Denoising and Compression in Wavelet Domainvia Projection on to Approximation Coefficients
We describe a new filtering approach in the wavelet domain for image denoising and compression, based on the projections of details subbands coefficients (resultants of the splitting procedure, typical in wavelet domain) onto the approximation subband coefficients (much less noisy). The new algorithm is called Projection Onto Approximation Coefficients (POAC). As a result of this approach, only the approximation subband coefficients and three scalars are stored and/or transmitted to the channel. Besides, with the elimination of the details subbands coefficients, we obtain a bigger compression rate. Experimental results demonstrate that our approach compares favorably to more typical methods of denoising and compression in wavelet domain.
[1] D. L. Donoho, "De-noising by soft-thresholding," IEEE Trans. Inform.
Theory, vol. 41, no. 3, pp. 613-627, 1995.
[2] D. L. Donoho, and I. M. Johnstone, "Adapting to unknown smoothness
via wavelet shrinkage," Journal of the American Statistical Assoc., vol.
90, no. 432, pp. 1200-1224., 1995.
[3] D. L. Donoho, and I. M. Johnstone, "Ideal spatial adaptation by wavelet
shrinkage," Biometrika, 81, 425-455, 1994.
[4] I. Daubechies. Ten Lectures on Wavelets, SIAM, Philadelphia, PA.
1992.
[5] I. Daubechies, "Different Perspectives on Wavelets," in Proceedings of
Symposia in Applied Mathematics, vol. 47, American Mathematical
Society, USA, 1993.
[6] S. Mallat, "A theory for multiresolution signal decomposition: The
wavelet representation," IEEE Trans. Pattern Anal. Machine Intell., vol.
11, pp. 674-693, July 1989.
[7] S. G. Mallat, "Multiresolution approximations and wavelet orthonormal
bases of L2 (R)," Transactions of the American Mathematical Society,
315(1), pp.69-87, 1989a.
[8] X.-P. Zhang, and M. Desai, "Nonlinear adaptive noise suppression based
on wavelet transform," Proceedings of the ICASSP98, vol. 3, pp. 1589-
1592, Seattle, 1998.
[9] X.-P. Zhang, "Thresholding Neural Network for Adaptive Noise
reduction," IEEE Transactions on Neural Networks, vol.12, no. 3,
pp.567-584, 2001.
[10] X.-P. Zhang, and M. Desai, "Adaptive Denoising Based On SURE
Risk," IEEE Signal Proc. Letters, vol.5, no. 10, 1998.
[11] X.-P. Zhang and Z.Q. Luo, "A new time-scale adaptive denoising
method based on wavelet shrinkage," in Proceedings of the ICASSP99,
Phoenix, AZ., March 15-19, 1999.
[12] M. Lang, H. Guo, J. Odegard, C. Burrus, and R. Wells, "Noise reduction
using an undecimated discrete wavelet transform," IEEE Signal Proc.
Letters, vol. 3, no. 1, pp. 10-12, 1996.
[13] H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian
wavelet shrinkage," J. Amer. Statist. Assoc., vol. 92, pp. 1413-1421,
1997.
[14] S. G. Chang, B. Yu, and M. Vetterli, "Spatially adaptive wavelet
thresholding with context modeling for image denoising," IEEE Trans.
Image Processing, vol. 9, pp. 1522-1531, Sept. 2000.
[15] S. G. Chang, B. Yu, and M. Vetterli, "Adaptive wavelet thresholding for
image denoising and compression," IEEE Trans. Image Processing, vol.
9, pp. 1532-1546, Sept. 2000.
[16] S. G. Chang and M. Vetterli, "Spatial adaptive wavelet thresholding for
image denoising," in Proc. ICIP, vol. 1, 1997, pp. 374-377.
[17] M. S. Crouse, R. D. Nowak, and R. G. Baraniuk, "Wavelet-based
statistical signal processing using hidden Markov models," IEEE
Trans.Signal Processing, vol. 46, pp. 886-902, Apr. 1998.
[18] M. Malfait and D. Roose, "Wavelet-based image denoising using a
Markov random field a priori model," IEEE Trans. Image Processing,
vol. 6, pp. 549-565, Apr. 1997.
