Application of cubic box spline wavelets in the analysis of signal singularities

Rakowski, W.

International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) no. 4, pp. 927-941.

Résumé

In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines which have compact and short support and which do not introduce a phase shift. The digital filters associated with cubic box wavelets that are applied in implementing the discrete dyadic wavelet transform are defined. The filters and the algorithme à trous of the discrete dyadic wavelet transform are used in detecting signal singularities and in calculating the measures of signal singularities in the form of a Lipschitz exponent. The article presents examples illustrating the use of cubic box spline wavelets in the analysis of signal singularities.

Détail

Keywords: cubic box spline, wavelets, dyadic wavelet transform, singularity detection
Mots-clés : falki, diadyczna transformacja falkowa, detekcja pojedyncza

Bibliographie

[1] Babaud, J., Witkin, A.P. and Baudin, M. (1986). Uniqueness of the Gaussian kernel for scale-space filtering, IEEE Transactions on Pattern Analysis and Machine Intelligence 8(1): pp. 26–33.

[2] Boor, C. (1978). A Practical Guide to Splines, Springer-Verlag, New York, NY.

[3] Holschneider, M., Kronland-Martinet, R., Morlet, J. and Tchamitchian, P. (1989). A real-time algorithm for signal analysis with help of the wavelet transform, in J.-M. Combes, A. Grossmann and A.P. Tchamitchian (Eds.), Wavelets, Time-Frequency Methods and Phase Space, Springer-Verlag, Berlin/Heidelberg, pp. 286–297.

[4] Mallat, S. (1991). Zero-crossings of a wavelet transform, IEEE Transactions on Information Theory 37(4): 1019–1033.

[5] Mallat, S. (2009). A Wavelet Tour of Signal Processing: The Sparce Way, Third Edition, Academic Press, Burlington, MA.

[6] Mallat, S. and Hwang, L.W. (1992). Singularity detection and processing with wavelets, IEEE Transactions on Information Theory 38(2): 617–643.

[7] Mallat, S. and Zhong, S. (1992). Characterization of signals from multiscale edges, IEEE Transactions on Pattern Analysis and Machine Intelligence 14(7): 710–732.

[8] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2007). Numerical Recipes The Art of Scientific Computing, Cambridge University Press, Cambridge.

[9] Rakowski, W. (2003). A proof of the necessary condition for perfect reconstruction of signals using the two-channel wavelet filter bank, Bulletin of the Polish Academy of Science: Technical Sciences 51(1): 14–23.

[10] Shensa, M.J. (1992). The discrete wavelet transform: Wedding the à trous and Mallat algorithms, IEEE Transactions on Signal Processing 40(10): 2464–2482.

[11] Skodras, A., Christopoulos, C. and Ebrahimi, T. (2001). The JPEG 2000 still image compression standard, IEEE Signal Processing Magazine 18(5): 36–58.

[12] Tu, C.-L. and Hwang, W.-L. (2005). Analysis of singularities from modulus maxima of complex wavelets, IEEE Transactions on Information Theory 51(3): 1049–1062.

[13] Unser, M. (1999). Splines: A perfect fit for signal and image processing, IEEE Signal Processing Magazine 16(6): 22–38.

[14] Unser, M. and Blu, T. (2003). Mathematical properties of the JPEG2000 wavelet filters, IEEE Transactions on Image Processing 12(9): 1080–1090.

[15] Witkin, A.P. (1983). Scale-space filtering, Proceedings of the International Conference on Artificial Intelligence, Karlsruhe, Germany, pp. 1019–1022.

[16] Witkin, A.P. (1984). Scale-space filtering: A new approach to multi-scale description, in S. Ullman and W. Richards (Eds.), Image Understanding, Ablex, Norwood, NJ.

[17] Zhao, Y., Hu, J., Zhang, L. and Liao, T. (2013). Mallat wavelet filter coefficient calculation, 5th International Conference on Computational and Information Sciences, Shiyan, Hubei, China, pp. 963–965.