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International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Effect of Rayleigh and Exponential noises on Sierpinski triangle with their diversified effects Vidushi Sukhwal1 and Richa Gupta2 1 Amity University, Department of Computer Science and Engineering, Sector - 125, Noida, Uttar Pradesh 2 Amity University, Department of Computer Science and Engineering, Sector - 125, Noida, Uttar Pradesh Abstract: This paper works upon some noises like Rayleigh and exponential which are induced on Sierpinski triangle and 2. PRELIMINARIES subsequently their effects have been examined and brought into picture. In current scenario Fractals like Sierpinski Due to the magnificent usage of fractals in some major triangle, Sierpinski carpet and Mandelbrot map [1] have application like image compression and in the creation of enormous application in astrophysics, computer graphics, realistic landscapes, we have assented to consider it our Networking and Image processing and mathematical fundamental model of research. They can be broadly modeling etc. All these diverse fields face common problem categorized into myriad kinds like IFS fractals, of occurrence of noise in generated fractal patterns. Hence Mandelbrot Sets, Fractal Canopies, Nonstandard fractals effect of noise on patterns needs consideration. etc[1]. Keywords: Fractal, Noise, Sierpinski Triangle, IFS The approach for the creation of fractals utilized in this paper is IFS (Iterated Function Systems). IFS fractals are 1. INTRODUCTION generated by starting with a figure and by imposing Sierpinski triangle also called the Sierpinski gasket or several geometric transforms smaller figures are created. Sierpinski Sieve is basically a fractal with four equal With the repetition of this method a fractal is given as an triangles inscribed in it. This is one of the basic examples end product[3]. of self-similar sets, i.e. mathematically generated pattern IFS consist of a collection of contractive affine that can be reproducible at any magnification or transformations. reduction. The Sierpinski triangle has Hausdorff n dimension log(3)/log(2) = 1.585 approx. Some W (*) wi (*) (1) applications where this fractal is quite evident are in i 1 astronomy where there are self similar galaxy structures, For an input set S, we can compute wi for each i, and then in weather forecasting where the fractal geometry helps take the union of these sets in order to get the new W(S). us in visually analyzing the weather model, in analyzing Sierpinski gasket is an IFS fractal which is formed by the the seismic patterns etc. self replicating triangles, contracting towards a fixed However, these fractals can be corrupted by certain noises image. which degrade their quality to quite an extent. One of the noises is the Rayleigh noise which is introduced as a consequence of wind velocity that seamlessly penetrates into an image during its generation or propagation. Similarly, the penetration of another type of noise named Exponential noise allows for the interdependence between additional waiting time and elapsed waiting time[8]. Such noises corrupt the image which can be visually identified by the means of their looks. The image which forms the seed for the Sierpinski triangle, gets faded Figure 1: Sierpinski Triangle when noise is introduced. Correspondingly with every level of iteration, the subjected image keeps on degrading. It is a product of three main transformations, which are Hence its quality is reduced [2]. represented by the matrices that follows- This study of variances is accomplished with the help of 0.5 0 0 histogram plotting, which form the basis of representation of results in this paper. T 1 0 0.5 0 (2) 0 0 1 Volume 1, Issue 4 November - December 2012 Page 120 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 0.5 0 0.5 3. ALGORITHM 3.1. Implementation of Sierpinski Triangle [6] T 2 0 0.5 0 (3) 0 Sierpinski triangle can be created by a number of 0 1 different approaches. The first step in the geometric construction of the Sierpinski Triangle involves splitting 0.5 0 0 up of a triangle into three transformed triangles. When T 3 0 0.5 0.5 (4) we look at the resultant Sierpinski Triangle, we can zoom 0 it into any of three sub-triangles, and it will look exactly 0 1 like the entire Sierpinski triangle itself. In T1, a scaling of s = 0.5 has been done, which places the triangle in the lower left part. T2 is scaled by s = 0.5, followed by a translation of 0.5, moving it in the x direction. T3, in the similar fashion is scaled by s = 0.5 and translated by 0.5 in the y direction[5]. At each level of iteration, the fractals come across number Figure 2: The standard process for Sierpinski Triangle of noises which degrade its quality to quite an extent. Noise may be present in an image due to diverse reasons The approach used to implement Sierpinski sieve in this and quality of the resultant image depends on the type of paper recalls the transformation equations (2), (3), (4) in noise subjected to the image[2]. Influence of different one of the above paragraph[3]. noises is quite diverse on same image taken. For example, Step 1: The process starts by taking up an image and a biomedical image corrupted due to noise cannot be used analyzing its size. for diagnosis of diseases. A satellite image which is Step 2: Affine transformations firstly scaling and then followed by translation are implemented on it. Scaling damaged by noise fails to represent the remote sensed reduces the input image to exactly half of its size by data of, say, a geographic terrain[8]. taking s = 0.5. Translation eventually moves the scaled Here, Rayleigh and exponential noises have been taken to image at its exact location in order to produce an image demonstrate this part of examination. Rayleigh noise is which somewhat looks like a triangle made up of three described to be the random process with the Rayleigh images. distribution of probability density function as[7] Step 3: Iterate steps 1 and step 2 until a contractive image 2 2 ( z a )e ( z a ) /b for z a is produced approaching towards a fixed point called p( z ) b (5) attractor of a Sierpinski gasket [7]. 