A Novel Inter-class Clustering Method for Image Reconstruction

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A Novel Inter-class Clustering Method for

Image Reconstruction

S. Suneetha* Siva Priya. T** Dr. C.NagaRaju*** M.S.R. Prasad****

*M.Tech (SE) L.B.R.C.E ,Mylavaram

**M.Tech (SE) L.B.R.C.E, Mylavaram

*** Professor and Head of IT, L.B.R.C.E, Mylavaram

**** Professor (IT), L.B.R.C.E, Mylavaram

Abstract- In this paper a novel inter class clustering

method is developed for image reconstruction. The aim

of image restoration is the removal of noise from

images. The simplest possible approach for noise

removal is various types of filters such as low-pass

filters or median filters. More sophisticated methods

assume a model of how the local image structures look

like, a model which distinguishes them from the noise.

The proposed method analyzed the image data in terms

of the local image structures, such as lines or edges, and

then controlling the filtering based on local information

from the analysis, a better level of noise removal is

usually obtained compared to the simpler approaches.

The fuzzy entropic technique produced better results

for simple and synthetic images and retains the

important features in multiresolution images and

extracting the features effectively, but it fails to stumpy

edged and noisy images. To overcome this limitation

here inter class clustering method is proposed for

reconstruction. The experimental result shows that the

proposed method has produce better results over the

entropic and fuzzy classification methods.

Keywords: Entropy, Classification, clustering, fuzzy set and DDA.

I. INTRODUCTION

As computational power increases, data-driven algorithms (DDA) have begun to gain in popularity in many fields. In image processing, data-driven

descriptions of structure are becoming increasingly important. Traditionally, many models used in applications such as de-noising and segmentation have been based on the assumption of piecewise smoothness [1]-[3], Unfortunately, this type of model is too simple to capture the textures present in a large percentage of real images. This drawback has limited the performance of such models, and motivated data driven representations. One data-driven strategy is to use image neighborhoods or patches as a feature vector for representing local structure. Image neighborhoods are rich enough to capture the local structures of real images, but do not impose an explicit model. This representation has been used as a basis for image de-noising [4]-[10].

Fuzzy set theory has been proven to be useful in many areas of image processing [11]-[13]. It is well known that images have some degrees of fuzziness such as indistinct borders, ill-defined shapes, and different densities. So, fuzzy logic would be a better choice to handle the fuzziness of image than traditional methods. In this study, fuzzy logic is employed to accomplish this objective. First, the intensities of an image are transformed to an interval

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87

the image. The selection of a cross-over point could be viewed as an object-background classification problem, and the thresholding techniques [15], [16] can be applied. Valley-seeking approaches are usually utilized to select a threshold if the histogram is bimodal. However, histograms would not always be bimodal, especially in medical images. The fuzzy

region of the  function is chosen as the range from the mean intensity to the maximum intensity of the image. For both de-noising and segmentation, it has been demonstrated that the accuracy of this strategy is comparable to state-of-the-art methods.

In this paper, section II describes about the feature extraction, section III gives the details about reconstruction of the image, section IV shows the results.

II. FUZZY ENTROPIC METHOD

An alternative approach to calculate the approximate mean of an image is to find the grey level corresponding to the peak of the histogram which represents the maximum number of occurrences. The experiments show that these two mean values are quite close.

Then, the entropies of the object and background distributions can be defined as

1 1

log

)

(

 







k t i e t k

i t k

i b

P

p

P

p

t

H

(1)

t i e N t i t i o P p P p t H    



  1 log 1 ) ( 1 (2)

Where Hb(t) stands for the entropy of background

pixels and Ho(t) stands for the entropy of object

pixels. The maximum information of the background and object distributions can be obtained by



(

)

(

)



max

H

t

H

t

Arg

t





_tN__k _b



_o (3)

Where t* is the optimal threshold. The value of

t*is employed as the crossover point of the 

function. The  function can be computed as follows: (4)           otherwise b c b c c g S c ifg c b c b c g S c b g ), , 2 / , ; ( 1 ), , 2 / , ; ( ) , ; (

 (5)

Where g is the intensity, c is the cross-over point and let c=t*, and b is the fuzzy bandwidth defined as:



(

),

(

)



max

c

k

N

c

b





(6)

Where k is the mean value, and N is the

maximum intensity of the image. We use  function

to fuzzify the original image. The Fuzzy  function is constructed to locate the intensities of image.

