A unifying parametric framework for 2D steerable wavelet transforms

SIAM Journal on Imaging Sciences

Volumn 6, Issue 1, 2013, Pages 102-135

  Unser, Michael ^a   Chenouard, Nicolas ^a  

^a EPFL   (Switzerland)

Author keywords

Frame; Riesz transform; Steerability; Wavelets

Indexed keywords

ANGULAR SELECTIVITY; DIRECTIONAL WAVELETS; FRAME; FREQUENCY PROFILES; RIESZ TRANSFORM; STEERABILITY; STEERABLE PYRAMIDS; WAVELETS;

FEATURE EXTRACTION; PRINCIPAL COMPONENT ANALYSIS; SIGNAL DENOISING;

WAVELET TRANSFORMS;

EID: 84875890422     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/120866014     Document Type: Article

References (48)

1. S. G. Mallat, A theory of multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), pp. 674-693.
2. E. Simoncelli and W. Freeman, The steerable pyramid: A flexible architecture for multi-scale derivative computation, in Proceedings of the International Conference on Image Processing, Vol. 3, 1995, pp. 444-447.
3. E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, Shiftable multiscale transforms, IEEE Trans. Inform. Theory, 38 (1992), pp. 587-607.
4. 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., 12 (2003), pp. 1338-1351.
5. F. Denis and A. Baskurt, Multidirectional curvilinear structures detection using steerable pyramid, J. Electron. Imag., 13 (2004), pp. 756-765.
6. J. Portilla and E. P. Simoncelli, A parametric texture model based on joint statistics of complex wavelet coefficients, Int. J. Comput. Vis., 40 (2000), pp. 49-70.
7. G. Tzagkarakis, B. Beferull-Lozano, and P. Tsakalides, Rotation-invariant texture retrieval with Gaussianized steerable pyramids, IEEE Trans. Image Process., 15 (2006), pp. 2702-2718.
8. C. Y. Su, Y. T. Zhuang, L. Huang, and F. Wu, Steerable pyramid-based face hallucination, Pattern Recognition, 38 (2005), pp. 813-824.
9. Z. Liu, K. Tsukada, K. Hanasaki, Y. K. Ho, and Y. P. Dai, Image fusion by using steerable pyramid, Pattern Recognition Lett., 22 (2001), pp. 929-939.
10. I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), pp. 1413-1457.
11. H. Rabbani, Image denoising in steerable pyramid domain based on a local Laplace prior, Pattern Recognition, 42 (2009), pp. 2181-2193.
12. G. Jacovitti and A. Neri, Multiresolution circular harmonic decomposition, IEEE Trans. Signal Process., 48 (2000), pp. 3242-3247.
13. M. Unser and D. Van De Ville, Wavelet steerability and the higher-order Riesz transforms, IEEE Trans. Image Process., 19 (2010), pp. 636-652.
14. S. Held, M. Storah, P. Massopust, and B. Forster, Steerable wavelet frames based on the Riesz transform, IEEE Trans. Image Process., 19 (2010), pp. 653-667.
15. d), IEEE Trans. Image Process., 20 (2011), pp. 2705-2721.
16. J. G. Daugman, Complete discrete 2-D Gabor transforms by neural networks for image-analysis and compression, IEEE Trans. Acoust. Speech Signal Process., 36 (1988), pp. 1169-1179.
17. 2 singularities, Comm. Pure Appl. Math., 57 (2004), pp. 219-266.
18. M. N. Do and M. Vetterli, The contourlet transform: An efficient directional multiresolution image representation, IEEE Trans. Image Process., 14 (2005), pp. 2091-2106.
19. V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. Dragotti, Directionlets: Anisotropic multidirectional representation with separable filtering, IEEE Trans. Image Process., 15 (2006), pp. 1916-1933.
20. K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal., 39 (2007), pp. 298-318.
21. A. da Cunha, J. Zhou, and M. Do, The nonsubsampled contourlet transform: Theory, design, and applications, IEEE Trans. Image Process., 15 (2006), pp. 3089-3101.
22. D. Donoho, G. Kutyniok, M. Shahram, and X. Zhuang, A rational design of a digital shearlet transform, in Proceedings of the 9th International Conference on Sampling Theory and Applications (SampTA), 2011.
