Alternative Vector Median Filter Algorithm
for Color Images Noise Reduction
Mahmoud Hassan
Electronics Engineering Department
Princess Sumaya University
Amman, Jordan
Raja’Al-Omari
King Abdullah II School for Information
Technology
Jordan University
Amman- Jordan
Abstract: -The vector median filter is a well known non-linear vector order statistics filter can reduce
noise while preserving image details. This paper presents a practical technique for noise reduction of
color images using an alternative vector median filter algorithm. This technique gives comparable
results to vector median filter with the advantage of high speed processing .A comparison with vector
median filter will be presented.
Key-words: Vector median filter, Color images, Noise, Color weights, Standard normalized mean
square error, L-norm
1 INTRODUCTION
Noise reduction is one of the major issues in
image processing [1]. Many concepts are
present now for noise suppression. Median filter
technique is one of most effective techniques. It
is a non-linear spatial domain filter and it has
many advantages over other spatial filters that
are used for noise reduction.
Major advantage of a median filter for grayscale
images is reducing noise while preserving image
details.
Median filtering technique is now perfectly
structured for grayscale images (i.e. one channel
image) where the pixel consists of only one
attribute.
Multi–channel images (color images) [2] are
now widely used in many applications. Median
filters have also been used to filter color images.
For color images the application of median
filters is not straight-forward.
Vector median filters (VMF) have been
introduced [3,4] to simulate the concept of
median filtering in order to reduce the color
image noise.
Several methods have been introduced to use
the median filters for noise removal such as:
An adaptive scalar median filter[5], a vector
median filter with weighting[6] and without
weighting[4[, a reduced vector median filter[7]
and other methods[5,8,9,10,11].
The vector median filters consider the pixel as a
vector not a scalar. It is suggesting a single
quantity that represents the three attributes of
the pixel. This resulting quantity is considered
as the ranking value, which later will
differentiate one pixel from the other in order to
find the median.
The major issue of a vector median filter is to
transform three perpendicular vectors, Red,
Green and Blue to one – dimensional quantity
The theory of vector median filter is based on
measuring the distance between some selected
pixels to all other pixels in the window. The
pixel that shows the least sum of distances to the
other pixels in the window will be chosen as the
median pixels.
The mathematical model of the vector median
filter is defined as follows [3,4,9]:
Assume N-sample vectors {X1, X2, ……, XN},
the output of L-norm operator is given by :
VM {X1, X2, …, XN} = XVM
where
XVM Є {X1, X2, …, XN}
and
N
 || Xvm  Xi || L ≤
i 1
N
 || Xj  Xi ||
i 1
th
VM is the vector median operator; Xi is the i
component of color vector (pixel) values.
The implementation of vector median filter is
based on L-norm to order vector (pixel)
according to their relative magnitude
differences. If more than one vector (pixel)
2
ALTERNATIVE VECTOR MEDIAN
ALGORITHM
The main goal of the vector median filter is to
produce a comparable quantity from three
vectors (pixels) of a color image. This paper
introduces another concept to calculate another
comparable quantity. It is the distance of the
vector (pixel) from the origin with a weight
multiplied by each attribute to make the result
unique for each vector (pixel).
The mathematical model for this alternative
algorithm is presented as follows. Assume P(R,
G, B) is a vector (pixel) that consists of three
components: Red, Green and Blue. The
mapping function D (P) is defined as follows:
D (P) =
WR * R 2  WG * G 2  WB * B 2
(2)
Where R, G and B are the attributes of vector
(pixel) P; WR, WG, and WB are color weights
attached to each attribute in order to distinguish
different vectors (pixels) who share the same
values of attributes but in different order. For
.
L
j=1,…N
(1)
satisfies the condition of above equation, then
the closer vector (pixel) to the center of the
window will be selected. Vector median filter
algorithm is inefficient due to the lengthy
calculations needed.
example, consider WR =1, WG =2 and WB = 3.
Without these weights, two vectors (pixels) such
as (20, 70, 90) and (70, 90, 20) will have the
same mapping values. So, the weight will
eliminate any ambiguity.
It follows that mapping function D (P) is unique
for each vector (pixel) and it has good indication
of the vector (pixel) value.
Implementation of above alternative vector
median filter is straight forward and as is
expected it takes much less processing time
when compared with the vector median filter.
3 RESULTS AND COMPARISON STUDY
The alternative vector media algorithm was
simulated using Java 1.4. An image processing
program was generated to achieve the tasks of
the algorithm. Possible flowcharts for the two
algorithms are shown in Figure 1.
The above two algorithms (vector median filter
and the alternative median filter) have been
applied on a sample image. Figure 2 and Figure
3 shows the filtered images which look very
close.
Noisy Input Image
Noisy Input Image
Split the image into windows w (square or
rectangle of number of rows and columns)
Split the image into windows w (square or
rectangle of number of rows and columns)
CompX = 0;

For each Pixel Xi in the window w DO:

For each Pixel Xj in the window find Xi
– Xj add absolute value to CompX.

For each Pixel Xi in the window w DO:
o
Calculate the value
D(Xi) =
W * Ri 2  W * Gi 2  W * Bi 2
R
G
B

Find The Median D(Xi) between the D(X)
values resulted.

For each Pixel X in the window w let
XVM = X

SUMVM = 0

Replace the central value in the window
with the value of XVM.

For each Pixel X in the window w find |
X-XVM and add its absolute value to
SUMVM

For all SUMVM resulted, consider the most
close and less from CompX. Take the XVM
that caused SUMVM to result as median
Pixel.

