Wavelet steerability and the higher-order Riesz transform

IEEE Transactions on Image Processing

Volumn 19, Issue 3, 2010, Pages 636-652

  Unser, Michael ^a   Van De Ville, Dimitri ^a,b  

^a EPFL   (Switzerland)

^b UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

Directional derivatives; Frames; Hilbert transform; Multiresolution decomposition; Riesz transform; Steerable filters; Wavelet transform

Indexed keywords

DIRECTIONAL DERIVATIVE; HILBERT TRANSFORM; MULTI RESOLUTION DECOMPOSITION; RIESZ TRANSFORM; STEERABLE FILTERS;

SIGNAL RECONSTRUCTION; WAVELET DECOMPOSITION;

MATHEMATICAL TRANSFORMATIONS;

EID: 77249087202     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2009.2038832     Document Type: Article

References (48)

1. S. G. Mallat, "A theory of multiresolution signal decomposition: The wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell., vol.11, no.7, pp. 674-693, Jul. 1989.
2. W. T. Freeman and E. H. Adelson, "The design and use of steerable filters," IEEE Trans. Pattern Anal. Mach. Intell., vol.13, no.9, pp. 891-906, Sep. 1991.
3. E. Simoncelli and H. Farid, "Steerable wedge filters for local orientation analysis," IEEE Trans. Image Process., vol.5, no.9, pp. 1377-1382, Sep. 1996.
4. P. Perona, "Deformable kernels for early vision," IEEE Trans. Pattern Anal. Mach. Intell., vol.17, no.5, pp. 488-499, May 1995.
5. M. Jacob and M. Unser, "Design of steerable filters for feature detection using Canny-like criteria," IEEE Trans. Pattern Anal. Mach. Intell., vol.26, no.8, pp. 1007-1019, Aug. 2004.
6. M. Michaelis and G. Sommer, "A Lie group approach to steerable filters," Pattern Recognit. Lett., vol.16, no.11, pp. 1165-1174, 1995.
7. Y. Hel-Or and P. C. Teo, "Canonical decomposition of steerable functions," J. Math. Imag. Vis., vol.9, no.1, pp. 83-95, 1998.
8. P. C. Teo and Y. Hel-Or, "Lie generators for computing steerable functions," Pattern Recognit. Lett., vol.19, no.1, pp. 7-17, 1998.
9. E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, "Shiftable multiscale transforms," IEEE Trans. Inf. Theory, vol.38, no.2, pp. 587-607, 1992.
10. A. Karasaridis and E. Simoncelli, "A filter design technique for steerable pyramid image transforms," in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, May 7-10, 1996, vol.4, pp. 2387-2390.
11. E. Simoncelli and W. Freeman, "The steerable pyramid: a flexible architecture for multi-scale derivative computation," in Proc. Int. Conf. Image Processing, Oct. 23-26, 1995, vol.3, pp. 444-447.
12. Y. Wiaux, L. Jacques, and P. Vandergheynst, "Correspondence principle between spherical and Euclidean wavelets," Astrophys. J., vol.632, p. 15, 2005.
13. J. G. Daugman, "Complete discrete 2-D Gabor transforms by neural networks for image-analysis and compression," IEEE Trans. Acoust. Speech Signal Process., vol.36, no.7, pp. 1169-1179, Jul. 1988.
14. E. J. Candès and D. L. Donoho, "Ridgelets: A key to higher-dimensional intermittency?," Phil. Trans. Roy. Soc. Lond. A., pp. 2495-2509, 1999.
15. N. Kingsbury, "Image processing with complex wavelets," Philos. Trans. Roy. Soc. Lond. A, Math. Phys. Eng. Sci., vol.357, no.1760, pp. 2543-2560, 1999.
16. I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, "The dual-tree complex wavelet transform," IEEE Signal Process. Mag., vol.22, no.6, pp. 123-151, 2005.
17. P. Vandergheynst and J.-F. Gobbers, "Directional dyadicwavelet transforms: Design and algorithms," IEEE Trans. Image Process., vol.11, no.4, pp. 363-372, Apr. 2002.
18. M. N. Do and M. Vetterli, "The contourlet transform: An efficient directional multiresolution image representation," IEEE Trans. Image Process., vol.14, pp. 2091-2106, 2005.
19. E. Le Pennec and S. Mallat, "Sparse geometric image representations with bandelets," IEEE Trans. Image Process., vol.14, pp. 423-438, 2005.
20. V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. Dragotti, "Directionlets: Anisotropic multidirectional representation with separable filtering," IEEE Trans. Image Process., vol.15, pp. 1916-1933, 2006.
