plot_filter {gasper}

Plot Tight-Frame Filters

Description

plot_filter provides a graphical representation of tight-frame filters as functions of the eigenvalues of the Laplacian matrix.

Usage

plot_filter(lmax, b, N = 1000, filter_func = zetav, filter_params = list())

Arguments

lmax

Largest eigenvalue of the Laplacian matrix (numeric scalar).

b

Parameter that controls the number of scales (numeric scalar).

N

Number of discretization points for the x-axis. By default, N is set to 1000.

filter_func

Function used to compute the filter values. By default, it uses the zetav function but other frame filters can be pass.

filter_params

List of additional parameters required by filter_func. Default is an empty list.

Details

The plotted functions represent the square root of the values given by the zetav function at different scales.

This function plots the square roots of the functions forming the partition of unity, corresponding to the construction of tight frames on the graph. The square root operation is essential as it ensures the Parseval identity, making the constructed frame "tight" and preserving the energy of signals on the graph when mapped to their frame representation.

plot_filter first determines the number of scales based on the largest eigenvalue \lambda_{\text{max}} and the parameter b as:

k_{\text{max}} = \left\lfloor \frac{\log(\lambda_{\text{max}})}{\log(b)} \right\rfloor + 2

The function then plots the square root of the values given by the zetav function over the range [0, \lambda_{\text{max}}] for each scale.

Note

plot_filter can be adapted for other filters by passing a different filter function to the filter_func parameter. The computation of k_{\text{max}} using \lambda_{\text{max}} and b applies primarily to the default zetav filter. It can be overridden by providing it in the filter_params list for other filters.

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.

See Also

zetav

Examples

plot_filter(6,2)