23. 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, 18 (2001), pp. 1862-1870.
24. 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, 18 (2001), pp. 1871-1881.
25. M. Unser, D. Sage, and D. Van De Ville, Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transforms, IEEE Trans. Image Process., 18 (2009), pp. 2402-2418.
26. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.
27. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
28. Q. Sun and M. Unser, Left-inverses of fractional Laplacian and sparse stochastic processes, Adv. Comput. Math., 36 (2012), pp. 399-441.
29. J. P. Ward, K. N. Chaudhury, and M. Unser, Decay properties of Riesz transforms and steerable wavelets, SIAM J. Imaging Sci., submitted.
30. M. Papadakis, G. Gogoshin, I. A. Kakadiaris, D. J. Kouri, and D. K. Hoffman, Nonseparable radial frame multiresolution analysis in multidimensions, Numer. Funct. Anal. Optim., 24 (2003), pp. 907-928.
31. J. R. Romero, S. K. Alexander, S. Baid, S. Jain, and M. Papadakis, The geometry and the analytic properties of isotropic multiresolution analysis, Adv. Comput. Math., 31 (2009), pp. 283-328.
32. W. T. Freeman and E. H. Adelson, The design and use of steerable filters, IEEE Trans. Pattern Anal. Mach. Intell., 13 (1991), pp. 891-906.
33. D. Van De Ville and M. Unser, Complex wavelet bases, steerability, and the Marr-like pyramid, IEEE Trans. Image Process., 17 (2008), pp. 2063-2080.
34. M. Felsberg and G. Sommer, The monogenic signal, IEEE Trans. Signal Process., 49 (2001), pp. 3136-3144.
35. J. W. Goodman, Introduction to Fourier Optics, Roberts and Company, Greenwood Village, CO, 2005.
36. D. Slepian, Prolate spheroidal wave-functions, Fourier-analysis, and uncertainty-V: The discrete case, Bell System Tech. J., 57 (1978), pp. 1371-1430.
37. D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., 25 (1983), pp. 379-393.
38. E. Simoncelli and H. Farid, Steerable wedge filters for local orientation analysis, IEEE Trans. Image Process., 5 (1996), pp. 1377-1382.
39. E. Candès, L. Demanet, D. Donoho, and L. Ying, Fast discrete curvelet transforms, Multiscale Model. Simul., 5 (2006), pp. 861-899.
40. C. S. Guo, Y. J. Han, J. B. Xu, and J. P. Ding, Radial Hilbert transform with Laguerre-Gaussian spatial filters, Optic Lett., 31 (2006), pp. 1394-1396.
41. P. Kittipoom, G. Kutyniok, and W.-Q. Lim, Construction of compactly supported shearlet frames, Constr. Approx., 35 (2012), pp. 21-72.
42. J.-L. Starck, D. Elad, and D. Donoho, Redundant multiscale transforms and their application for morphological component separation, in Advances in Imaging and Electron Physics, Vol. 132, Elsevier, Amsterdam, 2004, pp. 287-348.
43. N. Chenouard, S. Vernhettes, I. Bloch, and J.-C. Olivo-Marin, Morphological source separation for particle tracking in complex biological environments, in Proceedings of the 19th International Conference on Pattern Recognition (ICPR 2008), 2008, pp. 1-4.
44. K. Gousset, E. Schiff, C. Langevin, Z. Marijanovic, A. Caputo, D. T. Browman, N. Chenouard, F. de Chaumont, A. Martino, J. Enninga, J.-C. Olivo-Marin, D. Männel, and C. Zurzolo, Prions hijack tunneling nanotubes for intercellular spread, Nat. Cell Biol., 11 (2009), pp. 328-336.
45. G. Kutyniok and W.-Q. Lim, Image separation using wavelets and shearlets, in Curves and Surfaces, Lecture Notes in Comput. Sci. 6920, J.-D. Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, M.-L. Mazure, and L. Schumaker, eds., Springer, Berlin, Heidelberg, 2012, pp. 416-430.
46. M. Unser, Local linear transforms for texture measurements, Signal Process., 11 (1986), pp. 61-79.
47. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
48. J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math., 6 (1976), pp. 409-434.