inverse_sgwt: Compute Inverse Spectral Graph Wavelet Transform

In gasper: Graph Signal Processing

Compute Inverse Spectral Graph Wavelet Transform

Description

inverse_sgwt computes the inverse (adjoint) Spectral Graph Wavelet Transform (SGWT) for wavelet coefficients wc.

Usage

inverse_sgwt(
  wc,
  evalues,
  evectors,
  b = 2,
  filter_func = zetav,
  filter_params = list()
)

Arguments

wc	Numeric vector representing the spectral graph wavelet coefficients to reconstruct the graph signal from.
evalues	Numeric vector of eigenvalues of the Laplacian matrix.
evectors	Matrix of eigenvectors of the Laplacian matrix.
b	Numeric scalar that control 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 passed.
filter_params	List of additional parameters required by filter_func. Default is an empty list.

Details

The computation corresponds to the frame defined by the tight_frame function. Other filters can be passed as parameters. Given the tightness of the frame, the inverse is simply the application of the adjoint linear transformation to the wavelet coefficients.

Given wavelet coefficients wc, inverse_sgwt reconstructs the original graph signal using the inverse 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 reconstructed signal for that scale is computed as:

\mathbf{f}_j = (U \mathbf{wc}_j \odot g_j) U^T

where

g_j(\lambda) = \sqrt{\psi_j(\lambda)}

and \odot denotes element-wise multiplication.

The final result is the sum of \mathbf{f}_j across all scales to reconstruct the entire graph signal.

Value

f A graph signal obtained by applying the SGWT adjoint to wc.

Note

inverse_sgwt 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

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

forward_sgwt, tight_frame

Examples

## Not run: 
# 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)

# Reconstruct the graph signal using the inverse SGWT
f_rec <- inverse_sgwt(wc, decomp$evalues, decomp$evectors)

## End(Not run)

gasper documentation built on Oct. 27, 2023, 1:07 a.m.