This paper emphasizes on designing of a novice quantizer that is more suitable for compressing images through curvelet transform  This compression algorithm is tested on various images like plain  textured and building images  The results are compared with the existing techniques like curvelet with existing quatizer  wavelet with existing quantizer and wavelet with proposed quantzer  The proposed algorithm    curvelets with proposed quantizer    outperforms the existing techniques  The performance is evaluated through visual clarity   Peak Signal to Noise Ratio  PSNR  and compression metrics such as Compression ratio and Bit-rat

Dr. G.Jagadeeswar Reddy

English

Global Journal of Computer Science and Technology (GJCST)

© 2012. G.Jagadeeswar Reddy, T.Jayachandraprasad, M.N.Giriprasad, M. Madhavi Latha & T. Satya Savithri. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non-commercial use, distribution, and reproduction inany medium, provided the original work is properly cited.    Global Journal of Computer Science and Technology Volume 12 Issue 8 Version 1.0 April 2012 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 0975-4172 & Print ISSN: 0975-4350  Curvelets with New Quantizer for Image Compression By G.Jagadeeswar Reddy, T.Jayachandraprasad, M.N.Giriprasad, M. Madhavi Latha & T. Satya Savithri                                                                      EVMCET, Narasaraopet Guntur, AP, INDIA                                         Abstract - This paper emphasizes on designing of a novice quantizer that is more suitable for compressing images through curvelet transform. This compression algorithm is tested on various images like plain, textured and building images. The results are compared with the existing techniques like curvelet with existing quatizer, wavelet with existing quantizer and wavelet with proposed quantzer. The proposed algorithm “curvelets with proposed quantizer” outperforms theexisting techniques. The performance is evaluated through visual clarity , Peak Signal to Noise Ratio (PSNR) and compression metrics such as Compression ratio and Bit-rate.Keywords : Quantizer, Compression ratio, Bit-rate, curvelet, wavelet.GJCST Classification: I.4.2Curvelets with New Quantizer for Image CompressionStrictly as per the compliance and regulations of:Curvelets with New Quantizer for Image Compression G.Jagadeeswar Reddy α, T.Jayachandraprasad σ, M.N.Giriprasad ρ, M. Madhavi Latha Ѡ & T. Satya Savithri¥                                                with variety of elongated shapes with different aspect ratios and are oriented at variety of directions. In particular, it has been designed for representing edges and other singularities along curves efficiently compared to traditional transforms, i.e. using very few coefficients for a given accuracy of reconstruction. Approximately for representing an edge to squared error 1/N requires 1/N wavelets and only about curvelets. In this paper practical implementations of proposed compression method is focused. The next section discusses the quantizer design for proposed compression method, section 3 demonstrates the algorithm of new compression technique and section 4 explains the simulation results. Finally, the superiority of proposed technique over existing methods is demonstrated using PSNR and compression metrics. II. Quantizer Design and Coding The uniform quantizer is the most commonly used scalar quantizer for transform based image compression due to its simplicity. It is also known as a linear quantizer since its staircase input-output response lies along a straight line(with a unit slope). Two commonly used linear staircase quantizers are the midtread and the midrise quantizers. a) Existing Quantizer From MATLAB tool box, it is observed that the quantizer is defined as: y=Q _T(x)               where: Q _T(x) = 0 , if |x|<T                              Q _T(x) = sgn(x) * [x/T], otherwise     (1) The input values for quantizer are created with evenly spacing and bin size(T) is assumed as 0.1. The quantization curve is as shown in Fig. 1(a). From the quantization curve, it can be said that the above quantizer is midtread. For decompression, de-quantized values are computed using the following equation. Dq=sgn(Q) * (abs(Q)+0.5)*T                                          (2) Where Q are quantizer output values and Dq are de-quantized values. In the above equation 0.5 is chosen to get the de-quantized values at the mid-point of each quantization bin. De-quantized real values are shown in Fig. 1(b). © 2012 Global Journals Inc.  (US)Global Journal of Computer Science and Technology  Volume XII  Issue VIII  Version I        65  2012April1/ NAbstract - This paper emphasizes on designing of a novice quantizer that is more suitable for compressing images through curvelet transform. This compression algorithm is tested on various images like plain, textured and building images. The results are compared with the existing techniques like curvelet with existing quatizer, wavelet with existing quantizer and wavelet with proposed quantzer. The proposed algorithm “curvelets with proposed quantizer” outperforms the existing techniques. The performance is evaluated through visual clarity , Peak Signal to Noise Ratio (PSNR) and compression metrics such as Compression ratio and Bit-rate.Keywords: Quantizer, Compression ratio, Bit-rate, curvelet, wavelet.I. Introductionast few years have produced abundant results in compression techniques for images because they are being used widely in internet technologies etc. So far wavelet based compression of digital signals and images is a topic of interest. Throughout the scientific and engineering disciplines many hundreds of papers published [1-8] in journals, in which a wide range of tools and ideas have been proposed and studied. The major drawback associated with wavelet based techniques are their inability to capture geometry of two dimensional edges, though they have proven their worth in number of fields. Hence, the field of image processingrequiers a true 2-dimensional transform that can efficiently handle the intrinsic geometrical structure which is the key in visual information.Candes and Donoho [9] developed a new multi-scale transform which is called as the curvelet transform . It was the first proposed in the context of objects f(x1 ; x2) defined on the continuum plane (x1,x2)ÎR2, which is Motivated by the needs of image analysis. The new transform is believed to capture image information more efficiently than the wavelettransform by giving basis elements in addition to having the qualities of wavelet. Basis functions in  curvelets areAuthor α : Ece Dept., Evmcet, Narasaraopet-522601, Guntur, Ap, India. Email: Jagsuni@Yahoo.ComAuthor σ : Ece Dept., Rgmcet, Nandyal-518502, Kurnool, Ap, India. Email:Jp.Talari@Gmail.ComAuthor ρ : Ece Dept., Jntuce, Pulivendula-515002, Ap, India. Email:Mahendran_Gp@Rediffmail.ComAuthor Ѡ : Ece Dept., Jntuh, Hyderabad- 500085, A.P, India. Email: Mlmakkena@Yahoo.ComAuthor ¥ : Ece Dept., Jntuh, Hyderabad- 500085, A.P, India. Email: Tirumalasatya@Rediff.ComP              (a)                                           (b) Fig. 1:  Existing Quantizer (a) Quantization curve (b) De-quantized values It is known that the subinterval size of the midtread quantizer is larger than the midrise quantizer. Hence for the uniformly distributed input, the average quantization error of the midtread quantizer is larger than that of midrise quantizer. b) Proposed Quantizer The two constraints to be considered while designing quantizer are as follows: 1. Quantizer must have average quantization error as low as possible. 