MEDIAN FILTERING TECHNIQUES FOR VECTOR VALUED SIGNALS
Constantin Vertan, Marius Malciu, Vasile Buzuloiu, Viorel Popescu
Applied Electronics Department, Faculty of Electronics and Telecommunications, Bucuresti “Politehnica”
University
PO Box 35-57, Bucuresti 35, Romania
E-mail: vertan@alpha.imag.pub.ro
ABSTRACT
Vector (multichannel) signal processing has arisen
in the last decades as a major area of interest. The classical
single-channel (scalar valued) processing techniques
cannot deal with the interchannel dependencies, yielding
to the development of new methods. This paper
investigates two approaches to the median filtering of
vector valued signals: filtering based on the reduced
ordering in the space of signal samples and filtering with
channel decorrelation. A novel filter is proposed that uses
channel independentization by the central limit theorem.
This filter performs better than classical filters in terms of
mean square error measures and visual perception.
1. INTRODUCTION
In the last years, the area of vector valued
(multichannel, multispectral, multicomponent) signal (and
particularly: image) processing has dramatically increased.
The leading edge of development and interest is in the
domain of remote sensing (dealing with 3 to 18 channels
images), but the classical color images still remain the
preferred research domain. Typically, a color image is
represented in each pixel by a three component vector. The
vector components are in general (or at least in the
acquisition stage) the amounts of pure red, green and blue
that compose the local color.
These vector valued signals cannot be reduced to
a stack of separately processed scalar components, due to
the inherent (although not obvious) correlation between
these components. This is why standard (scalar)
processing techniques are not very appropriate.
In the following of the paper, Section 2 discusses
some median filters based on a reduced ordering in the
signal samples space and Section 3 presents some possible
ways of using the channel decorrelation in median filtering
(including the novel proposed NDMMF - Normalized
Decorrelated Marginal Median Filter). Finally, Section 4
summarizes some experimental results and Section 5
presents conclusions and further developments.
2. REDUCED ORDERING BASED MEDIAN FILTERING
The median filtering [1] is a nonlinear, local order
based processing technique, providing good results in
image enhancement (impulsive noise removal with detail
preservation). The main processing task behind this filter is
the ascending ordering of the selected pixel values.
Although natural and immediate in the scalar case (gray
scale images), the ordering becomes a problem in a vector
space. A classical paper in the domain of vector valued
data ordering is [2]. In [2] several ordering criteria are
discussed and classified as marginal, conditional, partial
and reduced (or aggregate) ordering.
The marginal ordering means the scalar ordering
of each component (recalling the model of stack of
separately processed components) and we shall refer to the
MMF (Marginal Median Filter) as the basic vector filter.
The main disadvantage of this ordering is the fact that the
ordered values can be significantly different from the
original data to be ordered.
The conditional ordering implies to order the data
according to a hierarchical order of the components; the
lexicographic ordering is the best example of such a
criterion; this ordering is a complete one (from the
mathematical point of view) but the space topology is not
preserved. The results of applying this ordering for the
RGB space are not good.
The partial ordering is based on the convex hull
like set decomposition of the data samples; the inner-most
set will be a kind of median. The implementation problems
are related to the time consuming convex hull determination
and to the inaccuracy due to the very sparse nature of the
cluster of data points. (We recall that the data points are
issued from an analysis window, typically a 3 by 3 window
containing only 9 samples).
Finally, the reduced ordering seems more
appropriate. The idea is to order some scalars, each scalar
being computed from the components of the vector from
the data set; the same ordering is assumed for the vectors.
A general case is the ordering with respect to a
fixed reference vector α and a generalized distance
determined by a positive semidefined matrix A. In this case,
the scalars are the distances from each data vector to the
T
reference vector d ( x i , α ) = ( xi − α ) ⋅ A ⋅ ( x i − α ) .
The classical VMF (Vector Median Filter) [3] is
introduced by means of the aggregate (cumulative)
distance (the sum of distances from a vector to all other
vectors):
N
D( xi ) =
λ ( xi − x j ) T ⋅ A ⋅ ( xi − x j ) .
j =1
The median is the vector with the minimum aggregate
distance. In [4] the same principle was used for angular
distances. In [5] a linear combination of distances to the
(local) marginal median, marginal mean and center located
vector for noise removal from color images was
successfully used (the AHM - Adaptive Hybrid Median).
If A=I (identity matrix), the Euclidean distance is
obtained. The idea of the MVMF (Modified Vector Median
Filter) approach is to use a weighted Euclidean distance.
