Realistic simulated elevation data in R / Perlin noise

Question

Does anyone know how to create a simulation raster elevation dataset - i.e. a 2d matrix of realistic elevation values - in R? R's jitter doesn't seem appropriate. In Java/Processing the noise() function achieves this with a Perlin noise algorithm e.g.:

size(200, 200);
float ns = 0.03; // for scaling
for (float i=0; i<200; i++) {
  for (float j=0; j<200; j++) {
    stroke(noise(i*ns, j*ns) * 255);
    point(i, j);
  }
}

But I've found no references to Perlin noise in R literature. Thanks in advance.

Answer 1

Here is an implementation in R, following the explanations in http://webstaff.itn.liu.se/~stegu/TNM022-2005/perlinnoiselinks/perlin-noise-math-faq.html

perlin_noise <- function( 
  n = 5,   m = 7,    # Size of the grid for the vector field
  N = 100, M = 100   # Dimension of the image
) {
  # For each point on this n*m grid, choose a unit 1 vector
  vector_field <- apply(
    array( rnorm( 2 * n * m ), dim = c(2,n,m) ),
    2:3,
    function(u) u / sqrt(sum(u^2))
  )
  f <- function(x,y) {
    # Find the grid cell in which the point (x,y) is
    i <- floor(x)
    j <- floor(y)
    stopifnot( i >= 1 || j >= 1 || i < n || j < m )
    # The 4 vectors, from the vector field, at the vertices of the square
    v1 <- vector_field[,i,j]
    v2 <- vector_field[,i+1,j]
    v3 <- vector_field[,i,j+1]
    v4 <- vector_field[,i+1,j+1]
    # Vectors from the point to the vertices
    u1 <- c(x,y) - c(i,j)
    u2 <- c(x,y) - c(i+1,j)
    u3 <- c(x,y) - c(i,j+1)
    u4 <- c(x,y) - c(i+1,j+1)
    # Scalar products
    a1 <- sum( v1 * u1 )
    a2 <- sum( v2 * u2 )
    a3 <- sum( v3 * u3 )
    a4 <- sum( v4 * u4 )
    # Weighted average of the scalar products
    s <- function(p) 3 * p^2 - 2 * p^3
    p <- s( x - i )
    q <- s( y - j )
    b1 <- (1-p)*a1 + p*a2
    b2 <- (1-p)*a3 + p*a4
    (1-q) * b1 + q * b2
  }
  xs <- seq(from = 1, to = n, length = N+1)[-(N+1)]
  ys <- seq(from = 1, to = m, length = M+1)[-(M+1)]
  outer( xs, ys, Vectorize(f) )
}

image( perlin_noise() )

You can have a more fractal structure by adding those matrices, with different grid sizes.

a <- .6
k <- 8
m <- perlin_noise(2,2,2^k,2^k)
for( i in 2:k )
  m <- m + a^i * perlin_noise(2^i,2^i,2^k,2^k)
image(m)
m[] <- rank(m) # Histogram equalization
image(m)