21. H. Greenspan, S. Belongie, R. Goodman, P. Perona, S. Rakshit, and C. Anderson, "Overcomplete steerable pyramid filters and rotation invariance," in Proc. CVPR, Jun. 21-23, 1994, pp. 222-228.
22. M. Do and M. Vetterli, "Rotation invariant texture characterization and retrieval using steerablewavelet-domain hidden markov models," IEEE Trans. Multimedia, vol.4, pp. 517-527, 2002.
23. G. Tzagkarakis, B. Beferull-Lozano, and P. Tsakalides, "Rotation-invariant texture retrieval with gaussianized steerable pyramids," IEEE Trans. Image Process., vol.15, pp. 2702-2718, 2006.
24. F. Denis and A. Baskurt, "Multidirectional curvilinear structures detection using steerable pyramid," J. Electron. Imag., vol.13, no.4, pp. 756-765, 2004.
25. A. Jepson, D. Fleet, and T. El-Maraghi, "Robust online appearance models for visual tracking," in Proc. IEEE CVPR, 2001, vol.1, pp. I-415-I-422.
26. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process., vol.12, no.11, pp. 1338-1351, Nov. 2003.
27. A. A. Bharath and J. Ng, "A steerable complex wavelet construction and its application to image denoising," IEEE Trans. Image Process., vol.14, no.7, pp. 948-959, Jul. 2005.
28. M. Riesz, "Sur les fonctions conjuguées," Math. Z., vol.27, pp. 218-244, 1927.
29. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton Univ. Press, 1971.
30. M. Felsberg and G. Sommer, "The monogenic signal," IEEE Trans. Signal Process., vol.49, pp. 3136-3144, 2001.
31. M. Felsberg and G. Sommer, "The monogenic scale-space: A unifying approach to phase-based image processing in scale-space," J. Math. Imag. Vis., vol.21, no.1, pp. 5-26, 2004.
32. G. Metikas and S. C. Olhede, "Multiple multidimensional Morse wavelets," IEEE Trans. Signal Process., vol.55, pp. 921-936, 2007.
33. K. G. Larkin, "Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Amer. A, vol.18, no.8, pp. 1871-1881, 2001.
34. K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform," J. Opt. Soc. Amer. A, vol.18, no.8, pp. 1862-1870, 2001.
35. 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.
36. S. Held, M. Storah, P. Massopust, and B. Forster, "Steerable wavelet frames based on the Riesz transform," IEEE Trans. Image Process., to be published.
37. I. Gelfand and G. Shilov, Generalized Functions. New York: Academic, 1964, vol.1.
38. S. L. Hahn, Hilbert Transforms in Signal Processing. Boston, MA: Artech House, 1996.
39. R. A. Adams, Sobolev Spaces. New York: Academic, 1975.
40. R. J. Duffin and A. C. Schaeffer, "A class of nonharmonic Fourier series," Trans. Amer. Math. Soc., vol.72, pp. 341-366, Mar. 1952.
41. A. Aldroubi, "Portraits of frames," Proc. Amer. Math. Soc., vol.123, no.6, pp. 1661-1668, 1995.
42. O. Christensen, An Introduction to Frames and Riesz Bases. New York: Birkhauser, 2003.
43. M. Unser and D. Van De Ville, "The pairing of a wavelet basis with a mildly redundant analysis via subband regression," IEEE Trans. Image Process., vol.17, no.11, pp. 2040-2052, Nov. 2008.
44. D. Van De Ville, T. Blu, and M. Unser, "Isotropic polyharmonic B-splines: Scaling functions and wavelets," IEEE Trans. Image Process., vol.14, no.11, pp. 1798-1813, Nov. 2005.
45. P. Burt and E. Adelson, "The Laplacian pyramid as a compact image code," IEEE Trans. Commun., vol.31, no.11, pp. 532-540, Apr. 1983.
46. P. Tafti, D. Van De Ville, and M. Unser, "Invariances, Laplacian-like wavelet bases, and the whitening of fractal processes," IEEE Trans. Image Process., vol.18, no.4, pp. 689-702, Apr. 2009.
47. D. Van De Ville and M. Unser, "Complex wavelet bases, steerability, and the Marr-like pyramid," IEEE Trans. Image Process., vol.17, no.11, pp. 2063-2080, Nov. 2008.
48. M. Feilner, D. Van De Ville, and M. Unser, "An orthogonal family of quincunx wavelets with continuously adjustable order," IEEE Trans. Image Process., vol.14, no.4, pp. 499-510, Apr. 2005.