Optimal Image Compression Based on Sign and Magnitude Coding of Wavelet Coefficients
Wavelet transforms is a very powerful tools for image compression. One of its advantage is the provision of both spatial and frequency localization of image energy. However, wavelet transform coefficients are defined by both a magnitude and sign. While algorithms exist for efficiently coding the magnitude of the transform coefficients, they are not efficient for the coding of their sign. It is generally assumed that there is no compression gain to be obtained from the coding of the sign. Only recently have some authors begun to investigate the sign of wavelet coefficients in image coding. Some authors have assumed that the sign information bit of wavelet coefficients may be encoded with the estimated probability of 0.5; the same assumption concerns the refinement information bit. In this paper, we propose a new method for Separate Sign Coding (SSC) of wavelet image coefficients. The sign and the magnitude of wavelet image coefficients are examined to obtain their online probabilities. We use the scalar quantization in which the information of the wavelet coefficient to belong to the lower or to the upper sub-interval in the uncertainly interval is also examined. We show that the sign information and the refinement information may be encoded by the probability of approximately 0.5 only after about five bit planes. Two maps are separately entropy encoded: the sign map and the magnitude map. The refinement information of the wavelet coefficient to belong to the lower or to the upper sub-interval in the uncertainly interval is also entropy encoded. An algorithm is developed and simulations are performed on three standard images in grey scale: Lena, Barbara and Cameraman. Five scales are performed using the biorthogonal wavelet transform 9/7 filter bank. The obtained results are compared to JPEG2000 standard in terms of peak signal to noise ration (PSNR) for the three images and in terms of subjective quality (visual quality). It is shown that the proposed method outperforms the JPEG2000. The proposed method is also compared to other codec in the literature. It is shown that the proposed method is very successful and shows its performance in term of PSNR.
[1] W.B Pennebaker, and J.L. Mitchel, JPEG Still Image Data Compression
Standard, Van Nostrand Reinhold, 1993.
[2] G.K. Walace, "The jpeg still image compression standard", Comm.
ACM, vol.34, no.4, pp30-44, Apr. 1994.
[3] K.R. Rao, and P. Yip, Discrete Cosine Transform- Algorithms,
Advantages, Applications, Academic Press, 1990.
[4] H.S. Malvar, Signal Processing with Lapped Transforms, Norwood,
MA, Artech House, 1992.
[5] R.Grisel et F. Astrade, "Compression d-images par modélisation des
coefficients tcd en lois mélange et quantification adaptative dans
l-environnement jpeg", Traitement du Signal, vol.14, no.3, pp.301-315,
1997.
[6] S.G. Mallat, "A theory for multi-resolution signal decomposition: The
wavelet representation", IEEE Trans. On Pattern Analysis and Machine
Intelligence, vol.11, no.7, pp.674-693, Jul. 1989.
[7] J.M. Shapiro, "Embedded image coding using zerotree of wavelet
coefficients", IEEE Trans. On Signal Processing, vol.41, no.12,
pp.3445-3462, Dec. 1993.
[8] A. Zandi, J.D. Allen, E.L. Schwartz, and M. Boliek, "CREW:
Compression with reversible embedded wavelet", IEEE Data
Compression Conference, Snowbird, pp.212-221, Mar. 1995.
[9] A. Said, and W.A. Pearlman, "An image multi-resolution representation,
for lossless and lossy compression", IEEE Trans. On Image Processing,
vol.5, no.6, pp.1303-1310, Sep. 1996.
[10] A. Said, and W.A. Pearlman, "A new fast and efficient image codec
based on set partitioning in hierarchical trees", IEEE Trans. On Circuits
and Systems for Video Technology, vol.6, no.3, pp.243-250, Jun. 1996.
[11] S.A Martucci, I. Sodager, T.H. Chiang, and Y.Q. Zhang, "A zerotree
wavelet coder", IEEE Trans. On Circuits and Systems for Video
Technology, vol.7, no.1, pp.109-118, Fev. 1997.
[12] J. Li, P. Cheng, and C. Kuo, "On the improvement of embedded zerotree
wavelet coding", Proc. SPIE, Visual Communications and Image
Processing, Orlando, pp.1490-1501, Apr. 1995.
[13] P.G. Sherwood, and K. Zeger, "Progressive image coding for noisy
channels", IEEE Signal Processing Letters, vol.4, no.7, pp.189-191, Jul.
1997.
