I. INTRODUCTION. Keywords - Anisotropic Diffusion, Digital Image Processing, Diffusion Technique, Diffusion Algorithm, MMSE

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1 PDE-Based De-Noising Approach for Efficient Noise Reduction with Edge Preservation in Digital Image Processing Hemant Kumar Sahu*, Mr. Gajendra Singh** *Research Scholar, Department of Information Technology, RGPV, Bhopal ** Department of Information Technology, RGPV, Bhopal ABSTRACT Anisotropic diffusion has widely been applied as a mechanism for intra region smoothing of images. The results of anisotropic diffusion can be used to obtain an enhanced image or as a precursor to higher-level processing such as shape description, edge detection, image segmentation and object identification and tracking, Although attractive in terms of edge localization and the ability to control scale, anisotropic diffusion may lead to the creation of false edges and false regions, among other ill effects. In this paper, the MMSE-based filtering performed on the finest scale, the optimized anisotropic diffusion model and the anisotropic diffusion is performed on the filtered scalespace. This research work is proposed an improved nonlinear PDE-based de-noising algorithm to achieve efficient noise reduction with edge preservation due to the construction of the wavelet-based linear scale-space. Keywords - Anisotropic Diffusion, Digital Image Processing, Diffusion Technique, Diffusion Algorithm, MMSE I. INTRODUCTION The term digital image processing generally refers to processing of a two-dimensional picture by a digital computer. It implies digital processing of any two dimensional data. A digital image is an array of real or complex number represented by a finite number of bits an image given in the digitized form and stored as a matrix of binary digits in computer memory. An image may be defined as a two-dimensional function, f(x, y), where x and y are spatial coordinates, and the amplitude of f at any pair of coordinates (x, y) is called the intensity or gray level of image at that point, Digital image processing is based on the conversion of analog image field to equivalent digital form. In digital image processing systems, one usually deals with arrays of numbers obtained by spatially sampling points of a physical image. After processing, another array of numbers is produced, and these numbers are then used to reconstruct a continuous image for viewing. Image samples nominally represent some physical measurements of a continuous image field, for example, measurements of the image intensity or photographic density. In the design and analysis of image processing systems, it is convenient and often necessary mathematically to characterize the image to be processed. PDE-based Linear and non-linear Diffusion methods are explained below: Linear Diffusion The simplest and best investigated PDE method for smoothing images is to apply a linear diffusion process. We shall focus on the relation between linear diffusion filtering and the convolution with a Gaussian, analyse its smoothing properties for the image as well as its derivatives, and review the fundamental properties of the Gaussian scale-space induced by linear diffusion filtering. Afterwards a survey on discrete aspects is given and applications and limitations of the linear diffusion paradigm are discussed. The section is concluded by sketching two linear generalizations which can incorporate a-priori knowledge: affine Gaussian scale-space and directed diffusion processes [13]. Nonlinear Diffusion Adaptive smoothing methods are based on the idea of applying a process which itself depends on local properties of the image. Although this concept is well known in the image processing and a corresponding PDE formulation was first given by Perona and Malik. We shall discuss this model in detail, especially its illposedness aspects. This gives rise to study regularizations. These techniques can be extended to anisotropic processes which make use of an adapted diffusion tensor instead of a scalar diffusivity. Nonlinear diffusion filters have been applied for post processing fluctuating data, for visualizing quality-relevant features

