The document proposes a new filtering scheme based on partial differential equations (PDEs) to remove random-valued impulse noise from images. It introduces a new "edge, noisy, interior" (ENI) parameter to distinguish different pixel types. Based on ENI, it redefines controlling speed and fidelity functions for PDE models. New PDE models are presented using the controlling functions, allowing selective diffusion and fidelity processes. Experimental results on standard test images show the new PDE models outperform other methods like median filtering, adaptive center-weighted median, three-state median, Luo filter, and genetic programming in denoising ability, especially at higher noise levels.

1. An Efficient Denoising Approach for Random-Valued
Impulse Noise using PDE Method
R.Padmanaban1, S.Saravanakumar2
1
PG scholar,2 Assistant Professor
1,2
Department of Electronics and Communication Engineering
Anna University of Technology, Coimbatore
Academic Campus, Jothipuram, Coimbatore-641047.
Padhu.ar@gmail.com, sskaucbe@gmail.com
Abstract: This paper is concerned about a new filtering scheme three-state median (TSM) [4] and the adaptive center-
based on contrast enhancement for removing the random weighted median (ACWM) [5] are two widely used filters.
valued impulse noise. An efficient PDE (partial differential The TSM filter uses the median and the CWM both for
equation) based algorithm for removal of random-valued detection and reduction. The ACWM uses the comparison
impulse noise from corrupted images is proposed in this paper.
of CWMs and adaptive thresholds for detection and the
The function for increasing the difference between noise-free
and noisy pixels is introduced. It denotes the number of simple median for reduction that consistently works well in
homogeneous pixels in a local neighborhood and is significantly suppressing both types of impulses. The Luo filter [6] is an
different for edge pixels, noisy pixels, and interior pixels. The efficient detail-preserving two-stage method that requires no
controlling speed function and the controlling fidelity function previous training. The genetic programming (GP) filter
is redefined to depend on noise and noise free pixels. According [7]employs two cascaded detectors for detection and two
to that two controlling functions, the diffusion and fidelity corre-sponding estimators for reduction. The first detector
process at edge pixels, noisy pixels, and interior pixels can be identifies the majority of noisy pixels. The second detector
selectively carried out. Furthermore, a class of second-order searches for the remaining noise missed by the first detector,
improved and edge-preserving PDE denoising models is
usually hidden in image details or with amplitudes close to
proposed based on the two new controlling functions in order
to deal with random valued impulse noise reliably. Two its local neigh-borhood. The core of two-stage filters is the
controlling functions are extended to automatically other PDE impulse detection process. In case of random-valued
models. Extensive simulation results exhibit that the proposed impulse noise, the detection of an impulse is relatively more
method significantly outperforms many other well-known difficult in comparison with salt-and-pepper impulse noise
techniques. [8]. Despite decades of research in this area, the effective
detection for random-valued impulse noise with high-noise
Keywords— Anisotropic diffusion, diffusion speeds, fidelity levels is still an open problem. Let us now present the
process, partial differential equation (PDE)-based image purpose of this paper. In this paper, we focus on the removal
denoising, random-valued impulse noise. of random-valued impulse noise by using partial differential
equation (PDE) methods. PDE-based image processing
I. INTRODUCTION methods have been studied extensively as a useful tool for
image denoising and enhancement. The basic idea of PDE-
Impulse noise is a common kind of signal noise that based methods is to deform a given image with a PDE and
can significantly corrupt images. The impulse noise can be obtain the desired result as the solution of this PDE with the
classified either as salt and pepper with noisy pixels taking noisy image as initial conditions. By using PDEs, one can
either maximum or minimum value, or as random-valued model the images in a continuous domain, see existing
impulse noise. A class of widely used nonlinear digital methods in a different viewpoint, and combine multiple
filters is median filters (MED). Median filters are known for algorithms together. Furthermore, high accuracy and
their capability to remove impulse noise while preserving stability can be naturally obtained with the help of the
the edges. The main drawback of a standard median filter is available extensive research on numerical analysis.
that it is effective only for low-noise densities [1]. The Although there have been different PDEs denoising models
switching scheme concept [2] or the two-stage method [3] is developed in the past two decades, as briefly described in
frequently used strategy for enhancing the performance of Section II, little has been done regarding anisotropic
impulse noise filters. The basic idea of the methods is that diffusion for filtering impulse noise. Here, we propose a
the noisy pixels are detected first and filtered afterward, class of second-order improved and edge-preserving PDE
whereas the undisturbed pixels are left unchanged. There are denoising models based on two new controlling functions in
four two-stage filters (or the switching schemes) that are order to deal with random-valued impulse noise reliably.
worth mentioning that will used later for comparison. The We will redefine the controlling speed function and the

