International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
The Component Median Filter for Noise Removal in
Digital Images
Harish 1 and M.R.Gowtham2
1
Department of Electronics and Communication Engineering,
Priyadarshini College of Engineering, NELLORE – 524 004, A. P., INDIA
2
Department of Electronics and Communication Engineering,
Priyadarshini College of Engineering, NELLORE – 524 004, A. P., INDIA
Abstract
The objective of the project is to develop a
Component filtering algorithm to reconstruct noise
affected images. The purpose of the algorithm is to
remove noise from a signal that might occur
through transmission of an image. This algorithm
can be applied to one dimensional as well as two
dimensional signals. In the component Median
Filter each scalar component is treated
independently. A filter mask is placed over a point
in the signal. For each component of each point
under the mask, a single median component is
determined. These components are then combined
to form to a new point, which is then used to
represent the point in the signal studied. When
working with color images, however, this filter
regularly out performs the simple Median filter.
When noise affects a point in a gray scale image,
this result is called “salt and pepper” noise. In color
images, this property of “salt and pepper” noise is
typical of noise models where only one scalar value
of point is affected. For this noise model, the
Component Median Filter is more accurate than the
Simple Median Filter.
Keywords— Image Enhancement, Component
Median Filter, Mean Square Error, Root Mean
Square Error, Peak to Signal Noise Ratio, Noise.
1. INTRODUCTION
In image processing it is usually necessary to perform
high degree of noise reduction in an image before
performing higher-level processing steps, such as
edge detection. In software, a smoothing filter is used
to remove noise from an image. Each pixel is
represented by three scalar values representing the
red, green and blue chromatic intensities. At each
pixel studied, a smoothing filter takes into account
the surrounding pixels to design a move accurate
version of this pixel. By taking neighboring pixels
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into considerations, extreme ‘noisy’ pixels can be
replaced. However, outlier pixels may represent in
corrupted fine details, which may be lost due to the
smoothing process.
The median filter [2] is a non-linear digital
filtering technique, often used to remove noise from
images or other signals. The idea is to examine a
sample of the input and decide if it is representative
of the signal. This is performed using a window
consisting of an odd number of samples. The values
in the window are sorted into numerical order; the
median value, the sample in the center of the
window, is selected as the output. The oldest sample
is discarded, a new sample acquired, and the
calculation repeats. The Component Median Filter
defined in (1) also relies on the statistical median
concept.
CMF(x1…,xN)=
( 1 ….
( 1 …
( 1 …
)
)
)
(1)
When transferring an image, sometimes transmission
problems cause a signal to spike, resulting in one of
the three point scalars transmitting an incorrect value.
This type of transmission error is called “salt and
pepper” noise due to the bright and dark spots that
appear on the image as a result of the noise. The ratio
of incorrectly transmitted points to the total number
of points is referred to as the noise composition of the
image. The goal of a noise removal filter is to take a
corrupted image as input and produce an estimation
of the original with no foreknowledge of the noise
composition of the image.
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2.
NOISE REMOVAL IN DIGITAL
IMAGES
The input to the filter is 3*3 sized mask noisy
images. In the next step it is applied to a respective
filter. By simulation we get the output of the filter is
noise eliminated image.
from the data instead of adding artificial
noise to the data.
4) Speckle: It is a type of multiplicative noise.
It is added to the image using the equation
J=I+n*I, where n is uniformly distributed
random noise with mean 0 and variance V.
Noise can generally be grouped in two classes:
1) independent noise, and
2) Noise which is dependent on the image data.
Image independent noise can often be described by
an additive noise model, where the recorded image
f(i,j) is the sum of the true image s(i,j) and the noise
n(i,j):
( , ) = ( , ) + ( , )(2)
Fig.1: Block Diagram
2.1 Noise Types
In common use the word noise means
unwanted sound or noise pollution. In electronics
noise can refer to the electronic signal corresponding
to acoustic noise (in an audio system) or the
electronic signal corresponding to the (visual) noise
commonly seen as 'snow' on a degraded television or
video image. In signal processing or computing it can
be considered data without meaning; that is, data that
is not being used to transmit a signal, but is simply
produced as an unwanted by-product of other
activities. In Information Theory, however, noise is
still considered to be information. In a broader sense,
film grain or even advertisements in web pages can
be considered noise.