Replace the central value in the window
with the value of XVM.

W = Next Window
No
Is w the last
window?
Yes
No
(a
Save Output Image
(Filtered Image)
Modified Vector
Median Filter
Algorithm
(b)
Is w the last
window?
(b)
Yes
Save Output Image
(Filtered Image)
Vector Median
Filter Algorithm
(a)
Figure 1: flowcharts for: (a) possible vector median filter. (b) modified vector median filter
Figure 2: (a) Image 1: Noisy Bridge image by salt. (b) Filtered bridge image using VMF.
Figure 3: (a) Image 1: Noisy Bridge image by salt. (b) Filtered bridge image using MVMF.
The filtering efficiency for noise removal was
measured by a standard approach; the standard
Normalized Mean Square Error (NMSE)[9].
Equation (3) represents NMSE.
Height W idth
NMSE 
  || f (i, j )  f (i, j ) ||
i 1
2
j 1
* 100 (3)
Height W idth
 ||
i 1
f (i, j ) ||
2
j 1
Equation 3 is very useful for comparison study.
Vector median filter requires a large amount of
Noisy Bridge (255 x 255)
Noisy e-board (Pepper). (464 x 448)
Noisy e-board (Salt). (464 x 448)
Noisy e-board (Salt & Pepper). (445 x 440)
Noisy Lena Image. (512 x 512)
time when applied to an image for noise
removal. The calculations needed to accomplish
the task of Equation 1 are lengthy. Table1
shows a comparison study applied on five
degraded images (Figures 3 through 7).
As it was expected, Table 1 confirms the
advantage of processing time needed to remove
the noise and enhance the image. Fortunately,
the reduction of processing time did not reduce
the expected enhancement as for the vector
median filter
VMF
NMSE (%)
Time (s)
24.96
9.87
15.87
29.64
10.12
29.17
21.59
29.16
7.36
38.63
MVMF
NMSE (%) Time (s)
24.95
0.73
15.86
2.14
10.10
2.09
21.59
2.01
7.82
2.44
Table 1: Comparison Study between VMF and MVMF
Figure 4: (a) Image 1: Noisy Lena Image. (b) Image 2: Filtered Lena Image by MVMF
Figure 5: (a) Image 1: Noisy e-board image by pepper. (b) Filtered e-board image using MVMF.
Figure 6: (a) Image 1: Noisy e-board image by salt. (b) Filtered e-board image using MVMF.
Figure 7: (a) Image 1: Noisy e-board image (salt & pepper). (b) Filtered e-board image using MVMF.
4 CONCLUSION
This paper has introduced an alternative median
filter algorithm for enhancing color images. The
initial comparisons with the vector median filter
are encouraging.
The alternative vector algorithm is offering two
advantages. The first one is a noise removal for
color images comparable to the vector median
results. The second one is the short processing
time needed to enhance an image.
The achieved results are based on specific
weighted factors.
More work will be done to investigate the issues
of the proposed color weights and the effect on
large images
5 REFERENCES
1. R.G. Gonzalez, and R.E. Woods, Digital
Iimage Processing, Second Edition,
Prentice Hall, 2002.
2. C. Vertan, M.Malciu,V.Buzuloiu, and
V.Popescu, “ Median Filtering
Techniques for Vector Valued Signals” ,
International Conference on Image
Processing ( ICIP96 ), pp. 17A4, 1996.
3. A.Koschan, and M.Abidi, “ A
Comparision of Median Filter
Techniques for Noise Removal in Color
Images”, Proc. 7 th German Workshop
on Color Image Processing, D. Paulus,
J.Denzler (Eds.), Erlangen Germany ,
Report University of Erlangen Nurnberg, Institute of Computer
Science, vol.34, no.15 , pp.69-79 ,
October 2001.
4. F.Argenti, M.Barni, V.Coppellini and
A.Mecocci, “Vector Median Deblurring
Filter for Color Image Restoration”,
Electronis Letters, vol.27, pp.1899 –
1900, 1991.
5. K.P.Valavanis, J. Zheng and J.M.Gauch,
On Impulse Noise Removal in Color
Images, Proc. Int. Conf. on Robotics and
Automation, Sacramento, Ca,
1991,pp144-149.
6. R. Wichman, K. Oistamo, Q. Liu, M.
Grundstrom and Y. Neuvo, Weighted
Vector Median Operation For Filtering
Multispectral Data, Proc.SPIE 1818,
Visual Communications and Image
Processing, 1992, pp 376-383.
7. C. S. Regazzoni, and A. Teschioni, A
New Approach to Vector Median
Filtering Based on Space Filling Curves,
IEEE Trans. On Image Processing 6,
1997, pp 1025-1037
8. K.N. Plataniotis and A.N.
Venetsanopoulos, Color Image
Processing and Applications, SpringerVerlag, , 2000
9. P. Trahanias, and A. Venetsenopoules,
Vector Directional Filters - a New Class
of Multichannel Image Processing
Filters, IEEE Trans . on Image
Processing, vol.2, pp. 528 534, 1993.
10. P. Bojarczaki , and S. Osowski,
Denoising of Images- a Comparison of
Different Filtering Approaches, WSEAS
Transactions on Computers, vol. 3,
2004,pp738-743.
11. . S. Manikandan, O. Uma Maheswari
and D. Ebenezer, An Adaptive Length
Recursive Weighted Median Filter with
Improved Performance in Impulsive
Noisy Environment, WSEAS
Transactions on Electronics, issue 3,
vol.1, 2004, pp443-448