Fuzzy entropy based optimal

International Journal on Soft Computing (IJSC) Vol.6, No. 2, May 2015
DOI:10.5121/ijsc.2015.6202 17
FUZZY ENTROPY BASED OPTIMAL
THRESHOLDING TECHNIQUE FOR IMAGE
ENHANCEMENT
U.Sesadri1
,B. Siva Sankar2
, C. Nagaraju3
Assistant.Professor&Head of CSE, Vaagdevi Institute of Technology and Science,
Proddature1
M.Tech student of Vaagdevi Institute of Technology and Science, Proddatur2
Assoc. Professor&Head of CSE, YSRCE of YVU, Proddatur3
ABSTRACT
Soft computing is likely to play aprogressively important role in many applications including
image enhancement. The paradigm for soft computing is the human mind. The soft computing
critique has been particularly strong with fuzzy logic. The fuzzy logic is facts representationas a
rule for management of uncertainty. Inthis paperthe Multi-Dimensional optimized problem is
addressed by discussing the optimal thresholding usingfuzzyentropyfor Image enhancement. This
technique is compared with bi-level and multi-level thresholding and obtained optimal
thresholding values for different levels of speckle noisy and low contrasted images. The fuzzy
entropy method has produced better results compared to bi-level and multi-level thresholding
techniques.
KEY WORDS
fuzzy entropy, segmentation, soft computing, MAD and optimal thresholding
1. INTRODUCTION
Soft computing approaches have been applied to numerous real-world problems. Applications can
be found in image segmentation, pattern recognition, image enhancement and industrial
inspection, speech processing, robotics, naturallanguage understanding, etc. Thethresholding
technique is a reckless, modern and computationally low-cost segmentation technique that is
always serious and decisive in image enhancementuses. The value of image thresholding is not
always satisfactory because of the presence of noise and vagueness and ambiguity among the
classes. In [1], a three-level thresholding method has been presented for image enhancement
based on probability partition, fuzzy partition and entropy concept. This concept can be easily
extended to N-level (N>3) thresholding. However it fails for multi-resolution real images. In [2]
the image thresholding by using cluster organization from the histogram of an imageisproposed
based on inter-class variance of the clusters to be merged and the intra-class variance of the new
merged cluster. In [3], multi-level thresholding technique is presented for color image
segmentation by maximizing the conditional entropy but Fuzziness in images due to noise poses a

18
great challengein image segmentation and thresholding.In future the abovemethod may be
comprehensive to deal with noisy images by use offuzzy tools etc. In [4] this paper presents a
new histogram thresholding methodology using fuzzy and rough set theories. The power of the
proposed methodology lies in the fact that it does not make any prior assumptions about the
histogram unlike many existing techniques.In [5], fuzzy c means threshold clustering method was
used for underwater images.This emphasizesthe necessity of image segmentation, which splits an
image intoparts that take strong correlations with objects to reflect theactual information collected
from the real world when compare tooriginal image fuzzy method gives cut down value. In [6]
Unsupervised Image Thresholding using Fuzzy Measures is presented. This method successfully
segmenting the images ofbimodal and multi-model histograms for the automatic selection of
seedsubsets to decide the effective ROI. In [7], Bacterial Foraging (BF) algorithm based on
Tsallis objective function is presented for multilevel thresholding in image segmentation. In [8],
two-stage fuzzy set theoretic approach to image thresholding that uses the measure of fuzziness to
evaluate the fuzziness of an image and to find an optimal threshold value is proposed. But this
method is not appropriate to color images. In [9], fast image segmentation methods based on
swarm intelligence and 2-D Fisher criteria thresholding were used for image segmentation. In
[10], was used a procedure like thresholding by fuzzy c-means (THFCM) algorithm for image
segmentation to find an automatic threshold value. In [11], An Automatic Multilevel
Thresholding Method for Image segmentation was proposed based on Discrete Wavelet
Transforms and Genetic Algorithm. It worksonly for synthetic and real images. In [12] used
thresholding technique with genetic algorithm to find optimal thresholds between the various
objects and the background. In [13], an image segmentation framework which applied automatic
thresholding selection by means of fuzzy set theory and density model. In [14],Otsu andcanny
edge detectionswere the two techniques used for image segmentation. In [15], image
segmentation was implemented based on thresholding on Gaussian and salt & pepper noises. It is
not appropriate for other noises.
2. BI-LEVEL THRESHOLDING
The below gray-levelbimodal histogram shown in Fig 1) corresponds to an image, f1(x, y),
composed of objects overlapping with background of theimage. One exact way to extract the
objects from the background image is to select a threshold ‘Th’ that separates the modes. And
then any point (x, y) for which f1(x, y)>Th, is called an object point; if not, the point is called a
background point.
In given image, let the gray levels are L and the range is from {0, 1, 2... (L-1)}. Then the gray
level occurrence probability k is defined by following equation:
‫݌‬௥௞ =
ℎ(݇)
ܰ
݂‫0( ݎ݋‬ ≤ ݇ ≤ (‫ܮ‬ − 1))
Where ℎ(݇)is corresponding gray level number of pixels, N is the image total number of pixels
that is equal to ∑ ℎ(݅).௅ିଵ
௜ୀ଴
The thresholding problem of bi-level can be described as follows
‫)ݐ(ܬ݁ݖ݅݉݅ݔܽܯ‬ = ‫ܧܪ‬଴ + ‫ܧܪ‬ଵ (1)

