Non-separable wavelet
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.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.
They offer a few important advantages. Notably, using non-separable filters leads to more parameters in design, and consequently better filters.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.
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 waveletsG. Uytterhoeven and A. Bultheel, "The Red-Black Wavelet Transform," in IEEE Signal Processing Symposium, pp. 191–194, 1998.
- ContourletsM. 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.
- ShearletsG. Kutyniok and D. Labate, "Shearlets: Multiscale Analysis for Multivariate Data," 2012.
- DirectionletsV. 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.
- Steerable pyramidsE. 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.
- Non-separable schemes for tensor-product waveletsD. 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.