zetav {gasper} | R Documentation |
Evaluate Localized Tight-Frame Filter Functions
Description
zetav
evaluates the filters associated with a specific tight-frame construction.
Usage
zetav(x, k, b = 2)
Arguments
x |
A vector representing the support on which to evaluate the filter |
k |
A scalar representing the scale index. |
b |
A scalar parameter that governs the number of scales (b=2 default). |
Details
The function zetav
evaluates the partition of unity functions following the methodology described in the references similar to the Littlewood-Paley type, based on a partition of unity, as proposed in the reference papers. This approach, inspired by frame theory, facilitates the construction of filter banks, ensuring effective spectral localization.
A finite collection is a finite partition of unity on the compact interval
. It satisfies:
Let be a function with support in [0,1]. It's defined as:
For a given . Based on this function
, the partition of unity functions
are defined as:
and for all :
where is defined by:
Given this finite partition of unity , the Parseval identity implies that the following set of vectors forms a tight frame:
Value
Returns a numeric vector of evaluated filter values.
References
Coulhon, T., Kerkyacharian, G., & Petrushev, P. (2012). Heat kernel generated frames in the setting of Dirichlet spaces. Journal of Fourier Analysis and Applications, 18(5), 995-1066.
Göbel, F., Blanchard, G., von Luxburg, U. (2018). Construction of tight frames on graphs and application to denoising. In Handbook of Big Data Analytics (pp. 503-522). Springer, Cham.
Leonardi, N., & Van De Ville, D. (2013). Tight wavelet frames on multislice graphs. IEEE Transactions on Signal Processing, 61(13), 3357-3367.
de Loynes, B., Navarro, F., Olivier, B. (2021). Data-driven thresholding in denoising with Spectral Graph Wavelet Transform. Journal of Computational and Applied Mathematics, Vol. 389.
Examples
## Not run:
x <- seq(0, 2, by = 0.1)
g <- zetav(x, 1, 2)
plot(x, g, type = "l")
## End(Not run)