zetav evaluates the filters associated with a specific tight-frame construction.
zetav(x, k, b = 2)Returns a numeric vector of evaluated filter values.
A vector representing the support on which to evaluate the filter
A scalar representing the scale index.
A scalar parameter that governs the number of scales (b=2 default).
The function zetav evaluates the partition of unity functions \(\psi\) 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 \((\psi_j)_{j=0, \ldots, J}\) is a finite partition of unity on the compact interval \([0, \lambda_{\mathrm{max}}]\). It satisfies: $$ \psi_j : [0,\lambda_{\mathrm{max}}] \rightarrow [0,1]~\textrm{for all}~ j \in \{1,\ldots,J\}~\textrm{and}~\forall \lambda \in [0,\lambda_{\mathrm{max}}],~\sum_{j=0}^J \psi_j(\lambda)=1. $$
Let \(\omega : \mathbb R^+ \rightarrow [0,1]\) be a function with support in [0,1]. It's defined as: $$ \omega(x) = \begin{cases} 1 & \text{if } x \in [0,b^{-1}] \\ b \cdot \frac{x}{1 - b} + \frac{b}{b - 1} & \text{if } x \in (b^{-1}, 1] \\ 0 & \text{if } x > 1 \end{cases} $$ For a given \(b > 1\). Based on this function \(\omega\), the partition of unity functions \(\psi\) are defined as: $$ \psi_0(x) = \omega(x) $$ and for all \(j \geq 1\): $$ \psi_j(x) = \omega(b^{-j} x) - \omega(b^{-j+1} x) $$ where \(J\) is defined by: $$ J = \left \lfloor \frac{\log \lambda_{\mathrm{max}}}{\log b} \right \rfloor + 2 $$
Given this finite partition of unity \((\psi_j)_{j=0, \ldots, J}\), the Parseval identity implies that the following set of vectors forms a tight frame: $$ \mathfrak F = \left \{ \sqrt{\psi_j}(\mathcal{L})\delta_i : j=0, \ldots, J, i \in V \right \}. $$
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.
if (FALSE) {
x <- seq(0, 2, by = 0.1)
g <- zetav(x, 1, 2)
plot(x, g, type = "l")
}
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