19
Where
‫ܧܪ‬଴ = − ෍
‫ݎ݌‬௜
‫ݓ‬଴
݈݊
‫ݎ݌‬௜
‫ݓ‬଴
, ‫ݓ‬଴ = ෍ ‫ݎ݌‬௜ܽ݊݀
௧ିଵ
௜ୀ଴
௧ିଵ
௜ୀ଴
‫ܧܪ‬ଵ = − ෍
‫ݎ݌‬௜
‫ݓ‬ଵ
݈݊
‫ݎ݌‬௜
‫ݓ‬ଵ
, ‫ݓ‬ଵ = ෍ ‫ݎ݌‬௜ܽ݊݀
௅ିଵ
௜ୀ଴
௅ିଵ
௜ୀ଴
And the eq. (1) maximizes optimal threshold in the gray level.
Fig1
This method produced better results for there the object and background pixels have gray levels
and grouped into two dominant modes.However bi-level thresholding technique fails for
overlapping image objects with background.
3. MULTILEVEL THRESHOLDING
The more general case of bi-level thresholding technique, where the multiple overlapping objects
characterizes the image histogram.Fig (2) shows thatuni-modelmultiple thresholding classifies a
point (x, y) as belonging to the one object class in Th1<f1(x, y) ≤Th2, to the other object class if
f1(x, y)> Th2, and to the background if f1(x, y)≤ Th1. The segmentation problems require
multiple thresholds for the quality enhancement. The multilevel threshold is the extension to the
Kapur’s entropy criterion and this method is treating as the optimal multilevel thresholding
problem.In the given image [th1, th2, th3…thm] the optimal threshold is calculated as follows and
it maximizes the following objective function:
‫(ܬ‬ሾ‫ݐ‬ℎ1, ‫ݐ‬ℎ2, … . , ‫ݐ‬ℎ݉ሿ) = ‫ܧܪ‬଴ + ‫ܧܪ‬ଵ + ‫ܧܪ‬ଶ + ⋯ + ‫ܧܪ‬௠
Where
‫ܧܪ‬଴ = − ෍
‫ݎ݌‬௜
‫ݓ‬଴
݈݊
‫ݎ݌‬௜
‫ݓ‬଴
, ‫ݓ‬଴ = ෍ ‫ݎ݌‬௜
௧௛ଵିଵ
௜ୀ଴
௧௛ଵିଵ
௜ୀ଴

