R/forward_gft.R

In gasper: Graph Signal Processing

#' Compute Forward Graph Fourier Transform
#'
#' \code{forward_gft} computes the Graph Fourier Transform (GFT) of a given graph signal \eqn{f}{f}.
#'
#' @export forward_gft
#' @param L Laplacian matrix of the graph.
#' @param f Numeric vector of the graph signal to analyze.
#' @param U Matrix of the Eigenvectors of the Laplacian matrix. If NULL (default), the function will compute the eigendecomposition of the Laplacian.
#' @return \code{hatf} Numeric vector. Graph Fourier Transform of \eqn{f}{f}.
#' @seealso \code{\link{inverse_gft}}
#' @details
#'
#' The GFT is the representation of the graph signal on an orthonormal basis of the  graph's Laplacian matrix. It allows to analyze the frequency content of signals defined on graphs. In this context, the "frequency" of a graph signal refers to its decomposition in terms of the graph's Laplacian eigenvectors, which are similar to the harmonics of classical Fourier analysis.
#'
#' The GFT of a graph signal \eqn{f}{f} is given by:
#' \deqn{
#' \hat{f} = U^T f
#' }{\hat{f} = U^T f}
#' where \eqn{U}{U} denotes the matrix of eigenvectors of the graph's Laplacian.
#'
#' When the eigenvectors \eqn{U}{U} are not provided, the function computes them using the Laplacian matrix \eqn{L}{L}.
#'
#' @examples
#' \dontrun{
#' # Extract the adjacency matrix from the grid1 and compute the Laplacian
#' L <- laplacian_mat(grid1$sA)
#'
#' # Create a sample graph signal
#' f <- rnorm(nrow(L))
#'
#' # Compute the forward GFT
#' hatf <- forward_gft(L, f)
#' }
#'
#' @references
#' Ortega, A., Frossard, P., Kovačević, J., Moura, J. M., & Vandergheynst, P. (2018). Graph signal processing: Overview, challenges, and applications. Proceedings of the IEEE, 106(5), 808-828.
#'
#' Shuman, D. I., Narang, S. K., Frossard, P., Ortega, A., & Vandergheynst, P. (2013). The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE signal processing magazine, 30(3), 83-98.

forward_gft <- function(L, f, U=NULL) {
  if (is.null(U)) {
    eigen_decomp <- eigensort(L)
    U <- eigen_decomp$evectors
  }
  hatf <- as.vector(t(U)%*%f)
  return(hatf)
}