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ashape3d

3D α-shape computation

Description

This function calculates the 3D α-shape of a given sample of points in the three-dimensional space for α>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 α.
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 α-complex (column 5) and for each α a value (1 or 0) indicating whether the tetrahedron belongs to the α-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 α-complex (columns 6 to 8) and for each α a value (0, 1, 2 or 3) indicating, respectively, that the triangle is not in the α-shape or it is interior, regular or singular (columns 9 to last). As defined in Edelsbrunner and Mucke (1994), a simplex in the α-complex is interior if it does not belong to the boundary of the α-shape. A simplex in the α-complex is regular if it is part of the boundary of the α-shape and bounds some higher-dimensional simplex in the α-complex. A simplex in the α-complex is singular if it is part of the boundary of the α-shape but does not bounds any higher-dimensional simplex in the α-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 α-complex (columns 5 to 7) and for each α a value (0, 1, 2 or 3) indicating, respectively, that the edge is not in the α-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 α-complex (columns 3 and 4) and for each α 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 α.
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)
alphashape3d

Implementation of the 3D Alpha-Shape for the Reconstruction of 3D Sets from a Point Cloud

v1.3.1
GPL-2
Authors
Thomas Lafarge, Beatriz Pateiro-Lopez
Initial release
2020-12-09

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