After considering the intensity brightness, the local information is also used as the enhancement criterion. Local geometrical information is employed to measure the non-uniformity of the image. First, local variances are computed









2 1

1

M j j xy

g

M



 

       





2

2 ) ( 1 xy j xy g M M   (7)

Where xyis the local mean, 2xy is the local variance

x and y are the coordinates of the current pixel, and M

is the dimension of the window, and gj is the

intensity. .

The local variance occurrence function (i.e., the histogram of local variances) is then calculated

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88 

 







V q xy

q

h

0 2

)

(

)

(













otherwise

t

,

0

0 ,

1 )

(



(8)

Where x and y are the coordinates of the locations where the variances were computed 1 < x, y < M and g is the gray level, g=0,1,---,V. Experiments show

that the local variance occurrence function contains nothing significance when V is greater than 100.

The threshold is found from the local variance occurrence function and determined by the minimum error thresholding technique. The minimum error thresholding technique is derived under a normal distribution. Here are two ways to implement this technique. One is to minimize the criterion function and the other is to search a threshold iteratively.

By considering a two class problem (C1 background and C2 object), a fitting technique can be applied to estimate the parameters from the histogram as follows:





T g

g

h

T

P

0

1

(

)

(

)

(9)



 



N T g

g

h

T

P

1

2

(

)

(

)

(10)

)

(

)

(

)

(

₁ 0

1

T

h

g

P

T

N g





















(11)

)

(

)

(

)

(

₂ 1

2

T

h

g

P

T

N T g

















 



(12)

)

(

)

(

))

(

)

(

₁ 0 2 1

1

T

g

T

h

g

P

T

T g

























(13)

)

(

)

(

))

(

)

(

₁ 1 2 2

2

T

g

T

h

g

P

T

N T g



















 





(14)

Where T is an arbitrary gray level h(g) is the probability of a gray level g , N is the maximum

intensity is the probability of the class, is the mean,

and  is the variance of the class.

Using the Baye’s classifier, the classification error

for two classes would be

)

(

log

2 )

(

log

2 ))

(

)

,

(

2 2

T

P

T

g

T

g

_e _i _e _i

i















 









(15)

Where













T

g

T

g

i

,

2 ,

1

(16)

Then, the minimum-error thresholding criterion would be





g

T

h

g

T

J

(

)

min

(

).



(

,

)

(17)

Where T is the optimum threshold.

Substituting the above parameters into the criterion J(T), we can find

)].

(

log

)

(

)

(

log

)

(

[

2 )]

(

log

)

(

)

(

log

)

(

[

2

1 )

(

2 2 1 1 2 2 1 1

T

P

T

P

T

P

T

P

T

P

T

P

T

J

e e e e











(18)

By employing a fitting technique, the optimum threshold T is found corresponding to the minimum

value of J(T). Then, the non-uniformity factor i may

be computed and transformed to the interval [0, 1] by classifying the local variance occurrence function to be background (uniformity) and object (non-uniformity) shown as follows:











otherwise

T

if

v

i i

i

,

1 ,

2 2



(19)

Where T is the optimum threshold determined in i2

is the local variance, and i is the pixel index. By Combining ∏ function and local variance occurrence

function, a new enhanced image is obtained:

N

v

c

b

g

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89

III. PROPOSED METHOD

This process can implement in two steps, in step1, normalization technique solves the dual purpose of removing the impulse noise from the image and reducing distortion in the image. It can achieve the filtering operation of an image corrupted with impulse noise. In step2, we present a new Euclidean distance for images, which we call Image Euclidean Distance (IMED). Unlike the traditional Euclidean distance, IMED takes into account the spatial relationships of pixels. IMED is then applied to image recognition. The proposed definition of the distance between two adjacent clusters in the histogram is based on both the difference between the means of the two clusters and the variance of the resultant cluster.