2. De-quantized values must be exactly at the mid-point of each quantization bin. Since the source considered for compression application is uniformly distributed, midrise quantizer is preferred, which satisfies the first constraint. To obtain the midrise quatizer and to satisfy second constraint, the midtread quantizer equations are restructured as: y=Q _T(x)              where: Q _T(x) = 0 , if |x|<T                      Q _T(x) = sgn(x) * ([x/T]+K)*T, otherwise (3) The input values for quantizer are created with evenly spacing and bin size(T) is assumed as 0.1. Here K is a constant. For decompression, de-quantized equation is modified as: Dq=sgn(Q) * ([abs(Q)/T]-M)*T                                        (4) Where Q are quantizer output values, Dq are de-quantized values and M is a constant. The above equations are tested for different values of K and M. Finally for K=0.8 and M=0.3 the quantization and de-quantizion curves are efficient andthe de-quantized values are exactly at the mid-point of each quantization bin as shown in Fig. 2.                  (a)                                                 (b) Fig. 2 :  Proposed Quantizer with K=0.8 and M=0.3 (a) Quantization curve (b) De-quantized values The quantization curve resembles midrise quantizer. Hence the proposed quantizer suits for the assumed uniformly distributed source, whose quantization equation becomes as: y=Q _T(x)               Q _T(x) = 0, if |x|<T               Q _T(x) = sgn(x) * ([x/T]+0.8)*T, otherwise     (5) and the de-quantization equation as: Dq=sgn(Q) * ([abs(Q)/T]-0.3)*T                                     (6) Where Q are quantizer output values and Dq are de-quantized values. The purpose of an entropy encoder (symbol encoder) is to reduce the number of bits required to represent each symbol at the quantizer output. For simplicity one of the entropy encoding technique i.e. an arithmetic coding, which is an integral part of MATLAB tool box has been used to serve the purpose. III. The algorithm The following steps are involved in the compression algorithm of Curvelet Transform. 1. Read input image f. 2. Apply the  2D FFT   and   obtain   Fourier   samples  3. For  each  scale j  and  angle l,  form  the  product,  4. Wrap this  product  around  the  origin  and  obtain, where the range for n1 and n2  is nowand      ( for       in    the   range  5. Apply the inverse 2D FFT to each  , hence collecting the discrete coefficients . 6. Quantize the coefficients with the proposed quantizer. 7. Entropy code the quantizer outputs. 8. Apply inverse operations to the result of step 7. Global Journal of Computer Science and Technology  Volume XII  Issue VIII  Version I          66    2012April©  2012 Global Journals Inc.  (US)Curvelets with New Quantizer for Image Compression1 2 1 2ˆ[ , ], / 2 , / 2f n n n n n n− ≤ <, 1 2 1 2ˆ[ , ] [ , ]j lU n n f n n, 1 2 , 1 2ˆ[ , ] ( )[ , ],j l j lf n n W U f n n= 1 1,0 jn L≤ <2 2,0 jn L≤ < θ,j lf( , , )Dc j l k( / 4, / 4)π π− ) .                                                                        © 2012 Global Journals Inc.  (US)Global Journal of Computer Science and Technology  Volume XII  Issue VIII  Version I        67  2012AprilCurvelets with New Quantizer for Image CompressionBased on the algorithm stated above, a compression technique has been developed for images of different sizes, which is tested using Curvlab[10] and MATLAB tools. The algorithm developed is working to a satisfaction and results are encouraging in case of curvelets. The performance of the algorithm is evaluated using the following metrics and index  (7) (8) (9) where M1 and M2 are the size of   the   image.    f (i, j ) is the original image, f (i, j) is the decompressed image. IV. Results and Discussion In the simulation, the proposed algorithm is tested on various images like plain (Barbara), Building and Textured images. The results are presented in Figs. 3 to 5 and in Table 1. The results indicate that there exists a qualitative difference between the methods considered for compression and reconstruction of image under test. The following observations are made from the results. 1. The curvelet with proposed quantizer algorithm enjoys superior compression ratio over wavelet with existing quantizer, curvelet with existing and wavelet with proposedquantizer algorithms. The performance is judged mainly through visual clarity and through PSNR. 2. The curvelet with proposed quantizer reconstruction displays higher sensitivity even at higher compression ratios. 