Effect of Rayleigh and Exponential noises on Sierpinski ... · including Exponential noise 4.3. Analysis of Results The effect of noise is evident from the histograms plot by us

International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: [email protected], [email protected]

Volume 1, Issue 4, November – December 2012 ISSN 2278-6856

Volume 1, Issue 4 November - December 2012 Page 120

Abstract: This paper works upon some noises like Rayleigh and exponential which are induced on Sierpinski triangle and subsequently their effects have been examined and brought into picture. In current scenario Fractals like Sierpinski triangle, Sierpinski carpet and Mandelbrot map [1] have enormous application in astrophysics, computer graphics, Networking and Image processing and mathematical modeling etc. All these diverse fields face common problem of occurrence of noise in generated fractal patterns. Hence effect of noise on patterns needs consideration. Keywords: Fractal, Noise, Sierpinski Triangle, IFS

1. INTRODUCTION Sierpinski triangle also called the Sierpinski gasket or Sierpinski Sieve is basically a fractal with four equal triangles inscribed in it. This is one of the basic examples of self-similar sets, i.e. mathematically generated pattern that can be reproducible at any magnification or reduction. The Sierpinski triangle has Hausdorff dimension log(3)/log(2) = 1.585 approx. Some applications where this fractal is quite evident are in astronomy where there are self similar galaxy structures, in weather forecasting where the fractal geometry helps us in visually analyzing the weather model, in analyzing the seismic patterns etc. However, these fractals can be corrupted by certain noises 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 when noise is introduced. Correspondingly with every level of iteration, the subjected image keeps on degrading. Hence its quality is reduced [2]. This study of variances is accomplished with the help of histogram plotting, which form the basis of representation of results in this paper.

2. PRELIMINARIES Due to the magnificent usage of fractals in some major application like image compression and in the creation of realistic landscapes, we have assented to consider it our fundamental model of research. They can be broadly categorized into myriad kinds like IFS fractals, Mandelbrot Sets, Fractal Canopies, Nonstandard fractals etc[1]. The approach for the creation of fractals utilized in this paper is IFS (Iterated Function Systems). IFS fractals are generated by starting with a figure and by imposing several geometric transforms smaller figures are created. With the repetition of this method a fractal is given as an end product[3]. IFS consist of a collection of contractive affine transformations.

1

(*) (*)n

ii

W w

(1)

For an input set S, we can compute wi for each i, and then take the union of these sets in order to get the new W(S). Sierpinski gasket is an IFS fractal which is formed by the self replicating triangles, contracting towards a fixed image.

Figure 1: Sierpinski Triangle

It is a product of three main transformations, which are represented by the matrices that follows-

1

0.5 0 00 0.5 00 0 1

T

(2)

Effect of Rayleigh and Exponential noises on Sierpinski triangle with their diversified effects

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




2

0.5 0 0.50 0.5 00 0 1

T

(3)

3

0.5 0 00 0.5 0.50 0 1

T

(4)

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 of noises which degrade its quality to quite an extent. Noise may be present in an image due to diverse reasons and quality of the resultant image depends on the type of noise subjected to the image[2]. Influence of different noises is quite diverse on same image taken. For example, a biomedical image corrupted due to noise cannot be used for diagnosis of diseases. A satellite image which is damaged by noise fails to represent the remote sensed data of, say, a geographic terrain[8]. Here, Rayleigh and exponential noises have been taken to demonstrate this part of examination. Rayleigh noise is described to be the random process with the Rayleigh distribution of probability density function as[7]

2( ) /2 ( )( )0

z a bb z a e for z ap z

for z a

(5)

The mean is given by / 4a b and the

variance is given by 2 (4 )4

b

Exponential noise is described by the probability density function as

0( )

0 0

azae for zp z

for z

(6)

The mean is given by 1a

and the variance is given

by 22

1a

.

The ramifications of these noises can be efficiently scrutinized with the means of histograms. Histograms are graphical representations which are used to give a visual impression of the entire data under inspection. Histograms are used in such a manner where an image is analyzed on the basis of its distribution of pixels in the entire range of gray scale. Results have been incorporated here with the use of these histograms.

3. ALGORITHM 3.1. Implementation of Sierpinski Triangle [6]

Sierpinski triangle can be created by a number of different approaches. The first step in the geometric construction of the Sierpinski Triangle involves splitting up of a triangle into three transformed triangles. When we look at the resultant Sierpinski Triangle, we can zoom it into any of three sub-triangles, and it will look exactly like the entire Sierpinski triangle itself.

Figure 2: The standard process for Sierpinski Triangle The approach used to implement Sierpinski sieve in this paper recalls the transformation equations (2), (3), (4) in one of the above paragraph[3]. Step 1: The process starts by taking up an image and analyzing its size. Step 2: Affine transformations firstly scaling and then followed by translation are implemented on it. Scaling reduces the input image to exactly half of its size by taking s = 0.5. Translation eventually moves the scaled image at its exact location in order to produce an image which somewhat looks like a triangle made up of three images. Step 3: Iterate steps 1 and step 2 until a contractive image is produced approaching towards a fixed point called attractor of a Sierpinski gasket [7]. The algorithm employs Mann iteration which is defined by

1( ) ( )n nf x f x (9) The outcome of this algorithm is observed to resemble that of a photocopying machine.

Figure 3: First three photocopies of an image generated

3.2. Induction of noise While these images are generated, noise is subsequently been introduced at every level of iteration. Rayleigh and exponential noises have been applied on the images on the basis of the equations presented in the paper. In this way, the actual degradation of the image is uniquely identified and studied. When corrupted images propagate, they continue to downgrade their quality to such an extent that it is impossible to retain their actual worth for further usage[2].

3.3. Analysis using histograms




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 large amount.

Figure 4: Original Image or seed without any noise

Figure 5: Histogram of the Original Image

Figure 6: First iteration with Rayleigh noise

Figure 7: Histogram for image for first iteration including Rayleigh noise

Figure 8: Second iteration with Rayleigh noise




Figure 9: Histogram for image for second iteration including Rayleigh noise

Figure 10: Third iteration with Rayleigh noise

Figure 11: Histogram for image for third iteration including Rayleigh noise

4.2. Exponential Noise[4] When a = 1 and z = random value, the following results are produced[9].

Figure 12: First iteration with Exponential noise

Figure 13: Histogram for image for first iteration including Exponential noise

Figure 14: Third iteration with Exponential noise




Figure 15: Histogram for image for third iteration including Exponential noise

4.3. Analysis of Results The effect of noise is evident from the histograms plot by us. From figure 5 and figure 7, we can clearly figure out that the number of pixels in the marked areas for a particular grey scale is very different. We see that the number of pixels in the original image is much more at the marked levels, while in the image with noise, the pixels at that same grey scale shows no pixels. 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 deteriorates the image quality.

5. CONCLUSION Sierpinski triangle is a fractal which is widely used in 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

[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, November 2008, Pages 921–924

AUTHOR

Vidushi Sukhwal (corresponding author) received the B.Tech degree in Computer Science and Technology from Amity University in 2012. Her research interests have been Digital Image Processing, Computer Architecture and

Computer Graphics.

Richa Gupta is a Professor at Amity University in the Department of Computer Science. She is a research scholar in Mahamaya Technical University. Her research area is Digital Image Processing.

Documents

Effect of Rayleigh and Exponential noises on Sierpinski ... · including Exponential noise 4.3. Analysis of Results The effect of noise is evident from the histograms plot by us