diff --git a/Polygonal_surface_reconstruction/doc/Polygonal_surface_reconstruction/Polygonal_surface_reconstruction.txt b/Polygonal_surface_reconstruction/doc/Polygonal_surface_reconstruction/Polygonal_surface_reconstruction.txt index 14ae6350a14..851a722a19d 100644 --- a/Polygonal_surface_reconstruction/doc/Polygonal_surface_reconstruction/Polygonal_surface_reconstruction.txt +++ b/Polygonal_surface_reconstruction/doc/Polygonal_surface_reconstruction/Polygonal_surface_reconstruction.txt @@ -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