Mesh Oriented datABase  (version 5.4.1)
Array-based unstructured mesh datastructure
HiReconstruction.hpp
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00001 /*! \file HiReconstruction.hpp
00002  * This class implements a high order surface/curve reconstruction method which takes a
00003  * surface/curve mesh as input and compute local polynomial fittings (in monomial basis) around user
00004  * specified vertices. Little noise is allowed and least square will be used in such case. This
00005  * method assumes the underlying geometry of input mesh is smooth. The local fitting results could
00006  * be used for estimate the exact geometry of the surface. For instance, if mesh refinement is
00007  * perform on the input mesh, then the position of new vertices introduced by refinement could be
00008  * estimated by the local fitting, rather than using linear interpolation.
00009  * Implementations are based on the WALF method in paper:
00010  * Jiao, Xiangmin, and Duo Wang. "Reconstructing high-order surfaces for meshing." Engineering with
00011  * Computers 28.4 (2012): 361-373.
00012  */
00013 
00014 #ifndef HI_RECONSTRUCTION_HPP
00015 #define HI_RECONSTRUCTION_HPP
00016 
00017 #include "moab/Range.hpp"
00018 #include "moab/HalfFacetRep.hpp"
00019 
00020 #ifdef MOAB_HAVE_MPI
00021 #include "moab/ParallelComm.hpp"
00022 #endif
00023 
00024 #include <vector>
00025 
00026 namespace moab
00027 {
00028 enum GEOMTYPE
00029 {
00030     HISURFACE,
00031     HI3DCURVE,
00032     HI2DCURVE
00033 };
00034 
00035 class Core;
00036 class HalfFaceRep;
00037 class ParallelComm;
00038 
00039 class HiReconstruction
00040 {
00041   public:
00042     HiReconstruction( Core* impl,
00043                       ParallelComm* comm  = 0,
00044                       EntityHandle meshIn = 0,
00045                       int minpnts         = 5,
00046                       bool recwhole       = true );
00047 
00048     ~HiReconstruction();
00049 
00050     ErrorCode initialize( bool recwhole );
00051 
00052     //! \brief Reconstruct a high order surface on given surface mesh
00053     /** Given a mesh, compute vertex based polynomial fittings for all vertices hosted by current
00054      * processor. The result will be stored interally for later usage of evalution. The inputs are:
00055      * a) degree, which is the order of polynomial used for vertex based fitting. b) interp, if it's
00056      * true, then interpolation will be applied for local fitting, otherwise it's least square
00057      * fitting. c) safeguard, specifies whether to use safeguarded numeric method. d) reset, if
00058      * fittings have been computed and stored in current object, then reset=true will recompute the
00059      * fittings based on user input and replace the existing one. \param degree Integer, order of
00060      * polynomials used for local fittings. \param interp Boolean, true=Interpolation, false=least
00061      * square fitting. \param safeguard Boolean, true=using safe guarded method in numerical
00062      * computing. \param reset Boolean, reset=true will recompute the fittings based on user input
00063      * and replace the existing one.
00064      */
00065     ErrorCode reconstruct3D_surf_geom( int degree, bool interp, bool safeguard, bool reset = false );
00066 
00067     //! \brief Reconstruct a high order surface on given surface mesh
00068     /** Given a mesh, compute vertex based polynomial fittings for all vertices hosted by current
00069      * processor. User could specify various degrees for different vertices. It assumes that the
00070      * input degrees for vertices stored in the same order as that this class stores vertices: 1)
00071      * reconstruction will be only performed at vertices hosted by current processor, thus input
00072      * npts should match the number of hosted vertices. 2) all hosted vertices will be stored in a
00073      * MOAB::Range object, degrees for all these vertices should be stored in degrees as the same
00074      * order in the MOAB::Range object The result will be stored interally for later usage of
00075      * evalution. \param npts Integer size of array pointed by degrees, used for check \param
00076      * degrees Integer arrray, order of polynomials for local fitting at all hosted vertices \param
00077      * interp Boolean, true=Interpolation, false=least square fitting. \param safeguard Boolean,
00078      * true=using safe guarded method in numerical computing. \param reset Boolean, reset=true will
00079      * recompute the fittings based on user input and replace the existing one.
00080      */
00081     ErrorCode reconstruct3D_surf_geom( size_t npts, int* degrees, bool* interps, bool safeguard, bool reset = false );
00082 
00083     //! \brief Reconstruct a high order curve on given curve mesh
00084     /** Given a curve mesh, compute vertex based polynomail fittings for all vertices hosted by
00085      * current processor. The vertex based fitting is done by perfoming three one-parameter fittings
00086      * along each axis, i.e. x,y,z. The result will be stored interally for later usage of
00087      * evalution. \param degree Integer, order of polynomials used for local fittings. \param interp
00088      * Boolean, true=Interpolation, false=least square fitting. \param safeguard Boolean, true=using
00089      * safe guarded method in numerical computing. \param reset Boolean, reset=true will recompute
00090      * the fittings based on user input and replace the existing one.
