Analytical footprints: Compact representation of elementary singularities in wavelet bases

IEEE Transactions on Signal Processing

Volumn 58, Issue 12, 2010, Pages 6105-6118

  Van De Ville, Dimitri ^a,b   Forster Heinlein, Brigitte ^c   Unser, Michael ^a   Blu, Thierry ^d  

^a EPFL   (Switzerland)

^b UNIVERSITY OF GENEVA   (Switzerland)

^c TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^d CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Elementary singularities; footprints; fractional derivatives; generalized fractional splines; wavelet bases

Indexed keywords

ELEMENTARY SINGULARITIES; FOOTPRINTS; FRACTIONAL DERIVATIVES; GENERALIZED FRACTIONAL SPLINES; WAVELET BASIS;

MATHEMATICAL OPERATORS; SPLINES; WAVELET ANALYSIS;

WAVELET DECOMPOSITION;

EID: 78649244369     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2010.2068295     Document Type: Article

References (46)

1. S. Mallat, "A theory for multiresolution signal decomposition: The wavelet decomposition", IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674-693, 1989.
2. I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis", IEEE Trans. Inf. Theory, vol. 36, no. 5, pp. 961-1005, 1990.
3. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way. San Diego, CA: Academic, 2009.
4. S. Mallat, "Zero-crossings of a wavelet transform", IEEE Trans. Inf. Theory, vol. 37, no. 4, pp. 1019-1033, Jul. 1991.
5. S. Mallat and S. Zhong, "Characterization of signals from multiscale edges", IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, no. 7, pp. 710-732, Jul. 1992.
6. S. Mallat and W. L. Hwang, "Singularity detection and processing with wavelets", IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 617-643, 1992.
7. Y.-P. Wang, "Image representations using multiscale differential operators", IEEE Trans. Image Process., vol. 8, no. 12, pp. 1757-1771, 1999.
8. J. F. Clearbout and F. Muir, "Robust modeling of erratic data", Geophys., vol. 38, no. 5, pp. 826-844, 1973.
9. F. Santosa and W. W. Symes, "Linear inversion of band-limited reflection seismograms", SIAM J. Sci. Statist. Comput., vol. 7, no. 4, pp. 1307-1330, 1986.
10. F. J. Herrmann, "Singularity characterization by monoscale analysis: Application to seismic imaging", Appl. Comput. Harmon. Anal., vol. 11, pp. 64-88, 2001.
11. M. Maysami, "Lithology constraints from seismic waveforms: Application to Opal-A to Opal-CT transition", Ph. D. dissertation, Dept. of Earth and Ocean Sciences, Univ. of British Columbia, Vancouver, BC, Canada, 2008.
12. F. J. Herrmann, "Seismic deconvolution by atomic decomposition: A parametric approach with sparseness constraints", Integr., vol. 12, no. 1, pp. 69-91, 2005.
13. C. Li, C. Zheng, and C. Tai, "Detection of ECG characteristic points using wavelet transforms", IEEE Trans. Biomed. Eng.bb, vol. 42, no. I, pp. 21-28, 1995.
14. B. M. Sadler and A. Swami, "Analysis of multiscale products for step detection and estimation", IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 1043-1051, Apr. 1999.
15. P. L. Dragotti and M. Vetterli, "Wavelet footprints: Theory, algorithms, and applications", IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1306-1323, 2003.
16. V. Bruni and D. Vitulano, "Wavelet based signal denoising via simple singularities approximation", Signal Process., vol. 86, pp. 859-876, 2006.
17. M. Unser and T. Blu, "Fractional splines and wavelets", SIAM Rev., vol. 42, pp. 43-67, 2000.
18. T. Blu and M. Unser, "A complete family of scaling functions: The (a, r)-fractional splines", in Proc. 28th IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP'03), Hong Kong, Apr. 2003, vol. VI, pp. 421-424.
19. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993.
20. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivative. Theory and Applications. New York: Gordon and Breach, 1993.
21. I. Podlubny, Fractional Differential Equations. New York: Academic, 1998.
22. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Philadelphia, PA: Saunders, 2006.
23. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Singapore: World Scientific, 2010.
24. R. Gorenflo and F. Mainardi, "Fractional calculus: Integral and differential equations of fractional order", in Fractals and Fractional Calculus in Continuum Mechanics. New York: Springer-Verlag, 1997, vol. 378, CISM Courses and Lectures, pp. 223-276.
25. M. D. Ortigueira, "On the fractional central differences and derivatives", J. Vibrat. Control, vol. 14, no. 9-10, pp. 1255-1266, 2008.
26. M. Unser and T. Blu, "Self-similarity: Part I-Splines and operators", IEEE Trans. Signal Process., vol. 55, no. 4, pp. 1352-1363, Apr. 2007.
27. I. M. Gel'fand and G. E. Shilov, Properties and Operations Transl.: E. Saletan. New York: Academic, 1964, vol. I, Generalized Functions.
28. L. Schwartz, Théorie des Distributions. Paris, Franceb: Hermann, 1998.
29. D. C. Champeney, A Handbook of Fourier Theorems. Cambridge, U. K.: Cambridge Univ. Press, 1987.
30. T. Blu and M. Unser, "Wavelets, fractals, and radial basis functions", IEEE Trans. Signal Process., vol. 50, no. 3, pp. 543-553, Mar. 2002.
31. M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms", Signal Process., vol. 30, pp. 141-162, 1993.
32. G. Battle, "A block spin construction of ondelettes. Part I: Lemarié functions", Commun. Math. Phys., vol. 110, pp. 601-615, 1987.
33. C. K. Chui and J.-Z. Wang, "A cardinal spline approach to wavelets", Proc. Amer. Math. Soc, vol. 113, no. 3, pp. 785-793, Nov. 1991.
34. C. Micchelli, C. Rabut, and F. I. Utreras, "Using the refinement equation for the construction of pre-wavelets III: Elliptic splines", Numer. Algorithms 1, no. 4, pp. 331-352, 1991.
35. I. Khalidov, D. Van De Ville, T. Blu, and M. Unser, "Construction of wavelet bases that mimic the behaviour of some given operator", in Proc. SPIE Conf. Mathematical Imaging: Wavelet XII, San Diego, CA, Aug. 26-29, 2007, vol. 6701, pp. 67010S-1-67010S-7.
36. D. Van De Ville and M. Unser, "Complex wavelet bases, steerability, and the Marr-like pyramid", IEEE Trans. Image Process., vol. 17, no. II, pp. 2063-2080, Nov. 2008.
37. M. Unser, D. Sage, and D. Van De Ville, "Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform", IEEE Trans. Image Process., vol. 18, no. 11, pp. 2402-2418, Nov. 2009.
38. M. Unser and D. Van De Ville, "Higher-order Riesz transforms and steerable wavelet frames", in Proc. 2009 IEEE Int. Conf. Image Processing (ICIP), Cairo, Egypt, Nov. 7-11, 2009, pp. 3801-3804.
39. T. Blu and M. Unser, "The fractional spline wavelet transform: Definition and implementation", in Proc. 25th IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Istanbul, Turkey, Jun. 5-9, 2000, vol. I, pp. 512-515.
40. A. Aldroubi, M. Unser, and M. Eden, "Cardinal spline filters: Stability and convergence to the ideal sinc interpolator", Signal Process., vol. 28, no. 2, pp. 127-138, Aug. 1992.
41. S. Mallat and Z. Zhang, "Matching pursuit with time-frequency dictionaries", IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397-3415, 1993.
42. J. A. Tropp, "Greed in good: Algorithmic results for sparse approximation", IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2231-2242, 2004.
43. R. Gribonval and M. Nielsen, "Beyond sparsity: Recovering structured representations by t1 minimization and greedy algorithms", Adv. Comput. Math., vol. 28, no. 1, pp. 23-41, 2008.
44. C. Dossal and S. Mallat, "Sparse spike deconvolution with minimum scale", presented at the SPARS05, Rennes, France, Nov. 16-18, 2005.
45. N. Kingsbury, "Complex wavelets for shift invariant analysis and filtering of signals", Appl. Comput. Harmon. Anal., vol. 10, no. 3, pp. 234-253, May 2001.
46. K. N. Chaudhury and M. Unser, "Construction of Hilbert transform pairs of wavelet bases and Gabor-like transforms", IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3411-3425, Sep. 2009.