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.firstname.lastname@example.org, email@example.comAbstract: This paper is concerned about a new filtering scheme three-state median (TSM)  and the adaptive center-based on contrast enhancement for removing the random weighted median (ACWM)  are two widely used filters.valued impulse noise. An efficient PDE (partial differential The TSM ﬁlter uses the median and the CWM both forequation) based algorithm for removal of random-valued detection and reduction. The ACWM uses the comparisonimpulse noise from corrupted images is proposed in this paper. of CWMs and adaptive thresholds for detection and theThe function for increasing the difference between noise-freeand noisy pixels is introduced. It denotes the number of simple median for reduction that consistently works well inhomogeneous pixels in a local neighborhood and is significantly suppressing both types of impulses. The Luo ﬁlter  is andifferent for edge pixels, noisy pixels, and interior pixels. The efﬁcient detail-preserving two-stage method that requires nocontrolling speed function and the controlling fidelity function previous training. The genetic programming (GP) ﬁlteris redefined to depend on noise and noise free pixels. According employs two cascaded detectors for detection and twoto that two controlling functions, the diffusion and fidelity corre-sponding estimators for reduction. The ﬁrst detectorprocess at edge pixels, noisy pixels, and interior pixels can be identiﬁes the majority of noisy pixels. The second detectorselectively 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 toproposed based on the two new controlling functions in orderto deal with random valued impulse noise reliably. Two its local neigh-borhood. The core of two-stage ﬁlters is thecontrolling functions are extended to automatically other PDE impulse detection process. In case of random-valuedmodels. Extensive simulation results exhibit that the proposed impulse noise, the detection of an impulse is relatively moremethod significantly outperforms many other well-known difﬁcult in comparison with salt-and-pepper impulse noisetechniques. . 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 theprocess, partial differential equation (PDE)-based image purpose of this paper. In this paper, we focus on the removaldenoising, 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 andcan significantly corrupt images. The impulse noise can be obtain the desired result as the solution of this PDE with theclassified either as salt and pepper with noisy pixels taking noisy image as initial conditions. By using PDEs, one caneither maximum or minimum value, or as random-valued model the images in a continuous domain, see existingimpulse noise. A class of widely used nonlinear digital methods in a different viewpoint, and combine multiplefilters is median filters (MED). Median filters are known for algorithms together. Furthermore, high accuracy andtheir capability to remove impulse noise while preserving stability can be naturally obtained with the help of thethe 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 . The Although there have been different PDEs denoising modelsswitching scheme concept  or the two-stage method  is developed in the past two decades, as briefly described infrequently used strategy for enhancing the performance of Section II, little has been done regarding anisotropicimpulse noise filters. The basic idea of the methods is that diffusion for filtering impulse noise. Here, we propose athe noisy pixels are detected first and filtered afterward, class of second-order improved and edge-preserving PDEwhereas the undisturbed pixels are left unchanged. There are denoising models based on two new controlling functions infour 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
controlling fidelity function from a completely different Where u(x,y;t) is the evolving image derived from thepoint of view. We introduce the notion of ENI (the original image at ‘t’ time, and “ ” and “div” areabbreviation for “edge pixels, noisy pixels, and interior the gradient and divergence operators.pixels”) to our controlling speed function. The ENI can beused to distinguish edge pixels, noisy pixels, and interior Catté et al.  improved the controlling speed function g(.)pixels, and can be calculated by utilizing the local by using instead of and proposed aneighbourhood statistics based on the number of the pixels selective smoothing modelwith similar intensity (or called homogeneous pixels). Ourcontrolling speed function is defined to depend on ENI. (2) Thus, the diffusion at edge pixels, noisy pixels, and Alvarez et al.  have made the significant improvementsinterior pixels is made with various speeds according to our through controlling the diffusion direction and proposed thecontrolling function. We also introduce the ENI to the degenerate diffusion PDE modelcontrolling fidelity function. A selective fidelity process canbe carried out according to the new controlling fidelityfunction, to reduce the smoothing effect near edges. We test (3)the proposed PDE denoising models on five standardimages degraded by random-valued impulse noise with Obviously, the controlling speed function g(.) and thevarious noise levels. We compare our PDEs models with the controlling fidelity coefficient have played an important rolerelated 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 andTSM, 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, webriefly describe related previous PDE denoising models. In III. CLASS OF SECOND-ORDER EDGE-PRESERVING PDESection III, we will