padhu.ar newconference
DESCRIPTION
image processing. image denoisingTRANSCRIPT
An Efficient Denoising Approach for Random-Valued
Impulse Noise using PDE Method
R.Padmanaban1, S.Saravanakumar
2
1 PG scholar,
2 Assistant Professor
1,2Department of Electronics and Communication Engineering
Anna University of Technology, Coimbatore
Academic Campus, Jothipuram, Coimbatore-641047.
[email protected], [email protected]
Abstract: This paper is concerned about a new filtering scheme
based on contrast enhancement for removing the random
valued impulse noise. An efficient PDE (partial differential
equation) based algorithm for removal of random-valued
impulse noise from corrupted images is proposed in this paper.
The function for increasing the difference between noise-free
and noisy pixels is introduced. It denotes the number of
homogeneous pixels in a local neighborhood and is significantly
different for edge pixels, noisy pixels, and interior pixels. The
controlling speed function and the controlling fidelity function
is redefined to depend on noise and noise free pixels. According
to that two controlling functions, the diffusion and fidelity
process at edge pixels, noisy pixels, and interior pixels can be
selectively carried out. Furthermore, a class of second-order
improved and edge-preserving PDE denoising models is
proposed based on the two new controlling functions in order
to deal with random valued impulse noise reliably. Two
controlling functions are extended to automatically other PDE
models. Extensive simulation results exhibit that the proposed
method significantly outperforms many other well-known
techniques.
Keywords— Anisotropic diffusion, diffusion speeds, fidelity
process, partial differential equation (PDE)-based image
denoising, random-valued impulse noise.
I. INTRODUCTION
Impulse noise is a common kind of signal noise that
can significantly corrupt images. The impulse noise can be
classified either as salt and pepper with noisy pixels taking
either maximum or minimum value, or as random-valued
impulse noise. A class of widely used nonlinear digital
filters is median filters (MED). Median filters are known for
their capability to remove impulse noise while preserving
the edges. The main drawback of a standard median filter is
that it is effective only for low-noise densities [1]. The
switching scheme concept [2] or the two-stage method [3] is
frequently used strategy for enhancing the performance of
impulse noise filters. The basic idea of the methods is that
the noisy pixels are detected first and filtered afterward,
whereas the undisturbed pixels are left unchanged. There are
four two-stage filters (or the switching schemes) that are
worth mentioning that will used later for comparison. The
three-state median (TSM) [4] and the adaptive center-
weighted median (ACWM) [5] are two widely used filters.
The TSM filter uses the median and the CWM both for
detection and reduction. The ACWM uses the comparison
of CWMs and adaptive thresholds for detection and the
simple median for reduction that consistently works well in
suppressing both types of impulses. The Luo filter [6] is an
efficient detail-preserving two-stage method that requires no
previous training. The genetic programming (GP) filter
[7]employs two cascaded detectors for detection and two
corre-sponding estimators for reduction. The first detector
identifies the majority of noisy pixels. The second detector
searches for the remaining noise missed by the first detector,
usually hidden in image details or with amplitudes close to
its local neigh-borhood. The core of two-stage filters is the
impulse detection process. In case of random-valued
impulse noise, the detection of an impulse is relatively more
difficult in comparison with salt-and-pepper impulse noise
[8]. Despite decades of research in this area, the effective
detection for random-valued impulse noise with high-noise
levels is still an open problem. Let us now present the
purpose of this paper. In this paper, we focus on the removal
of random-valued impulse noise by using partial differential
equation (PDE) methods. PDE-based image processing
methods have been studied extensively as a useful tool for
image denoising and enhancement. The basic idea of PDE-
based methods is to deform a given image with a PDE and
obtain the desired result as the solution of this PDE with the
noisy image as initial conditions. By using PDEs, one can
model the images in a continuous domain, see existing
methods in a different viewpoint, and combine multiple
algorithms together. Furthermore, high accuracy and
stability can be naturally obtained with the help of the
available extensive research on numerical analysis.
Although there have been different PDEs denoising models
developed in the past two decades, as briefly described in
Section II, little has been done regarding anisotropic
diffusion for filtering impulse noise. Here, we propose a
class of second-order improved and edge-preserving PDE
denoising models based on two new controlling functions in
order to deal with random-valued impulse noise reliably.
We will redefine the controlling speed function and the
controlling fidelity function from a completely different
point of view. We introduce the notion of ENI (the
abbreviation for “edge pixels, noisy pixels, and interior
pixels”) to our controlling speed function. The ENI can be
used to distinguish edge pixels, noisy pixels, and interior
pixels, and can be calculated by utilizing the local
neighbourhood statistics based on the number of the pixels
with similar intensity (or called homogeneous pixels). Our
controlling speed function is defined to depend on ENI.
