Error Model in Wavelet-compressed Images
Main Article Content
Keywords
wavelet, compression of images, observed image, compression linear and nonlinear.
Abstract
In this paper we study image compression as a way to compare Wavelet and Fourier models, by minimizing the error function. The particular problem we consider is to determine basis {ei} minimizing the error function between the original image and the recovered one after compression. It is to be noted or remarked that there are many applications in such diverse fields as for example medicine and astronomy, where no image deteriorating is acceptable since even noise is considered essential.
MSC: 30H25, 65Txx, 65T60
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References
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[41] C. Sidney Burrus, Ramesh A. Gopinath and Haitao Guo. Introduction to wavelets and wavelet transforms, A Primer, ISBN 0–13–489600–9. Prentice Hall, New Jersey, 1998.
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[43] Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks, ISBN–13 978– 0961408879.Wellesley-Cambridge Press, Cambridge, MA, 1996.
[44] Stephane G. Mallat. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, II(7), 674–693 (1989).
[45] Wenchang Sun and Xingwei Zhou. Sampling theorem for wavelet subspaces: error estimate and irregular sampling. IEEE Transactions on Signal Processing, ISSN 1053–587X, 48(1), 223–226 (2000).
[46] Hans Triebel. Fourier Analysis and Function Spaces. Teubner-Texte Math. 7, Leipzig: Teubner, 1977.
[47] Yves Meyer. Wavelets and functions with bounded variation from image processing to pure mathematics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 77–105 (2000).
[48] E. Quak and N.Weyrich. Decomposition and reconstruction algorithms for spline wavelets on a bounded inverval. Applied and Computational Harmonic Analysis (ACHA), I(3), 217–231 (1994).
[49] M. Unser, A. Aldroubi and M. Eden. B-spline processing I: Theory and II: Efficient design and application. IEEE Transactions on Signal Processing, ISSN 1053-587X, 41(2), 821–848 (1993).
[50] R. A. DeVore, B. Jawerth and V. Popov. Compression of wavelet decomposi- tion. American Journal of Mathematics, ISSN 0002–9327, 114, 737–785 (1992).
[51] S. Mallat. Multiresolution approximations and wavelet orthonormal bases for L2(Rd). Transactions of the American Mathematical Society, ISSN 0002–9947, 315, 69–87 (1989).
7, 315, 69–87 (1989).
[2] Roger. J. Clarke. Transform Coding of Images, ISBN–10 0121757307. Academic Press, San Diego, CA, 1985.
[3] Roger. J. Clarke. Digital Compression of Still Images and Video, ISBN–10 012175720X. Academic Press, San Diego, CA, 1995.
[4] Allen Gersho and Robert M. Gray. Vector Quantization and Signal Compression, ISBN 0–7923–9181–0. Kluwer Academic Publishers, Boston, 1992.
[5] Rafael C. Gonz´alez and Richard E. Woods. Digital Image Processing, ISBN-13 978–0201508031. Addison-Wesley, New York, 1992.
[6] Akram Aldroubi. The wavelet transform: A surfing guide, inWavelets in Medicine and Biology, ISBN–10 084939483X. Editors Akram Aldroubi and Michael Unser, CRC Press, New York, 3–36 (1996).
[7] Charles K. Chui.Wavelets: A Mathematical Tool for Signal Analysis, ISBN–10 0– 89871–384–6. SIAM Monographs on Mathematical Modeling and Computation, Philadelphia, 1997.
[8] Michael Unser. A practical guide to implementation of the wavelet transform, in Wavelets in Medicine and Biology, ISBN–10 084939483X. Editors Akram Aldroubi and Michael Unser, CRC Press, New York, 37–76 (1996).
[9] Michio Yamada and Koji Ohkitani. Orthonormal wavelet expansion and its application to turbulence. Progress of Theoretical Physics, ISSN 1347–4081, 83(5), 819–823 (1990).
[10] Michio Yamada and Koji Ohkitani. An identification of energy cascade in turbulence by orthonormal wavelet analysis, Progress of Theoretical Physics, ISSN 1347–4081, 86(4), 799–815 (1991).
[11] Michio Yamada and Koji Ohkitani. Orthonormal wavelet analysis of turbulence. Fluid Dynamics Research, ISSN 0169–5983, 8, 101–115 (1991).
[12] A. Chambolle, R. A. De Vore, Nam-Yong Lee and B. J. Lucier. Nonlinear wavelet image processing: variational problems,compression, and noise removal through wavelet shrinkage. IEEE Transactions on Image Processing, ISSN 1057–7149, 7(3), 319–335 (1998).
[13] R. A. De Vore, B. Jawerth and B. J. Lucier. Image Compression throughWavelet transform Coding. IEEE Transactions on Information Theory, ISSN 0018–9448, 38(2), 719–746 (1992).
[14] John Miano. Compressed Image File Formats: JPEG, PNG, GIF, XBM, BPM, ISBN 0201604434. Addison–Wesley, New York, 1999.
[15] Albert Boggess and Francis J. Narcowich. A First Course in Wavelets with Fourier Analysis, ISBN–10 0130228095. Prentice Hall, New Jersey, 2001.
[16] Ingrid Daubechies. Ten Lectures on Wavelets, ISBN–10 0898712742. SIAM, Philadelphia, 1992.
[17] Eugenio Hern´andez and Guido Weiss. A First Course on Wavelets, ISBN–10 0849382742. CRC Press, Boca Raton, FL, 1996.
[18] Yves Meyer. Ondelettes et op´erateurs, I: Ondelettes, ISBN 2–7056–6125–0. Editeur Hermann, Paris, 1990.
