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Diffstat (limited to 'src/Tangential_complex/doc/Intro_tangential_complex.h')
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1 files changed, 12 insertions, 8 deletions
diff --git a/src/Tangential_complex/doc/Intro_tangential_complex.h b/src/Tangential_complex/doc/Intro_tangential_complex.h index f4fc8ac7..501f4a8b 100644 --- a/src/Tangential_complex/doc/Intro_tangential_complex.h +++ b/src/Tangential_complex/doc/Intro_tangential_complex.h @@ -35,9 +35,11 @@ namespace tangential_complex { \section tangentialdefinition Definition -A Tangential Delaunay complex is a <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> +A Tangential Delaunay complex is a +<a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> designed to reconstruct a \f$k\f$-dimensional smooth manifold embedded in \f$d\f$-dimensional Euclidean space. -The input is a point sample coming from an unknown manifold, which means that the points lie close to a structure of "small" intrinsic dimension. +The input is a point sample coming from an unknown manifold, which means that the points lie close to a structure of +"small" intrinsic dimension. The running time depends only linearly on the extrinsic dimension \f$ d \f$ and exponentially on the intrinsic dimension \f$ k \f$. @@ -46,17 +48,19 @@ An extensive description of the Tangential complex can be found in \cite tangent \subsection whatisthetc What is a Tangential Complex? Let us start with the description of the Tangential complex of a simple example, with \f$ k=1 \f$ and \f$ d=2 \f$. -The input data is 4 points \f$ P \f$ located on a curve embedded in 2D. +The point set \f$ \mathscr P \f$ is located on a closed curve embedded in 2D. +Only 4 points will be displayed (more are required for PCA) to simplify the figures. \image html "tc_example_01.png" "The input" -For each point \f$ p \f$, estimate its tangent subspace \f$ T_p \f$ (e.g. using PCA). +For each point \f$ P \f$, estimate its tangent subspace \f$ T_P \f$ using PCA. \image html "tc_example_02.png" "The estimated normals" -Let us add the Voronoi diagram of the points in orange. For each point \f$ p \f$, construct its star in the Delaunay triangulation of \f$ P \f$ restricted to \f$ T_p \f$. +Let us add the Voronoi diagram of the points in orange. For each point \f$ P \f$, construct its star in the Delaunay +triangulation of \f$ \mathscr P \f$ restricted to \f$ T_P \f$. \image html "tc_example_03.png" "The Voronoi diagram" The Tangential Delaunay complex is the union of those stars. In practice, neither the ambient Voronoi diagram nor the ambient Delaunay triangulation is computed. -Instead, local \f$ k \f$-dimensional regular triangulations are computed with a limited number of points as we only need the star of each point. -More details can be found in \cite tangentialcomplex2014. +Instead, local \f$ k \f$-dimensional regular triangulations are computed with a limited number of points as we only +need the star of each point. More details can be found in \cite tangentialcomplex2014. \subsection inconsistencies Inconsistencies @@ -65,7 +69,7 @@ An inconsistency occurs when a simplex is not in the star of all its vertices. Let us take the same example. \image html "tc_example_07_before.png" "Before" -Let us slightly move the tangent subspace \f$ T_q \f$ +Let us slightly move the tangent subspace \f$ T_Q \f$ \image html "tc_example_07_after.png" "After" Now, the star of \f$ Q \f$ contains \f$ QP \f$, but the star of \f$ P \f$ does not contain \f$ QP \f$. We have an inconsistency. \image html "tc_example_08.png" "After" |