inverse_sgwt.Rd

% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/inverse_sgwt.R
\name{inverse_sgwt}
\alias{inverse_sgwt}
\title{Compute Inverse Spectral Graph Wavelet Transform}
\usage{
inverse_sgwt(
  wc,
  evalues,
  evectors,
  b = 2,
  filter_func = zetav,
  filter_params = list()
)
}
\arguments{
\item{wc}{Numeric vector representing the spectral graph wavelet coefficients to reconstruct the graph signal from.}

\item{evalues}{Numeric vector of eigenvalues of the Laplacian matrix.}

\item{evectors}{Matrix of eigenvectors of the Laplacian matrix.}

\item{b}{Numeric scalar that control the number of scales in the SGWT. It must be greater than 1.}

\item{filter_func}{Function used to compute the filter values. By default, it uses the \code{\link{zetav}} function but other frame filters can be passed.}

\item{filter_params}{List of additional parameters required by filter_func. Default is an empty list.}
}
\value{
\code{f} A graph signal obtained by applying the SGWT adjoint to \code{wc}.
}
\description{
\code{inverse_sgwt} computes the pseudo-inverse Spectral Graph Wavelet Transform (SGWT) for wavelet coefficients \code{wc}.
}
\details{
The computation corresponds to the frame defined by the \code{\link{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 \code{wc}, \code{inverse_sgwt} reconstructs the original graph signal using the inverse SGWT.

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

For each scale \eqn{j = 0,\ldots, J}{j = 0,..., J}, where
\deqn{J = \left\lfloor \frac{\log(\lambda_{\text{max}})}{\log(b)} \right\rfloor + 2}{J = floor(log(lambda_max)/log(b)) + 2} the reconstructed signal for that scale is computed as:
\deqn{
\mathbf{f}_j = (U \mathbf{wc}_j \odot g_j) U^T
}{\mathbf{f}_j = (U wc_j * g_j) U^T}
where \deqn{g_j(\lambda) = \sqrt{\psi_j(\lambda)}}{g_j(lambda) = sqrt(psi_j(lambda))} and \eqn{\odot}{*} denotes element-wise multiplication.

The final result is the sum of \eqn{\mathbf{f}_j}{f_j} across all scales to reconstruct the entire graph signal.
}
\note{
\code{inverse_sgwt} can be adapted for other filters by passing a different filter function to the \code{filter_func} parameter.
The computation of \eqn{k_{\text{max}}}{k_max} using \eqn{\lambda_{\text{max}}}{lambda_max} and \eqn{b}{b} applies primarily to the default \code{zetav} filter. It can be overridden by providing it in the \code{filter_params} list for other filters.
}
\examples{
\dontrun{
# 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)
}

}
\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.
}
\seealso{
\code{\link{forward_sgwt}}, \code{\link{tight_frame}}
}