Modelo de error en imágenes comprimidas con wavelets
Main Article Content
Keywords
wavelet, compresión de imágenes, imagen observada, compresión lineal y no lineal.
Resumen
En este artículo se presenta la compresión de imágenes a través de la comparación entre el modelo Wavelet y el modelo Fourier, utilizando la minimización de la función de error. El problema que se estudia es específico, consiste en determinar una base {ei} que minimice la función de error entre la imagen original y la recuperada después de la compresión. Es de resaltar que existen muchas aplicaciones, por ejemplo, en medicina o astronomía, en donde no es aceptable ningún deterioro de la imagen porque toda la información contenida, incluso la que se estima como ruido, se considera imprescindible.
MSC: 30H25, 65Txx, 65T60
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Referencias
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[32] Ferrel G. Stremler. Introduction to Comunication Systems, 2nd edition, ISBN 968–500009–3. Addison–Wesley Publishing Company, Inc. Massachusetts, 1982.
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[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.
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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).
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[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).