2 in computer aided quality control, and for enhancing textures such as fingerprints. They have proved to be useful for improving sub sampling and line detection, for blind image restoration, for scale-space based segmentation algorithms for segmentation of textures and remotely sensed data, and for target tracking in infrared images. Most applications, however, are concerned with the filtering of medical images [13]. II. FOURIER VS WAVELET TRANSFORM The Fourier transform is a useful tool to analyze the frequency components of the signal. However, if we take the Fourier transform over the whole time axis, we cannot tell at what instant a particular frequency rises. Fourier transform gives the information of both time and frequency. But still another problem exists: limits the resolution in frequency. Wavelet transform seems to be a solution to the problem above. Wavelet transforms are based on small wavelets with limited duration. The translated-version wavelets locate where we concern. Whereas the scaled-version wavelets allow us to analyze the signal in different scale. The main difference between wavelet and Fourier is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies of a finite length or fast decaying oscillating waveform. the wavelet transform are classified in to discrete wavelet transform (DWT) and continuous wavelet transform(cwt).both are continuous time domain(analog) transforms, CWT operates over every possible scale and translation whereas DWT use a specific subset of scale and translation values or representation. One major advantage afforded by wavelets is the ability to perform local analysis, that is, to analyze a localized area of a larger signal. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss, aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. Furthermore, because it affords a different view of data than those presented by traditional techniques, wavelet analysis can often compress or de-noise a signal without appreciable degradation. Indeed, in their brief history within the signal processing field, wavelets have already proven themselves to be an indispensable addition to the analyst s collection of tools and continue to enjoy a burgeoning popularity today. As with any diffusion technique, processing highresolution imagery via anisotropic diffusion usually requires a significant number of iterations, precluding real-time processing. Depending upon the realization of the diffusion process, high-frequency error, or noise, can be rapidly eliminated Applications of Anisotropic Diffusion Anisotropic diffusion filtering is use to remedy difficulties in analysis of electronic portal images, stemming from their low contrast and high noise levels. Anisotropic diffusion is a nonlinear filter based on the numerical solution to the partial differential equation describing the process of diffusion. In this study we show that this filter is capable of greatly reducing noise in homogeneous areas of portal images while preserving the edges and contrast associated with anatomical features. We also demonstrate that the application of anisotropic diffusion leads to more consistent and reproducible visual extraction of features from portal images. Anisotropic diffusion can be used to remove noise from digital images without blurring edges. With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the heat equation which is equivalent to Gaussian blurring. This is ideal for removing noise but also indiscriminately blurs edges too. When the diffusion coefficient is chosen as an edge seeking function, such as in Perona and Malik, the resulting equations encourage diffusion (hence smoothing) within regions and prohibit it across strong edges. Hence the edges can be preserved while removing noise from the image. By running the diffusion with an edge seeking diffusion coefficient for a certain number of iterations, the image can be evolved towards a piecewise constant image with the boundaries between the constant components being detected as edges. III. LITERATURE SURVEY H. S. Kim, J. M. Yoo M. S. Park T. N. Dinh G.S Lee presents in [8] Anisotropic diffusion is one of the widely used techniques in the field of image process. Tremendous applications show promising results by using anisotropic diffusion algorithm or its revisions. In conventional anisotropic diffusions usually 4- neighborhood directions are used: north, south, west and east except diagonal directions of the image. These

3 methods result in the loss of image details and cause false contours. To overcome the shortcoming of these conventional anisotropic diffusion methods, a new anisotropic diffusion method based on diagonal edges is proposed. Experiments results show that the process time of the proposed method including diagonal edges over conventional methods can be faster and the image quality improvement can be better. Jeny Rajan, M.R Kaimal presents: in [9] A PDE based hybrid method for image denoising is introduces. The method is a bi-stage filter with anisotropic diffusion followed by wavelet based Bayesian Shrinkage. Here efficient denoising is achieved by reducing the convergence time of anisotropic diffusion. As the convergence time decreases, image blurring can be restricted and will produce a better denoised image than anisotropic or wavelet based methods. Experimental results based on PSNR, SSIM and edge analysis shows excellent performance of the proposed method. J.bai and X. chu Feng presents in [10] This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal. These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function, so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations. T. Zhang presents in [11] Local Winer flitering in the wavelet domain is an effective image denoising method of low complexity. In this letter, we propose a image denoising method based on Dual-Tree complex wavelet with ellipse windows thresholding combining anisotropic diffusion in image denoising algorithm, where the elliptic windows are used for different oriented subbands in order to estimate the signal variances of noisy wavelet coefficients. Authors use the complex wavelet which has stronger directional ability and local 6 directional Wiener filter to get a clearer image, then use clearer image guidance the diffusion function of anisotropic diffusion to reduce noise in the image. The experimental results show that the proposed algorithm improves the denosing performance significantly. W. Zhiming and B. H. Zhang Li presents in [12] An image denoising algorithm based on anisotropic diffusion in wavelet domain with adaptive regularize filter size and time step is proposed. After stationary wavelet transform (SWT), detail coefficients were smoothed by regularized P-M diffusion with different size of Gaussian kernel and time step on different level. On high level, wavelet coefficient were smoother and more stable, so the regularize kernel size was smaller and large time step was used. On the other hand, large kernel size and small time step was used for low level wavelet coefficients. Finally, denoised image was obtained by inverse SWT with diffused detail coefficients and unchanged approximate coefficients. Experimental results show that proposed algorithm obtained better results on images degraded by both Gaussian white noise and hybrid noise. Therefore, the main challenge for noise reduction is how to preserve the information-bearing structures such as edges and textures to get satisfactory visual quality when improving the signal-to-noise-ratio (SNR). Edge-preserving noise reduction has become a very intensive research topic. Traditional Gaussian smoothing is not effective for preserving image edges The anisotropic diffusion techniques are in general efficient to preserve image edges when they are used to reduce noise. However, they are not very effective to denoise those images that are corrupted by a high level of noise mainly for the lack of a reliable edge-stopping criterion in the partial differential equation (PDE).To preserve the edges we use wavelet based anisotropic diffusion for noise reduction. IV. DIFFUSION ALGORITHM It is very necessary for the anisotropic diffusion techniques to reduce the influence of noise on the edgestopping criterion and gradient measurements. For this end, we propose to decompose the noisy image using the DWT [21, 22] so as to construct a linear scale-space representation. After DWT, noise originally in the spatial noisy image is amplified into high frequency information, and in the wavelet-based scale-space, noise is mostly located in the finest scale. Furthermore, due to the smoothing functionality of the scaling function in the wavelet transform, noise in the detail sub bands tends to decrease as the scale increases. The wavelet components at each scale are the decomposition results of the approximation component at the next finer scale, while the approximation component at that scale is a smoothed version of the original image. Thus, the linear scalespace representation is more stationary than the raw noisy image. Chan et al. developed another wavelet