The weighted Euclidean distance is defined by a
nonidentity matrix A, in the simpler case a diagonal matrix.
Since in the analysis window the components (red, green,
blue for the particular case of color images) are not equally
distributed or proportional, it seems natural to use some
weights that emphasize the local relative importance of
each component.
There are several choices for the assignment of
weights: a component is considered important according to
its range (the difference between the local maximum and
minimum), its normalized range (range to average ratio), its
variance or contrast scaling (variance to average ratio, as
defined in [6]). Another possible choice is the use of the
covariance matrix for A, yielding to the Mahalanobis
distance, wellknown in the pattern recognition problems.
As the experimental results have shown (see
Section 4), the best results were obtained using the
weighting by the range. This seems normally, since some
difference between the components of two vectors could
be important if the range is small or relatively neglectable if
the range is big.
3. DECORRELATION BASED MEDIAN FILTERING
As suggested by [7], a component wise
processing of multichannel images could be successfully
performed if the components are decorrelated first. The
best choice for this decorrelation procedure is the
Karhunen Loeve transform, but other (faster) transforms
could be applied, if allowed by the statistical model of the
image.
The DMMF (Decorrelated Marginal Median Filter)
is composed from a decorrelation block (direct integral
transform), a scalar median filter for each decorrelated
component and a recorrelation block (inverse integral
transform).
Since the Karhunen Loeve transform (and the
Discrete Cosine transform DCT) concentrates most of the
spectrum energy (relevant feature) in a few coefficients, it
is natural to assume that the median filtering is not
necessary to be performed on every component, but just
on the most significant one. This approach is used in the
TDMMF (Truncated Decorrelated Marginal Median Filter),
reducing the computational amount. The component to be
processed is chosen as the most important decorrelated
component with respect to a given importance criterion
(range, normalized range, variance, contrast scaling, “dccoefficient”).
We may notice that the truncated filtering
approach can also be applied in the initial RGB space (with
the same local importance criteria). As the results show, a
significant improvement can be obtained only for small
noise amounts.
Further improvements are obtained by the better
NDMMF (Normalized Decorrelated Marginal Median
Filter). The idea of this filter is that normal random
variables, KL transformed, become independent (so
decorrelated). The point is to obtain from the original RGB
components distribution a normal distribution. It seems
two methods are at hand for this task: the histogram
specification technique [6] and the central limit theorem.
The histogram specification leads to a Gaussian shaped
distribution that includes “gaps” and is not invertible (after
the median filtering, another histogram specification must
be performed to obtain an approximation of the original
distribution).
According to the central limit theorem, the sum of
a few independent, identical distributed random variables
has approximately a normal distribution. The sum can be
introduced by a local linear invertible transform. This
transform replaces each pixel value within the analysis
window by the sum of the other 8 values in the window;
the result of a median filtering of these sums is assigned to
the central value and the inverse transform is performed.
Using this technique, some averaging properties are
introduced in the median filter; not surprisingly, this filter
performs very good in the RGB space too (without any
decorrelation).
H W
λλ
MCRE =
4. EXPERIMENTAL RESULTS
The tests were conducted on two classical 256 by
256 true color (24 bits per pixel) images, “fish” and
“waterfall”. Both were corrupted with impulsive (salt &
pepper) noise with 1% to 4% noise affected pixels in each
component, with no correlation in between, Gaussian
(normal) zero-mean, variance 25 additive noise and a
mixture of Gaussian and 2% impulsive noise.
The filtering efficiency for noise removal was
measured by the standard [4] NMSE (Normalized Mean
Square Error) and MCRE (Mean Chromaticity Error),
defined by:
H W
λλ
NMSE =
f (i , j ) − f ( i , j )
i =1 j =1
⋅100 and
H W
λλ
2
f (i , j )
2
i =1 j =1
i = 1 j =1
2
f ( i, j )
f ( i, j )
−
f (i , j )
f (i , j )
⋅ 100 .
HW
In the definitions above, H and W are the image dimensions
2
1
in pixels,
is the L norm and
is the L norm, f(i,j)
and f (i , j ) are the original and filtered values of the pixel
at location (i,j).
A subjective quality mark was accorded by visual
inspection of the results.