Noise can block, distort, or change the
meaning of a message in both human and electronic
communication. In many of these areas, the special
case of thermal noise arises, which sets a
fundamental lower limit to what can be measured or
signaled and is related to basic physical processes at
the molecular level described by well known simple
formulae.
2.1.1
Types of noise
1) Salt & Pepper: As the name suggests, this
noise looks like salt and pepper. It gives the
effect of "On and off" pixels.
2) Gaussian: This is Gaussian White Noise. It
requires mean and variance as the additional
inputs.
3) Poisson: Poisson noise is not an artificial
noise. It is a type of noise which is added
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The noise n(i,j) is often zero-mean and described by
its variance . The impact of the noise on the image is
often described by the signal to noise ratio (SNR),
which is given by
=
=
− 1(3)
Where and
are the variances of the true image
and the recorded image, respectively.
In many cases, additive noise is evenly
distributed over the frequency domain (i.e. white
noise), whereas an image contains mostly low
frequency information. Hence, the noise is dominant
for high frequencies and its effects can be reduced
using some kind of low pass filter. This can be done
either with a frequency filter or with a spatial filter.
(Often a spatial filter is preferable, as it is
computationally less expensive than a frequency
filter.)
In the second case of data dependent noise,
(e.g. arising when monochromatic radiation is
scattered from a surface whose roughness is of the
order of a wavelength, causing wave interference
which results in image speckle), it can be possible to
model noise with a multiplicative, or non-linear,
model. These models are mathematically more
complicated, hence, if possible, the noise is assumed
to be data independent.
2.2 Salt and Pepper Noise
Another common form of noise is data drop-out
noise (commonly referred to as intensity spikes,
speckle or salt and pepper noise). Here, the noise is
caused by errors in the data transmission. The
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corrupted pixels are either set to the maximum value
(which looks like snow in the image) or have single
bits flipped over. In some cases, single pixels are set
alternatively to zero or to the maximum value, giving
the image a `salt and pepper' like appearance.
Unaffected pixels always remain unchanged. The
noise is usually quantified by the percentage of pixels
which are corrupted.
3.
IMAGE ENHANCEMENT IN THE
SPATIAL DOMAIN
The principal objective of enhancement is to process
an image so that the result is more suitable than the
original image for a specific application.
Image enhancement approaches fall into two
broad categories: spatial domain methods[1] and
frequency domain methods. The term spatial domain
refers to the image plane itself, and approaches in this
category are based on direct manipulation of pixels in
an image. Frequency domain processing techniques
are based on modifying the Fourier transform of an
image. Enhancement techniques based on various
combinations of methods from these two categories
are not unusual. Visual evaluation of image quality is
a highly subjective process, thus making the
definition of a “good image” an elusive standard by
which to compare algorithm performance.
The term spatial domain refers
aggregate of pixels composing an image.
domain methods are procedures that operate
on these pixels. Spatial domain processes
denoted by the expression
g (x,y)=T[f (x,y)]
to the
Spatial
directly
will be
(4)
Where f (x,y) is the input image, g (x,y) is
the processed image, and T is an operator on f,
defined over some neighborhood of (x,y). In addition,
T can operate on a set of input images, such as
performing the pixel-by- pixel sum of K images for
noise reduction.
The principle approach in defining a
neighborhood about a point (x,y) is to use a square or
rectangular sub image area centered at (x,y). The
center of the sub image is moved from pixel to pixel
starting, say, at the top left corner. The operator T is
applied at each location (x,y) to yield the output, g, at
that location. The process utilizes only the pixels in
the area of the image spanned by the neighborhood.
Although other neighborhood shapes, such as
approximations to a circle, sometimes are used,
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square and rectangular arrays are by far the most
predominant because of their ease of implementation.