20
‫ܧܪ‬ଵ = − ෍
‫ݎ݌‬௜
‫ݓ‬ଵ
݈݊
‫ݎ݌‬௜
‫ݓ‬ଵ
, ‫ݓ‬ଵ = ෍ ‫ݎ݌‬௜
௧௛ଶିଵ
௜ୀ଴
௧௛ଶିଵ
௜ୀ଴
‫ܧܪ‬ଶ = − ෍
‫ݎ݌‬௜
‫ݓ‬ଶ
݈݊
‫ݎ݌‬௜
‫ݓ‬ଶ
, ‫ݓ‬ଶ = ෍ ‫ݎ݌‬௜
௧௛ଷିଵ
௜ୀ଴
௧௛ଷିଵ
௜ୀ଴
‫ܧܪ‬௠ = − ෍
‫ݎ݌‬௜
‫ݓ‬௠
݈݊
‫ݎ݌‬௜
‫ݓ‬௠
, ‫ݓ‬௠ = ෍ ‫ݎ݌‬௜ܽ݊݀
௅ିଵ
௜ୀ଴
௅ିଵ
௜ୀ଴
Fig 2
This method also works for overlapping images by considering multiple threshold values but
however it is not suitable for low contrasted images.
4. FUZZY ENTROPY TECHNIQUE
The fuzzy entropy is a statistical measure of randomness in animage, the pixel values, and Dmi
occur with probabilitiespr(Dmi), which are given by the bin heights of the normalizedhistogram.
The maximum fuzzy entropy principle based on probability partition is defined as Let Dm = {
(i,j) : i=0,1,….M-1; j = 0,1,..N-1}, Gr = {0, 1… l-1}, where M, N and l are 3 positive integers.
Thus a digitized picture defines a map‫:ܫ‬ ‫݉ܦ‬ → ‫.ݎܩ‬ Let In(x,y) be the gray level value of the
image at the pixel (x,y)
‫,ݔ(݊ܫ‬ ‫)ݕ‬ ∈ ‫,ݔ(∀ݎܩ‬ ‫)ݕ‬ ∈ ‫݉ܦ‬
‫݉ܦ‬௞ = { (‫,ݔ‬ ‫,ݔ(݊ܫ :)ݕ‬ ‫)ݕ‬ = ݇, (‫,ݔ‬ ‫)ݕ‬ ∈ ‫݉ܦ‬
K= 0, 1, 2... M-1
ℎ௞ =
௡ೖ
ே∗ெ
, ݇ = 0,1, … . . , ݉ − 1
Where݊௞ denotes the number of pixels in‫݉ܦ‬௞. The following conclusions can be easily formed:
ራ ‫݉ܦ‬௞ = ‫݉ܦ ,݉ܦ‬௝ ∩
௠ିଵ
௞ୀ଴
‫݉ܦ‬௞ = ∅(݇ ≠ ݆)
0 ≤ ℎ௞ ≤ 1, ෍ ℎ௞ = 1, ݇ = 0,1, … . ݉ − 1
௠ିଵ
௞ୀ଴