To measure the above two characteristics, we regard the histogram as a probability density function. Let h(z),z = 0, 1, . . ., L _ 1, be the histogram of the target image, where z indicates the gray level and L is the number of available gray levels including empty ones. The histogram h(z) indicates the occurrence frequency of the pixel with gray level z. If we define p(z) as p(z) = h (z)/N, where N is the total number of pixels in the image, p(z) is

regarded as the probability of the occurrence of the pixel with gray level z. We also define a function P (Ck) of a cluster Ck as follows:





k

x

K

p

x

C

P

1

)

(

)

(

(21)

This function indicates the occurrence probability of pixels belonging to the cluster Ck. The

distance between the clusters Ck1 and Ck2 is Euclidian

distance and is defined as

(22) (22)

The two parameters in the definition correspond to the inter-class variance and the intra-class variance, respectively. The inter-intra-class variance,

)

(

₁ ₂

2

1

C

k



C

k



, is the sum of the square distances

between the means of the two clusters and the total mean of both clusters, and defined as follows:

)

(

₁ ₂

2

1

C

k



C

k



=





2

2 2 1 1 2 2 1 2

1

_*

₍

₎

₍

₎

))

(

)

(

)

(

).

(

K K K K K

K

_C

C

P

C

P

C

P

C

P

_





(23)

Where μ1 (Ck1) is the mean of the cluster Ck1, is

defined as follows:





k

x K

K

p

x

C

P

C

1 1

1

*

(

)

(

1 )

(



(24)

The intra-class variance



_T2

(

C

_k₁



C

_k₂

)

is the variance of all pixel values is defined as follows:

)

(

₁ ₂

2

k k

T

C



C





 









n x K K T K K

x

p

C

x

C

P

C

P

(

)

(

₁

)

*

₁

(

1 2

)

*

(

)

1 _

(25)

Where



_T

(

C

_K₁



C

_K₂

)

denotes the global mean of the clusters Ck1 and Ck2, is defined as follows:

)

(

)

(

)

(

)

(

)

(

)

(

)

(

2 1 1 1 1 2 1 k K K K K K K K T

C

P

C

P

C

P

C

P

C









(26)

IV. EXPERIMENTAL RESULTS

In order to evaluate the performance of the proposed method, it has been tested on synthetic and multi-resolution images like Lena, cameraman, crow, rice and blood cells but only some of images are kept in this paper. The proposed method is tested with salt & pepper, Gaussian and speckle noises and is summarized in the Table1. The results corresponding to the Image with human observations are good. The minimum mean square error method is applied and

)

(

)

(

₁ ₂ 2 ₁ ₂

2

1

C

K

C

K T

C

K

C

K

(5)

90

got the table of values and drawn the graph for quality measures. The graph shows the clear difference between the three methods with three types of noise. The proposed method can remove the salt & pepper, gaussian and speckle noise. Here the evaluations obtained by the proposed method are better than those of two other methods which are applied to image without noise.

Observations:

Gaussian Noise Crow Image

Dense Image

Pen Image

Salt & Pepper Noise

Crow Image

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Dense Image

Pen Image

Speckle Noise

Crow Image

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92

Pen Image

Evaluation Table and Graph

Crowimage

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93

Pen Image

V. CONCLUSION

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VI. REFERENCES

[1] D. Mumford and J. Shah, “Optimal

approximations by piecewise smooth functions and associated variational problems,” Commun. Pure

Appl. Math., vol. 42, no. 4, pp. 577–685, 1989. [2] P. Perona and J. Malik, “Scale-space edge

detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 12, no. 7, pp. 629–

639, Jul. 1990.

[3] L. I. Rudin, S. J. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D, vol. 60, pp. 259–268, 1992.