Thus, the diffusion at edge pixels, noisy pixels, and
interior pixels is made with various speeds according to our
controlling function. We also introduce the ENI to the
controlling fidelity function. A selective fidelity process can
be carried out according to the new controlling fidelity
function, to reduce the smoothing effect near edges. We test
the proposed PDE denoising models on five standard
images degraded by random-valued impulse noise with
various noise levels. We compare our PDEs models with the
related PDE models and other special filtering methods for
random-valued impulse noise, including MED, ACWM,
TSM, Luo, and GP.
This paper is organized as follows. In Section II, we
briefly describe related previous PDE denoising models. In
Section III, we will first describe the ENI of an image, our
controlling speed function, and controlling fidelity function
in detail, respectively. Then, we present a class of second-
order improved and edge-preserving PDE denoising models
for random-valued impulse noise removal based on our two
controlling functions. Section IV shows the experiments and
discussion, and is followed by conclusion in Section V.
II.REVIEW OF PDE METHOD
Many different PDE models have been proposed for
image denoising in the past years. Since it is not feasible to
discuss all the models here, we briefly describe some
representative PDE models that are related to our study. The
original PDE filtering model, proposed by Witkin [9], is the
linear heat equation that diffuses in all directions and
destroys edges. To overcome this problem, many
researchers have corrected this limitation from various
points of view, mainly including:
1) from controlling the speed of the diffusion;
2) from controlling the direction of the diffusion;
3) from adding a fidelity term; and
4) from their combinations.
Perona and Malik (PM) were the first to try such an
approach through controlling the speed of the diffusion and
proposed a nonlinear adaptive diffusion process [10],
termed as anisotropic diffusion. The PM nonlinear diffusion
equation is of the form
(1)
Where u(x,y;t) is the evolving image derived from the
original image at ‘t’ time, and “ ” and “div” are
the gradient and divergence operators.
Catté et al. [11] improved the controlling speed function g(.)
by using instead of and proposed a
selective smoothing model
(2)
Alvarez et al. [15] have made the significant improvements
through controlling the diffusion direction and proposed the
degenerate diffusion PDE model
(3)
Obviously, the controlling speed function g(.) and the
controlling fidelity coefficient have played an important role
in the performances of PDE denoising models.
Unfortunately, the previous controlling speed functions and
the controlling fidelity coefficient have some drawbacks,
especially when they are used to remove impulse noise.
III. CLASS OF SECOND-ORDER EDGE-PRESERVING PDE
DENOISING MODELS
In this section, the ENI of an image is defined that can
be used to distinguish edge pixels, noisy pixels, and interior
pixels. Then, the controlling speed function g(.) and the
controlling fidelity function are redefined. Based on the
two new controlling functions, we present a class of second-
order edge-preserving PDE denoising models for random-
valued impulse noise removal.
A. Definition of the ENI of an Image
The ENI denote the number of homogeneous pixels in
a local neighbourhood, so the ENI is significantly different
for edge pixels, noisy pixels, and interior pixels.
Subsequently, we describe in detail how to calculate the
ENI of an image.
Let be the location of a pixel under
consideration,
(4)
Denote the neighbour points of the central pixel p with
window size of (2w+1)× (2w+1)(w>0) while let be
a set of neighbour pixels centred at ‘p’ but exclude ‘p’ for
each q ε , defined d(p,q) as the absolute difference in
intensity of the pixels between p and q, i.e,
(5)
Then, the gray intensity of each q ε is classified into
two groups by a predefined threshold T.
(6)
Finally, the ENI of the pixel is defined as
(7)
B. Our Controlling Speed Function
The ENI of an image is significantly different for edge
pixels, noisy pixels, and interior pixels. The ENI for impulse
noise pixels is minimum the ENI for edge pixels takes
intermediate value, and the ENI for interior pixels is
maximum, so it is reasonable that the controlling speed
function is defined to depend on ENI. The shape of the
controlling speed functions should be like either Fig. 3(a) or
(b) in order to achieve reduced diffusion at and around edge
pixels while allowing diffusion at impulse-corrupted noisy
pixels and interior pixels. Here, we redefine the controlling
speed function as
(8)
Like the previous controlling speed functions, the value of
our controlling speed function is between 0 and 1, i.e.,
According to the new controlling speed
function the diffusion speed at interior pixels.
Moreover, the values of at
noisy pixels are larger than these at edges, so the diffusion
speeds at noisy pixels are faster than these at edges. Thus,
noisy pixels can be effectively removed while preserving
edges very well.
C. Our Controlling Fidelity Function
Similar to our controlling speed function we
also introduce the ENI to the fidelity term. We propose a
controlling fidelity function as
(9)
The function is monotone increasing and its value range is
[0,0.5] The values of (.) at noise pixels are minimum or
near to zero, at edges pixels take intermediate value, and at
interior pixels are maximum and close to 0.5. Thus,
according to our controlling function (.) the fidelity
process at noisy pixels is inhibited, while fidelity processes
at edge pixels and interior pixels are encouraged, which are
also desired.