21
‫ݎ݌‬௞ௗ = ‫݉ܦ(ݎ݌‬௞ௗ) = ‫ݎ݌‬௞ ∗ ‫ݎ݌‬ௗ
௞ൗ
‫ݎ݌‬௞௠ = ‫݉ܦ(ݎ݌‬௞௠) = ‫ݎ݌‬௞ ∗ ‫ݎ݌‬௠
௞ൗ
‫ݎ݌‬௞௕ = ‫݉ܦ(ݎ݌‬௞௕) = ‫ݎ݌‬௞ ∗ ‫ݎ݌‬௕
௞ൗ
‫ݎ݌‬ௗ
௞ൗ + ‫ݎ݌‬௠
௞ൗ + ‫ݎ݌‬௕
௞ൗ = 1
ߖௗ(݇) =
‫ە‬
ۖۖ
‫۔‬
ۖۖ
‫ۓ‬
1 ݇ ≤ ‫1݌‬
1 −
(݇ − ‫)1݌‬ଶ
(‫1ݎ‬ − ‫)1݌‬ ∗ (‫1ݍ‬ − ‫)1݌‬
‫1݌‬ < ݇ ≤ ‫1ݍ‬
(݇ − ‫)1ݎ‬ଶ
(‫1ݎ‬ − ‫)1݌‬ ∗ (‫1ݎ‬ − ‫)1ݍ‬
‫1ݍ‬ < ݇ ≤ ‫1ݎ‬
0 ݇ > ‫1ݎ‬
ߖ௠(݇) =
‫ە‬
ۖ
ۖ
ۖ
ۖ
ۖ
‫۔‬
ۖ
ۖ
ۖ
ۖ
ۖ
‫ۓ‬
0 ݇ ≤ ‫1݌‬
(݇ − ‫)1݌‬ଶ
(‫1ݎ‬ − ‫)1݌‬ ∗ (‫1ݍ‬ − ‫)1݌‬
‫1݌‬ < ݇ ≤ ‫1ݍ‬
1 −
(݇ − ‫)1ݎ‬ଶ
(‫1ݎ‬ − ‫)1݌‬ ∗ (‫1ݎ‬ − ‫)1ݍ‬
‫1ݍ‬ < ݇ ≤ ‫1ݎ‬
1 ‫1ݎ‬ < ݇ ≤ ‫2݌‬
1 −
(݇ − ‫)2݌‬ଶ
(‫2ݎ‬ − ‫)2݌‬ ∗ (‫2ݍ‬ − ‫)2݌‬
‫2݌‬ < ݇ ≤ ‫2ݍ‬
(݇ − ‫)2ݎ‬ଶ
(‫2ݎ‬ − ‫)2݌‬ ∗ (‫2ݎ‬ − ‫)2ݍ‬
‫2ݍ‬ < ݇ ≤ ‫2ݎ‬
0 ݇ > ‫2ݎ‬
ߖ௕(݇) =
‫ە‬
ۖۖ
‫۔‬
ۖۖ
‫ۓ‬
0 ݇ ≤ ‫2݌‬
(݇ − ‫)2݌‬ଶ
(‫2ݎ‬ − ‫)2݌‬ ∗ (‫2ݍ‬ − ‫)2݌‬
‫2݌‬ < ݇ ≤ ‫2ݍ‬
1 −
(݇ − ‫)2ݎ‬ଶ
(‫2ݎ‬ − ‫)2݌‬ ∗ (‫2ݎ‬ − ‫)2ݍ‬
‫2ݍ‬ < ݇ ≤ ‫2ݎ‬
1 ݇ > ‫2ݎ‬
Where six parameters p1, q1, r1, p2, q2, r2 satisfy the following condition:
0 < ‫1݌‬ ≤ ‫1ݍ‬ ≤ ‫1ݎ‬ ≤ ‫2݌‬ ≤ ‫2ݍ‬ ≤ ‫2ݎ‬ < 255
The fuzzy entropy function of every class is specifiedunder:
‫ܧܪ‬ௗ = − ෍
‫ݎ݌‬௞ ∗ ߖௗ(݇)
‫ݎ݌‬ௗ
ଶହହ
௞ୀ଴
∗ ln ൜
‫ݎ݌‬௞ ∗ ߖௗ(݇)
‫ݎ݌‬ௗ
ൠ
‫ܧܪ‬௠ = − ෍
‫ݎ݌‬௞ ∗ ߖ௠(݇)
‫ݎ݌‬௠
ଶହହ
௞ୀ଴
∗ ln ൜
‫ݎ݌‬௞ ∗ ߖ௠(݇)
‫ݎ݌‬௠
ൠ

22
‫ܧܪ‬௕ = − ෍
‫ݎ݌‬௞ ∗ ߖ௕(݇)
‫ݎ݌‬௕
ଶହହ
௞ୀ଴
∗ ln ൜
‫ݎ݌‬௞ ∗ ߖ௕(݇)
‫ݎ݌‬௕
ൠ
Then the sum fuzzy entropy function is specified as follow:
‫,1݌(ܧܪ‬ ‫,1ݍ‬ ‫,1ݎ‬ ‫,2݌‬ ‫,2ݍ‬ ‫)2ݎ‬ = ‫ܧܪ‬ௗ + ‫ܧܪ‬௠ + ‫ܧܪ‬௕
Along with six variables p1, q1, r1, p2, q2, r2 the total fuzzy entropy is varied. An optimal
combination of (p1, q1, r1, p2, q2, r2) can be found.So that the total fuzzy entropy HE (p1, q1, r1,
p2, q2, r2) has the maximum value.
Fig 3
5. QUALITY PARAMETERS
5.1. Mean:mean is defined in terms of theaverage gray levels of the imagewith human
observations. In gray image the mean is calculated as
‫)ߤ(݊ܽ݁ܯ‬ =
ଵ
௉ொ
∑ ∑ ݃1(‫,ݔ‬ ‫)ݕ‬
ொ
௬ୀଵ
௉
௫ୀଵ
Where P, Q are the width and height in terms of the gray level pixels the image and g1(x, y) is
gray value.
5.2. Standard deviation: the standard deviation of gray level image iscalculated as follows
ܵ‫)ߪ(ݐ‬ = ට
ଵ
௉ொ
∑ ∑ (݃1(‫,ݔ‬ ‫)ݕ‬ − ߤ)ொ
௬ୀଵ
௉
௫ୀଵ
2
Where P, Q are the width and height of the image,ߤ is mean of the image, g1(x,y) is gray level
value of the image,ܵ‫)ߪ(ݐ‬ is standard deviation
5.3. Mean Absolute deviation: The gray levels Mean Absolute deviation (MAD) from the
median is calculated as follows
‫ܦܣܯ‬ =
ଵ
௉ொ
∑ ∑ ⃒݃1(‫,ݔ‬ ‫)ݕ‬ − ⃒݉݁݀݅ܽ݊
ொ
௬ୀଵ
௉
௫ୀଵ
Where M, N are the width and height of the image in terms of pixels, median is the median gray
value of the image mask.
5.4. The Manhattan distance function computes the distance that would be traveled to get
from one data point to theother point. The Manhattan distance between two images is the sum of
the differences of their corresponding components.
Manhattan Distance = Abs (a-x) + Abs (b-y) +Abs(c-z)

