9n13 ijaet0313548 revised

International Journal of Advances in Engineering & Technology, Mar. 2013. ŠIJAET ISSN: 2231-1963

EFFICIENT IMAGE COMPRESSION TECHNIQUE USING FULL, COLUMN AND ROW TRANSFORMS ON COLOUR IMAGE H. B. Kekre1, Tanuja Sarode2 and Prachi Natu3 1

Sr. Professor, MPSTME, Deptt. of Computer Engg., NMIMS University, Mumbai, India Associate Professor Department of Computer Engg., TSEC, Mumbai University, India 3 Ph. D. Research Scholar, MPSTME, NMIMS University, Mumbai, India

2

ABSTRACT This paper presents image compression technique based on column transform, row transform and full transform of an image. Different transforms like, DFT, DCT, Walsh, Haar, DST, Kekre’s Transform and Slant transform are applied on colour images of size 256x256x8 by separating R, G, and B colour planes. These transforms are applied in three different ways namely: column, row and full transform. From each transformed image, specific number of low energy coefficients is eliminated and compressed images are reconstructed by applying inverse transform. Root Mean Square Error (RMSE) between original image and compressed image is calculated in each case. From the implementation of proposed technique it has been observed that, RMSE values and visual quality of images obtained by column transform are closer to RMSE values given by full transform of images.

Row transform gives quite high RMSE values as compared to column and full transform at higher compression ratio. Aim of the proposed technique is to achieve compression with acceptable image quality and lesser computations by using column transform.

KEYWORDS:

I.

Image compression, Full transform, Column transform, Row transform.

INTRODUCTION

Rapid increase in multimedia applications has been observed since last few years. It leads to higher use of images and videos as compared to text data. Use of these applications play important role in communication, educational tools, gaming applications, entertainment industry and many other areas.

When images and videos come into picture, issue of their storage, processing and transmission cannot be neglected. Images contain considerable amount of redundancies. Hence storage and transmission of compressed images instead of uncompressed images has been proved to be advantageous. Image compression has the added advantage of being tolerant to distortion due to peculiar characteristics of human visual system [1]. Major aim of image compression is to reduce the storage space or transmission bandwidth and maintain acceptable image quality simultaneously. Image compression techniques are broadly classified into two categories: lossless compression and lossy compression. In lossless image compression exact replica of original image can be obtained from compressed image; however it gives low compression ratio, which is not the case in lossy image compression. Wide research has been done in this area and it includes compression using Discrete Fourier Transform (DFT) [11] and Discrete Cosine Transform (DCT) [2].Compression using warped DCT is proposed in [16]. Recent work includes wavelet based image compression using orthogonal wavelet transform[12] and hybrid wavelet transform[17].Fractal image compression is discussed by Veenadevi and Ananth in [18]. This paper presents transform based image compression that uses column transform, row transform and full transform of an image.

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II.

TRANSFORM BASED IMAGE COMPRESSION

Image compression plays a vital role in several important and diverse applications including televideo conferencing, remote sensing, medical imaging [2,3] and magnetic resonance imaging[4].

Transform based coding is major component of image and video processing applications. It is based on the fact that pixels in an image exhibit a certain level of correlation with their neighbouring pixels.

A transformation is, therefore defined to map this spatial (correlated) data into transformed (uncorrelated) coefficients [5]. It means that the information content of an individual pixel is relatively small and to a large extent visual contribution of a pixel can be predicted using its neighbours [1, 6].Transform based compression techniques use a reversible linear mathematical transform to map the pixel values onto a set of coefficients which are then quantized and encoded. It is lossy compression technique. Previously, Discrete Fourier Transform (DFT) is used to change the pixels in the original image into frequency domain coefficients. Discrete Cosine Transform (DCT) is most widely used approach in image and video compression, as the performance approaches to that of Karhunen-Loeve transform (KLT) for first order Morkov process[16].

2.1. Discrete Cosine Transform (DCT) Discrete Cosine Transform (DCT) is widely used transformation technique for image compression.

Other transforms like Haar, Walsh, Slant, Discrete sine transform (DST) can also be used for image compression. DCT converts the spatial image representation into frequency components. Low frequency components appear at the topmost left corner of the block that contains maximum information of the image.

