the following
similarity functions, which can he treated as kernels ofnon-
parametric density estimation, 17-91, (.see Fig. 4).

If this condition is satisfied, then the central pixel is con-

sidered a,,not disturhed by the noise process, otherwise the
pixel Fi for which the cumulative similarity valuc achieves
maximum, replaces the central noisy pixel. In this way the
filter replaces the central pixcl only when it is really noisy
and preserves the original undistorted image structures.
The parameter / I can he set experimentally or can be
determined adaptively using the technique described in [71.

a') b)
Fig. 2. Illustration of the construction of the new filtering
technique. First the cumulativesimilarityvalue MObetween
the central pixel F0 and its neighbors is calculated (.a), then
the pixel Fo is rejected from the filter window and the cu-
mulative similarity values M I : k = 1: . . . , n of the pixels
Fig. 1. Illustration of the efficiency ofthc new algorithm of FI,. . . , F, are determined, (b).
impulsive noise reduction in color images: a ) tcst iinase, h)
image compted by 4% impulsive salt & pepper noise, e)
new filter output, d ) effect of median filtering (3 x 3 mask).
It is interesting to note, that the best results were achieved
for the simplest similarity function p?(:c),which allows to
construct a Sast noise reduction algorithm. In the multichan-
nel case, we have

j= I j=l.j#"
Table 1. I'ilters taken Soor comparisons, [l-31

wherep{F;.Fe} = llFk - Ft)11 and 11 . 11 is the LZ n o m .

I - 106
 30 - ;-~...\.

37.5
37
,I/'
,/,' '
~ :\
\
36.5
\\
36
I
i
/ \\,
35.5
i
!

Table 2. Comparison of the new algorithm based on differ-

ent kernel functions with the standard techniques, using the
LEN4 color image contaminated by 5% of impulsive noise.

3. RESUIXS AND CONCLUSION

The new algorithm presented i n this paper can he seen as
a modification and improvement of the Vector Median Fil-
ter. The comparison with standard color image processing
filters, ('Tah. 2, Fig. 1 and 3) shows that the new filter out-
performs the standard procedures used in color image pro-
cessing, when the impulse noise is to he eliminated. The
new filter class based on the similarity functions and ker- I 2 3 4 5 8 7 8 0 I011 12431415 1817181Q20
nel dcnsity estimation is significantly faster than VMF and X Noise
therefore it can be applied in various applications, in which
the computational speed plays a crucial role.
4. REFERENCES
[ I ] 1. Pitas, A. N. Venetsanopoulos, 'Nonlinesr Digital Filters : Princi-
ples and Applications', Kluwer Academic hhlishen, Bos$oo, MA,
1990.
[2] K.N. Plataniotis, A.N. Venetsanopoulos, 'Color Image Processing
and Applications', Springer Verlag, 2000.
131 1. Pitas, P. l'sakalidcs, Multivariate ordering in color image process-
ing, IEEE Trans.on Circuits and Systems for Video lechnology, I ,
3, 247-256, 1991. 33
[4] J. Astola, i? Haavisto, Y. Neuovo, Vector median fillen, IEEE Pro-
csedings, 78,678-689, 1990.
[SI K.N. Plataniotis, D. Androutsos, A.N. Venelsanopoulos. Colourlm-
age Processing Using Fuzzy Vector Directional Filten, Pmceed- 25
ings of the IEEE Workshop on Nonlinear Signd/lmage Processing.
Greece, 535-538,1995.
161 K.N. Plataniolir, D. Andmmutsos. V. Sri, A.N. V * ~ ~ I % D O ~ O U I O S ,
A nearesl neighbour multichannel filter. Electronic Letters, 1910- 0 2 4 6 8 10
1911, 1995. % Noke
171 B. S m l k a , A. Chydzinski, K. Wojciechowslj, K. Plataniotis, A.N. Fig. 3. a) Dependence of PSNR on the h parameter for
Venelsanoposlos. On the reduction of impulsive noise in multichan-
nel image processing, Optical Engineennp. ~ 1 40, . no. 6, pp. 902-
LENA image with 12% of corrupted pixels, b) efficiency of
9ox.2001. the new algorithm in terms of PSNR in comparison with
181 B.W. Silver"; "Density Estimation for Slatistics and Data Aod- standard filters. LENA image was contaminated by impul-
ysis", London. Chapman and Hall, 1986. sive noise with p from 1% to 20% and independently on
191 D.W. ScotI. "Multwariate Density Estimation". New York John Wi- each channel, p from I % to IO%, e).
ley. 1992.

-
I 107
 a) b) 0
Fig. 4. Cumulative similarity values dcpendence on the pixel pray scale value for a window containing a set of pixels with
intensities {15: 24: 33: 41: 4.5_55, 72, 90: 951 using thc po function (a')and {17 function (b). Plot ( e ) shows the comparison of
the total similarity functions MOwhen using two ditferent kernels.

FO F.
'F d) e)
I

Fig. 5. Illustration of the new filter construction. The supporting window 1V of size 3 x 3 contains 9 pixels of intensities
{15,24:33,41,45,55,7%:90,95}~(Fiy.4). Eachofthegrdphsfrom,a)to i)showsthedependence'of Ado andM,.i/o, [MI@ <
M O ) ,where A[,, denotes the cumulative similarity value with rejected central pixel FO on the pixel pray scale value. Graph
a) shows the plot of MOand for Fo = 15, plot b) for FO = 24 and so on till plot plot i)which shows the graphs of A4,
and Af,o for FO = 95. The arrangement of pixels surrounding Ihe central pixel FO is not relevant. The central pixel will he
replaced in cases: (a), (b), 1.0- (i),as in those c a e s there exists a pixel Fi for which MO< A,.