Formulas shown as code

Some formulas are shown as code due to the fact that the indentation of the start of the formula is 4 or more compared to the paragraph it is part of (handling of code sections has been improved from e.g. 1.8.13 to master).
the formula construct `\f$ .. \f$` should not be used when using e.g. the `equation` environment. With MathJax this works but in standard LaTeX an error is thrown (not yet the focus of CGAL, but might be of interest in the future / newer implementations of MathJax when more compatible).
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albert-github 2019-08-27 11:55:53 +02:00
parent 551313ac5c
commit 32dfb2e66f
1 changed files with 19 additions and 27 deletions

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@ -70,12 +70,10 @@ data-fitting, model complexity, and point coverage.
- Data-fitting. This term is intended to evaluate the fitting quality of the faces to the point cloud. It is
defined to measure a confidence-weighted percentage of points that do not contribute to the final reconstruction.
\f$
\begin{equation}
E_f = 1 - \frac{1}{|P|} \sum_{i=1}^N x_i \cdot support(f_i),
\label{eq:datafitting}
\end{equation}
\f$
\f{equation}{
E_f = 1 - \frac{1}{|P|} \sum_{i=1}^N x_i \cdot support(f_i),
\label{eq:datafitting}
\f}
\f$|P|\f$ is the total number of points in \f$P\f$. \f$support(f_i)\f$ is the number of supporting points
accounting for an appropriate notion of confidence.
@ -84,29 +82,25 @@ defined to measure a confidence-weighted percentage of points that do not contri
- Model complexity. This term is to encourage simple structures (i.e., large planar regions).
It is defined as the ratio of sharp edges in the model.
\f$
\begin{equation}
E_m = \frac{1}{|E|}\sum_{i=1}^{|E|} corner(e_i),
\label{eq:complexity}
\end{equation}
\f$
\f{equation}{
E_m = \frac{1}{|E|}\sum_{i=1}^{|E|} corner(e_i),
\label{eq:complexity}
\f}
\f$|E|\f$ denotes the total number of pairwise intersections in the candidate face set.
\f$corner(e_i)\f$ is an indicator function whose value is determined by the configuration of the two selected
faces connected by \f$e_i\f$. \f$corner(e_i)\f$ will have a value of 1 if the faces associated with \f$e_i\f$
introduce a sharp edge in the final model. Otherwise, \f$corner(e_i)\f$ has a zero value meaning the two faces
are coplanar.
\f$|E|\f$ denotes the total number of pairwise intersections in the candidate face set.
\f$corner(e_i)\f$ is an indicator function whose value is determined by the configuration of the two selected
faces connected by \f$e_i\f$. \f$corner(e_i)\f$ will have a value of 1 if the faces associated with \f$e_i\f$
introduce a sharp edge in the final model. Otherwise, \f$corner(e_i)\f$ has a zero value meaning the two faces
are coplanar.
- Point coverage. This term is intended to handle missing data. The idea is to keep the unsupported regions
(i.e., regions not covered by points) of the final model as small as possible. This term is defined as the ratio of
uncovered regions in the model.
\f$
\begin{equation}
E_c = \frac{1}{area(M)}\sum_{i=1}^N x_i \cdot (area(f_i) - area(M_i^\alpha)),
\label{eq:coverage}
\end{equation}
\f$
\f{equation}{
E_c = \frac{1}{area(M)}\sum_{i=1}^N x_i \cdot (area(f_i) - area(M_i^\alpha)),
\label{eq:coverage}
\f}
Here \f$area(M)\f$, \f$area(f_i)\f$, and \f$area(M_i^\alpha)\f$ denote the surface areas of the final model,
a candidate face \f$f_i\f$, and the \f$\alpha\f$-shape mesh \f$M_i^\alpha\f$ of \f$f_i\f$, respectively.
@ -121,8 +115,7 @@ It is defined as the ratio of sharp edges in the model.
With the above energy terms, the optimal set of faces can be obtained by minimizing a weighted sum of these terms
under certain hard constraints.
\f$
\begin{equation}
\f{equation}{
\begin{aligned}
\underset{\mathbf{X}}{\text{min}} \quad & \lambda_f \cdot E_f + \lambda_m \cdot E_m + \lambda_c \cdot E_c \\
\text{s.t.} \quad & \begin{cases}
@ -133,8 +126,7 @@ under certain hard constraints.
\end{cases}
\end{aligned}
\label{eq:optimization}
\end{equation}
\f$
\f}
Here \f$ \sum_{j \in \mathcal{N}(e_i)} {x_j} \f$ counts the number of faces connected by an edge \f$ e_i \f$.
This value is enforced to be either 0 or 2, meaning none or two of the faces can be selected. These hard constraints