Uniform sampling in a convex hull

As an illustration of the uniformly package, we will show how to uniformly sample some points in a convex hull.

We give an illustration in dimension 3 (in dimension 2, use the function runif_in_polygon).

Let’s store the vertices of an icosahedron in a matrix vs:

vs <- t(rgl::icosahedron3d()$vb[1:3,])
head(vs)
#>          [,1]      [,2] [,3]
#> [1,] 0.000000  0.618034    1
#> [2,] 0.000000  0.618034   -1
#> [3,] 0.000000 -0.618034    1
#> [4,] 0.000000 -0.618034   -1
#> [5,] 0.618034  1.000000    0
#> [6,] 0.618034 -1.000000    0

The icosahedron is convex, therefore its convex hull is itself.

The delaunayn function of the geometry package calculates a “triangulation” (tetrahedralization in dimension 3) of the convex hull of a set of points. We use it to get a tetrahedralization of our icoshaedron:

library(geometry)
tetrahedra <- delaunayn(vs, options="Qz")
head(tetrahedra)
#>      [,1] [,2] [,3] [,4]
#> [1,]    8    7    1   10
#> [2,]    8    3    1   10
#> [3,]    8    3    1    9
#> [4,]    8    3    6    9
#> [5,]    8   11    4    6
#> [6,]    8   11    4    2

Each row of the tetrahedra matrix is a vector of four identifiers of the vertices defining a tetrahedron.

Now, we calculate the volumes of each of these tetrahedra with the volume_tetrahedron function:

library(uniformly)
volumes <- 
  apply(tetrahedra, 1, 
        function(t){
          volume_tetrahedron(vs[t[1],], vs[t[2],], vs[t[3],], vs[t[4],])
        })

We normalize the volumes:

probs <- volumes/sum(volumes)

Now, here is the algorithm to uniformly sample a point in the icosahedron:

select a tetrahedron at random, with probability given by the normalized volumes;
uniformly sample a point in the picked tetrahedron.

That is:

i <- sample.int(nrow(tetrahedra), 1, prob=probs)
th <- tetrahedra[i,]
runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
#>           [,1]       [,2]      [,3]
#> [1,] -0.240087 0.01407167 0.5155659

Let’s use the algorithm to sample 100 random points:

nsims <- 100
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
  th <- tetrahedra[sample.int(nrow(tetrahedra), 1, prob=probs),]
  sims[k,] <- runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
}

library(rgl)
open3d(windowRect=c(100,100,600,600))
#> null 
#>    1
shade3d(icosahedron3d(), color="red", alpha=0.3)
points3d(sims)
rglwidget()

We can proceed in the same way in higher dimension, using the functions volume_simplex and runif_in_simplex instead of the functions volume_tetrahedron and runif_in_tetrahedron.

Sampling from a triangulated object

Note that the convexity is not the sine qua non condition to apply the above procedure: the ingredient we need is the “triangulation” of the object. We took a convex shape because delaunayn provides the triangulation of a convex shape.

Let’s give an example for a 3D star. Here is the star:

vs <- rbind(
  c(7.889562, 1.150329, -2.173651),
  c(2.212808, 1.150329, -2.230414),
  c(0.068023, 1.150328, -7.923502),
  c(-2.151306, 1.150329, -2.254857),
  c(-7.817406, 1.150328, -2.261558),
  c(-3.523133, 1.150328, 1.888122),
  c(-4.869315, 1.150328, 6.987552),
  c(-0.006854, 1.150329, 4.473047),
  c(4.838127, 1.150328, 7.041885),
  c(3.538153, 1.150329, 1.927652),
  c(0.033757, 0.000000, -0.314657),
  c(0.035668, 2.269531, -0.312831)
)
faces <- rbind(
  c(1, 11, 2),
  c(2, 11, 3),
  c(3, 11, 4),
  c(4, 11, 5),
  c(5, 11, 6),
  c(6, 11, 7),
  c(7, 11, 8),
  c(8, 11, 9),
  c(9, 11, 10),
  c(10, 11, 1),
  c(1, 12, 10),
  c(10, 12, 9),
  c(9, 12, 8),
  c(8, 12, 7),
  c(7, 12, 6),
  c(6, 12, 5),
  c(5, 12, 4),
  c(4, 12, 3),
  c(3, 12, 2),
  c(2, 12, 1)
)
open3d(windowRect=c(100,100,600,600))
#> null 
#>    2
for(i in 1:nrow(faces)){
 triangles3d(rbind(
   vs[faces[i,1],], 
   vs[faces[i,2],], 
   vs[faces[i,3],]), 
   color="red", alpha=0.4)
}
rglwidget()

This star is not convex but it is star-shaped with respect to its centroid, and its faces are triangular. Therefore we get a tetrahedralization by joining the centroid to each of the triangular faces.

Let’s calculate the volumes of these tetrahedra:

centroid <- colMeans(vs)
volumes <- apply(faces, 1,function(f){
  volume_tetrahedron(vs[f[1],], vs[f[2],], vs[f[3],], centroid)
})
probs <- volumes/sum(volumes)

Now we pick a face at random, with probability given by the normalized volumes, and we sample in the corresponding tetrahedron:

nsims <- 500
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
  f <- faces[sample.int(nrow(faces), 1, prob=probs),]
  sims[k,] <- runif_in_tetrahedron(1, vs[f[1],], vs[f[2],], vs[f[3],], centroid)
}

And now, let’s add the sampled points:

points3d(sims)
rglwidget()

- Sampling from a triangulated object