first describe the ENI of an image, our DENOISING MODELScontrolling speed function, and controlling fidelity functionin detail, respectively. Then, we present a class of second- In this section, the ENI of an image is defined that canorder improved and edge-preserving PDE denoising models be used to distinguish edge pixels, noisy pixels, and interiorfor random-valued impulse noise removal based on our two pixels. Then, the controlling speed function g(.) and thecontrolling functions. Section IV shows the experiments and controlling fidelity function are redefined. Based on thediscussion, 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 Imageimage denoising in the past years. Since it is not feasible todiscuss all the models here, we briefly describe some The ENI denote the number of homogeneous pixels inrepresentative PDE models that are related to our study. The a local neighbourhood, so the ENI is significantly differentoriginal PDE filtering model, proposed by Witkin , 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 thedestroys edges. To overcome this problem, many ENI of an image.researchers have corrected this limitation from various Let be the location of a pixel underpoints 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 with4) 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’ forapproach through controlling the speed of the diffusion and each q ε , defined d(p,q) as the absolute difference inproposed a nonlinear adaptive diffusion process , intensity of the pixels between p and q, i.e,termed as anisotropic diffusion. The PM nonlinear diffusionequation is of the form (5) Then, the gray intensity of each q ε is classified into (1) two groups by a predefined threshold T.
(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, asFinally, the ENI of the pixel is defined as (7)B. Our Controlling Speed Function The ENI of an image is significantly different for edgepixels, noisy pixels, and interior pixels. The ENI for impulsenoise pixels is minimum the ENI for edge pixels takesintermediate value, and the ENI for interior pixels is Equations (10)–(11) use the new controlling speed functionmaximum, so it is reasonable that the controlling speed and the new controlling fidelity function withfunction is defined to depend on ENI. The shape of the better performance. Therefore, we would expect to seecontrolling 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 thepixels while allowing diffusion at impulse-corrupted noisy performance of these models.pixels and interior pixels. Here, we redefine the controllingspeed 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 theLike 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 correspondingfunction 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 arenoisy pixels are larger than these at edges, so the diffusion capable of removing random-valued impulse noise. Thespeeds at noisy pixels are faster than these at edges. Thus, performances of various methods are quantitativelynoisy 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 standardC. 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 variousalso 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 theThe 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 betternear to zero, at edges pixels take intermediate value, and at performance by trial. For showing the filtered results withinterior 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 variousprocess at noisy pixels is inhibited, while fidelity processes window sizes and thresholds In Fig. 5, we plot theat edge pixels and interior pixels are encouraged, which are PSNR values of the restored images by our PDEs foralso desired. various window sizes and thresholds .Where T is variable from 5 to 60 pixels with an increment of 5, andD. Class of Second-Order PDE Denoising Models Based =2 and =3. In this test, we use the appropriate iterationon 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 thefunctions to the widely used PM model (1), the SDD model
noise level is, the larger value is and the smaller value . TABLE IThe 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 19.97Fig.1 . Restoration results by the different filters. (a) Noisy free Peppersimage (b) Noisy peppers image corrupted by 30% random-valued impulsenoise. (c) MED (7×7) filter. (d) TSM (7×7) filter. (e) ACWM filter. (f) Luofilter. (g) GP filter. (h) PPDE model.
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 –. The GP filter has no parameter . 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-valuedFig.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 filteringMED (7×7) filter. (d) TSM (7×7) filter. (e) Luo filter. (f) GP filter. (g) methods, including MED, TSM, ACWM, Luo, and GP. ThePPDE model. experimental results have demonstrated the performance of our PDEs. In addition, the new controlling functions can beB. 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 . Applications of the PDEcorrupted by random-valued impulse noise with three high models are in a broad range of image processing tasks suchnoise levels-40%, 50%, and 60% as test images. From Table as inpainting, image segmentation, and skeletonization, andI, one can find that the values of PSNR by PPDE and NTVD so on. Our two controlling functions can also be applied toare better than those by NPM. One can also find that the these PDE models, which is a progress on PDE-based imagePPDE requires fewer iteration times as compared with processing. In this section the architecture of proposed CSDNTVD. Therefore, here apply our PPDE to the these images
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