[14] H. Man, F. Kossentini, and M. Smith, "Robust EZW image coding for
nosy channels, IEEE Signal Processing Letters, vol.4, no.8, pp.227-229,
Aug. 1997.
[15] C.D Creusere, "A new method for robust image compression based on
the embedded zerotree wavelet algorithm", IEEE Trans. On Image
Processing Letters, vol.6, no.10, pp.1436-1442, Oct. 1997.
[16] J.K. Rogers, and P.C. Cosman, "Wavelet zerotree image compression
with packetization", IEEE Signal Processing Letters, vol.5, no.5,
pp.105-107, May. 1998.
[17] Z. Xiong, K. Ramchandran, and M.T. Orchard, "Space frequency
quantization for wavelet image coding" IEEE Tran. On Image
Processing, vol.6, no.5, pp.677-693, May. 1997.
[18] Z. Xiong, K. Ramchandran, and M.T. Orchard, "Wavelet packets image
coding using space frequency quantization", IEEE Trans. On Image
Processing, vol.7, pp.892-898, Jun. 1998.
[19] S. Joo, H. Kikuchi, S. Sasaki, and J. Shin, "Flexible zerotree coding of
wavelet coefficients", IEICE Trans. Fundamentals, vol.E82-A, no.4,
Apr. 1999.
[20] M.W. Marcellin, M.J. Gormis, A. Bilgin and M.P. Boliek, "An
overview JPEG2000", Proc. IEEE Data Compression Conference,
pp.523-541, 2000.
[21] J.D Villasenor, B. Belzer, and J. Lio, "Wavelet filter evaluation for
image compression", IEEE Tran. On Image Processing, vol.6, no.11,
pp.289-290, Aug. 1995.
[22] M. Antoni, M. Barlaud, P. Mathieu, and I. Daubechies, "Image coding
using wavelet transform", IEEE trans. On Image Processing, vol.1,
no.2, pp.205-220, Apr. 1992.
[23] E. Schwartz, A. Zandik, "Implementation of compression with reversible
embedded wavelets", SPIE, no. 2564, pp.32-43, 1995.
[24] X. Wu, "High-order context modelling and embedded conditional
entropy coding of wavelet coefficients for image compression", Proc.
31th Asilomor Conf. on Signals, Systems and Computers, pp.1378-1382,
1997.
[25] A. Bigin, P. Sementilli, and M. Marcellin, "Progressive image coding
using treillis coded quantization", IEEE Tran. On Image Processing,
vol.11, no.8, pp.1638-1643, Nov. 1999.
[26] D. Taubman, "High performance scalable image compression with
EBCOT", Proc. ICIP, pp.344-348, 1999.
[27] A. Deever, and S.S. Hemami, "What-s your sign: Efficient sign coding
for embedded wavelet image coding", Proc. Data Compression
Conference, 2000, Snowbird, Utah, March 2000.
[28] C. Tion, and S.S. Hemami, "An embedded image coding system based
on tar filter with classification", Proc. ICASSP, Montreal, Quebec,
Canada, May 2004.
[29] M. Jér├┤me, "Optimal Image coding based on probability distribution of
embedded zerotree wavelet symbols", Tunisian-German Conference on
Smart Systems and Devices, pp.666-671, Hammamet, Tunisia, Mar. 27-
30, 2001.
[30] M. Jér├┤me, Compression d-Images par Ondelettes, PhD Thesis, Ecole
Nationale d-ingénieurs de Tunis, Jul. 2002.
[31] M. Jér├┤me, and N. Ellouze, "Very low bit rate coding of image sequence
using embedded zerotree wavelet and symbol probability distributions",
Proc. IEEE 3th Inter. Conf. on Information, Communication & Signal
Processing, Singapore, Oct.15-18, 2001.
[32] M. Jér├┤me and N. Ellouze, "Embedded zerotree wavelet coding of image
sequence", Proc. Inter. Conf. on Wavelet Analysis and Its Applications,
Springer Publisher, Lecture Notes in Computer Science, vol. 2251, ISBN
3-540-43034-2, Hong Kong, pp.65-75, Dec.18-20, 2001.
[33] www.aware/jpeg2000 J2K-Tool, JPEG2000
Compression/Decompression Tool 2 5.1, Aware Inc.
[1] W.B Pennebaker, and J.L. Mitchel, JPEG Still Image Data Compression
Standard, Van Nostrand Reinhold, 1993.