2.2. Walsh Transform Walsh transform is non-sinusoidal orthogonal transform that decomposes a signal into a set of orthogonal rectangular waveforms called Walsh functions. The transformation has no multipliers and is real because the amplitude of Walsh functions has only two values, +1 or -1. Walsh functions are rectangular or square waveforms with values of -1 or +1. An important characteristic of Walsh functions is sequency which is determined from the number of zero-crossings per unit time interval.

Every Walsh function has a unique sequency value. The Walsh-Hadamard transform involves expansion using a set of rectangular waveforms, so it is useful in applications involving discontinuous signals that can be readily expressed in terms of Walsh functions.

2.3. Haar Transform Haar transform was proposed in 1910 by a Hungarian mathematician Alfred HaarError! Reference source not found.. The Haar transform is one of the earliest transform functions proposed.

Attracting feature of Haar transform is its ability to analyse the local features. This property makes it applicable in electrical and computer engineering applications. The Haar transform uses Haar function for its basis. The Haar function is an orthonormal, varies in both scale and position [8]. Haar transform matrix contains ones and zeros. Hence it requires no multiplications and less number of additions as compared to Walsh transform which makes it computationally efficient, fast and simple.

2.4. Discrete Sine Transform (DST) Discrete Sine Transform (DST) is a complementary transform of DCT. DCT is an approximation of KLT for large correlation coefficients whereas DST performs close to optimum KLT in terms of energy compaction for small correlation coefficients. DST is used as low-rate image and audio coding and in compression applications [9,10].

2.5. Fourier Transform In conventional Fourier transform, it is not easy to detect local properties of the signal. Hence Short Term Fourier Transform (STFT) was introduced. But it gives local properties at the cost of global properties [11].

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International Journal of Advances in Engineering & Technology, Mar. 2013. ©IJAET ISSN: 2231-1963 2.6. Kekre’s Transform Most of the transform matrices have to be in powers of two. This condition is not required in Kekre transform [12, 13] matrix. In Kekre transform matrix, all diagonal elements and the upper diagonal elements are one. Lower diagonal elements except the one exactly below the diagonal are zero.

2.7. Slant Transform Slant transform matrix is orthogonal with a constant function for the first row. The elements in other rows are defined by linear functions of the column index. Properties of Slant transform are: It has orthonormal set of basis vectors. First basis vector is constant basis vector, one slant basis vector, the sequency property, variable size transformation, fast computational algorithm and high energy compaction. Definition of slant transform and its properties are given in [14, 15].

III.

PROPOSED TECHNIQUE

In proposed compression technique, seven different transforms namely DFT, DCT, DWT, DST, DHT, DKT and Slant transform are applied on each 256x256 size colour image to obtain transformed image content. These transforms are applied in three different ways: column transform, row transform and full transform. Let ‘T’ denotes the transformation matrix, ‘f’ denotes an image and ‘F’ indicates transformed image. Then, Column transform of an image ‘f’ is [F] = [T]*[f] Row transform is written as: [F] = [f]*[T’] where, T’= Transpose of T And full transform is given by: [F] = [T]*[f]*[T’] In each of these three cases, low energy coefficients are eliminated from transformed image content.

Then image is reconstructed by applying inverse transform on it. In column transform, number of coefficients is reduced by eliminating some rows of transformed image. In row transform, it is done by eliminating some columns of transformed image whereas in full transform some rows as well as some columns are eliminated such that number of coefficients reduced is equal as that of column or row transform. Image is then reconstructed by applying inverse transform on the image which contains reduced number of coefficients than original image. Root mean square error and compression ratio is calculated in each case till acceptable image quality is obtained. Average of these RMSE values and compression ratio is used for performance analysis.

IV.

EXPERIMENTAL RESULTS

Experimentation is done on 12 sample colour images. Images of 256x256 sizes from different classes are selected. Experiments are performed in MATLAB 7.2 on a computer with dual core processor and 4 GB RAM. Test images are shown in figure 1.

