Non-separable wavelets are multi-dimensional wavelets that are not directly implemented as tensor products of wavelets on some lower-dimensional space. They have been studied since 1992.[1] They offer a few important advantages. Notably, using non-separable filters leads to more parameters in design, and consequently better filters.[2] The main difference, when compared to the one-dimensional wavelets, is that multi-dimensional sampling requires the use of lattices (e.g., the quincunx lattice). The wavelet filters themselves can be separable or non-separable regardless of the sampling lattice. Thus, in some cases, the non-separable wavelets can be implemented in a separable fashion. Unlike separable wavelet, the non-separable wavelets are capable of detecting structures that are not only horizontal, vertical or diagonal (show less anisotropy).
Examples
- Red-black wavelets[3]
- Contourlets[4]
- Shearlets[5]
- Directionlets[6]
- Steerable pyramids[7]
- Non-separable schemes for tensor-product wavelets[8]
References
- ^ J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn," IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 533–555, Mar. 1992.
- ^ J. Kovacevic and M. Vetterli, "Nonseparable two- and three-dimensional wavelets," IEEE Transactions on Signal Processing, vol. 43, no. 5, pp. 1269–1273, May 1995.
- ^ G. Uytterhoeven and A. Bultheel, "The Red-Black Wavelet Transform," in IEEE Signal Processing Symposium, pp. 191–194, 1998.
- ^ M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Transactions on Image Processing, vol. 14, no. 12, pp. 2091–2106, Dec. 2005.
- ^ G. Kutyniok and D. Labate, "Shearlets: Multiscale Analysis for Multivariate Data," 2012.
- ^ V. Velisavljevic, B. Beferull-Lozano, M. Vetterli and P. L. Dragotti, "Directionlets: anisotropic multi-directional representation with separable filtering," IEEE Trans. on Image Proc., Jul. 2006.
- ^ E. P. Simoncelli and W. T. Freeman, "The Steerable Pyramid: A Flexible Architecture for Multi-Scale Derivative Computation," in IEEE Second Int'l Conf on Image Processing. Oct. 1995.
- ^ D. Barina, M. Kula and P. Zemcik, "Parallel wavelet schemes for images," J Real-Time Image Proc, vol. 16, no. 5, pp. 1365–1381, Oct. 2019.
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