0 for z a The algorithm employs Mann iteration which is defined by The mean is given by a b / 4 and the f ( xn 1) f ( xn) (9) 2 b(4 ) The outcome of this algorithm is observed to resemble variance is given by 4 that of a photocopying machine. Exponential noise is described by the probability density function as ae az for z 0 p( z ) (6) 0 for z 0 Figure 3: First three photocopies of an image 1 generated The mean is given by and the variance is given a 1 3.2. Induction of noise by 2 . While these images are generated, noise is subsequently a2 been introduced at every level of iteration. Rayleigh and The ramifications of these noises can be efficiently exponential noises have been applied on the images on scrutinized with the means of histograms. Histograms are the basis of the equations presented in the paper. In this graphical representations which are used to give a visual way, the actual degradation of the image is uniquely impression of the entire data under inspection. identified and studied. When corrupted images propagate, Histograms are used in such a manner where an image is they continue to downgrade their quality to such an extent analyzed on the basis of its distribution of pixels in the that it is impossible to retain their actual worth for further entire range of gray scale. Results have been incorporated usage[2]. here with the use of these histograms. 3.3. Analysis using histograms Volume 1, Issue 4 November - December 2012 Page 121 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 These corrupted images are well demonstrated with the help of histograms that assist us in studying the variance in the original and the new images. The number of pixels for each grey scale level is examined and thus a proper comparison is done. 4. RESULTS Initial image (seed) is captured from standard camera. To obtain the faster results we have taken grey scale image. But algorithm is applicable to any colored image as well. The influence of noise on the Sierpinski triangle is shown for the following parameters – 4.1. Rayleigh noise[4] When a = 0, b = 1 and z = random value, the following results are generated [9]. This noise is introduced consecutively during the fractal generation which as a result deteriorates the image to a Figure 6: First iteration with Rayleigh noise large amount. Figure 7: Histogram for image for first iteration including Rayleigh noise Figure 4: Original Image or seed without any noise Figure 5: Histogram of the Original Image Figure 8: Second iteration with Rayleigh noise Volume 1, Issue 4 November - December 2012 Page 122 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Figure 9: Histogram for image for second iteration including Rayleigh noise Figure 12: First iteration with Exponential noise Figure 13: Histogram for image for first iteration including Exponential noise Figure 10: Third iteration with Rayleigh noise Figure 11: Histogram for image for third iteration including Rayleigh noise 4.2. Exponential Noise[4] Figure 14: Third iteration with Exponential noise When a = 1 and z = random value, the following results are produced[9]. Volume 1, Issue 4 November - December 2012 Page 123 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 [6] Zuoling Zhou, Li Feng, A new estimate of the Hausdorff measure of the Sierpinski gasket, Nonlinearity, 13(2000), 479-491. [7] Rafael C. Gonzalez and Richard E. Woods, Digital image processing. [8] Kazutoshi Gohara and Arata Okuyama, Fractal Transition: Hierarchical Structure And Noise Effect, Fractals 07, 313 (1999). DOI: 0.1142/S0218348X99000311 [9] Demitre Serletis, Effect of noise on fractal structure , Chaos, Solitons & Fractals , Volume 38, Issue 4, Figure 15: Histogram for image for third iteration November 2008, Pages 921–924 including Exponential noise AUTHOR 4.3. Analysis of Results The effect of noise is evident from the histograms plot by Vidushi Sukhwal (corresponding us. From figure 5 and figure 7, we can clearly figure out author) received the B.Tech degree in that the number of pixels in the marked areas for a Computer Science and Technology from particular grey scale is very different. We see that the Amity University in 2012. Her research number of pixels in the original image is much more at interests have been Digital Image the marked levels, while in the image with noise, the Processing, Computer Architecture and pixels at that same grey scale shows no pixels. Computer Graphics. Similarly, such variances can be clearly seen while comparing figure 5 and figure 11, figure 5 and figure 13, and figure 5 and figure 15. All these comparisons clearly conclude that noise Richa Gupta is a Professor at Amity deteriorates the image quality. University in the Department of Computer Science. She is a research scholar in Mahamaya Technical 5. CONCLUSION University. Her research area is Digital Sierpinski triangle is a fractal which is widely used in Image Processing. many practical applications like weather forecasting, image compression, Seismology etc [5]. When these fractals are corrupted with noises like Rayleigh and Exponential, their effects on the functioning of these applications turns out to be very appalling. Along with these noises, there are some more kinds of noises like Gaussian, speckle, Erlang etc[9]. All of these shows different types of degradations which are yet to be analyzed. References [1] Argyris J, Andreadis I, Karakasidis TE. On perturbation of the Mandelbrot map. Chaos, Solitons & Fractals 2000; 11:1131-6. [2] Argyris J, Andreadis I. On the influence of noise on the coexistence of chaotic attractors. Chaos, Solitons & Fractals 2000;11(6):941-6. MR1737636 (2000i: 37037). [3] John E. Hutchinson, Fractals and Self Similarity. Indiana University Mathematics Journal, Vol. 35, No. 5. 1981 [4] M.S. Alani, Digital Image Processing using Matlab, University Bookshop, Sharqa, URA, 2008 [5] K. Falconer, Fractal Geometry: Mathematical foundations and applications, seconded, Wiley, 2003 Volume 1, Issue 4 November - December 2012 Page 124

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Vidushi Sukhwal1 and Richa Gupta2
1Amity University, Department of Computer Science and Engineering, Sector - 125, Noida, Uttar Pradesh
2Amity University, Department of Computer Science and Engineering, Sector - 125, Noida, Uttar Pradesh

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International Journal of Emerging Trends & Technology in Computer Science (IJETTCS)
Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com
Volume 1, Issue 4, November – December 2012, ISSN 2278-6856, Impact Factor of IJETTCS for year 2012: 2.524

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