3D Alpha Shapes

Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec

Assume we are given a set S of points in 2D or 3D and we'd like to have something like ``the shape formed by these points.'' This is quite a vague notion and there are probably many possible interpretations, the alpha shape being one of them. Alpha shapes can be used for shape reconstruction from a dense unorganized set of data points. Indeed, an alpha shape is demarcated by a frontier, which is a linear approximation of the original shape [BB97].

As mentioned in Edelsbrunner's and Mücke's paper [EM94],
one can intuitively think of an alpha shape as the
following. Imagine a huge mass of ice-cream making up the space ℝ^{3}
and containing the points as ``hard'' chocolate pieces. Using one of
those sphere-formed ice-cream spoons we carve out all parts of the
ice-cream block we can reach without bumping into chocolate pieces,
thereby even carving out holes in the inside (e.g. parts not reachable
by simply moving the spoon from the outside). We will eventually end
up with a (not necessarily convex) object bounded by caps, arcs and
points. If we now straighten all ``round'' faces to triangles and line
segments, we have an intuitive description of what is called the
alpha shape of S. Here's an example for this process in 2D (where
our ice-cream spoon is simply a circle):

Alpha shapes depend on a parameter α from which they
are named.
What is α in the ice-cream game? α is the squared radius of the
carving spoon. A very small value will allow us to eat up all of the
ice-cream except the chocolate points themselves. Thus we already see
that the alpha shape degenerates to the point-set S for
α → 0. On the other hand, a huge value of α
will prevent us even from moving the spoon between two points since
it's way too large. So we will never spoon up ice-cream lying in the
inside of the convex hull of S, and hence the alpha shape for
α → ∞ is the convex hull of S.^{1}

More precisely, the definition of alpha shapes is based on an underlying triangulation that may be a Delaunay triangulation in case of basic alpha shapes or a regular triangulation (cf. 38.3) in case of weighted alpha shapes.

Let us consider the basic case with a Delaunay triangulation. We first define the alpha complex of the set of points S. The alpha complex is a subcomplex of the Delaunay triangulation. For a given value of α, the alpha complex includes all the simplices in the Delaunay triangulation which have an empty circumscribing sphere with squared radius equal or smaller than α. Here ``empty'' means that the open sphere do not include any points of S. The alpha shape is then simply the domain covered by the simplices of the alpha complex (see [EM94]).

In general, an alpha complex is a disconnected and non-pure complex: This means in particular that the alpha complex may have singular faces. For 0 ≤ k ≤ d-1, a k-simplex of the alpha complex is said to be singular if it is not a facet of a (k+1)-simplex of the complex. CGAL provides two versions of alpha shapes. In the general mode, the alpha shapes correspond strictly to the above definition. The regularized mode provides a regularized version of the alpha shapes. It corresponds to the domain covered by a regularized version of the alpha complex where singular faces are removed (See Figure 42.1 for an example).

**Figure 42.1: **Comparison of general and regularized alpha-shape.
**Left:** Some points are taken on the surface of a torus,
three points being taken relatively
far from the surface of the torus;
**Middle:** The general alpha-shape (for a large enough alpha value)
contains the singular triangle facet of the three isolated points;
**Right:** The regularized version (for the same value of alpha) does not contains any singular facet.

The alpha shapes of a set of points S form a discrete family, even though they are defined for all real numbers α. The entire family of alpha shapes can be represented through the underlying triangulation of S. In this representation each k-simplex of the underlying triangulation is associated with an interval that specifies for which values of α the k-simplex belongs to the alpha complex. Relying on this fact, the family of alpha shapes can be computed efficiently and relatively easily. Furthermore, we can select the optimal value of α to get an alpha shape including all data points and having less than a given number of connected components. Also, the alpha-values allows to define a filtration on the faces of the triangulation of a set of points. In this filtration, the faces of the triangulation are output in increasing order of the alpha value for which they appear in the alpha complex. In case of equal alpha value lower dimensional faces are output first.

The definition is analog in the case of weighted alpha shapes.
The input set is now a set of weighted points (which can be regarded
as spheres) and the underlying triangulation
is the regular triangulation of this set.
Two spheres, or two weighted points , with centers C_{1}, C_{2}
and radii r_{1}, r_{2} are said to be orthogonal iff
C_{1}C_{2} ^{2} = r_{1}^{2} + r_{2}^{2} and suborthogonal
iff C_{1}C_{2} ^{2} < r_{1}^{2} + r_{2}^{2}.
For a given value of α
the weighted alpha complex is formed with the simplices of the
regular triangulation triangulation
such that there is a sphere orthogonal to the weighted points associated
with the vertices of the simplex and suborthogonal to all the other
input weighted points. Once again the alpha shape is then defined as
the domain covered by a the alpha complex and comes in general and
regularized versions.