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Figure 1: Set of twelve test images of different classes used for experimental purpose namely (from left to right and top to bottom) Mandrill, Peppers, Lord Ganesha, Flower, Cartoon, dolphin, Birds, Waterlili, Bud, Bear, Leaves and Lenna

For each transform, comparison of three cases i.e. RMSE in Full, column and row transform is shown in figure 2 to 8. Figure 2 shows this comparison for DFT. RMSE values for full and column transform are almost same in this case. But row transform gives slight high values of RMSE.

Figure 2. Performance comparison of Average RMSE for Full DFT, column DFT and Row DFT against different Compression Ratios

Figure 3. Performance comparison of Average RMSE for Full Haar, column Haar and Row Haar against different Compression Ratios

Figure 3 shows comparison for Haar transform. Here, up to compression ratio 4, RMSE in full and column transform are almost same. Afterwards RMSE in column transform is approximately same as that of full transform.

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Figure 4. Performance comparison of Average RMSE for Full DCT, column DCT and Row DCT against different Compression Ratios

Figure 5. Performance comparison of Average RMSE for Full Walsh, column Walsh and Row Walsh against different Compression Ratios

As found in figure 4 and 5, RMSE values of column and full transform are closer. Row transform RMSE values are slightly higher in both DCT and Walsh transform.

Figure 6. Performance comparison of Average RMSE for Full Slant, column Slant and Row Slant against different Compression Ratios

Figure 7. Performance comparison of Average RMSE for Full Kekre transform, column Kekre and Row Kekre transform against different Compression Ratios

Graph plotted in figure 6 and 7 shows RMSE values obtained by applying Slant transform and Kekre transform respectively. These values are higher than the values obtained in DFT, DCT, Haar and Walsh. But difference between Full transform values and column transform values is again very small. Comparison of RMSE values for DST is shown in figure 8. Here also there is slight difference in column transform RMSE values and the values in Full transform.

Figure 8. Performance comparison of Average RMSE for Full DST, Column DST and Row DST against different compression ratios

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Figure 9.Performance comparison of Average RMSE for Full DFT, Haar, DCT, Walsh, Slant, Kekre Transform, and DST against compression ratio 1 to 5

Graph plotted in figure 9 shows comparison of RMSE values for seven different full transforms namely DFT, Haar, DCT, Walsh, Slant, Kekre Transform and DST. From the graph it can be observed that, full DFT gives least RMSE value among all other full transforms.

Figure 10.Performance comparison of Average RMSE for Column DFT, Haar, DCT, Walsh, Slant, Kekre Transform, and DST against compression ratio 1 to 5.

By observing and comparing Figure 10 with Figure 9, it is found that RMSE values of column transform for compression ratio 1 to 5 are close to the values obtained by full transform. Since in column transform we use [F] = [T]x[f] and not [F] = [T]x[f]x [T’] like in full transform, it saves half number of computations.

Figure 11. Performance comparison of Average RMSE for Row DFT, Haar, DCT, Walsh, Slant, Kekre Transform, DST against compression ratio 1 to 5.

It can be seen from Figure 11 that RMSE values for row transform are slight higher than column and full transforms.

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International Journal of Advances in Engineering & Technology, Mar. 2013. ©IJAET ISSN: 2231-1963 Table 1 presents the summary of Average RMSE and PSNR for full transforms. It can be observed that, good PSNR upto32 dB is obtained by DFT, DCT and DST at compression ratio 2.

Table 1. Summary of Average RMSE and PSNR for various ‘Full Transforms’

Full Transform DFT Haar DCT Walsh Slant Kekre Transform DST

AVG RMSE 6.325 10.828 6.012 9.273 33.628

Compression Ratio 4 8 AVG AVG PSNR PSNR RMSE RMSE 32.21 10.47 27.73 14.71 27.44 14.932 24.64 18.843 32.55 10.241 27.92 14.665 28.78 13.195 25.72 16.950 17.59 40.301 16.02 42.536

28.666

18.98

39.332

16.23

44.867

15.09

6.229

32.24

10.661

27.57

15.135

24.53

2

PSNR 24.77 22.62 24.8 23.54 15.55

Table 2 gives average RMSE and PSNR summary for column transform. Average RMSE in column transform is closer to that of full transform. Better PSNR is obtained for DFT. Table 2. Summary of Average RMSE and PSNR for various ‘Column Transforms’