[19] M. K. Mihcak, I. Kozintsev, K. Ramchandran, and P. Moulin, "Low
complexity image denoising based on statistical modeling of wavelet
coefficients," IEEE Trans. Signal Processing Lett., vol. 6, pp. 300-303,
Dec. 1999.
[20] E. P. Simoncelli, "Bayesian denoising of visual images in the wavelet
domain," in Bayesian Inference in Wavelet Based Models. New York:
Springer-Verlag, 1999, pp. 291-308.
[21] E. Simoncelli and E. Adelson, "Noise removal via Bayesian wavelet
coring," in Proc. ICIP, vol. 1, 1996, pp. 379-382.
[22] M. Belge, M. E. Kilmer, and E. L. Miller, "Wavelet domain image
restoration with adaptive edge-preserving regularization," IEEE Trans.
Image Processing, vol. 9, pp. 597-608, Apr. 2000.
[23] J. Liu and P. Moulin, "Information-theoretic analysis of interscale and
intrascale dependencies between image wavelet coefficients," IEEE
Trans. Image Processing, vol. 10, pp. 1647-1658, Nov. 2000.
[24] H. Guo, J. E. Odegard, M. Lang, R. A. Gopinath, I. Selesnick, and C. S.
Burrus, "Speckle reduction via wavelet shrinkage with application to
SAR based ATD/R," Technical Report CML TR94-02, CML, Rice
University, Houston, 1994.
[25] R.R. Coifman, and D.L. Donoho, Translation-invariant de-noising. A.
Antoniadis & G. Oppenheim (eds), Lecture Notes in Statistics, vol. 103.
Springer-Verlag, pp 125-150, 1995.
[26] M. Misiti, Y. Misiti, G. Oppenheim, and J.M. Poggi. (2001, June).
Wavelet Toolbox, for use with MATLAB®, User-s guide, version 2.1.
(Online). Available: http://www.rrz.unihamburg.
de/RRZ/Software/Matlab/Dokumentation/help/pdf_doc/wavele
t/wavelet_ug.pdf
[27] C.S. Burrus, R.A. Gopinath, and H. Guo, Introduction to Wavelets and
Wavelet Transforms: A Primer, Prentice Hall, New Jersey, 1998.
[28] B.B. Hubbard, The World According to Wavelets: The Story of a
Mathematical Technique in the Making, A. K. Peter Wellesley,
Massachusetts, 1996.
[29] A. Grossman and J. Morlet, "Decomposition of Hardy Functions into
Square Integrable Wavelets of Constant Shape," SIAM J. App Math, 15:
pp.723-736, 1984.
[30] C. Valens. (2004). A really friendly guide to wavelets. (Online).
Available: http://perso.wanadoo.fr/polyvalens/
clemens/wavelets/wavelets.html
[31] G. Kaiser, A Friendly Guide To Wavelets, Boston: Birkhauser, 1994.
[32] J.S. Walker, A Primer on Wavelets and their Scientific Applications,
Chapman & Hall/CRC, New York, 1999.
[33] E. J. Stollnitz, T. D. DeRose, and D. H. Salesin, Wavelets for Computer
Graphics: Theory and Applications, Morgan Kaufmann Publishers, San
Francisco, 1996.
[34] J. Shen and G. Strang, "The zeros of the Daubechies polynomials," in
Proc. American Mathematical Society, 1996.
[35] R. Yu, A.R. Allen, and J. Watson, An optimal wavelet thresholding for
speckle noise reduction, in Summer School on Wavelets: Papers,
Publisher: Silesian Technical University (Gliwice, Poland), pp77-81,
1996.
[36] H.Y. Gao, and A.G. Bruce, "WaveShrink with firm shrinkage,"
Statistica Sinica, 7, 855-874, 1997.
[37] L. Gagnon, and F.D. Smaili, "Speckle noise reduction of air-borne SAR
images with Symmetric Daubechies Wavelets," in SPIE Proc. #2759, pp.
1424, 1996.
[38] M.S. Crouse, R. Nowak, and R. Baraniuk, "Wavelet-based statistical
signal processing using hidden Markov models," IEEE Trans. Signal
Processing, vol 46, no.4, pp.886-902, 1998.