The class *CGAL::Alpha_shape_3<Dt>* represents the whole
family of alpha shapes for a given set of points.
The class includes the underlying triangulation *Dt*
of the set, and associates to each k-face of this triangulation
an interval specifying
for which values of α the face belongs to the
alpha complex.

The class provides functions to set and get the current α-value, as well as an iterator that enumerates the α values where the alpha shape changes.

Also the class has a filtration member function that, given
an output iterator with *CGAL::object*
as value type, outputs the faces of the triangulation
according to the
order of apparition in the alpha complex when alpha increases.

Finally, it provides a function to determine
the smallest value α
such that the alpha shape satisfies the following two properties

(ii) all data points are either on the boundary or in the interior
of the regularized version of the alpha shape (no singular faces).

(i) The number of components is equal or less than a given number .

The current implementation is static, that is after its construction
points cannot be inserted or removed.

The current implementation of this class is dynamic, that is after its construction
points can be inserted or removed.

**Figure 42.2: **Classification of simplices, a 2D example.
**Left:** The 2D Delaunay triangulation of a set of points;
**Right:** Classification of simplices for a given alpha (the squared radius of the red circle).
*INTERIOR*, *REGULAR* and *SINGULAR* simplices are depicted in black, green and blue
respectively. *EXTERIOR* simplices are not depicted. The vertex *s* and the edge *tu* are *SINGULAR*
since all higher dimension simplices they are incident to are *EXTERIOR*.
The facet *pqr* is *EXTERIOR* because the squared radius of its circumscribed circle is larger
than alpha.

The classes provide also output iterators to get for a given alpha value
the vertices, edges, facets and cells of the different types
(*EXTERIOR*, *SINGULAR*, *REGULAR* or
*INTERIOR*).

We currently do not specify concepts for the underlying triangulation
type. Models that work for a familly alpha-shapes are the instantiations
of the classes *CGAL::Delaunay_triangulation_3* and
*CGAL::Periodic_3_Delaunay_triangulation_3* (see
example 42.5.5). A model that works for a fixed alpha-shape are the instantiations
of the class *CGAL::Delaunay_triangulation_3*.
A model that works for a weighted alpha-shape is
the class *CGAL::Regular_triangulation_3*. The triangulation needs a geometric traits class
and a triangulation data structure as template parameters.

For the class *CGAL::Alpha_shape_3<Dt>*, the requirements of
the traits class are described in the concepts *CGAL::AlphaShapeTraits_3*
in the non-weighted case and *CGAL::WeightedAlphaShapeTraits_3* in the weighted case.
The Cgal kernels are models in the non-weighted case, and
the class *CGAL::Regular_triangulation_euclidean_traits_3* is a model
in the weighted case.
The triangulation data structure of the triangulation
has to be a model of the concept *CGAL::TriangulationDataStructure_3*,
and it must be parameterized with vertex and cell classes, which are model of the concepts
*AlphaShapeVertex_3* and *AlphaShapeCell_3*.
The package provides by default the classes
*CGAL::Alpha_shape_vertex_base_3<Gt>* and
*CGAL::Alpha_shape_cell_base_3<Gt>*. When using
*CGAL::Periodic_3_Delaunay_triangulation_3* as underlying
triangulation the vertex and cell classes need to be models to both
*AlphaShapeVertex_3* and
*Periodic_3TriangulationDSVertexBase_3* as well as
*AlphaShapeCell_3* and *Periodic_3TriangulationDSCellBase_3*
(see example 42.5.5).

For the class *CGAL::Fixed_alpha_shape_3<Dt>*, the requirements of
the traits class are described in the concepts *CGAL::FixedAlphaShapeTraits_3*
in the non-weighted case and *CGAL::FixedWeightedAlphaShapeTraits_3* in the weighted case.
The Cgal kernels are models in the non-weighted case, and
the class *CGAL::Regular_triangulation_euclidean_traits_3* is a model
in the weighted case.
The triangulation data structure of the triangulation
has to be a model of the concept *CGAL::TriangulationDataStructure_3*,
and it must be parameterized with vertex and cell classes, which are model of the concepts
*FixedAlphaShapeVertex_3* and *FixedAlphaShapeCell_3*.
The package provides models *CGAL::Fixed_alpha_shape_vertex_base_3<Gt>*
and *CGAL::Fixed_alpha_shape_cell_base_3<Gt>*, respectively.