D. Class of Second-Order PDE Denoising Models Based
on Our Two Controlling Functions
In this section, we introduce our two controlling
functions to the widely used PM model (1), the SDD model
(10), and the TVD model (14), and propose a class of
second-order edge preserving PDE denoising models. The
new PM model (NPM), the new SDD model (NSDD), and
the new TVD model (NTVD) are expressed, respectively, as
Equations (10)–(11) use the new controlling speed function
and the new controlling fidelity function with
better performance. Therefore, we would expect to see
something that shows the uniqueness of our models. In fact,
the experimental results, as shown later, demonstrate the
performance of these models.
IV. EXPERIMENTAL RESULTS AND DISCUSSION
The filtered results of the PDE models are relative to
the parameters in PDE models, and discrete time step and iteration time. In this section, we first discuss the
choices of the parameters w and T our PDE models. Then,
for evaluating the real performance of our PDE models, we
compare our PDE models with the previously corresponding
PDE models. Furthermore, we compare with other filters,
including MED, ACWM, TSM, Luo, and GP that are
capable of removing random-valued impulse noise. The
performances of various methods are quantitatively
measured by the peak SNR (PSNR). In all implementation,
For comparison, we take the same discrete schemes in the
compared PDE pair. Here the 8-bit 512× 512 standard
images: Lena, Peppers, Boat, and Airplane are chosen as
tested images that have distinctly different features and are
corrupted by random-valued impulse noise with various
noise levels.
A. Choices of the Parameters
Like the parameter in the previous controlling speed
function g(.) defined in (2) and (3), there is also not the
explicit formula or a method to determine the parameters
and in (8) and (9). They are chosen based on the better
performance by trial. For showing the filtered results with
respect to the parameters and the noisy Lena with 20%
random-valued impulse noise is tested by using various
window sizes and thresholds In Fig. 5, we plot the
PSNR values of the restored images by our PDEs for
various window sizes and thresholds .Where T is
variable from 5 to 60 pixels with an increment of 5, and
=2 and =3. In this test, we use the appropriate iteration
time of each model, that is, respectively, 40, 5, and 300 in
NPM, PPDE, and NTVD, as shown later. According to our
own experience in this method, generally, the higher the
noise level is, the larger value is and the smaller value .
The appropriate value of is 2 or 3 and the appropriate
value of is somewhere between 10 and 35.
Fig.1 . Restoration results by the different filters. (a) Noisy free Peppers
image (b) Noisy peppers image corrupted by 30% random-valued impulse noise. (c) MED (7×7) filter. (d) TSM (7×7) filter. (e) ACWM filter. (f) Luo
filter. (g) GP filter. (h) PPDE model.
TABLE I
COMPARISONS OF RESTORATION RESULTS IN PSNR (in Decibels)
OBTAINED BY VARIOUS FILTERS
40% Random valued impulse noise
Filters Lena Pepper Boat Airplane
MED 20.83 21.34 21.13 20.48
TSM 22.15 22.87 21.96 21.67
ACWM 22.91 23.07 22.47 22.03
LUO 23.37 23.65 23.32 22.98
GP 24.78 24.87 23.78 23.73
PPDE 26.56 26.89 25.63 24.86
50% Random valued impulse noise
Filters Lena Pepper Boat Airplane
MED 18.77 19.63 18.76 18.47
TSM 19.95 20.03 19.24 19.05
ACWM 20.42 21.41 20.02 19.79
LUO 21.14 21.97 20.96 21.38
GP 22.43 23.52 22.33 21.53
PPDE 24.18 25.37 24.84 23.69
60% Random valued impulse noise
Filters Lena Pepper Boat Airplane
MED 15.59 16.27 16.48 17.39
TSM 15.96 17.06 16.89 17.84
ACWM 16.67 17.85 17.58 18.57
LUO 18.04 18.45 18.02 18.83
GP 19.63 19.97 18.93 19.06
PPDE 21.78 20.86 20.29 19.97
(a) (b)
(c) (d)
(e) (f)
(g)
Fig.2 Restoration results by the different filters. (a) Noisy free lena image (b) Noisy lena image corrupted by 50% random-valued impulse noise. (c)
MED (7×7) filter. (d) TSM (7×7) filter. (e) Luo filter. (f) GP filter. (g)
PPDE model.
