mirror of https://github.com/CGAL/cgal
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$` should not be used when using e.g. the `equation` environment. With MathJax this works but in standard LaTeX an error is thrown (not yet the focus of CGAL, but might be of interest in the future / newer implementations of MathJax when more compatible).
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@ -70,12 +70,10 @@ data-fitting, model complexity, and point coverage.
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- Data-fitting. This term is intended to evaluate the fitting quality of the faces to the point cloud. It is
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defined to measure a confidence-weighted percentage of points that do not contribute to the final reconstruction.
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\f$
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\begin{equation}
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\f{equation}{
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E_f = 1 - \frac{1}{|P|} \sum_{i=1}^N x_i \cdot support(f_i),
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\label{eq:datafitting}
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\end{equation}
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\f$
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\f}
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\f$|P|\f$ is the total number of points in \f$P\f$. \f$support(f_i)\f$ is the number of supporting points
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accounting for an appropriate notion of confidence.
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@ -84,12 +82,10 @@ defined to measure a confidence-weighted percentage of points that do not contri
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- Model complexity. This term is to encourage simple structures (i.e., large planar regions).
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It is defined as the ratio of sharp edges in the model.
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\f$
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\begin{equation}
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\f{equation}{
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E_m = \frac{1}{|E|}\sum_{i=1}^{|E|} corner(e_i),
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\label{eq:complexity}
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\end{equation}
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\f$
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\f}
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\f$|E|\f$ denotes the total number of pairwise intersections in the candidate face set.
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\f$corner(e_i)\f$ is an indicator function whose value is determined by the configuration of the two selected
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@ -101,12 +97,10 @@ It is defined as the ratio of sharp edges in the model.
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(i.e., regions not covered by points) of the final model as small as possible. This term is defined as the ratio of
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uncovered regions in the model.
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\f$
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\begin{equation}
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\f{equation}{
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E_c = \frac{1}{area(M)}\sum_{i=1}^N x_i \cdot (area(f_i) - area(M_i^\alpha)),
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\label{eq:coverage}
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\end{equation}
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\f$
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\f}
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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,
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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.
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@ -121,8 +115,7 @@ It is defined as the ratio of sharp edges in the model.
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With the above energy terms, the optimal set of faces can be obtained by minimizing a weighted sum of these terms
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under certain hard constraints.
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\f$
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\begin{equation}
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\f{equation}{
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\begin{aligned}
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\underset{\mathbf{X}}{\text{min}} \quad & \lambda_f \cdot E_f + \lambda_m \cdot E_m + \lambda_c \cdot E_c \\
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\text{s.t.} \quad & \begin{cases}
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@ -133,8 +126,7 @@ under certain hard constraints.
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\end{cases}
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\end{aligned}
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\label{eq:optimization}
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\end{equation}
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\f$
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\f}
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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$.
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This value is enforced to be either 0 or 2, meaning none or two of the faces can be selected. These hard constraints
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