The class *CGAL::Alpha_shape_3<Dt>* represents the whole family
of alpha shapes for a given set of points while the class *CGAL::Fixed_alpha_shape_3<Dt>*
represents only one alpha shape (for a fixed alpha). When using the same kernel,
*CGAL::Fixed_alpha_shape_3<Dt>* being a lighter version, it is naturally much more efficient
when the alpha-shape is needed for a single given value of alpha.
In addition note that the class *CGAL::Alpha_shape_3<Dt>*
requires constructions (squared radius of simplices) while the
class *CGAL::Fixed_alpha_shape_3<Dt>* uses only predicates.
This implies that a certified construction of one (several)
alpha-shape, using the *CGAL::Alpha_shape_3<Dt>* requires a kernel
with exact predicates and exact constructions while using a kernel
with exact predicates is sufficient for the class *CGAL::Fixed_alpha_shape_3<Dt>*.
This makes the class *CGAL::Fixed_alpha_shape_3<Dt>* even more efficient in this setting.
In addition, note that the *Fixed* version is the only of the
two that supports incremental insertion and removal of points.

We give the time spent while computing the alpha shape of a protein (considered
as a set of weighted points) featuring 4251 atoms (using *gcc 4.3* under Linux with *-O3*
and *-DNDEBUG* flags, on a 2.27GHz Intel(R) Xeon(R) E5520 CPU):
Using *CGAL::Exact_predicates_inexact_constructions_kernel*, building
the regular triangulation requires 0.06s, then the class *CGAL::Fixed_alpha_shape_3<Dt>*
required 0.05s while the class *CGAL::Alpha_shape_3<Dt>* requires 0.35s.

This example builds a basic alpha shape using a Delaunay triangulation as underlying triangulation.

File:examples/Alpha_shapes_3/ex_alpha_shapes_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_3.h> #include <CGAL/Alpha_shape_3.h> #include <fstream> #include <list> #include <cassert> typedef CGAL::Exact_predicates_inexact_constructions_kernel Gt; typedef CGAL::Alpha_shape_vertex_base_3<Gt> Vb; typedef CGAL::Alpha_shape_cell_base_3<Gt> Fb; typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds; typedef CGAL::Delaunay_triangulation_3<Gt,Tds> Triangulation_3; typedef CGAL::Alpha_shape_3<Triangulation_3> Alpha_shape_3; typedef Gt::Point_3 Point; typedef Alpha_shape_3::Alpha_iterator Alpha_iterator; int main() { std::list<Point> lp; //read input std::ifstream is("./data/bunny_1000"); int n; is >> n; std::cout << "Reading " << n << " points " << std::endl; Point p; for( ; n>0 ; n--) { is >> p; lp.push_back(p); } // compute alpha shape Alpha_shape_3 as(lp.begin(),lp.end()); std::cout << "Alpha shape computed in REGULARIZED mode by default" << std::endl; // find optimal alpha value Alpha_iterator opt = as.find_optimal_alpha(1); std::cout << "Optimal alpha value to get one connected component is " << *opt << std::endl; as.set_alpha(*opt); assert(as.number_of_solid_components() == 1); return 0; }

File:examples/Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_3.h> #include <CGAL/Alpha_shape_3.h> #include <fstream> #include <list> #include <cassert> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Alpha_shape_vertex_base_3<K> Vb; typedef CGAL::Alpha_shape_cell_base_3<K> Fb; typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds; typedef CGAL::Delaunay_triangulation_3<K,Tds,CGAL::Fast_location> Delaunay; typedef CGAL::Alpha_shape_3<Delaunay> Alpha_shape_3; typedef K::Point_3 Point; typedef Alpha_shape_3::Alpha_iterator Alpha_iterator; typedef Alpha_shape_3::NT NT; int main() { Delaunay dt; std::ifstream is("./data/bunny_1000"); int n; is >> n; Point p; std::cout << n << " points read" << std::endl; for( ; n>0 ; n--) { is >> p; dt.insert(p); } std::cout << "Delaunay computed." << std::endl; // compute alpha shape Alpha_shape_3 as(dt); std::cout << "Alpha shape computed in REGULARIZED mode by defaut." << std::endl; // find optimal alpha values Alpha_shape_3::NT alpha_solid = as.find_alpha_solid(); Alpha_iterator opt = as.find_optimal_alpha(1); std::cout << "Smallest alpha value to get a solid through data points is " << alpha_solid << std::endl; std::cout << "Optimal alpha value to get one connected component is " << *opt << std::endl; as.set_alpha(*opt); assert(as.number_of_solid_components() == 1); return 0; }

The following examples build a weighted alpha shape requiring a
regular triangulation as underlying triangulation.
The alpha shape is built in *GENERAL* mode.

