synthesis.Rd

% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/synthesis.R
\name{synthesis}
\alias{synthesis}
\title{Compute the Synthesis Operator for Transform Coefficients}
\usage{
synthesis(coeff, tf)
}
\arguments{
\item{coeff}{Numeric vector/matrix. Transformed coefficients of the graph signal.}

\item{tf}{Numeric matrix. Frame coefficients.}
}
\value{
\code{y} Numeric vector/matrix. Synthesized graph signal.
}
\description{
\code{synthesis} computes the graph signal synthesis from its transform coefficients using the provided frame coefficients.
}
\details{
The \code{synthesis} operator uses the frame coefficients to retrieve the graph signal from its representation in the transform domain. It is the adjoint of the analysis operator \eqn{T_{\mathfrak F}}{T_F} and is defined by the linear map \eqn{T_{\mathfrak F}^\ast : \mathbb R^I \rightarrow \mathbb R^V}{T_F* : R^I -> R^V}. For a vector of coefficients \eqn{(c_i)_{i \in I}}{(c_i) for all i in I}, the synthesis operation is defined as:
\deqn{T^\ast_{\mathfrak F}(c_i)_{i \in I}=\sum_{i \in I} c_i r_i}{T*_F(c_i) for all i in I = sum of c_i * r_i for all i in I}

The synthesis is computed as:
\deqn{\code{y} = \code{coeff}^T\code{tf}}{y = t(coeff) \%*\% tf}
}
\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)

# Generate the tight frame coefficients using the tight_frame function
tf <- tight_frame(decomp$evalues, decomp$evectors)

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

# Compute the transform coefficients using the analysis operator
coef <- analysis(f, tf)

# Retrieve the graph signal using the synthesis operator
f_rec <- synthesis(coef, tf)
}
}
\seealso{
\code{\link{analysis}}, \code{\link{tight_frame}}
}