gasper (version 1.1.6)

forward_sgwt: Compute Forward Spectral Graph Wavelet Transform

Description

forward_sgwt computes the forward Spectral Graph Wavelet Transform (SGWT) for a given graph signal \(f\).

Usage

forward_sgwt(
  f,
  evalues,
  evectors,
  b = 2,
  filter_func = zetav,
  filter_params = list()
)

Value

wc A concatenated vector of wavelet coefficients.

Arguments

f

Numeric vector representing the graph signal to analyze.

evalues

Numeric vector of eigenvalues of the Laplacian matrix.

evectors

Matrix of eigenvectors of the Laplacian matrix.

b

Numeric scalar that controls the number of scales in the SGWT. It must be greater than 1.

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 transform is constructed based on the frame defined by the tight_frame function, without the need for its explicit calculation. Other filters can be passed as parameters. The SGWT provides a multi-scale analysis of graph signals.

Given a graph signal \(f\) of length \(N\), forward_sgwt computes the wavelet coefficients using SGWT.

The eigenvalues and eigenvectors of the graph Laplacian, are denoted as \(\Lambda\) and \(U\) respectively. The parameter \(b\) controls the number of scales, and \(\lambda_{\text{max}}\) is the largest eigenvalue.

For each scale \(j = 0, \ldots, J\), where $$J = \left\lfloor \frac{\log(\lambda_{\text{max}})}{\log(b)} \right\rfloor + 2$$ the wavelet coefficients are computed as: $$ \mathbf{w}_j = U \left( g_j \odot (U^T f) \right) $$ where $$g_j(\lambda) = \sqrt{\psi_j(\lambda)}$$ and \(\odot\) denotes element-wise multiplication.

The final result is a concatenated vector of these coefficients for all scales.

References

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.

Hammond, D. K., Vandergheynst, P., & Gribonval, R. (2011). Wavelets on graphs via spectral graph theory. Applied and Computational Harmonic Analysis, 30(2), 129-150.

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

inverse_sgwt, tight_frame

Examples

if (FALSE) {
# Extract the adjacency matrix from the grid1 and compute the Laplacian
L <- laplacian_mat(grid1$sA)

# Compute the spectral decomposition of L
decomp <- eigensort(L)

# Create a sample graph signal
f <- rnorm(nrow(L))

# Compute the forward Spectral Graph Wavelet Transform
wc <- forward_sgwt(f, decomp$evalues, decomp$evectors)
}