23
Where (a,b,c) and (x,y,z) are two referenced points to bematched.
6. EXPERIMENTAL RESULTS
In this paper, we have presented a new gray level thresholdingalgorithm based on the close
relationship betweenthe image thresholding problem and the fuzzylogic. In order to evaluate the
performance of the proposed fuzzy entropy technique, it has been tested using imageswith low
contrast andspeckle noise.The parameters like mean, standard deviation, MAD and Manhattan
distance function are calculated and results are shown in the form tables and graphs. Table1 to
table3 show the bi-level, multi-level and fuzzy entropy technique’s mean standard deviation,
mean-absolute-deviation and Manhattan distance function values respectively. The fuzzy entropy
technique is applied on low contrasted name plate with altered noise levels. Fuzzy entropy
technique is better for low contrasted name plate with noise up to 65% of speckle noise. From the
evaluation of the resultant images,were solved that the fuzzy entropy thresholding technique
yields betterimages than those obtained by the widely used Bi-level thresholding and multi-level
thresholding methods.
Fig 4: Comparison of Three Methods with Speckle Noise

24
Tabe1: bi-level thresholdingTable2: Multi-level thresholdingTable3: fuzzy entropy thresholding
Graph1: Mean Graph2: standard deviation
Graph3: mean absolute deviationGraph 4: Manhattan distance
7. CONCLUSION
In this paper the optimal threshold values are determined by three thresholding techniques of bi-
level, multi-level and fuzzy entropy and they are tested with a variety of representing low
contrasted as well as natural images with noise for Image enhancement. In proposed method,
fuzzy entropy thresholding is used for image enhancement and compared results with existing bi-
level and multi-level thresholding methods and proved that proposed method best fit for low
contrasted with speckle noise images. The tables and graphs are constructed by standard
deviation, mean, MAD and Manhattan distance function. Experiments on low contracted, noisy
and real images have proved the robustness of the proposed technique. Yet the fuzzy entropy
technique is not appropriate for salt & pepper and Gaussian noises.