[2] G.K. Walace, "The jpeg still image compression standard", Comm.
ACM, vol.34, no.4, pp30-44, Apr. 1994.
[3] K.R. Rao, and P. Yip, Discrete Cosine Transform- Algorithms,
Advantages, Applications, Academic Press, 1990.
[4] H.S. Malvar, Signal Processing with Lapped Transforms, Norwood,
MA, Artech House, 1992.
[5] R.Grisel et F. Astrade, "Compression d-images par modélisation des
coefficients tcd en lois mélange et quantification adaptative dans
l-environnement jpeg", Traitement du Signal, vol.14, no.3, pp.301-315,
1997.
[6] S.G. Mallat, "A theory for multi-resolution signal decomposition: The
wavelet representation", IEEE Trans. On Pattern Analysis and Machine
Intelligence, vol.11, no.7, pp.674-693, Jul. 1989.
[7] J.M. Shapiro, "Embedded image coding using zerotree of wavelet
coefficients", IEEE Trans. On Signal Processing, vol.41, no.12,
pp.3445-3462, Dec. 1993.
[8] A. Zandi, J.D. Allen, E.L. Schwartz, and M. Boliek, "CREW:
Compression with reversible embedded wavelet", IEEE Data
Compression Conference, Snowbird, pp.212-221, Mar. 1995.
[9] A. Said, and W.A. Pearlman, "An image multi-resolution representation,
for lossless and lossy compression", IEEE Trans. On Image Processing,
vol.5, no.6, pp.1303-1310, Sep. 1996.
[10] A. Said, and W.A. Pearlman, "A new fast and efficient image codec
based on set partitioning in hierarchical trees", IEEE Trans. On Circuits
and Systems for Video Technology, vol.6, no.3, pp.243-250, Jun. 1996.
[11] S.A Martucci, I. Sodager, T.H. Chiang, and Y.Q. Zhang, "A zerotree
wavelet coder", IEEE Trans. On Circuits and Systems for Video
Technology, vol.7, no.1, pp.109-118, Fev. 1997.
[12] J. Li, P. Cheng, and C. Kuo, "On the improvement of embedded zerotree
wavelet coding", Proc. SPIE, Visual Communications and Image
Processing, Orlando, pp.1490-1501, Apr. 1995.
[13] P.G. Sherwood, and K. Zeger, "Progressive image coding for noisy
channels", IEEE Signal Processing Letters, vol.4, no.7, pp.189-191, Jul.
1997.
[14] H. Man, F. Kossentini, and M. Smith, "Robust EZW image coding for
nosy channels, IEEE Signal Processing Letters, vol.4, no.8, pp.227-229,
Aug. 1997.
[15] C.D Creusere, "A new method for robust image compression based on
the embedded zerotree wavelet algorithm", IEEE Trans. On Image
Processing Letters, vol.6, no.10, pp.1436-1442, Oct. 1997.
[16] J.K. Rogers, and P.C. Cosman, "Wavelet zerotree image compression
with packetization", IEEE Signal Processing Letters, vol.5, no.5,
pp.105-107, May. 1998.
[17] Z. Xiong, K. Ramchandran, and M.T. Orchard, "Space frequency
quantization for wavelet image coding" IEEE Tran. On Image
Processing, vol.6, no.5, pp.677-693, May. 1997.
[18] Z. Xiong, K. Ramchandran, and M.T. Orchard, "Wavelet packets image
coding using space frequency quantization", IEEE Trans. On Image
Processing, vol.7, pp.892-898, Jun. 1998.
[19] S. Joo, H. Kikuchi, S. Sasaki, and J. Shin, "Flexible zerotree coding of
wavelet coefficients", IEICE Trans. Fundamentals, vol.E82-A, no.4,
Apr. 1999.
[20] M.W. Marcellin, M.J. Gormis, A. Bilgin and M.P. Boliek, "An
overview JPEG2000", Proc. IEEE Data Compression Conference,
pp.523-541, 2000.
[21] J.D Villasenor, B. Belzer, and J. Lio, "Wavelet filter evaluation for
image compression", IEEE Tran. On Image Processing, vol.6, no.11,
pp.289-290, Aug. 1995.
[22] M. Antoni, M. Barlaud, P. Mathieu, and I. Daubechies, "Image coding
using wavelet transform", IEEE trans. On Image Processing, vol.1,
no.2, pp.205-220, Apr. 1992.
