Non-separable wavelets are multi-dimensional
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s that are not directly implemented as
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
s of wavelets on some lower-dimensional space.
They have been studied since 1992.
They offer a few important advantages. Notably, using non-separable filters leads to more parameters in design, and consequently better filters.
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
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
).
Examples
* Red-black wavelets
*
Contourlet
Contourlets form a multiresolution directional tight frame designed to efficiently approximate images made of smooth regions separated by smooth boundaries. The contourlet transform has a fast implementation based on a Laplacian pyramid decompositi ...
s
*
Shearlet In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation o ...
s
* Directionlets
*
Steerable pyramids
* Non-separable schemes for tensor-product wavelets
[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.]
References
Wavelets
Multidimensional signal processing
Image processing
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