mirror of https://github.com/CGAL/cgal
polish Jet_fitting
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Andreas Fabri
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03c6c56549
commit
b71421e326
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@ -6,7 +6,7 @@ namespace CGAL {
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The class `Eigen_svd` provides an algorithm to solve in the least
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square sense a linear system with a singular value decomposition using
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\ref thirdpartyEigen Eigen. The field type is `double`.
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\ref thirdpartyEigen. The field type is `double`.
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\cgalModels `SvdTraits`
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@ -93,7 +93,7 @@ Monge_form();
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Point on the fitted surface where
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differential quantities are computed.
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*/
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Point_3 origin();
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Point_3 origin() const;
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/// @}
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@ -103,17 +103,17 @@ Point_3 origin();
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/*!
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*/
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Vector_3 maximal_principal_direction();
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Vector_3 maximal_principal_direction() const;
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/*!
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*/
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Vector_3 minimal_principal_direction();
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Vector_3 minimal_principal_direction() const;
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/*!
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*/
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Vector_3 normal_direction();
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Vector_3 normal_direction() const;
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/// @}
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@ -123,17 +123,17 @@ Vector_3 normal_direction();
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/*!
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\f$ i=0\f$ for the maximum and \f$ i=1\f$ for the minimum.
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*/
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FT principal_curvatures(size_t i);
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FT principal_curvatures(size_t i)const;
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/*!
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\f$ 0 \leq i \leq3\f$
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*/
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FT third_order_coefficients(size_t i);
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FT third_order_coefficients(size_t i) const;
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/*!
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\f$ 0 \leq i \leq4\f$
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*/
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FT fourth_order_coefficients(size_t i);
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FT fourth_order_coefficients(size_t i) const;
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/// @}
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@ -152,9 +152,9 @@ vector may not be good. The nearer this direction to the tangent
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plane the worse the estimation.
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<LI>As observed in \cite cgal:cp-edqpf-05 (section 3.1), the
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interpolating problem is not <I>poised</I> if the points project,
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interpolating problem is not well posed if the points project,
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into the fitting frame, onto an algebraic curve of degree \f$ d\f$. More
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generally, the problem is ill poised if the condition number is too
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generally, the problem is ill posed if the condition number is too
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large.
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</UL>
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In these cases, even if a result is provided, the estimation may not
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@ -219,14 +219,14 @@ In addition, the class `Monge_via_jet_fitting` stores
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\subsection Jet_fitting_3TemplateParameters Template Parameters
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\subsection Jet_fitting_3TemplateparameterDataKernel Template parameter DataKernel
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\subsubsection Jet_fitting_3TemplateparameterDataKernel Template parameter DataKernel
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This concept provides the types for the input sample points, together
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with \f$ 3d\f$ vectors and a number type. It is used as template for the
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class `Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>`. Typically, one can use
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`Cartesian<double>`.
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\subsection Jet_fitting_3TemplateparameterLocalKernel Template parameter LocalKernel
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\subsubsection Jet_fitting_3TemplateparameterLocalKernel Template parameter LocalKernel
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This is a parameter of the class
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`Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>`.
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@ -240,14 +240,14 @@ these types are converted back to `DataKernel` ones. Typically,
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one can use
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`Cartesian<double>` which is the default.
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\subsection Jet_fitting_3TemplateparameterSvdTraits Template parameter SvdTraits
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\subsubsection Jet_fitting_3TemplateparameterSvdTraits Template parameter SvdTraits
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This concept provides the number, vector and matrix types for algebra
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operations required by the fitting method in
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`Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>`
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. The main method is a linear solver using a singular value decomposition.
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\subsection Jet_fitting_3CompatibilityRequirements Compatibility Requirements
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\subsubsection Jet_fitting_3CompatibilityRequirements Compatibility Requirements
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To solve the fitting problem, the sample points are first converted
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from the `DataKernel` to the `LocalKernel` (this is done using
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@ -279,7 +279,7 @@ vertices of a given mesh. The neighborhood of a given vertex is
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computed using rings on the triangulation. Results are twofold:
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<UL>
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<LI>a human readable text file featuring the `Monge_via_jet_fitting::Monge_form` and
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numerical informations on the computation : condition number and the
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numerical informations on the computation: condition number and the
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PCA basis;
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<LI>another text file that records raw data (better for a vizualization
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post-processing).
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@ -308,8 +308,8 @@ The three bases involved in the estimation.
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\subsection Jet_fitting_3ComputingaBasisfortheFitting Computing a Basis for the Fitting
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<B>Input : samples
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Output : fitting-basis</B>
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<B>Input: samples
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Output: fitting-basis</B>
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Performing a PCA requires diagonalizing a symmetric matrix. This
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analysis gives an orthonormal basis whose \f$ z\f$-axis is provided by the
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@ -331,8 +331,8 @@ direction.
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\subsection secsolving Solving the Interpolation / Approximation Problem
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<B>Input : samples, fitting-basis
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Output : coefficients \f$ A_{i,j}\f$
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<B>Input: samples, fitting-basis
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Output: coefficients \f$ A_{i,j}\f$
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of the bivariate fitted polynomial in the fitting-basis </B>
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Computations are done in the fitting-basis and the origin is the point
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@ -393,7 +393,7 @@ D_r & 0_{r,\ N_d-r}\\
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\f$
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The number \f$ r\f$, which is the number of non zero singular values, is
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strictly lower than \f$ N_d\f$ if the system is under constrained. In any
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case, the unique solution which minimize \f$ ||A||_2\f$ is given by :
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case, the unique solution which minimize \f$ ||A||_2\f$ is given by:
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\f[
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\begin{equation}
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@ -433,10 +433,10 @@ condition number of the system. For more on these alternatives, see
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\subsection Jet_fitting_3PrincipalCurvatureDirections Principal Curvature / Directions
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<B>Input : coefficients of the fit \f$ A_{i,j}\f$,
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<B>Input: coefficients of the fit \f$ A_{i,j}\f$,
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fitting-basis
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Output : Monge basis wrt fitting-basis and world-basis
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Output: Monge basis wrt fitting-basis and world-basis
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</B>
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In the fitting basis, we have determined a height function expressed
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@ -482,10 +482,10 @@ translated by \f$ p\f$, i.e. the coordinates of the origin point are
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\subsection Jet_fitting_3ComputingHigherOrderMongeCoefficients Computing Higher Order Monge Coefficients
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<B>Input : coefficients of the fit, Monge basis wrt fitting-basis (\f$ P_{F
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<B>Input: coefficients of the fit, Monge basis wrt fitting-basis (\f$ P_{F
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\rightarrow M}\f$)
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Output : third and fourth order coefficients of Monge</B>
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Output: third and fourth order coefficients of Monge</B>
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We use explicit formula. The implicit equation of the fitted
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polynomial surface in the fitting-basis with origin the point
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@ -518,7 +518,7 @@ and \f$ g\f$ by \f$ f_{i,j,k}=\frac{\partial^{i+j+k}f}{\partial^ix
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\f$ g_{0,2}=k_2\f$. The partial derivative of order \f$ n\f$ of \f$ f\f$ depends on
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the matrix \f$ P_{F\rightarrow M}\f$ and the partial derivatives of order
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at most \f$ n\f$ of \f$ J_{A,d}\f$. The third and fourth order coefficients of are
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computed with the implicit function theorem. For instance :
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computed with the implicit function theorem. For instance:
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\f{eqnarray*}{
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&b_0=g_{3,0}=-{\frac { f_{3,0,0} f_{0,0,1} -3\, f_{1,0,1} f_{2,0,0} }{
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