[19] David F. Walnut. An Introduction to Wavelets Analysis, ISBN 0–8176–3962–4. Birkhauser, Boston, 2001.
[20] Gilbert G. Walter. Approximation with impulse trains. Results in Mathematics, ISSN 0378–6218, 34(1/2), 185–196 (1998).
[21] Gilbert G. Walter and Xiaoping Shen. A substitute for summability in wavelet expansions in Analysis of Divergence, ISBN 0817640584. Applied and Numerical Harmonic Analysis, Birkh¨auser, Boston, MA, 51–63 (1999).
[22] Gilbert G. Walter and Xiaoping Shen. Deconvolution using Meyer wavelets. Journal Integral Equations and Applications, ISSN 0897–3962, 11(4), 515–534 (1999).
[23] Ingrid Daubechies. Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, ISSN 0010–3640, 41(7), 909–996 (1988).
[24] Alan V. Oppenheim and Ronald W. Schafer. Discrete–Time Signal Processing, ISBN–13 978–0132162920. Prentice-Hall, Englewood Cliffs, NJ, 1989.
[25] Aline Bonami, Sylvain Durand and Guido Weiss.Wavelets obtained by continuos deformations of the Haar wavelet . Revista Matem´atica Iberoamericana, 12(1), 1–18 (1996).
[26] C. K. Chui. An Introduction to Wavelets, ISBN–10 0121745848.Academic Press, Boston, 1992.
[27] S. Mallat. A Wavelets Tour of Signal Processing. Academic Press, New York, 1998.
[28] Elwyn R. Berlekamp. Algebraic Coding Theory, ISBN–13 978–0894120633. Aegean Park Press, 1984.
[29] R. W. Hamming. Error detecting and error correcting codes. Bell System Technical Journal, 29, 147–160 (1950).
[30] Alian Poli and Llorenc Huguet. Error Correcting Codes: Theory and Applications, ISBN–10 0132848945. Prentice Hall, Paris, 1992.
[31] John G. Proakiss and Dimitris G. Manolakis. Signal Processing: Principles, Algorithms and Applications, ISBN–10 002396815X.Macmillan, New York, 1992.
[32] Ferrel G. Stremler. Introduction to Comunication Systems, 2nd edition, ISBN 968–500009–3. Addison–Wesley Publishing Company, Inc. Massachusetts, 1982.
[33] Roberto Togneri and Christopher J. S. deSilva. Fundamentals of Information Theory and Coding Design, ISBN–10 1584883103. Chapman & Hall/CRC, Boca Raton, 2003.
[34] C. E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27, 379–423, 623–656 (1948).
[35] Donald B. Percival and Andrew T. Walden. Wavelet Methods for Time Series Analysis, ISBN–10 0521640687, Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2000.
[36] M. Unser and T. Blu. Mathematical Properties of the JPEG2000 Wavelet Filters. IEEE Transactions on Image Processing, ISSN 1057–7149, 12(9), 1080– 1090 (September 2003).
[37] M. A. Pinsky. Introduction to Fourier Analysis and wavelet, Brooks/Cole, NJ, 2001.
[38] Gerald B. Folland. Real Analysis. Modern Techniques and Their Applications 2nd edition, ISBN-13 978–0–471–31716–6. JohnWiley & Sons, 1999.
[39] Gilbert G. Walter and Xiaoping Shen. Wavelets and Other Orthogonal Systems, 2nd edition, ISBN–13 9781584882275. Chapman & Hall/CRC, Boca Raton, FL, 2000.
[40] Denis Gabor. Theory of communications. J. Inst. Elect. Eng, London, 93(III), 429–457 (1946).
[41] C. Sidney Burrus, Ramesh A. Gopinath and Haitao Guo. Introduction to wavelets and wavelet transforms, A Primer, ISBN 0–13–489600–9. Prentice Hall, New Jersey, 1998.
[42] Ingrid Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory, ISSN 0018–9448, 36(5), 961–1005 (1990).
[43] Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks, ISBN–13 978– 0961408879.Wellesley-Cambridge Press, Cambridge, MA, 1996.
[44] Stephane G. Mallat. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, II(7), 674–693 (1989).
[45] Wenchang Sun and Xingwei Zhou. Sampling theorem for wavelet subspaces: error estimate and irregular sampling. IEEE Transactions on Signal Processing, ISSN 1053–587X, 48(1), 223–226 (2000).
[46] Hans Triebel. Fourier Analysis and Function Spaces. Teubner-Texte Math. 7, Leipzig: Teubner, 1977.
[47] Yves Meyer. Wavelets and functions with bounded variation from image processing to pure mathematics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 77–105 (2000).
[48] E. Quak and N.Weyrich. Decomposition and reconstruction algorithms for spline wavelets on a bounded inverval. Applied and Computational Harmonic Analysis (ACHA), I(3), 217–231 (1994).
[49] M. Unser, A. Aldroubi and M. Eden. B-spline processing I: Theory and II: Efficient design and application. IEEE Transactions on Signal Processing, ISSN 1053-587X, 41(2), 821–848 (1993).
[50] R. A. DeVore, B. Jawerth and V. Popov. Compression of wavelet decomposi- tion. American Journal of Mathematics, ISSN 0002–9327, 114, 737–785 (1992).
[51] S. Mallat. Multiresolution approximations and wavelet orthonormal bases for L2(Rd). Transactions of the American Mathematical Society, ISSN 0002–9947, 315, 69–87 (1989).
7, 315, 69–87 (1989).