Column Transform DFT Haar DCT Walsh Slant Kekre Transform DST

Compression Ratio 4 8 AVG AVG AVG PSNR PSNR RMSE RMSE RMSE 2.541 40.03 4.4072 35.24 6.288 9.728 28.37 15.440 24.35 20.886 7.386 30.76 12.915 25.91 18.343 9.728 28.37 15.440 24.35 20.886 35.900 17.02 42.512 15.56 44.686 2

PSNR 32.16 21.73 22.86 21.73 15.12

31.232

18.23

43.213

15.41

47.717

14.55

8.046

30.01

14.770

24.74

21.893

21.32

Table 3 shows performance of different row transforms in terms of RMSE and PSNR. DFT, DCT and DST show good average RMSE. Better PSNR is obtained for DFT. Table 3. Summary of Average RMSE and PSNR for various ‘Row Transforms’

Row Transform

DFT Haar DCT Walsh Slant Kekre Transform DST

Compression Ratio 4 8 AVG AVG AVG PSNR PSNR RMSE RMSE RMSE 2.559 39.96 4.458 35.14 6.410 9.910 28.2 15.869 24.12 21.705 7.530 30.59 13.168 25.74 18.874 9.910 28.2 15.869 24.12 21.705 36.765 16.82 43.484 15.36 45.761 2

PSNR 31.99 21.4 22.61 21.4 14.92

32.164

17.98

42.313

15.6

46.788

14.72

8.260

29.79

15.124

24.53

22.458

21.1

From twelve different query images with different colour and texture combination, ‘Mandrill’ image is selected as representative image for perceptual comparison. It contains different colour

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International Journal of Advances in Engineering & Technology, Mar. 2013. ©IJAET ISSN: 2231-1963 combinations and edges. Compressed images obtained by applying full, column and row transforms are shown below with corresponding compression ratio and RMSE value for each image.

Compression Ratio= 2 RMSE= 2.685373

Compression Ratio= 4 RMSE=4.641359

Compression Ratio= 8 RMSE=6.248745

Figure 12: Compressed ‘Mandrill’ images by applying full DFT

Compression Ratio= 2 RMSE=3.615713

Compression Ratio= 4 RMSE=5.167002

Compression Ratio= 8 RMSE=6.27264

Figure 13: Compressed ‘Mandrill’ images by applying column DFT

Compression Ratio= 2 RMSE=8.59163

Compression Ratio= 4 RMSE=13.5814

Compression Ratio= 8 RMSE=17.0931

Figure 14: Compressed ‘Mandrill’ images by applying Row DFT

Figures 12, 13, 14shows compressed ‘Mandrill’ image using full, column and row DFT respectively.

In each of the three cases compression ratio 2, 4 and 8 is considered. It is observed that RMSE value of column DFT at compression ratio 8 is very closer to one obtained by total DFT at same compression ratio.

Compression Ratio= 2 RMSE=9.224652

Compression Ratio= 4 RMSE=14.47318

Compression Ratio= 8 RMSE=18.31172

Figure 15: Compressed ‘Mandrill’ images by applying full DCT

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Compression Ratio= 2 RMSE= 11.8693776

Compression Ratio= 4 RMSE= 17.16473991

Compression Ratio= 8 RMSE= 20.90614907

Figure 16: Compressed ‘Mandrill’ images by applying column DCT

Compression Ratio= 2 RMSE= 9.88663

Compression Ratio= 4 RMSE= 16.16006

Compression Ratio= 8 RMSE= 21.11612

Figure 17: Compressed ‘Mandrill’ images by applying row DCT

Figures 15,16,17 show compressed ‘Mandrill’ image using full, column and row DCT for compression ratio 2,4 and 8. Again it is observed that RMSE value of column DCT at compression ratio 8 is very closer to one obtained by total DCT at same compression ratio.