[39] M. Mastriani y A. Giraldez, "Smoothing of coefficients in wavelet
domain for speckle reduction in Synthetic Aperture Radar images,"
ICGST International Journal on Graphics, Vision and Image Processing
(GVIP), Volume 6, pp. 1-8, 2005. (Online). Available:
http://www.icgst.com/gvip/v6/P1150517003.pdf
[40] M. Mastriani y A. Giraldez, "Despeckling of SAR images in wavelet
domain," GIS Development Magazine, Sept. 2005, Vol. 9, Issue 9,
pp.38-40. (Online). Available:
http://www.gisdevelopment.net/magazine/years/2005/sep/wavelet_1.htm
[41] M. Mastriani y A. Giraldez, "Microarrays denoising via smoothing of
coefficients in wavelet domain," WSEAS Transactions on Biology and
Biomedicine, 2005. (Online). Available: http://www.wseas.org/online
[42] M. Mastriani y A. Giraldez, "Fuzzy thresholding in wavelet domain for
speckle reduction in Synthetic Aperture Radar images," ICGST
International on Journal of Artificial Intelligence and Machine Learning,
Volume 5, 2005. (Online). Available:
http://www.icgst.com/aiml/v3/index.html
[43] M. Mastriani, "Denoising based on wavelets and deblurring via selforganizing
map for Synthetic Aperture Radar images," ICGST
International on Journal of Artificial Intelligence and Machine Learning,
Volume 5, 2005. (Online). Available:
http://www.icgst.com/aiml/v3/index.html
[44] M. Mastriani, "Systholic Boolean Orthonormalizer Network in Wavelet
Domain for Microarray Denoising," ICGST International Journal on
Bioinformatics and Medical Engineering, Volume 5, 2005. (Online).
Available: http://www.icgst.com/bime/v1/bimev1.html
[45] M. Mastriani, "Denoising based on wavelets and deblurring via selforganizing
map for Synthetic Aperture Radar images," International
Journal of Signal Processing, Volume 2, Number 4, pp.226-235, 2005.
(Online). Available: http://www.waset.org/ijsp/v2/v2-4-33.pdf
[46] M. Mastriani, "Systholic Boolean Orthonormalizer Network in Wavelet
Domain for Microarray Denoising," International Journal of Signal
Processing, Volume 2, Number 4, pp.273-284, 2005. (Online).
Available: http://www.waset.org/ijsp/v2/v2-4-40.pdf
[47] M. Mastriani y A. Giraldez, "Microarrays denoising via smoothing of
coefficients in wavelet domain," International Journal of Biomedical
Sciences, Volume 1, Number 1, pp.7-14, 2006. (Online). Available:
http://www.waset.org/ijbs/v1/v1-1-2.pdf
[48] M. Mastriani y A. Giraldez, "Kalman- Shrinkage for Wavelet-Based
Despeckling of SAR Images," International Journal of Intelligent
Technology, Volume 1, Number 3, pp.190-196, 2006. (Online).
Available: http://www.waset.org/jit/v1/v1-3-21.pdf
[49] M. Mastriani y A. Giraldez, "Neural Shrinkage for Wavelet-Based SAR
Despeckling," International Journal of Intelligent Technology, Volume
1, Number 3, pp.211-222, 2006. (Online). Available:
http://www.waset.org/jit/v1/v1-3-24.pdf
[50] M. Mastriani, "Fuzzy Thresholding in Wavelet Domain for Speckle
Reduction in Synthetic Aperture Radar Images," International Journal of
Intelligent Technology, Volume 1, Number 3, pp.252-265, 2006.
(Online). Available: http://www.waset.org/jit/v1/v1-3-30.pdf
[51] M. Mastriani, "New Wavelet-Based Superresolution Algorithm for
Speckle Reduction in SAR Images ," International Journal of Computer
Science, Volume 1, Number 4, pp.291-298, 2006. (Online). Available:
http://www.waset.org/ijcs/v1/v1-4-39.pdf
[52] S. Haykin. Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs,
New Jersey, 1986.
[53] Leon S. J., Linear Algebra with Applications, MacMillan, 1990, New
York.
[54] A.K. Jain, Fundamentals of Digital Image Processing, Englewood
Cliffs, New Jersey, 1989.
[1] D. L. Donoho, "De-noising by soft-thresholding," IEEE Trans. Inform.