File:examples/Alpha_shapes_3/ex_weighted_alpha_shapes_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Regular_triangulation_euclidean_traits_3.h> #include <CGAL/Regular_triangulation_3.h> #include <CGAL/Alpha_shape_3.h> #include <list> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Regular_triangulation_euclidean_traits_3<K> Gt; typedef CGAL::Alpha_shape_vertex_base_3<Gt> Vb; typedef CGAL::Alpha_shape_cell_base_3<Gt> Fb; typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds; typedef CGAL::Regular_triangulation_3<Gt,Tds> Triangulation_3; typedef CGAL::Alpha_shape_3<Triangulation_3> Alpha_shape_3; typedef Alpha_shape_3::Cell_handle Cell_handle; typedef Alpha_shape_3::Vertex_handle Vertex_handle; typedef Alpha_shape_3::Facet Facet; typedef Alpha_shape_3::Edge Edge; typedef Gt::Weighted_point Weighted_point; typedef Gt::Bare_point Bare_point; int main() { std::list<Weighted_point> lwp; //input : a small molecule lwp.push_back(Weighted_point(Bare_point( 1, -1, -1), 4)); lwp.push_back(Weighted_point(Bare_point(-1, 1, -1), 4)); lwp.push_back(Weighted_point(Bare_point(-1, -1, 1), 4)); lwp.push_back(Weighted_point(Bare_point( 1, 1, 1), 4)); lwp.push_back(Weighted_point(Bare_point( 2, 2, 2), 1)); //build alpha_shape in GENERAL mode and set alpha=0 Alpha_shape_3 as(lwp.begin(), lwp.end(), 0, Alpha_shape_3::GENERAL); //explore the 0-shape - It is dual to the boundary of the union. std::list<Cell_handle> cells; std::list<Facet> facets; std::list<Edge> edges; as.get_alpha_shape_cells(std::back_inserter(cells), Alpha_shape_3::INTERIOR); as.get_alpha_shape_facets(std::back_inserter(facets), Alpha_shape_3::REGULAR); as.get_alpha_shape_facets(std::back_inserter(facets), Alpha_shape_3::SINGULAR); as.get_alpha_shape_edges(std::back_inserter(edges), Alpha_shape_3::SINGULAR); std::cout << " The 0-shape has : " << std::endl; std::cout << cells.size() << " interior tetrahedra" << std::endl; std::cout << facets.size() << " boundary facets" << std::endl; std::cout << edges.size() << " singular edges" << std::endl; return 0; }

Same example as previous but using a fixed value of alpha.

File:examples/Alpha_shapes_3/ex_fixed_weighted_alpha_shapes_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Regular_triangulation_3.h> #include <CGAL/Regular_triangulation_euclidean_traits_3.h> #include <CGAL/Fixed_alpha_shape_3.h> #include <CGAL/Fixed_alpha_shape_vertex_base_3.h> #include <CGAL/Fixed_alpha_shape_cell_base_3.h> #include <list> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Regular_triangulation_euclidean_traits_3<K> Gt; typedef CGAL::Fixed_alpha_shape_vertex_base_3<Gt> Vb; typedef CGAL::Fixed_alpha_shape_cell_base_3<Gt> Fb; typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds; typedef CGAL::Regular_triangulation_3<Gt,Tds> Triangulation_3; typedef CGAL::Fixed_alpha_shape_3<Triangulation_3> Fixed_alpha_shape_3; typedef Fixed_alpha_shape_3::Cell_handle Cell_handle; typedef Fixed_alpha_shape_3::Vertex_handle Vertex_handle; typedef Fixed_alpha_shape_3::Facet Facet; typedef Fixed_alpha_shape_3::Edge Edge; typedef Gt::Weighted_point Weighted_point; typedef Gt::Bare_point Bare_point; int main() { std::list<Weighted_point> lwp; //input : a small molecule lwp.push_back(Weighted_point(Bare_point( 1, -1, -1), 4)); lwp.push_back(Weighted_point(Bare_point(-1, 1, -1), 4)); lwp.push_back(Weighted_point(Bare_point(-1, -1, 1), 4)); lwp.push_back(Weighted_point(Bare_point( 1, 1, 1), 4)); lwp.push_back(Weighted_point(Bare_point( 2, 2, 2), 1)); //build one alpha_shape with alpha=0 Fixed_alpha_shape_3 as(lwp.begin(), lwp.end(), 0); //explore the 0-shape - It is dual to the boundary of the union. std::list<Cell_handle> cells; std::list<Facet> facets; std::list<Edge> edges; as.get_alpha_shape_cells(std::back_inserter(cells), Fixed_alpha_shape_3::INTERIOR); as.get_alpha_shape_facets(std::back_inserter(facets), Fixed_alpha_shape_3::REGULAR); as.get_alpha_shape_facets(std::back_inserter(facets), Fixed_alpha_shape_3::SINGULAR); as.get_alpha_shape_edges(std::back_inserter(edges), Fixed_alpha_shape_3::SINGULAR); std::cout << " The 0-shape has : " << std::endl; std::cout << cells.size() << " interior tetrahedra" << std::endl; std::cout << facets.size() << " boundary facets" << std::endl; std::cout << edges.size() << " singular edges" << std::endl; return 0; }