Compression Ratio= 2 RMSE= 11.5486

Compression Ratio= 4 RMSE= 16.14666

Compression Ratio= 8 RMSE= 19.64522

Figure 18: Compressed ‘Mandrill’ images by applying full Haar Transform

Compression Ratio= 2 RMSE=12.93917685

Compression Ratio= 4 RMSE= 18.34300842

Compression Ratio= 8 RMSE= 22.14955424

Figure 19: Compressed ‘Mandrill’ images by applying column Haar Transform

Compression Ratio= 2 RMSE=11.84323

Compression Ratio= 4 RMSE= 18.0668

Compression Ratio= 8 RMSE= 22.9688

Figure 20: Compressed ‘Mandrill’ images by applying row Haar Transform

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International Journal of Advances in Engineering & Technology, Mar. 2013. ©IJAET ISSN: 2231-1963 Similarly, figures 18, 19, 20 present compressed images for full, column and row Haar transform respectively. At compression ratio 8, it gives acceptable compressed image but RMSE is higher than DFT and DCT.

Compression Ratio= 2 RMSE= 11.38372

Compression Ratio= 4 RMSE= 16.26495

Compression Ratio= 8 RMSE= 19.62808

Figure 21: Compressed ‘Mandrill’ images by applying full Walsh Transform

Compression Ratio= 2 RMSE= 12.93918

Compression Ratio= 4 RMSE= 18.34301

Compression Ratio= 8 RMSE= 22.14955

Figure 22: Compressed ‘Mandrill’ images by applying column Walsh Transform

Compression Ratio= 2 RMSE=11.8432

Compression Ratio= 4 RMSE= 18.0668

Compression Ratio= 8 RMSE= 22.9688

Figure 23: Compressed ‘Mandrill’ images by applying row Walsh Transform

Same results regarding RMSE values are observed for Walsh transform in figure 21, 22 and 23. For full, column and row Walsh transforms, image quality is acceptable but at the cost of higher RMSE values.

Compression Ratio= 2 RMSE= 9.331971

Compression Ratio= 4 RMSE= 14.69161

Compression Ratio= 8 RMSE= 18.56635

Figure 24: Compressed ‘Mandrill’ images by applying full DST

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Compression Ratio= 2 RMSE= 12.2453

Compression Ratio= 4 RMSE= 18.74562

Compression Ratio= 8 RMSE= 24.4524

Figure 25: Compressed ‘Mandrill’ images by applying column DST

Compression Ratio= 2 RMSE=10.29123

Compression Ratio= 4 RMSE= 17.4362

Compression Ratio= 8 RMSE= 23.74187

Figure 26: Compressed ‘Mandrill’ images by applying row DST

As shown in figures 24, 25 and 26 DST also gives good image quality with less error in three different cases i.e. full column and row DST. Slant and Kekre’s transform show poor performance in terms of RMSE for comp ratio greater than two. As compressed image quality is not perceptible, these transforms are not recommended.

V.

CONCLUSIONS

Here performance of column transform, row transform and full transform is compared using Root Mean Square Error (RMSE) as a performance measure with respect to compression ratio. RMSE values are calculated for compression ratio 1 to 5. Experimental results prove that RMSE values obtained for various compression ratios in column transform are closer to those obtained in full transform of an image. Hence instead of full transform of an image, column transform can be used for image compression, saving half number of computations. RMSE obtained in row transform is quite higher than column and full transform at higher values of compression ratio. Hence it is not recommended. Good PSNR is obtained using column transform. Among all the seven transforms used, DFT, DCT and DST give better results in terms of RMSE and reconstructed image quality than other transforms. Walsh and Haar transforms also give acceptable results with an advantage of less computation whereas Slant and Kekre transform do not give good results. Hence they are not recommended.

VI.

FUTURE WORK

Future work includes application of orthogonal wavelet transforms on colour images. Change in the RMSE values if any, can be compared. Also PSNR values and quality of reconstructed image can be studied to compare their performance against the one in this paper.

REFERENCES [1]. Dipti Bhatnagar, Sumit Budhiraja, “Image Compression using DCT based Compressive Sensing and Vector Quantization”, IJCA, Vol50 (20), pp. 34-38, July 2012.

[2]. Ahmed, N., Natarajan T., Rao K. R.: Discrete cosine transform. In: IEEE Transactions on Computers, Vol. 23, 90-93, 1974.

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International Journal of Advances in Engineering & Technology, Mar. 2013. ©IJAET ISSN: 2231-1963 [5]. Bullmore, E., J. Fadili, V. Maxim, L. Sendur, J. Suckling, B. Whitcher, M. Brammer and M. Breakspear, “Wavelets and Functional Magnetic Resonance Imaging of the Human Brain NeuroImage”, 23(1): 234-249, 2004.