Theory, vol. 41, no. 3, pp. 613-627, 1995.
[2] D. L. Donoho, and I. M. Johnstone, "Adapting to unknown smoothness
via wavelet shrinkage," Journal of the American Statistical Assoc., vol.
90, no. 432, pp. 1200-1224., 1995.
[3] D. L. Donoho, and I. M. Johnstone, "Ideal spatial adaptation by wavelet
shrinkage," Biometrika, 81, 425-455, 1994.
[4] I. Daubechies. Ten Lectures on Wavelets, SIAM, Philadelphia, PA.
1992.
[5] I. Daubechies, "Different Perspectives on Wavelets," in Proceedings of
Symposia in Applied Mathematics, vol. 47, American Mathematical
Society, USA, 1993.
[6] S. Mallat, "A theory for multiresolution signal decomposition: The
wavelet representation," IEEE Trans. Pattern Anal. Machine Intell., vol.
11, pp. 674-693, July 1989.
[7] S. G. Mallat, "Multiresolution approximations and wavelet orthonormal
bases of L2 (R)," Transactions of the American Mathematical Society,
315(1), pp.69-87, 1989a.
[8] X.-P. Zhang, and M. Desai, "Nonlinear adaptive noise suppression based
on wavelet transform," Proceedings of the ICASSP98, vol. 3, pp. 1589-
1592, Seattle, 1998.
[9] X.-P. Zhang, "Thresholding Neural Network for Adaptive Noise
reduction," IEEE Transactions on Neural Networks, vol.12, no. 3,
pp.567-584, 2001.
[10] X.-P. Zhang, and M. Desai, "Adaptive Denoising Based On SURE
Risk," IEEE Signal Proc. Letters, vol.5, no. 10, 1998.
[11] X.-P. Zhang and Z.Q. Luo, "A new time-scale adaptive denoising
method based on wavelet shrinkage," in Proceedings of the ICASSP99,
Phoenix, AZ., March 15-19, 1999.
[12] M. Lang, H. Guo, J. Odegard, C. Burrus, and R. Wells, "Noise reduction
using an undecimated discrete wavelet transform," IEEE Signal Proc.
Letters, vol. 3, no. 1, pp. 10-12, 1996.
[13] H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian
wavelet shrinkage," J. Amer. Statist. Assoc., vol. 92, pp. 1413-1421,
1997.
[14] S. G. Chang, B. Yu, and M. Vetterli, "Spatially adaptive wavelet
thresholding with context modeling for image denoising," IEEE Trans.
Image Processing, vol. 9, pp. 1522-1531, Sept. 2000.
[15] S. G. Chang, B. Yu, and M. Vetterli, "Adaptive wavelet thresholding for
image denoising and compression," IEEE Trans. Image Processing, vol.
9, pp. 1532-1546, Sept. 2000.
[16] S. G. Chang and M. Vetterli, "Spatial adaptive wavelet thresholding for
image denoising," in Proc. ICIP, vol. 1, 1997, pp. 374-377.
[17] M. S. Crouse, R. D. Nowak, and R. G. Baraniuk, "Wavelet-based
statistical signal processing using hidden Markov models," IEEE
Trans.Signal Processing, vol. 46, pp. 886-902, Apr. 1998.
[18] M. Malfait and D. Roose, "Wavelet-based image denoising using a
Markov random field a priori model," IEEE Trans. Image Processing,
vol. 6, pp. 549-565, Apr. 1997.
[19] M. K. Mihcak, I. Kozintsev, K. Ramchandran, and P. Moulin, "Low
complexity image denoising based on statistical modeling of wavelet
coefficients," IEEE Trans. Signal Processing Lett., vol. 6, pp. 300-303,
Dec. 1999.
[20] E. P. Simoncelli, "Bayesian denoising of visual images in the wavelet
domain," in Bayesian Inference in Wavelet Based Models. New York:
Springer-Verlag, 1999, pp. 291-308.
[21] E. Simoncelli and E. Adelson, "Noise removal via Bayesian wavelet
coring," in Proc. ICIP, vol. 1, 1996, pp. 379-382.
[22] M. Belge, M. E. Kilmer, and E. L. Miller, "Wavelet domain image
restoration with adaptive edge-preserving regularization," IEEE Trans.
