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improve doc

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Sébastien Loriot 2015-06-18 09:56:36 +02:00
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Advancing_front_surface_reconstruction/doc/Advancing_front_surface_reconstruction/Advancing_front_surface_reconstruction.txt
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@ -30,10 +30,10 @@ In most surface based Delaunay algorithms the triangles are
selected independently, that is in parallel \cgalCite{agj-lcsr-00}\cgalCite{ab-srvf-98}. selected independently, that is in parallel \cgalCite{agj-lcsr-00}\cgalCite{ab-srvf-98}.
This chapter presents a surface-based Delaunay surface This chapter presents a surface-based Delaunay surface
reconstruction algorithm by selecting the triangles sequentially, that reconstruction algorithm that sequentially selects the triangles, that
is by using previous selected triangles to select a new triangle for is it uses previously selected triangles to select a new triangle for
advancing the front. At each advancing step the most plausible advancing the front. At each advancing step the most plausible
triangle is selected, and the triangles are selected in a way that triangle is selected, and such that the triangles selected
generates an orientable manifold triangulated surface. generates an orientable manifold triangulated surface.
Two other examples of this greedy approach are the ball pivoting Two other examples of this greedy approach are the ball pivoting
@ -53,16 +53,16 @@ We describe next the algorithm and provide examples.
A detailed description of the algorithm and the underlying theory are provided A detailed description of the algorithm and the underlying theory are provided
in \cgalCite{cgal:csd-gdbsra-04}. in \cgalCite{cgal:csd-gdbsra-04}.
The first step of the algorithm is the construction of a 3D Delaunay The first step of the algorithm is the construction of a 3D Delaunay
triangulation of the point set. The Delaunay triangle with the triangulation of the point set.
smallest radius is the starting point for the greedy algorithm. The The radius of a triangle \f$ t \f$ is the radius of the smallest sphere
radius of a triangle \f$ t \f$ is the radius of the smallest sphere
passing through the vertices of \f$ t\f$ and enclosing no sample passing through the vertices of \f$ t\f$ and enclosing no sample
point. In other words, the radius \f$ r_t\f$ is the distance from any point. In other words, the radius \f$ r_t\f$ is the distance from any
vertex of \f$ t\f$ to the Voronoi edge dual to \f$ t\f$. This triangle with vertex of \f$ t\f$ to the Voronoi edge dual to \f$ t\f$. This triangle with
three boundary edges is the initial triangulated surface, and its three boundary edges is the initial triangulated surface, and its
boundary is the advancing front. boundary is the advancing front.
The Delaunay triangle with the smallest radius is the starting point
for the greedy algorithm.
The algorithm maintains a priority queue of candidate triangles, that The algorithm maintains a priority queue of candidate triangles, that
is of valid triangles incident to the boundary edges of the current is of valid triangles incident to the boundary edges of the current
@ -77,7 +77,7 @@ which are explained next.
\subsection AFSR_Topology Topological Constraints \subsection AFSR_Topology Topological Constraints
Any triangle \f$t\f$ considered as next potential candidate shares an Any triangle \f$t\f$ considered as the next potential candidate shares an
edge \f$e\f$ with the front of the current reconstruction. Let \f$b\f$ edge \f$e\f$ with the front of the current reconstruction. Let \f$b\f$
be the vertex of \f$t\f$ opposite to \f$e\f$. There are four be the vertex of \f$t\f$ opposite to \f$e\f$. There are four
configurations where \f$t\f$ is added to the surface. configurations where \f$t\f$ is added to the surface.
@ -105,15 +105,15 @@ radius. While the radius is a good criterion in the case of 2D smooth
curve reconstruction \cgalCite{b-cccda-94}, we need another criterion curve reconstruction \cgalCite{b-cccda-94}, we need another criterion
for 3D surface reconstruction, namely the dihedral angle between for 3D surface reconstruction, namely the dihedral angle between
triangles on the surface, that is the angle between the normals of the triangles on the surface, that is the angle between the normals of the
triangles. triangles. There are two bounds namely \f$ \alpha_\mathrm{sliver} \f$
and \f$ \beta \f$.
We denote by \f$ \beta_t\f$ the angle between the normal of a triangle We denote by \f$ \beta_t\f$ the angle between the normal of a triangle
\f$ t\f$ incident on a boundary edge \f$ e \f$ and the normal of the \f$ t\f$ incident on a boundary edge \f$ e \f$ and the normal of the
triangle on the surface incident to \f$ e \f$. triangle on the surface incident to \f$ e \f$.
The *candidate* triangle of an edge \f$ e \f$ is the triangle The *candidate* triangle of an edge \f$ e \f$ is the triangle
with the smallest radius, that is valid for \f$ e \f$ with the smallest radius that is valid for \f$ e \f$
and that has \f$ \beta_t < \alpha_\mathrm{sliver} \f$. and that has \f$ \beta_t < \alpha_\mathrm{sliver} \f$.
There may be no such triangle. In the implementation There may be no such triangle. In the implementation
of the algorithm \f$ \alpha_\mathrm{sliver} = 5\pi/6 \f$. of the algorithm \f$ \alpha_\mathrm{sliver} = 5\pi/6 \f$.