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[12]. H.B.Kekre, Tanuja Sarode, Sudeep Thepade, “Inception of hybrid wavelet Transform using Two orthogonal Transforms and It’s use for Image compression”, IJCSIS, vol 9, no. 6, 2011.

[13]. H. B. Kekre, Sudeep Thepade, “Image retrieval using Non – Involutional Orthogonal Kekre’s Transform”, International Journal of Multidisciplinary Research and Advances in Engineering (IJMRAE), Ascent Publication House, 2009, Volume 1, No. I, 2009. Abstract available online at www.Ascent-journals.com [14]. Mourence M. Anguh and Ralph R. Martin, “A truncation method for computing slant transforms with applications to image processing”, IEEE Trans. on Communications, vol. 43, no. 6, pp.

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[17]. H.B.Kekre, Tanuja Sarode, sudeep Thepade, Sonal Shroff, “ Instigation of Orthogonal Wavelet Transforms using Walsh, Cosine, Hartley, Kekre Transforms and their use in Image Compression”, International Journal of Computer Science and Information Security (IJCSIS), Vol 9, No. 6, pp. 125-133, 2011.

[18]. Veensdevi S.V., A. G. Ananth, “Fractal Image compression using Quadtree Decomposition and Huffman Coding”, Signal and Image Processing: An International Journal (SIPIJ), Vol. 3, No.2, pp. 207-212, April 2012.

AUTHORS H. B. Kekre has received B.E. (Hons.) in Telecomm. Engg. from Jabalpur University in 1958, M.Tech (Industrial Electronics) from IIT Bombay in 1960, M.S.Engg. (Electrical Engg.) from University of Ottawa in 1965 and Ph.D. (System Identification) from IIT Bombay in 1970. He has worked Over 35 years as Faculty of Electrical Engineering and then HOD Computer Science and Engg. at IIT Bombay. After serving IIT for 35 years, he retired in 1995. After retirement from IIT, for 13 years he was working as a professor and head in the department of computer engineering and Vice principal at Thadomal Shahani Engg. College, Mumbai. Now he is senior professor at MPSTME, SVKM’s NMIMS University. He has guided 17 Ph.Ds., more than 100 M.E./M.Tech and several B.E. / B.Tech projects, while in IIT and TSEC. His areas of interest are Digital Signal processing, Image Processing and Computer Networking. He has more than 450 papers in National / International Journals and Conferences to his credit. He was Senior Member of IEEE. Presently He is Fellow of IETE, Life Member of ISTE and Senior Member of International Association of Computer Science and Information Technology (IACSIT). Recently fifteen students working under his guidance have received best paper awards. Currently eight research scholars working under his guidance have been awarded Ph. D. by NMIMS (Deemed to be University). At present seven research scholars are pursuing Ph.D. program under his guidance.

Tanuja K. Sarode has received M.E. (Computer Engineering) degree from Mumbai University in 2004, Ph.D. from Mukesh Patel School of Technology, Management and Engg. SVKM’s NMIMS University, Vile-Parle (W), Mumbai, INDIA. She has more than 11 years of experience in teaching. Currently working as Assistant Professor in Dept. of Computer Engineering at Thadomal Shahani Engineering College, Mumbai. She is member

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International Journal of Advances in Engineering & Technology, Mar. 2013. ŠIJAET ISSN: 2231-1963 of International Association of Engineers (IAENG) and International Association of Computer Science and Information Technology (IACSIT). Her areas of interest are Image Processing, Signal Processing and Computer Graphics. She has 137 papers in National /International Conferences/journal to her credit.

Prachi Natu has received B.E. (Electronics and Telecommunication) degree from Mumbai University in 2004. Currently pursuing Ph.D. from NMIMS University. She has 08 years of experience in teaching. Currently working as Assistant Professor in Department of Computer Engineering at G. V. Acharya Institute of Engineering and Technology, Shelu (Karjat). Her areas of interest are Image Processing, Database Management Systems and Operating Systems. She has 12 papers in International Conferences/journal to her credit.

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