3. In case of plain (Barbara) image the curvelet with proposed quantizer performs better at all compression ratios compared to other algorithms. When it is tested on Building and Textured images at low compression ratios curvelet with existing quantizer performs better compared to other algorithms, but at higher compression ratios curvelet with proposed quantizer outperforms the other algorithms irrespective of image.  (a)        (b)                                      (c)                                                (d)                                       (e) Fig. 3 :  Barbara image for compression ratio of 1:20 (a) Original image (b) Wavelet with existing quantizer (c) Curvelet with existing quantizer (d) Wavelet with proposed quantizer (e) Curvelet with proposed quantizer ( )(%)( )Output file size bytesCompression ratioInput file size bytes=8*( ( ))( )( )Output file size bytesRate bppInput file size bytes=1 221 221 1max( ( , ))10 log10[ ( , ) ( , )]M Mi jM M f i jpsnr dBf i j f i j= =× ×=′−∑∑              Fig. 4 :  PSNR Vs Compression ratio corresponding to Wavelet with existing quantizer, Curvelet with existing quantizer, Wavelet with proposed quantizer and Curvelet with proposed quantizer for Barbara image               Fig. 5:  Bit rate Vs PSNR corresponding to Wavelet with existing quantizer, Curvelet with existing quantizer, Wavelet with proposed quantizer and Curvelet with proposed quantizer for Barbara image Table 1: PSNR Vs Compression ratio for various algorithms w.r.t. Barbara image                 V. Conclusion In this paper practical implementations of proposed compression method is focused, tested on various types of images and it is proved that curvelet with proposed compression algorithm outperforms the existing techniques even at higher compression ratios.    Global Journal of Computer Science and Technology  Volume XII  Issue VIII  Version I          68    2012April©  2012 Global Journals Inc.  (US)Curvelets with New Quantizer for Image CompressionS.No Compression ratio PSNR in dB(Barbara image) Wavelet With existing quantizerCurvelet With existing quantizer Wavelet With proposed quanizer Curvelet With proposed quantizer 1 10 18.5611 25.9800 28.7499 32.6106 2 20 15.0924 20.8571 25.5371 29.4896 3 30 14.4269 19.2967 24.2425 27.8719 4 40 14.1493 18.6013 23.4729 26.8112 5 50 13.5877 18.2636 22.9075 26.0476 6 60 13.6517 18.1034 22.4776 25.4763 7 70 13.6520 17.8048 22.1051 25.0048 8 80 13.6514 17.4066 21.7726 24.6282 9 90 13.6510 16.6934 21.5213 24.2929 10 100 13.6500 15.6489 21.2750 24.0100 Applied and Computational Harmonic Analysis, pp.147-187;1997. 2. Colm Mulcahy, Ph.D., “Image compression using the Haar wavelet transform” Spelman Science and Math journal, pp:22-31. 3. J. Rissanen, “Stochastic Complexity in Statistical Inquiry,” World Scientific, 189. 4. G.J. Sullivan, “Efficient Scalar Quantization of Exponential and Laplacian Random Variables,” IEEE Trans. On Information theory, vol.42, no.5, pp.1365-1374; Sept 1996. 5. M.N.Do and M.Vetterli, “Orthonormal finite ridgelet transform for image compression,” in proc.IEEE Int. Conf. Image Processing (ICIP); sept.2000. 6. Pat Yip, “The Transform and Data Compression Handbook”, CRC Press ; sept. 2000. 7. M.L.Hilton “Compressing still and Moving Images with wavelets”, Multimedia systems, vol2,no3;1994. 8. Anil K. Jain, “Fundamentals of Digital Image Processing”, Prentice Hall; 2000. 9. E.J. Candes and D.L. Donoho, “Curvelets.” Manuscript. http: //www.stat.stanford.edu/~ donoho/ Reports/1998/curvelets.zip; 1999. 10. Candes, E., Demanet, L.,Donoho, D. and Ying, L. “Fast Discrete Curvelet Transforms,” Applied and Comp.Math., Caltech; July 2005. © 2012 Global Journals Inc.  (US)Global Journal of Computer Science and Technology  Volume XII  Issue VIII  Version I        69  2012AprilCurvelets with New Quantizer for Image CompressionReferences références referencias1. F.G.Meyer and R.R. Coifman. “Brushlets: a tool for directional image analysis and image compression,” 

Curvelets With New Quantizer For Image Compression

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Curvelets With New Quantizer For Image Compression

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