get.real.dft.mat: Matrix applying the two-dimensional real Fourier transform.

In spate: Spatio-Temporal Modeling of Large Data Using a Spectral SPDE Approach

Matrix applying the two-dimensional real Fourier transform.

Description

Returns the matrix that applies the two-dimensional real Fourier transform.

Usage

get.real.dft.mat(wave, indCos, ns = 4, n)

Arguments

wave	Matrix of size 2 x NF with spatial wavenumbers. NF is the number of Fourier functions.
indCos	Vector of integers indicating the position of columns in 'wave' of wavenumbers of cosine terms.
ns	Number of real Fourier functions that have only a cosine and no sine term. 'ns' is maximal 4.
n	Number of grid points on each axis. n x n is the total number of spatial points.

Value

A matrix that applies the two-dimensional real Fourier transform.

Author(s)

Fabio Sigrist

Examples

##Example nr. 1: sampling from a Matern field
n <- 50
spateFT <- spate.init(n=n,T=1)
spec <- matern.spec(wave=spateFT$wave,n=n,rho0=0.05,sigma2=1,norm=TRUE)
Phi <- get.real.dft.mat(wave=spateFT$wave, indCos=spateFT$indCos, n=n)
sim <- Phi %*% (sqrt(spec)*rnorm(n*n))
image(1:n,1:n,matrix(sim,nrow=n),main="Sample from Matern field",xlab="",ylab="")

##Example nr. 2: image reconstruction
n <- 50##Number of points on each axis
##Low-dimensional: only 41 Fourier functions
spateFT <- spate.init(n=n,T=17,NF=45)
Phi.LD <- get.real.dft.mat(wave=spateFT$wave, indCos=spateFT$indCos, ns=spateFT$ns, n=n)
##Mid-dimensional: 545 (of potentially 2500) Fourier functions
spateFT <- spate.init(n=n,T=17,NF=101)
Phi.MD <- get.real.dft.mat(wave=spateFT$wave, indCos=spateFT$indCos, ns=spateFT$ns, n=n)
##High-dimensional: all 2500 Fourier functions
spateFT <- spate.init(n=n,T=17,NF=2500)
Phi.HD <- get.real.dft.mat(wave=spateFT$wave, indCos=spateFT$indCos, ns=spateFT$ns, n=n)
##Define image
image <- rep(0,n*n)
for(i in 1:n){
  for(j in 1:n){    
    image[(i-1)*n+j] <- cos(5*(i-n/2)/n*pi)*sin(5*(j)/n*pi)*(1-abs(i/n-1/2)-abs(j/n-1/2))
  }
}
opar <- par(no.readonly = TRUE)
par(mfrow=c(2,2),mar=c(2,3,2,1))
image(1:n, 1:n, matrix(image, nrow = n),col = cols(),xlab="",ylab="",main="Original image")
##Aply inverse Fourier transform, dimension reduction, and Fourier transform
spec.LD <- t(Phi.LD) %*% image
image.LD <- Phi.LD %*% spec.LD
spec.MD <- t(Phi.MD) %*% image
image.MD <- Phi.MD %*% spec.MD
spec.HD <- t(Phi.HD) %*% image
image.HD <- Phi.HD %*% spec.HD
image(1:n, 1:n, matrix(image.LD, nrow = n),col = cols(),
      xlab="",ylab="",main="45 of 2500 Fourier terms")
image(1:n, 1:n, matrix(image.MD, nrow = n),col = cols(),
      xlab="",ylab="",main="101 of 2500 Fourier terms")
image(1:n, 1:n, matrix(image.HD, nrow = n),col = cols(),
      xlab="",ylab="",main="All 2500 Fourier terms")
par(opar) # Reset par() settings

spate documentation built on Oct. 3, 2023, 5:09 p.m.