Tables 1 and 2 and Figures 1 to 4 summarize the
results of some typical filters for the two test images:
original noisy images (1), Marginal Median Filter (2), Vector
Median Filter (3), Modified Vector Median Filter with
weighted Euclidean distance by local range (4),
Decorrelated Marginal Median Filter by KL transform (5)
and DCT (6), Truncated Decorrelated Marginal Median
Filter by KL transform and median filtering of the
component with biggest local range (7), Normalized
Decorrelated Marginal Median Filter by KL transform (8),
DCT (9) and Normalized Marginal Median Filter in the RGB
space (10).
Table 1: Median filtering of “waterfall” image
Noise
1%
4%
Mixt.
Gauss
Noise
1%
4%
Mixt.
Gauss
Filter type 1
MCRE NMSE
1.049
15.365
4.072
31.049
19.330 29.495
18.604 21.396
Filter type 2
MCRE NMSE
1.880
9.316
2.146
9.697
8.013
13.476
8.048
13.422
Filter type 3
MCRE
NMSE
2.136
10.941
2.476
12.401
10.627
17.049
10.311
16.036
Filter type 4
MCRE
NMSE
2.056
10.636
2.264
11.272
10.352
16.791
10.307
16.102
Filter type 5
MCRE
NMSE
1.750
9.429
1.887
10.447
6.762
13.780
7.758
13.639
Filter type 6
MCRE NMSE
1.758
9.646
1.859
10.556
7.647
13.666
7.697
13.302
Filter type 7
MCRE NMSE
1.302
15.234
2.908
27.311
15.582
31.785
15.559
27.902
Filter type 8
MCRE
NMSE
1.754
8.521
2.541
10.180
7.802
13.276
7.700
12.703
Filter type 9
MCRE
NMSE
1.781
8.773
2.600
10.327
7.800
13.133
7.657
12.472
Filter type 10
MCRE
NMSE
1.831
8.434
2.790
9.503
8.168
12.976
8.021
12.606
Filter type 3
MCRE
NMSE
6.734
21.003
7.395
22.898
19.341
25.908
18.568
24.543
Filter type 4
MCRE
NMSE
6.567
20.678
7.007
21.969
19.060
25.593
18.527
24.495
Filter type 5
MCRE
NMSE
5.621
19.861
5.955
20.963
11.932
24.162
12.015
23.793
Table 2: Median filtering of “fish” image
Noise
1%
4%
Mixt.
Gauss
Filter type 1
MCRE NMSE
1.409
18.975
4.996
36.112
23.462 32.143
22.556 21.833
Filter type 2
MCRE NMSE
5.991
19.184
6.749
19.563
16.412
21.922
16.561
21.717
Noise
1%
4%
Mixt.
Gauss
Filter type 6
MCRE NMSE
5.635
20.288
6.022
21.481
11.270 22.891
11.188 22.618
Filter type 7
MCRE NMSE
5.749
33.682
7.381
42.107
21.182
52.292
21.330
48.563
Filter type 8
MCRE
NMSE
5.533
17.602
7.182
19.317
12.178
22.178
11.819
21.382
By analyzing the results, we can notice the
improvement obtained by the weighted distance technique
(MVMF) and the decorrelation based techniques (DMMF).
The NDMMF shows the best results in terms of NMSE and
visual accuracy of the image. The computational amount is
almost the same for all the filters; just the implementation of
the KL transform is more annoying, but the DCT can be
successfully used instead.
Filter type 9
MCRE
NMSE
5.541
18.034
7.246
19.849
11.773
21.152
11.264
20.614
Filter type 10
MCRE
NMSE
5.856
17.036
7.935
17.966
16.261
20.188
15.856
19.699
The presented color median filtering techniques
give encouraging results. With the same computational
amount as the VMF, or even less, the DMMF and the
NDMMF obtain significant better enhancement of the
images, both visual and measured.
Fig. 3: MVMF filtering of Fig. 1 with range weighting,
MCRE=2.264%, NMSE=11.272%
Fig. 1: 4% impulsive noise degraded “waterfall”,
MCRE=4.072%, NMSE=31.079%
Further developments may consider the
processing in some other color representation space that
either emphasize the color characteristics (like HSI - Hue
Saturation Intensity) or maximizes the relative intercolor
distances, enabling better color discrimination (the CIE
1964 U*V*W * representation [6]).
6. REFERENCES
Fig. 2: NDMMF filtering of Fig. 1 using DCT,
MCRE=2.600%, NMSE=10.327%
5. CONCLUSIONS
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Englewood Cliffs NJ, Prentice Hall Intl., 1989
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