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

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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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Description: 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