Image Processing, vol. 9, pp. 597-608, Apr. 2000.
[23] J. Liu and P. Moulin, "Information-theoretic analysis of interscale and
intrascale dependencies between image wavelet coefficients," IEEE
Trans. Image Processing, vol. 10, pp. 1647-1658, Nov. 2000.
[24] H. Guo, J. E. Odegard, M. Lang, R. A. Gopinath, I. Selesnick, and C. S.
Burrus, "Speckle reduction via wavelet shrinkage with application to
SAR based ATD/R," Technical Report CML TR94-02, CML, Rice
University, Houston, 1994.
[25] R.R. Coifman, and D.L. Donoho, Translation-invariant de-noising. A.
Antoniadis & G. Oppenheim (eds), Lecture Notes in Statistics, vol. 103.
Springer-Verlag, pp 125-150, 1995.
[26] M. Misiti, Y. Misiti, G. Oppenheim, and J.M. Poggi. (2001, June).
Wavelet Toolbox, for use with MATLAB®, User-s guide, version 2.1.
(Online). Available: http://www.rrz.unihamburg.
de/RRZ/Software/Matlab/Dokumentation/help/pdf_doc/wavele
t/wavelet_ug.pdf
[27] C.S. Burrus, R.A. Gopinath, and H. Guo, Introduction to Wavelets and
Wavelet Transforms: A Primer, Prentice Hall, New Jersey, 1998.
[28] B.B. Hubbard, The World According to Wavelets: The Story of a
Mathematical Technique in the Making, A. K. Peter Wellesley,
Massachusetts, 1996.
[29] A. Grossman and J. Morlet, "Decomposition of Hardy Functions into
Square Integrable Wavelets of Constant Shape," SIAM J. App Math, 15:
pp.723-736, 1984.
[30] C. Valens. (2004). A really friendly guide to wavelets. (Online).
Available: http://perso.wanadoo.fr/polyvalens/
clemens/wavelets/wavelets.html
[31] G. Kaiser, A Friendly Guide To Wavelets, Boston: Birkhauser, 1994.
[32] J.S. Walker, A Primer on Wavelets and their Scientific Applications,
Chapman & Hall/CRC, New York, 1999.
[33] E. J. Stollnitz, T. D. DeRose, and D. H. Salesin, Wavelets for Computer
Graphics: Theory and Applications, Morgan Kaufmann Publishers, San
Francisco, 1996.
[34] J. Shen and G. Strang, "The zeros of the Daubechies polynomials," in
Proc. American Mathematical Society, 1996.
[35] R. Yu, A.R. Allen, and J. Watson, An optimal wavelet thresholding for
speckle noise reduction, in Summer School on Wavelets: Papers,
Publisher: Silesian Technical University (Gliwice, Poland), pp77-81,
1996.
[36] H.Y. Gao, and A.G. Bruce, "WaveShrink with firm shrinkage,"
Statistica Sinica, 7, 855-874, 1997.
[37] L. Gagnon, and F.D. Smaili, "Speckle noise reduction of air-borne SAR
images with Symmetric Daubechies Wavelets," in SPIE Proc. #2759, pp.
1424, 1996.
[38] M.S. Crouse, R. Nowak, and R. Baraniuk, "Wavelet-based statistical
signal processing using hidden Markov models," IEEE Trans. Signal
Processing, vol 46, no.4, pp.886-902, 1998.
[39] M. Mastriani y A. Giraldez, "Smoothing of coefficients in wavelet
domain for speckle reduction in Synthetic Aperture Radar images,"
ICGST International Journal on Graphics, Vision and Image Processing
(GVIP), Volume 6, pp. 1-8, 2005. (Online). Available:
http://www.icgst.com/gvip/v6/P1150517003.pdf
[40] M. Mastriani y A. Giraldez, "Despeckling of SAR images in wavelet
domain," GIS Development Magazine, Sept. 2005, Vol. 9, Issue 9,
pp.38-40. (Online). Available:
http://www.gisdevelopment.net/magazine/years/2005/sep/wavelet_1.htm
[41] M. Mastriani y A. Giraldez, "Microarrays denoising via smoothing of
coefficients in wavelet domain," WSEAS Transactions on Biology and
Biomedicine, 2005. (Online). Available: http://www.wseas.org/online
[42] M. Mastriani y A. Giraldez, "Fuzzy thresholding in wavelet domain for
speckle reduction in Synthetic Aperture Radar images," ICGST
International on Journal of Artificial Intelligence and Machine Learning,
Volume 5, 2005. (Online). Available:
http://www.icgst.com/aiml/v3/index.html
[43] M. Mastriani, "Denoising based on wavelets and deblurring via selforganizing
map for Synthetic Aperture Radar images," ICGST
International on Journal of Artificial Intelligence and Machine Learning,
Volume 5, 2005. (Online). Available:
http://www.icgst.com/aiml/v3/index.html
[44] M. Mastriani, "Systholic Boolean Orthonormalizer Network in Wavelet
Domain for Microarray Denoising," ICGST International Journal on
Bioinformatics and Medical Engineering, Volume 5, 2005. (Online).