[4] D. D. Muresan and T. W. Parks, “Adaptive principal components and image denoising,” in Proc.

Int. Conf. Image Processing, 2003, vol. 1, pp. 101–

104.

[5] A. Buades, B. Coll, and J.-M.Morel, “A non-local algorithm for image denoising,” in Proc. IEEE Conf.

Computer Vision and Pattern Recognition,

2005, pp. 60–65.

[6] C. Kervrann and J. Boulanger, “Optimal spatial adaptation for patchbased image denoising,” IEEE

Trans. Image Process., vol. 15, no. 10, pp. 2866–

2878, Oct. 2006.

[7] S. P. Awate and R. T. Whitaker, “Unsupervised,

information-theoretic, adaptive image filtering for image restoration,” IEEE Trans. Pattern Anal. Mach.

Intell., vol. 28, no. 3, pp. 364–376, Mar. 2006.

[8] N. Azzabou, N. Paragios, and F. Guichard, “Image denoising based on adapted dictionary computation,” in Proc. Int. Conf. Image Processing,

2007, vol. III, pp. 109–112.

[9] T. Tasdizen, “Principal components for non-local means image denoising,”in Proc. Int. Conf. Image

Processing, 2008.

[10] J. Orchard, M. Ebrahim, and A. Wang, “Efficient nonlocal-means denoising using the SVD,”

in Proc. Int. Conf. Image Processing, 2008.

[11] H. D. Cheng, C. H. Chen, and H. H. Chiu, “Fuzzy homogeneity approach to image thresholding and segmentation,” Inform. Sci.: An Int. J., vol.98,

nos. 1–4, pp. 237–262, 1997.

[12] X. Li, Z. Zhao, and H. D. Cheng, “Fuzzy

entropy threshold approach to breast cancer detection,” Inform. Sci., Applicat.: An Int. J., vol. 4,

no.1, pp. 49–56, 1995.

[13] W. Pedrycz, “Fuzzy sets in pattern recognition: Methodology and methods,” Pattern Recogn., vol.

23, no. 1/2, pp. 121–146, 1990.

[14] S. K. Pal and D. K. D. Majumder, Fuzzy Mathematical Approach to Pattern Recognition. New

York: Wiley, 1986.

[15] P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision,

Graphics, Image Processing, vol. 41, pp.233–260, 1988.

AUTHORS

S. Sunitha received her B.Tech

Degree in Information Technology from V.R. Siddhartha Engineering College, Vijayawada,her M.Tech in Software Engineering from LBRCollege of Engineering, Mylavaram. Currently, she is working as a assistant professor in V.R.Siddhartha Engineering College. She has six years of teaching experience. She attended two conferences and seven workshops. She has published one research paper in international journal.

Siva Priya.Tummala received her B.Tech degree in

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Institute of Technology, Vijayawada, her M.Tech degree in Software Engineering from LakiReddy BaliReddy College of Engineering, Mylavaram. Currently, she is working as a assistant professor in LakiReddy BaliReddy College of Engineering, Mylavaram. She has got 2 years of teaching experience from Paladugu Parvathi Devi Institute of Engineering And Technology.She has published three research papers in international journal.

Dr.C. NagaRaju received his B.Tech

degree in Computer Science from J.N.T.University Anantapur, M.Tech degree in Computer Science from J.N.T.University Hyderabad and PhD in digital Image processing from J.N.T.University Hyderabad. Currently, he is working as a professor & Head of IT in LakiReddy Bali reddy College of Engineering, Vijayawada. He is professor incharge for systems department. He has got 16 years of teaching experience. He has published thirty research papers in various national and international journals and about twenty eight research papers in various national and international conferences. He has attended twenty seminars and workshops. He is member of various professional societies like IEEE, ISTE and CSI.

Prof. M.S.R.Prasad received his

B.Tech degree in Computer Science from Mysore University Mysore, ME degree in Computer Science from Vinayaka Missions University Salem. Currently, he is working as a professor in the department of IT, LakiReddy Bali reddy College of Engineering,