[23] E. Schwartz, A. Zandik, "Implementation of compression with reversible
embedded wavelets", SPIE, no. 2564, pp.32-43, 1995.
[24] X. Wu, "High-order context modelling and embedded conditional
entropy coding of wavelet coefficients for image compression", Proc.
31th Asilomor Conf. on Signals, Systems and Computers, pp.1378-1382,
1997.
[25] A. Bigin, P. Sementilli, and M. Marcellin, "Progressive image coding
using treillis coded quantization", IEEE Tran. On Image Processing,
vol.11, no.8, pp.1638-1643, Nov. 1999.
[26] D. Taubman, "High performance scalable image compression with
EBCOT", Proc. ICIP, pp.344-348, 1999.
[27] A. Deever, and S.S. Hemami, "What-s your sign: Efficient sign coding
for embedded wavelet image coding", Proc. Data Compression
Conference, 2000, Snowbird, Utah, March 2000.
[28] C. Tion, and S.S. Hemami, "An embedded image coding system based
on tar filter with classification", Proc. ICASSP, Montreal, Quebec,
Canada, May 2004.
[29] M. Jér├┤me, "Optimal Image coding based on probability distribution of
embedded zerotree wavelet symbols", Tunisian-German Conference on
Smart Systems and Devices, pp.666-671, Hammamet, Tunisia, Mar. 27-
30, 2001.
[30] M. Jér├┤me, Compression d-Images par Ondelettes, PhD Thesis, Ecole
Nationale d-ingénieurs de Tunis, Jul. 2002.
[31] M. Jér├┤me, and N. Ellouze, "Very low bit rate coding of image sequence
using embedded zerotree wavelet and symbol probability distributions",
Proc. IEEE 3th Inter. Conf. on Information, Communication & Signal
Processing, Singapore, Oct.15-18, 2001.
[32] M. Jér├┤me and N. Ellouze, "Embedded zerotree wavelet coding of image
sequence", Proc. Inter. Conf. on Wavelet Analysis and Its Applications,
Springer Publisher, Lecture Notes in Computer Science, vol. 2251, ISBN
3-540-43034-2, Hong Kong, pp.65-75, Dec.18-20, 2001.
[33] www.aware/jpeg2000 J2K-Tool, JPEG2000
Compression/Decompression Tool 2 5.1, Aware Inc.
@article{"International Journal of Information, Control and Computer Sciences:55799", author = "Mbainaibeye Jérôme and Noureddine Ellouze", title = "Optimal Image Compression Based on Sign and Magnitude Coding of Wavelet Coefficients", abstract = "Wavelet transforms is a very powerful tools for image compression. One of its advantage is the provision of both spatial and frequency localization of image energy. However, wavelet transform coefficients are defined by both a magnitude and sign. While algorithms exist for efficiently coding the magnitude of the transform coefficients, they are not efficient for the coding of their sign. It is generally assumed that there is no compression gain to be obtained from the coding of the sign. Only recently have some authors begun to investigate the sign of wavelet coefficients in image coding. Some authors have assumed that the sign information bit of wavelet coefficients may be encoded with the estimated probability of 0.5; the same assumption concerns the refinement information bit. In this paper, we propose a new method for Separate Sign Coding (SSC) of wavelet image coefficients. The sign and the magnitude of wavelet image coefficients are examined to obtain their online probabilities. We use the scalar quantization in which the information of the wavelet coefficient to belong to the lower or to the upper sub-interval in the uncertainly interval is also examined. We show that the sign information and the refinement information may be encoded by the probability of approximately 0.5 only after about five bit planes. Two maps are separately entropy encoded: the sign map and the magnitude map. The refinement information of the wavelet coefficient to belong to the lower or to the upper sub-interval in the uncertainly interval is also entropy encoded. An algorithm is developed and simulations are performed on three standard images in grey scale: Lena, Barbara and Cameraman. Five scales are performed using the biorthogonal wavelet transform 9/7 filter bank. The obtained results are compared to JPEG2000 standard in terms of peak signal to noise ration (PSNR) for the three images and in terms of subjective quality (visual quality). It is shown that the proposed method outperforms the JPEG2000. The proposed method is also compared to other codec in the literature. It is shown that the proposed method is very successful and shows its performance in term of PSNR.
", keywords = "Image compression, wavelet transform, sign coding, magnitude coding.", volume = "1", number = "7", pages = "2036-9", }