The simplest form of T is when the
neighborhood is of size of 1X1 (that is, a single
pixel). In this case, g depends only on the value of f
at (x,y), and T becomes a gray level ( also called an
intensity or mapping) transformation function of the
form
s = T(r)
(5)
Where, for simplicity in notation, r and s are
variables denoting, respectively, the gray level of f
(x,y) and g (x,y) at any point (x,y). For example, if
T(r) has the form shown in Fig.(a), the effect of this
transformation would be to produce an image of
higher contrast than the original by darkening the
levels below m and brightening the levels above m in
the original image. In this technique, known as
contrast stretching, the values of r below m are
compressed by the transformation function into a
narrow range of s, toward black. The opposite effect
takes place for values of r above m. In the limiting
case shown in Fig.(b), T (r) produces a two-level
(binary) image. A mapping of this form is called a
thresholding function. Some fairly simple, yet
powerful, processing approaches can be formulated
with
gray-level
transformations.
Because
enhancement at any point in an image depends only
on the gray level at that point, techniques in this
category often are referred to as point processing.
Larger neighborhoods allow considerable more
flexibility. The general approach is to use a function
of the values of f in a predefined neighborhood of (x,
y) to determine the values of g at (x, y). One of the
principle approaches in this formulation is based on
the use of so-called masks (also referred to as filters,
kernels, templates, or windows). Basically, a mask is
a small (say, 3X3) 2-D array, in which the values of
mask coefficients determine the nature of the process,
such as image sharpening. Enhancement techniques
based on this type of approach often are referred to as
mask processing or filtering.
3.1 Local enhancement
The two histogram processing methods discussed in
the previous two sections are global in the sense that
pixels are modified by a transformation function
based on the gray-level distribution over an entire
image. Although this global approach is suitable for
overall enhancement, it is often necessary to enhance
details small over small areas. The number of pixels
in these areas may have negligible influence on the
computation of a global transformation, so the use of
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this type of transformation does not necessarily
guarantee the desired local enhancement. The
solution is to devise transformation functions based
on the gray-level distribution or other properties in
the neighborhood of every pixel in the image. We
discuss local histogram processing here for the sake
of clarity and continuity.
The histogram processing techniques previously
described are easily adaptable to local enhancement.
The procedure is to define a square or rectangular
neighborhood and move the centre of this area from
pixel to pixel. At each location, the histogram of the
points in the neighborhood is computed and either a
histogram equalization or histogram specification
transformation function is obtained. This function is
finally used to map the gray level of the pixel
centered in the neighborhood. The centre of the
neighborhood region is then moved to an adjacent
pixel location and the procedure is repeated. Since
only one new row or column of the neighborhood
changes during a pixel-to-pixel translation of the
region, updating the histogram obtained in the
previous location with the new data introduced at
each motion step is possible. This approach has
obvious advantages over repeatedly computing the
histogram over all pixels in the neighborhood region
each time the region is moved pixel location. Another
approach often used to reduce computation is to
utilize non-overlapping regions, but this method
usually produces an undesirable checkerboard effect.
3.2 Enhancement
Operations
Using
Arithmetic/Logic
Arithmetic/Logic operations involving images are
performed on a pixel-by-pixel basis between two or
more images (this excludes the logic operation NOT,
which is performed on a single image). As an
example, subtraction of two images results in a new
image whose pixel at coordinates (x, y) is the
difference between the pixels in that same location in
the two images being subtracted. Depending on the
hardware and/or software being used, the actual
mechanics
of
implementing
arithmetic/logic
operations can be done sequentially, one pixel at a
time, or in parallel, where all operations are
performed simultaneously.
Logic operations similarly operate on a
pixel-by-pixel basis. We need only be concerned with
the ability to implement the AND, OR and NOT
logic operators because these three operators are
functionally complete. In other words, any other logic
operator can be implemented by using only these
three basic functions. When dealing with logic
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operations on gray-scale images, pixel values are
processed as strings of binary numbers. For example,
performing the NOT operation on a black, 8-bit pixel
(a string of eight 0’s) produces a white pixel (a string
of eight 1’s). Intermediate values are processed the
same way, changing all1’s to 0’s and vice versa.