2. controlling fidelity function from a completely different Where u(x,y;t) is the evolving image derived from the
point of view. We introduce the notion of ENI (the original image at ‘t’ time, and “ ” and “div” are
abbreviation for “edge pixels, noisy pixels, and interior the gradient and divergence operators.
pixels”) to our controlling speed function. The ENI can be
used to distinguish edge pixels, noisy pixels, and interior Catté et al. [11] improved the controlling speed function g(.)
pixels, and can be calculated by utilizing the local by using instead of and proposed a
neighbourhood statistics based on the number of the pixels selective smoothing model
with similar intensity (or called homogeneous pixels). Our
controlling speed function is defined to depend on ENI. (2)
Thus, the diffusion at edge pixels, noisy pixels, and
Alvarez et al. [15] have made the significant improvements
interior pixels is made with various speeds according to our
through controlling the diffusion direction and proposed the
controlling function. We also introduce the ENI to the
degenerate diffusion PDE model
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 (3)
the proposed PDE denoising models on five standard
images degraded by random-valued impulse noise with Obviously, the controlling speed function g(.) and the
various noise levels. We compare our PDEs models with the controlling fidelity coefficient have played an important role
related PDE models and other special filtering methods for in the performances of PDE denoising models.
random-valued impulse noise, including MED, ACWM, Unfortunately, the previous controlling speed functions and
TSM, Luo, and GP. the controlling fidelity coefficient have some drawbacks,
especially when they are used to remove impulse noise.
This paper is organized as follows. In Section II, we
briefly describe related previous PDE denoising models. In III. CLASS OF SECOND-ORDER EDGE-PRESERVING PDE
Section III, we will first describe the ENI of an image, our DENOISING MODELS
controlling speed function, and controlling fidelity function
in detail, respectively. Then, we present a class of second- In this section, the ENI of an image is defined that can
order improved and edge-preserving PDE denoising models be used to distinguish edge pixels, noisy pixels, and interior
for random-valued impulse noise removal based on our two pixels. Then, the controlling speed function g(.) and the
controlling functions. Section IV shows the experiments and controlling fidelity function are redefined. Based on the
discussion, and is followed by conclusion in Section V. two new controlling functions, we present a class of second-
order edge-preserving PDE denoising models for random-
II.REVIEW OF PDE METHOD valued impulse noise removal.
Many different PDE models have been proposed for A. Definition of the ENI of an Image
image denoising in the past years. Since it is not feasible to
discuss all the models here, we briefly describe some The ENI denote the number of homogeneous pixels in
representative PDE models that are related to our study. The a local neighbourhood, so the ENI is significantly different
original PDE filtering model, proposed by Witkin [9], is the for edge pixels, noisy pixels, and interior pixels.
linear heat equation that diffuses in all directions and Subsequently, we describe in detail how to calculate the
destroys edges. To overcome this problem, many ENI of an image.
researchers have corrected this limitation from various Let be the location of a pixel under
points of view, mainly including: consideration,
1) from controlling the speed of the diffusion; (4)
2) from controlling the direction of the diffusion;
3) from adding a fidelity term; and
Denote the neighbour points of the central pixel p with
4) from their combinations.
window size of (2w+1)× (2w+1)(w>0) while let be
Perona and Malik (PM) were the first to try such an a set of neighbour pixels centred at ‘p’ but exclude ‘p’ for
approach through controlling the speed of the diffusion and each q ε , defined d(p,q) as the absolute difference in
proposed a nonlinear adaptive diffusion process [10], intensity of the pixels between p and q, i.e,
termed as anisotropic diffusion. The PM nonlinear diffusion
equation is of the form (5)
Then, the gray intensity of each q ε is classified into
(1)
two groups by a predefined threshold T.