The following example shows how to use the periodic Delaunay triangulation (Chapter 40) as underlying triangulation for the alpha shape computation.

In order to define the original domain and to benefit from the built-in heuristic optimizations of the periodic Delaunay triangulation computation it is recommended to first construct the triangulation and then construct the alpha shape from it. The alpha shape constructor that takes a point range can be used as well but in this case the original domain cannot be specified and the default unit cube will be chosen and no optimizations will be used.

It is also recommended to switch the triangulation to 1-sheeted covering if possible. Note that a periodic triangulation in 27-sheeted covering space is degenerate. In this case an exact constructions kernel needs to be used to compute the alpha shapes. Otherwise the results will suffer from round-off problems.

File:examples/Alpha_shapes_3/ex_periodic_alpha_shapes_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Periodic_3_triangulation_traits_3.h> #include <CGAL/Periodic_3_Delaunay_triangulation_3.h> #include <CGAL/Alpha_shape_3.h> #include <CGAL/Random.h> #include <CGAL/point_generators_3.h> // Traits typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Periodic_3_triangulation_traits_3<K> PK; // Vertex type typedef CGAL::Periodic_3_triangulation_ds_vertex_base_3<> DsVb; typedef CGAL::Triangulation_vertex_base_3<PK,DsVb> Vb; typedef CGAL::Alpha_shape_vertex_base_3<PK,Vb> AsVb; // Cell type typedef CGAL::Periodic_3_triangulation_ds_cell_base_3<> DsCb; typedef CGAL::Triangulation_cell_base_3<PK,DsCb> Cb; typedef CGAL::Alpha_shape_cell_base_3<PK,Cb> AsCb; typedef CGAL::Triangulation_data_structure_3<AsVb,AsCb> Tds; typedef CGAL::Periodic_3_Delaunay_triangulation_3<PK,Tds> P3DT3; typedef CGAL::Alpha_shape_3<P3DT3> Alpha_shape_3; typedef PK::Point_3 Point; int main() { typedef CGAL::Creator_uniform_3<double, Point> Creator; CGAL::Random random(7); CGAL::Random_points_in_cube_3<Point, Creator> in_cube(1, random); std::vector<Point> pts; // Generating 1000 random points for (int i=0 ; i < 1000 ; i++) { Point p = *in_cube++; pts.push_back(p); } // Define the periodic cube P3DT3 pdt(PK::Iso_cuboid_3(-1,-1,-1,1,1,1)); // Heuristic for inserting large point sets (if pts is reasonably large) pdt.insert(pts.begin(), pts.end(), true); // As pdt won't be modified anymore switch to 1-sheeted cover if possible if (pdt.is_triangulation_in_1_sheet()) pdt.convert_to_1_sheeted_covering(); std::cout << "Periodic Delaunay computed." << std::endl; // compute alpha shape Alpha_shape_3 as(pdt); std::cout << "Alpha shape computed in REGULARIZED mode by default." << std::endl; // find optimal alpha values Alpha_shape_3::NT alpha_solid = as.find_alpha_solid(); Alpha_shape_3::Alpha_iterator opt = as.find_optimal_alpha(1); std::cout << "Smallest alpha value to get a solid through data points is " << alpha_solid << std::endl; std::cout << "Optimal alpha value to get one connected component is " << *opt << std::endl; as.set_alpha(*opt); assert(as.number_of_solid_components() == 1); return 0; }

^{1} | ice cream, ice cream!!! The wording of this introductory paragraphs is borrowed from Kaspar Fischer's `` Introduction to Alpha Shapes'' which can be found at http://people.inf.ethz.ch/fischerk/pubs/as.pdf. The picture has been taken from Walter Luh's homepage at http://www.stanford.edu/&wtilde;luh/cs448b/alphashapes.html. |

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CGAL Open Source Project.
Release 3.9.
26 September 2011.