B. Comparison With Other Filters
Here, we take Lena, Peppers, Boat, and Airplane,
corrupted by random-valued impulse noise with three high
noise levels-40%, 50%, and 60% as test images. From Table
I, one can find that the values of PSNR by PPDE and NTVD
are better than those by NPM. One can also find that the
PPDE requires fewer iteration times as compared with
NTVD. Therefore, here apply our PPDE to the these images
and compare with other special filters for random-valued
impulse noise, including MED, TSM, ACWM, Luo, and
GP. In this test, the window size, threshold, and iteration
time in our PPDE are chosen, respectively, as
The parameters in the TSM, ACWM, and Luo filters
are chosen according to the suggestions given by the authors
[4]–[6]. The GP filter has no parameter [7]. Table II lists the
PSNR values of all methods for Lena, Peppers, Boat, and
Airplane corrupted by random-valued impulse noise with
40%, 50%, and 60% noise levels. Generally, the PSNR
performance of the proposed PDE filter is comparable to
those of the Luo and GP filters, but the proposed PDE filter
performs better than MED, TSM, and ACWM.
Furthermore, a subjective visual result of the noise
reduction is presented in Fig. 1(a) is the noisy Peppers
image with 30% random-valued impulse noise. The
restoration results in Fig. 1(a) obtained by MED, TSM,
ACWM, Luo, and GP and our PPDE are given in Fig. 1(b)–
(h). The enlarged details of the noise-free and the filtered
results produced by the several filters are given in Fig. 2(a)–
(g), respectively. The desired visual result is produced by
our PPDE filter. Obviously, our PPDE can preserve edges
better as compared with MED, TSM, ACWM, Luo, and GP.
V. CONCLUSION
We have considered PDE-based image denoising
algorithms for random-valued impulse noise. This paper has
redefined the controlling speed function and the controlling
fidelity function. The diffusion and fidelity process at edge
pixels, noisy pixels, and interior pixels is selectively carried
out according to our two controlling functions to remove
random-valued impulse noise effectively while preserving
edges well. Furthermore, we present a class of second-order
edge-preserving PDE denoising models based on the two
new controlling functions. We test the proposed PDE
models on five standard images corrupted by random-valued
impulse noise with various noise levels and compare with
the related second-order PDE models and the other filtering
methods, including MED, TSM, ACWM, Luo, and GP. The
experimental results have demonstrated the performance of
our PDEs. In addition, the new controlling functions can be
extended automatically to any other PDE denoising models
such as the coupled PDEs [14]. Applications of the PDE
models are in a broad range of image processing tasks such
as inpainting, image segmentation, and skeletonization, and
so on. Our two controlling functions can also be applied to
these PDE models, which is a progress on PDE-based image
processing. In this section the architecture of proposed CSD
CS shift-and-add multiplier is presented shown in figure 4.
Our architecture works on the concept of shifting and
adding of partial products to realize the multiplied result.
The functions of different blocks are explained below.
REFERENCES
[1] K. S. Srinivasan and D. Ebenezer (2007), “A new
fast and efficient decisionbased algorithm for
removal of high- density impulse noises,” IEEE.
[2] T. Sun and Y. Neuvo (1994), “Detail-preserving
median-based filters in image processing,” Pattern
Recognit. Lett.
[3] R. H. Chan, C.-W. Ho, and M. Nikolova (2005),
“Salt-and-pepper noise removal by median-type
noise detectors and detail-preserving
regularization,” IEEE.
[4] T. Chen, K.-K. Ma, and L.-H. Chen (1994), “Tri-
state median-based filters in image denoising,”
IEEE.
[5] T. Chen and H.Wu (2009), “Adaptive impulse
detection using center-weighted median filters,”
IEEE.
[6] W. Luo (2007), “An efficient algorithm for the
removal of impulse noise from corrupted images,”
Int. J. Electron. Communication.
[7] N. Petrovic and V. Crnojevic (2008), “Universal
impulse noise filter based on genetic
programming,” IEEE.
[8] U. Ghanekar, A. K. Singh, and R. Pandey (2010),
“A contrast enhancement based filter for removal
of random valued impulse noise,” IEEE.
[9] A. P.Witkin (1983), “Scale-space filtering,” in
Proc. 8th Int. Joint Conf. Artif. Intell.
[10] P. Perona and J. Malik (1990) , “Scale-space and
edge detection using anisotropic diffusion,” IEEE.
[11] F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll
(1992), “Image selective smoothing and edge-
detection by nonlinear diffusion,” SIAM J. Numer.
[12] Y.You and M. Kaveh (2000), “Fourth-order partial
differential equations for noise removal,” IEEE.
[13] M. J. Black, G. Sapiro, D. H. Marimont, and D.
Heeger (1998) , “Robust anisotropic diffusion,”
IEEE.
[14] Y. Chen, C. Barcelos, and B. Mair (2001),
“Smoothing and edge detection by time-varying
coupled nonlinear diffusion equations,” Compt.
Vis. Image.
[15] L. Alvarez, P. Lions, and J. M. Morel (1992),
“Image selective smoothing and edge-detection by
nonlinear diffusion. II,” SIAM J. Numer.