Thus, the NOT logic operator performs the same
function as the negative transformation. The AND or
OR operations are used for masking; that is, for
selecting sub images in an image. In the AND and
OR image masks, light represents a binary 1 and dark
represents a binary 0. Masking sometimes is referred
to as region of interest (ROI) processing. In terms of
enhancement, masking is used primarily to isolate an
area for processing. This is done to highlight that area
and differentiate it from the rest of the image. Logic
operations also are used frequently in conjunction
with morphological operations.
Of the four arithmetic operations, subtraction and
addition (in that order) are the most useful for image
enhancement. We consider division of two images
simply as multiplication of one image by the
reciprocal of the other. Aside from the obvious
operation of multiplying an image by a constant to
increase its average gray level, image multiplication
finds use in enhancement primarily as a masking
operation that is more general than the logical masks
discussed in the previous paragraph. In other words,
multiplication of one image by another can be used to
implement gray-level, rather than binary, masks.
4. COMPONENT MEDIAN FILTER
The Component Median Filter [3] also relies on the
statistical median concept. In the Simple Median
Filter, each point in the signal is converted to a single
magnitude. In the Component Median Filter each
scalar component is treated independently. A filter
mask is placed over a point in the signal. For each
component of each point under the mask, a single
median component is determined.
These components are then combined to
form a new point, which is then used to represent the
point in the signal studied. When working with color
images, however, this filter regularly outperforms the
Simple Median Filter. When noise affects a point in a
grayscale image, the result is called “salt and pepper”
noise. In color images, this property of “salt and
pepper” noise is typical of noise models where only
one scalar value of a point is affected. For this noise
model, the Component Median Filter is more
accurate than the Simple Median Filter. The
disadvantage of this filter is that it will create a new
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signal point that did not exist in the original signal,
which may be undesirable in some applications.
( , … . . ,
)= ∑
(6)
The linear MSE estimator is the estimator achieving
minimum MSE among all estimators of the form AY
+ b. If the measurement Y is a random vector, A is a
matrix and b is a vector. (Such an estimator would
more correctly be termed an affine MSE estimator,
but the term linear estimator is widely used.
5.2 Root Mean Square Error (RMSE)
4.1 Algorithm of Component Median Filter
Step 1: Select a noisy image.
Step 2: If the noisy image is color, separate each
plane using MATLAB commands.
Step 2(a): Each scalar component is treated in
dependently.
Step 3: Generate zero arrays around an image based
on image mask size using pad array command.
Step 4: select 3 * 3 masks from an image and the
mask should be odd sized
Step 5: Then sort the pixel values within the mask in
ascending order.
Step 6: For each component of each point under the
mask a single median component is determined.
Step 7: These components are then combined to form
a new point which is then used to represent the point
in the signal studied.
Step 8: Restore the output image and calculate the
Mean Square Error and Peak Signal to Noise Ratio
value.
5. ESTIMATION OF QUALITY OF
RECONSTRUCTED IMAGES.
In mathematics, the root mean square (abbreviated
RMS or rms), also known as the quadratic mean, is a
statistical measure of the magnitude of a varying
quantity. It is especially useful when variations are
positive and negative, e.g., sinusoidal. RMS is used
in various fields, including electrical engineering; one
of the more prominent uses of RMS is in the field of
signal amplifiers.
It can be calculated for a series of discrete
values or for a continuously varying function. The
name comes from the fact that it is the square root of
the mean of the squares of the values. It is a special
case of the generalized mean with the exponent p = 2.
The RMS value of a set of values (or a
continuous-time waveform) is the square root of the
arithmetic mean (average) of the squares of the
original values (or the square of the function that
defines the continuous waveform).
{ ,
In the case of a set of n values,
, … , }the RMS value is given by:
5.1 Mean Square Error (MSE)
+
=
In statistics and signal processing, a Mean
square error (MSE) estimator describes the approach
which minimizes the mean square error (MSE),
which is a common measure of estimator quality.
