Publication TypeJournal Article
Year of Publication2006
AuthorsSchmeelk, J
Start Page29
Date Published2006

The notion of frames is introduced in Hilbert Spaces. Wavelet analysis together with a brief description for the stability of images is introduced using wavelet transforms. The “control” is measured by the notion of a frame. Two wavelet transform examples are then presented and illustrated on a two dimensional image.

Refereed DesignationRefereed
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[1] E. Aboufadel and S. Schlicker, John Wiley and Sons, Inc., NY, 1999.
[2] H.C. Andrews and B.R. Hunt, Digital Image Restoration, Prentice Hall, NJ, 1977.
[3] D.H. Ballard, Parameter Nets, Artificial Intelligence, 22 (1984), 235-267.
[4] D.H. Ballard and C.M. Brown, Computer Vision, Prentice Hall, NJ, 1982.
[5] B.G. Batchelor, Pattern Recognition, Plenum Press, NY, 1978.
[6] F.W. Campbell and J.G. Robson, Application of Fourier analysis to the visibility of gratings,
J. Physiol., 197 (1968), 551-566.
[7] O. Christensen and D.T. Stoeva, p-frames in separable Banach spaces, Advances in Computational
Mathematics, 18 (2003), 117-126.
[8] C. Chui (Editor), An Introduction to Wavelets, Volume 1, Academic Press Inc., NY, 1992.
[9] C. Chui (Editor), A Tutorial in Theory and Application, Academic Press Inc., NY, 1992.
[10] I. Daubechies, The wavelet transform, time frequency localization and signal analysis, IEEE
Transactions on Information Theory, 36 (1990), no. 5, 961-1005.
[11] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, PA, 1992.
[12] L. Debnath, Wavelet Transforms and their Applications, Birkh¨auser, MA, 2002.
[13] R.C. Gonzalez and R. Woods, Digital Image Processing, Addison-Wesley Pub. Co., MA, 1992.
[14] C.E. Heil and D.F. Walnut, Continuous and discrete wavelet transforms, Siam Review, 31 (1989), no. 4, 628-666.
[15] B.B. Hubbard, The World According to Wavelets, A.K. Peters, MA, 1996.
[16] A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, NJ, 1989.
[17] T.H. Koornwinder, Wavelets: An Elementary Treatment of Theory and Applications, World Scientific, NJ, 1993.
[18] J.S. Lim, Two-Dimensional Signal and Image Processing, Prentice Hall, NJ, 1990.
[19] S. Mallat and S. Zhong, Characterization of signals from multiscale edges, IEEE Transactions
on Pattern Analysis and Machine Intelligence, 14 (1992), no. 7, 710-732.
[20] Y. Meyer, Wavelet Algorithms and Application, SIAM (Translated by R. Ryan), PA, 1993.
[21] G. Nagy, State of the art in pattern recognition, Proc. IEEE, 56 (1968), 836-862.
[22] W. Pedrycz, Fuzzy sets in pattern recognition; methodology and methods, Pattern Recognition,
20 (1990), no. 1-2, 121-146.
[23] W.K. Pratt, Digital Image Processing, John Wiley and Sons, NY, 1991.
[24] M. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and I. Raphael
(Editors), Wavelets and their Applications, Jones and Bartlett, MA, 1992.
[25] R.J. Schalkoff, Digital Image Processing and Computer Vision, John Wiley and Sons, NY, 1989.
[26] J. Schmeelk, Wavelet transforms not Fourier transforms on two-dimensional images, International
Journal of Computational and Numerical Analysis and Applications, 4 (2003), no. 2, 131-155.
[27] G. Strange, Wavelet transforms versus Fourier transforms, Bulletin of the AMS, 28 (1993), no. 2, 288-305.
[28] Y.Y. Tang, L.H. Yang, J. Liu, and H. Ma, Wavelet Theory and its Application to Pattern
Recognition, World Scientific, NJ, 2000.
[29] D.F. Walnut, An Introduction to Wavelet Analysis, Birkh¨auser, MA, 2002.