Available: http://www.icgst.com/bime/v1/bimev1.html
[45] M. Mastriani, "Denoising based on wavelets and deblurring via selforganizing
map for Synthetic Aperture Radar images," International
Journal of Signal Processing, Volume 2, Number 4, pp.226-235, 2005.
(Online). Available: http://www.waset.org/ijsp/v2/v2-4-33.pdf
[46] M. Mastriani, "Systholic Boolean Orthonormalizer Network in Wavelet
Domain for Microarray Denoising," International Journal of Signal
Processing, Volume 2, Number 4, pp.273-284, 2005. (Online).
Available: http://www.waset.org/ijsp/v2/v2-4-40.pdf
[47] M. Mastriani y A. Giraldez, "Microarrays denoising via smoothing of
coefficients in wavelet domain," International Journal of Biomedical
Sciences, Volume 1, Number 1, pp.7-14, 2006. (Online). Available:
http://www.waset.org/ijbs/v1/v1-1-2.pdf
[48] M. Mastriani y A. Giraldez, "Kalman- Shrinkage for Wavelet-Based
Despeckling of SAR Images," International Journal of Intelligent
Technology, Volume 1, Number 3, pp.190-196, 2006. (Online).
Available: http://www.waset.org/jit/v1/v1-3-21.pdf
[49] M. Mastriani y A. Giraldez, "Neural Shrinkage for Wavelet-Based SAR
Despeckling," International Journal of Intelligent Technology, Volume
1, Number 3, pp.211-222, 2006. (Online). Available:
http://www.waset.org/jit/v1/v1-3-24.pdf
[50] M. Mastriani, "Fuzzy Thresholding in Wavelet Domain for Speckle
Reduction in Synthetic Aperture Radar Images," International Journal of
Intelligent Technology, Volume 1, Number 3, pp.252-265, 2006.
(Online). Available: http://www.waset.org/jit/v1/v1-3-30.pdf
[51] M. Mastriani, "New Wavelet-Based Superresolution Algorithm for
Speckle Reduction in SAR Images ," International Journal of Computer
Science, Volume 1, Number 4, pp.291-298, 2006. (Online). Available:
http://www.waset.org/ijcs/v1/v1-4-39.pdf
[52] S. Haykin. Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs,
New Jersey, 1986.
[53] Leon S. J., Linear Algebra with Applications, MacMillan, 1990, New
York.
[54] A.K. Jain, Fundamentals of Digital Image Processing, Englewood
Cliffs, New Jersey, 1989.
@article{"International Journal of Electrical, Electronic and Communication Sciences:62641", author = "Mario Mastriani", title = "Denoising and Compression in Wavelet Domainvia Projection on to Approximation Coefficients", abstract = "We describe a new filtering approach in the wavelet domain for image denoising and compression, based on the projections of details subbands coefficients (resultants of the splitting procedure, typical in wavelet domain) onto the approximation subband coefficients (much less noisy). The new algorithm is called Projection Onto Approximation Coefficients (POAC). As a result of this approach, only the approximation subband coefficients and three scalars are stored and/or transmitted to the channel. Besides, with the elimination of the details subbands coefficients, we obtain a bigger compression rate. Experimental results demonstrate that our approach compares favorably to more typical methods of denoising and compression in wavelet domain.
", keywords = "Compression, denoising, projections, wavelets.", volume = "3", number = "11", pages = "2172-11", }