−
∫ [ ( )]
=
(9)
And the RMS for a function over all time is
(7)
Where the expectation is taken over both X
(8)
The corresponding formula for a continuous function
(or waveform) f(t) defined over the interval
≤
≤ is
Let X is an unknown random variable, and
let Y be a known random variable (the measurement).
An estimator
is any function of the
measurement Y, and its MSE is given by
=
+… … … +
= lim
→
∫ [ ( )]
(10)
and Y.
The MSE estimator is then defined as the
estimator achieving minimal MSE. In many cases, it
is not possible to determine a closed form for the
MMSE estimator. In these cases, one possibility is to
seek the technique minimizing the MSE within a
particular class, such as the class of linear estimators.
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The RMS over all time of a periodic function is
equal to the RMS of one period of the function. The
RMS value of a continuous function or signal can be
approximated by taking the RMS of a series of
equally spaced samples.
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
5.3 Peak to Signal Noise Ratio (PSNR)
6.1 Images
The phrase peak signal-to-noise ratio, often
abbreviated PSNR, is an engineering term for the
ratio between the maximum possible power of a
signal and the (or codec type) and same content.
It is most easily defined via the mean square
error (MSE) which for two m×n monochrome images
I and K where one of the images is considered a
noisy approximation of the other is defined as:
=
∑
∑
[ ( , ) − ( , )]
Fig.2: Original image.
(11)
The PSNR is defined as:
= 10. log
=20. log
√
(12)
Here, MAXI is the maximum possible pixel
value of the image. When the pixels are represented
using 8 bits per sample, this is 255. More generally,
when samples are represented using linear PCM with
B bits per sample, MAXI is 2B−1. For color images
with three RGB values per pixel, the definition of
PSNR is the same except the MSE is the sum over all
squared value differences divided by image size and
by three.
Typical values for the PSNR in lossy image
and video compression are between 30 and 50 dB,
where higher is better. Acceptable values for wireless
transmission quality loss are considered to be about
20 dB to 25 dB. When the two images are identical
the MSE will be equal to zero, resulting in an infinite
PSNR.
6. SIMULATION AND RESULTS
Fig.3: Noise corrupted image.
Fig.4: Noise removed image.
6.2 Results
The results are obtained by calculating the Mean
Square Error performance, Peak Signal to Noise
Ratio performance and Peak Signal to Noise Ratio
performance. These values are to be obtained for
Gaussian noise, salt and pepper noise, Speckle noise.
Gaussian noise corrupted images have better PSNR.
The original image is generally is to be corrupted
by adding the noise to it in order to get the noisy
image. Filtering technique is applied to the noisy
image to obtain the noise eliminated image.
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7. Conclusion
We have introduced Component Median
Filter for removing impulse noise from images and
shown how they compare to four well-known
techniques for noise removal. In the comparison of
noise removal filters, it was concluded that
Component Median Filter performed the best overall
noise compositions tested. By using the quantitative
analysis we proved that most of the in noise digital
images is removed with component median filter.
8. References
Fig.5: Mean Square Error performance.
Fig.6: Peak Signal to Noise Ratio performance.
[1] R.Nathan,” Spatial Frequency Filtering,” in
Picture Processing and Psychopictrotics,B.S
.Lipkin and A.Roswnfeld,Eds., Academic
Press, New York.
[2] G.A.Mastin ,” Adaptive Filters for Digital
Image Noise Smoothing : An Evaluation,
“Computer Vision, Graphics , and Image
Processing.
[3] T.A.Nodes and N.C.Gallagher , Jr.,” Median
Filters : Some Manipulations and Their
Properties,”IEEE Trans. Acoustics, Speech,
and Signal Processing.
[4] “Digital image processing” by Rafael C.
Gonzalez and Richard E.Woods.
[5] “Digital image restoration” by Andrews and
Hunt.
[6] “Digital image processing” by B.Chanda
and D.Dutta Majumder.
Fig.7: Root Mean Square Error to Noise Ratio
performance
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