3. (10), and the TVD model (14), and propose a class of
second-order edge preserving PDE denoising models. The
(6)
new PM model (NPM), the new SDD model (NSDD), and
the new TVD model (NTVD) are expressed, respectively, as
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 Equations (10)–(11) use the new controlling speed function
maximum, so it is reasonable that the controlling speed and the new controlling fidelity function with
function is defined to depend on ENI. The shape of the better performance. Therefore, we would expect to see
controlling speed functions should be like either Fig. 3(a) or something that shows the uniqueness of our models. In fact,
(b) in order to achieve reduced diffusion at and around edge the experimental results, as shown later, demonstrate the
pixels while allowing diffusion at impulse-corrupted noisy performance of these models.
pixels and interior pixels. Here, we redefine the controlling
speed function as IV. EXPERIMENTAL RESULTS AND DISCUSSION
The filtered results of the PDE models are relative to
(8) the parameters in PDE models, and discrete time step
and iteration time. In this section, we first discuss the
Like the previous controlling speed functions, the value of choices of the parameters w and T our PDE models. Then,
our controlling speed function is between 0 and 1, i.e., for evaluating the real performance of our PDE models, we
According to the new controlling speed compare our PDE models with the previously corresponding
function the diffusion speed at interior pixels. PDE models. Furthermore, we compare with other filters,
Moreover, the values of at including MED, ACWM, TSM, Luo, and GP that are
noisy pixels are larger than these at edges, so the diffusion capable of removing random-valued impulse noise. The
speeds at noisy pixels are faster than these at edges. Thus, performances of various methods are quantitatively
noisy pixels can be effectively removed while preserving measured by the peak SNR (PSNR). In all implementation,
edges very well. For comparison, we take the same discrete schemes in the
compared PDE pair. Here the 8-bit 512× 512 standard
C. Our Controlling Fidelity Function images: Lena, Peppers, Boat, and Airplane are chosen as
tested images that have distinctly different features and are
Similar to our controlling speed function we corrupted by random-valued impulse noise with various
also introduce the ENI to the fidelity term. We propose a noise levels.
controlling fidelity function as
A. Choices of the Parameters
(9)
Like the parameter in the previous controlling speed
function g(.) defined in (2) and (3), there is also not the
The function is monotone increasing and its value range is explicit formula or a method to determine the parameters
[0,0.5] The values of (.) at noise pixels are minimum or and in (8) and (9). They are chosen based on the better
near to zero, at edges pixels take intermediate value, and at performance by trial. For showing the filtered results with
interior pixels are maximum and close to 0.5. Thus, respect to the parameters and the noisy Lena with 20%
according to our controlling function (.) the fidelity random-valued impulse noise is tested by using various
process at noisy pixels is inhibited, while fidelity processes window sizes and thresholds In Fig. 5, we plot the
at edge pixels and interior pixels are encouraged, which are PSNR values of the restored images by our PDEs for
also desired. various window sizes and thresholds .Where T is
variable from 5 to 60 pixels with an increment of 5, and
D. Class of Second-Order PDE Denoising Models Based =2 and =3. In this test, we use the appropriate iteration
on Our Two Controlling Functions time of each model, that is, respectively, 40, 5, and 300 in
NPM, PPDE, and NTVD, as shown later. According to our
In this section, we introduce our two controlling own experience in this method, generally, the higher the
functions to the widely used PM model (1), the SDD model

4. noise level is, the larger value is and the smaller value . TABLE I
The appropriate value of is 2 or 3 and the appropriate COMPARISONS OF RESTORATION RESULTS IN PSNR (in Decibels)
value of is somewhere between 10 and 35. 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
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.

5. 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
(a) (b)
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.
(c) (d)
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.
(e) (f)
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
(g)
new controlling functions. We test the proposed PDE
models on five standard images corrupted by random-valued
Fig.2 Restoration results by the different filters. (a) Noisy free lena image
impulse noise with various noise levels and compare with
(b) Noisy lena image corrupted by 50% random-valued impulse noise. (c) the related second-order PDE models and the other filtering
MED (7×7) filter. (d) TSM (7×7) filter. (e) Luo filter. (f) GP filter. (g) methods, including MED, TSM, ACWM, Luo, and GP. The
PPDE model. experimental results have demonstrated the performance of
our PDEs. In addition, the new controlling functions can be
B. Comparison With Other Filters extended automatically to any other PDE denoising models
Here, we take Lena, Peppers, Boat, and Airplane, such as the coupled PDEs [14]. Applications of the PDE
corrupted by random-valued impulse noise with three high models are in a broad range of image processing tasks such
noise levels-40%, 50%, and 60% as test images. From Table as inpainting, image segmentation, and skeletonization, and
I, one can find that the values of PSNR by PPDE and NTVD so on. Our two controlling functions can also be applied to
are better than those by NPM. One can also find that the these PDE models, which is a progress on PDE-based image
PPDE requires fewer iteration times as compared with processing. In this section the architecture of proposed CSD
NTVD. Therefore, here apply our PPDE to the these images

6. CS shift-and-add multiplier is presented shown in figure 4. coupled nonlinear diffusion equations,” Compt.
Our architecture works on the concept of shifting and Vis. Image.
adding of partial products to realize the multiplied result. [15] L. Alvarez, P. Lions, and J. M. Morel (1992),
The functions of different blocks are explained below. “Image selective smoothing and edge-detection by
nonlinear diffusion. II,” SIAM J. Numer.
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