R: 3D alpha-shape computation

ashape3d {alphashape3d}

R Documentation

3D `\alpha`-shape computation

Description

This function calculates the 3D \alpha-shape of a given sample of points in the three-dimensional space for \alpha>0.

Usage

ashape3d(x, alpha, pert = FALSE, eps = 1e-09)

Arguments

`x`	A 3-column matrix with the coordinates of the input points. Alternatively, an object of class `"ashape3d"` can be provided, see Details.
`alpha`	A single value or vector of values for `\alpha`.
`pert`	Logical. If the input points are not in general position and `pert` it set to TRUE the observations are perturbed by adding random noise, see Details.
`eps`	Scaling factor used for data perturbation when the input points are not in general position, see Details.

Details

If x is an object of class "ashape3d", then ashape3d does not recompute the 3D Delaunay triangulation (it reduces the computational cost).

If the input points x are not in general position and pert is set to TRUE, the function adds random noise to the data. The noise is generated from a normal distribution with mean zero and standard deviation eps*sd(x).

Value

An object of class "ashape3d" with the following components (see Edelsbrunner and Mucke (1994) for notation):

`tetra`	For each tetrahedron of the 3D Delaunay triangulation, the matrix `tetra` stores the indices of the sample points defining the tetrahedron (columns 1 to 4), a value that defines the intervals for which the tetrahedron belongs to the `\alpha`-complex (column 5) and for each `\alpha` a value (1 or 0) indicating whether the tetrahedron belongs to the `\alpha`-shape (columns 6 to last).
`triang`	For each triangle of the 3D Delaunay triangulation, the matrix `triang` stores the indices of the sample points defining the triangle (columns 1 to 3), a value (1 or 0) indicating whether the triangle is on the convex hull (column 4), a value (1 or 0) indicating whether the triangle is attached or unattached (column 5), values that define the intervals for which the triangle belongs to the `\alpha`-complex (columns 6 to 8) and for each `\alpha` a value (0, 1, 2 or 3) indicating, respectively, that the triangle is not in the `\alpha`-shape or it is interior, regular or singular (columns 9 to last). As defined in Edelsbrunner and Mucke (1994), a simplex in the `\alpha`-complex is interior if it does not belong to the boundary of the `\alpha`-shape. A simplex in the `\alpha`-complex is regular if it is part of the boundary of the `\alpha`-shape and bounds some higher-dimensional simplex in the `\alpha`-complex. A simplex in the `\alpha`-complex is singular if it is part of the boundary of the `\alpha`-shape but does not bounds any higher-dimensional simplex in the `\alpha`-complex.
`edge`	For each edge of the 3D Delaunay triangulation, the matrix `edge` stores the indices of the sample points defining the edge (columns 1 and 2), a value (1 or 0) indicating whether the edge is on the convex hull (column 3), a value (1 or 0) indicating whether the edge is attached or unattached (column 4), values that define the intervals for which the edge belongs to the `\alpha`-complex (columns 5 to 7) and for each `\alpha` a value (0, 1, 2 or 3) indicating, respectively, that the edge is not in the `\alpha`-shape or it is interior, regular or singular (columns 8 to last).
`vertex`	For each sample point, the matrix `vertex` stores the index of the point (column 1), a value (1 or 0) indicating whether the point is on the convex hull (column 2), values that define the intervals for which the point belongs to the `\alpha`-complex (columns 3 and 4) and for each `\alpha` a value (1, 2 or 3) indicating, respectively, if the point is interior, regular or singular (columns 5 to last).
`x`	A 3-column matrix with the coordinates of the original sample points.
`alpha`	A single value or vector of values of `\alpha`.
`xpert`	A 3-column matrix with the coordinates of the perturbated sample points (only when the input points are not in general position and `pert` is set to TRUE).

References

Edelsbrunner, H., Mucke, E. P. (1994). Three-Dimensional Alpha Shapes. ACM Transactions on Graphics, 13(1), pp.43-72.

Examples


T1 <- rtorus(1000, 0.5, 2)
T2 <- rtorus(1000, 0.5, 2, ct = c(2, 0, 0), rotx = pi/2)
x <- rbind(T1, T2)
# Value of alpha
alpha <- 0.25
# 3D alpha-shape
ashape3d.obj <- ashape3d(x, alpha = alpha)
plot(ashape3d.obj)

# For new values of alpha, we can use ashape3d.obj as input (faster)
alpha <- c(0.15, 1)
ashape3d.obj <- ashape3d(ashape3d.obj, alpha = alpha)
plot(ashape3d.obj, indexAlpha = 2:3)

[Package alphashape3d version 1.3.2 Index]

3D \alpha-shape computation