4 function to represent the piecewise-smooth functions [23], which can be used as an alternative tool for multiscale representation. Afterwards, we perform the MMSEbased filtering on the finest scale. Since for typical images the real signals are mostly located at the coarser scales and only a small fraction of them, corresponding to the sharpest edges is located at the finest scale the MMSE-based filtering can significantly reduce noise without affecting edges. As a result, the linear scalespace becomes even more stationary. Finally, we perform the anisotropic diffusion on the more stationary linear scale-space rather than on the raw noisy image domain. Since at each scale, less noise has influence on the PDE than that in the raw noisy image, the anisotropic diffusion coefficients and gradient measurements become more reliable and the anisotropic diffusion is more efficient. Furthermore, the more stationary wavelet-based scale-space makes it possible to optimize the PDE by removing the regularization component such as Gaussian smoothing and other methods [15,16], making the PDE more robust. This means that the filtered wavelet-based scale-space essentially works as a new regularization method for the PDE and it is unnecessary to smooth the wavelet-based scale-space using the Gaussian filtering at each step of the anisotropic diffusion process. Gradients can be directly calculated from the wavelet coefficients at the same scale. Zhu et al. proposed a multiscale reactiondiffusion method for texture simulation and noise reduction [20], but in their method, the reactiondiffusion is still performed in the spatial domain not in the wavelet transform domain. V. PROPOSED ALGORITHM The proposed algorithm for noise reduction can be summarized as follows- Decompose the noisy image into a scale-space with four levels using the DWT to obtain the components W I W 2, d I and S d I. j j j4' 2 1 j4' 2 Classify wavelet coefficients for each wavelet transform component at each scale into two categories by singularity detection. For the wavelet transform components of W I 2, d and W2 I, perform the MMSE-based filtering as described in This step can be optional. 2 Perform the anisotropic diffusion algorithm described in on the wavelet transform 2, d W I and W I to components j j 2 1 j4' 2 1 j4' obtain the denoised components 2, d 2, d W 2 j I and W 2 j I Two 1 j 4 1 j 4 different values can be set with the classification result obtained (in step 2). 2, d W I and W I The components of 2 j 2 j j1 j1 and may have been denoised in (step 3). The lowpass component is kept without doing any modifications. Perform the inverse DWT on the denoised wavelet transform components, W j j 2 I W 2 I and the low-pass 1 j4' 1 j4' component S24I image. to reconstruct the denoised The proposed wavelet-based anisotropic diffusion algorithm with all five steps is called WBAD and that without including (step 3) is called WMSAD. The scheme that only includes the MMSE-based filtering on the finest scale is called WT_MMSE. VI. SIMULATION RESULTS AND COMPARISON The proposed algorithm is first compared with the counterparts of robust anisotropic diffusion (ROI) the SDWT_BSF algorithm and in detail from PSNR values and visual quality of the denoised images. Both SDWT_BSF and EWID are the non iterative waveletbased denoising techniques. When testing the RAD and proposed algorithm, with respect to different noise variance are listed in Table 1 and 2. Performance (PSNR IN DB) of the Proposed WMSAD algorithm Compared with that of the RAD, SDWT_BSF, and EWID Algorithms for the images of Bird and rigid art with respect to different Noise Variances is given below:

5 Table 1: Comparison the RAD, SDWT_BSF, and EWID Algorithms for the images of Bird Scheme PROPOSED METHOD PSNR (db) vs. Noise Variance SDWT_BSF SDWT_BSF RAD PROPOSED METHOD SDWT_BSF EWID Graph 1: The graph ploted above represents the comparison of the image of bird between proposed method and different algorithm on various noise varriance values, the respective values are given in Table 1. RAD Table 2: Comparison the RAD, SDWT_BSF, and EWID Algorithms for the images of Rigid Art PSNR (db) vs. Noise Variance Scheme PROPOSED METHOD SDWT_BSF SDWT_BSF RAD PROPOSED METHOD SDWT_BSF EWID RAD Graph 2: The graph ploted above represents the comparison of the image of rigid art for proposed method by different algorithm on various noise varriance values, the respective values are given in Table 2. VII. CONCLUSION This above proposed research work is an improved nonlinear PDE-based de-noising algorithm to achieve efficient noise reduction with edge preservation due to the construction of the wavelet-based linear scale-space. We use the discrete Fourier trans-form to implement the numerical algorithm and give an iterative scheme in the frequency domain. It is one important aspect of the algorithm that it considers the input image as a periodic image. To overcome this problem, we use a folded algorithm by extending the image symmetrically about its borders. Finally, we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio. From PSNR values, we can see that the proposed algorithm achieves the best denoising performance on comparison of others. VIII. FUTURE SCOPE However the future perspective could also be to extend the proposed algorithm to the regularization of surfaces of higher dimensional images. REFERENCES [1] H. S. Kim, J. M. Yoo, M. S. Park, T. N. Dinch and G. S. Lee, An Anisotropic Diffusion Based On Diagonal Edges, Advanced Communication Technology, 9th International Conference., vol.1, pp , [2] J. Ranjan and M.R. Kaimal, Image Denoising Using Wavelet Embeded Anisotropic Diffusion, IEE International Conference on Visual Information Engineering., pp [3] J. Bai and X. chu feng, Fractional-Order Anisotropic Diffusion For Image Denoising, IEEE Transaction on Image processing, vol. 16, pp , [4] T. Zhang, Image Denoising Algorithm Via Wiener Filtering With Elliptic Directional Windows Combine Anisotropic Diffusion In Complex Wavelet Domain, International conference on Computer Science and service system., pp , [5] W. Zhiming, B. Hong and Z.Li, Image Denoising By Anisotropic Diffusion In Wavelet Domain, Measuring Technology and Mechatronics Automation, 3rd International Conference. vol.2, pp , 2011.

6 [6] J. Weickert, Anisotropic Diffusion in Image Processing. Stuttgart, Germany: Teubner-Verlag, [7] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., vol.12, pp , [8] F. Catte, P. L. Lions, J. M. Morel, and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., vol.29, pp , [9] F. Torkamani-Azar and K. E. Tait, Image recovery using the anisotropic diffusion equation, IEEE Trans. Image Process., vol.5, pp , [10] T. H. B. Romeny and M., Eds., Geometry-Driven Diffusion in Computer Vision. Norwell, MA: Kluwer, 1994, vol. 1, Computational Imaging and Vision. [11] S. C. Zhu and D. Mumford, Prior learning and gibbs reaction-diffusion, IEEE Transaction Pattern Anal. Mach. Intell., vol.19, pp , [12] S. Mallat and S. Zhong, Characterization of signals from multiscale edges, IEEE Transaction Pattern Anal. Mach. Intel., vol.14, pp , [13] S. Mallat and W. L. Hwang, Singularity detection and processing with wavelets, IEEE Transaction. Inf. Theory., vol.38, pp , [14] T. Chan and H. M. Zhou, ENO-Wavelet Transforms for piecewise smooth functions, SIAM J. Numer Anal., vol.40, pp , 2002.

Denoising and Edge Detection Using Sobelmethod

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