25
REFERENCES
[1]Wen-Bing Tao, Jin-Wen Tian, Jian Liu, “Image segmentation by three-level thresholding based on
maximum fuzzy entropy and genetic algorithm” 2003 Pattern Recognition Letters 24, pp. 3069–3078
[2]AgusZainalArifin, Akira Asano, “Image segmentation by histogram thresholdingusing hierarchical
cluster analysis” 2006 Pattern Recognition Letters xxx, pp. xxx–xxx
[3]R.Sukesh Kumar, Abhisek Verma and Jasprit Singh, “Color Image Segmentation and Multi-Level
Thresholding by Maximization of Conditional Entropy” 2007 World Academy of Science,
Engineering and Technology on International Journal of Computer, Control, Quantum and
Information Engineering, Vol: 1, No: 6, pp. 1607-1615
[4]Debashis Sen and Sankar K. Pal, “Histogram Thresholding Using Fuzzy and Rough Measures of
Association Error” 2009 ieee transactions on image processing, VOL. 18, NO. 4, pp. 879-889
[5]Dr.G.Padmavathi, Mr.M.Muthukumar and Mr. Suresh Kumar Thakur, “Nonlinear Image segmentation
using fuzzy c means clustering method with thresholding for underwater images” 2010 IJCSI
International Journal of Computer Science Issues, Vol. 7, Issue 3, No 9,pp. 35-40
[6]M Seetharama Prasad, T Divakar, B Srinivasa Rao and Dr C Naga Raju, “Unsupervised Image
Thresholding using Fuzzy Measures” 2011 International Journal of Computer Applications (0975 –
8887),Volume 27– No.2, pp. 32-41
[7]P.D. Sathya, and R. Kayalvizhi, “modified bacterial foraging algorithm based multi-level thresholding
for image segmentation” 2011 Engineering Applications of Artificial Intelligence 24, pp. 595-615
[8]Ambar Dutta, AvijitKar and B. N. Chatterji, “fuzzy set theoretic approach to image thresholding” 2011
International Journal of Computer Science, Engineering and Applications (IJCSEA) Vol.1,
No.6,pp.63-72
[9]Zhiwei Ye, Zhengbing Hu, Xudong Lai and Hongwei Chen, “Image Segmentation Using Thresholding
and Swarm Intelligence” 2012 journal of software, VOL. 7, NO. 5, pp. 1074-1082
[10]Firas A. Jassim, “Hybrid Image Segmentation using Discerner Cluster in FCM and Histogram
Thresholding” 2012 International Journal of Graphics & Image Processing |Vol 2|issue 4|, pp. 241-
244
[11]RakeshKumar, TapeshParashar,Gopal Verma, “a multilevel automatic thresholding for image
segmentation using genetic algorithm and dwt”International Journal of Electronics and Computer
Science Engineering, ISSN- 2277-1956/V1N1-153-160, pp. 153-160
[12]S. Vijayakumar,A.obulasu, Dr.L.S.S. Reddy and C. Nagaraju “reconstruction of digital images by
combining Multiple gradients” © 2005 – 2009,Journal of Theoretical and Applied Information
Technology,vol:10 pp.no.29-34
[13]Jianli Li, Bingbin Dai, Kai Xiao, and Aboul Ella Hassanien, “Density Based Fuzzy Thresholding for
Image Segmentation” 2012 A. Ell Hassanien et al. (Eds.): AMLTA 2012, CCIS 322, pp. 118–127
[14]Jamil A. M. Saif , Ali Abdo Mohammed Al-Kubati, AbdultawabSaifHazaa, Mohammed Al- Moraish,
“Image Segmentation Using Edge Detection and Thresholding” 2013 The 13th International Arab
Conference on Information Technology ACIT, ISSN : 1812-0857, PP.473-476
[15]S.BanuChitra, V.Gayathri and M.Chanda Mona, “a study of image segmentation using thresholding
technique on a noisy image” 2014 International Journal of Human Computer Interaction (IJHCI)
Singaporean Journal of Scientific Research (SJSR) Vol.6.No.6 2014 Pp.

26
AUTHORS
U. Sesadri is currently working as Assistant. Professor and HOD in the Department of
CSE in Vaagdevi Institute of Technology & science, Proddatur, KadapaDistrict,Andhra
Pradesh, India. He received his M.Sc. in Mathematics from SV University, Tirupati, M.E
in Computer Science and Engineering from Sathyabama University, Chennai and pursing
PhD in Digital Image Processing from V.T. University, Belgaum. He got 7 years of
teaching experience. He attended five National Level workshops and two international
level conferences. He organized 10 National levelworkshops and two National level paper
presentations.
Bodicherla Siva Sankar is doing his M. tech in Vaagdevi Institute of Technology,
Proddatur, Kadapa, A.P, India. He received his B.Tech degree in Information Technology
from J.N.T. University Anantapur and he has attended two workshops on cloud computing
& Big Data and Research challenges in Digital Image Processing.
Dr. C. Naga Raju is currently working as Associate Professor and Head of the
Department of Computer Science and Engineering at YSR Engineering College of
Yogivemana University, Proddatur, Kadapa District, and Andhra Pradesh, India. He
received his B.Tech Degree in Computer Science and Engineering from J.N.T.University,
Anantapur, and M.Tech Degree in Computer Science from J.N.T.University Hyderabad
and PhD in digital Image processing from J.N.T.University Hyderabad. He has got 18
years of teaching experience. He received research excellence award, teaching excellence award and
Rayalaseemavidhyaratna award for his credit. He wrote text book on C & Data structures and Pattern
Recognition. He has six PhD scholars. He has published fifty six research papers in various National and
International Journals and about thirty research papers in various National and International Conferences.
He has attended twenty seminars and workshops. He delivered 10 keynote addresses. He is member of
various professional societies like IEEE, ISTE and CSI.

Fuzzy entropy based optimal

More Related Content

What's hot (17)

Viewers also liked (19)

Similar to Fuzzy entropy based optimal (20)

Recently uploaded (20)

Fuzzy entropy based optimal