00091      */
00092     ErrorCode reconstruct3D_curve_geom( int degree, bool interp, bool safeguard, bool reset = false );
00093 
00094     //! \brief Reconstruct a high order curve on given curve mesh
00095     /** Given a curve mesh, compute vertex based polynomail fittings for all vertices hosted by
00096      * current processor. The vertex based fitting is done by perfoming three one-parameter fittings
00097      * along each axis, i.e. x,y,z. User could specify various degrees for different vertices. It
00098      * assumes that the input degrees for vertices stored in the same order as that this class
00099      * stores vertices: 1) reconstruction will be only performed at vertices hosted by current
00100      * processor, thus input npts should match the number of hosted vertices. 2) all hosted vertices
00101      * will be stored in a MOAB::Range object, degrees for all these vertices should be stored in
00102      * degrees as the same order in the MOAB::Range object The result will be stored interally for
00103      * later usage of evalution. \param npts Integer size of array pointed by degrees, used for
00104      * check \param degrees Integer arrray, order of polynomials for local fitting at all hosted
00105      * vertices. \param interp Boolean, true=Interpolation, false=least square fitting. \param
00106      * safeguard Boolean, true=using safe guarded method in numerical computing. \param reset
00107      * Boolean, reset=true will recompute the fittings based on user input and replace the existing
00108      * one.
00109      */
00110     ErrorCode reconstruct3D_curve_geom( size_t npts, int* degrees, bool* interps, bool safeguard, bool reset = false );
00111 
00112     //! \brief Construct vertex based polynomial fitting on a surface mesh
00113     /** Given a vertex on a surface mesh, construct a local fitting around this vertex. Stencils
00114      * around this vertex will be selected according to input degree and if data is noise. Local
00115      * uv-plane will be the estimated tangent plane at this vertex. minpnts will be used to specify
00116      * the minimum number allowed in the local stencil. The result will be returned to user by
00117      * preallocated memory coords, degree_out, coeffs. \param vid EntityHandle, the fitting will be
00118      * performed around this vertex for the local height function over the uv-plane. \param interp
00119      * Boolean, true=Interpolation, false=least square fitting. \param degree Integer, order of
00120      * polynomials used for local fittings. \param minpnts Integer, the allowed minimum number of
00121      * vertices in local stencil. If too small, the resulted fitting might be low order accurate. If
00122      * too large, it may introduce overfitting. \param safeguard Boolean, true=using safe guarded
00123      * method in numerical computing. \param coords Pointer to double, preallocated memory by user,
00124      * should have at least 9 doubles; stores the global coordinates of local coordinates system uvw
00125      * directions. \param degree_out Pointer to integer, used to store the degree of resulted
00126      * fitting \param coeffs, Pointer to double, preallocated memory for coefficients of local
00127      * fittings, should have at least (degree+2)(degree+1)/2 doubles.
00128      */
00129     ErrorCode polyfit3d_walf_surf_vertex( const EntityHandle vid,
00130                                           const bool interp,
00131                                           int degree,
00132                                           int minpnts,
00133                                           const bool safeguard,
00134                                           const int ncoords,
00135                                           double* coords,
00136                                           int* degree_out,
00137                                           const int ncoeffs,
00138                                           double* coeffs );
00139 
00140     //! \brief Construct vertex based polynomial fitting on a curve mesh
00141     /** Given a vertex on a curve mesh, construct three one-parameter local fittings for each
00142      * coordinates axis around this vertex. Stencils around this vertex will be selected according
00143      * to input degree and if data is noise. Local u-line, or the single parameter will be the
00144      * estimated tangent line at this vertex. On each axis of xyz, a polynomial fitting will be
00145      * performed according to user input. minpnts will be used to specify the minimum number allowed
00146      * in the local stencil. The result will be returned to user by preallocated memory coords,
00147      * degree_out, coeffs. \param vid EntityHandle, the fittings will be performed around this
00148      * vertex. \param interp Boolean, true=Interpolation, false=least square fitting. \param degree
00149      * Integer, order of polynomials used for local fittings. \param minpnts Integer, the allowed
00150      * minimum number of vertices in local stencil. If too small, the resulted fitting might be low
00151      * order accurate. If too large, it may introduce overfitting. \param safeguard Boolean,
00152      * true=using safe guarded method in numerical computing. \param coords Pointer to double,
00153      * preallocated memory by user, should have at least 3 doubles; stores the global coordinates of
00154      * local coordinate system u direction. \param degree_out Pointer to integer, used to store the
00155      * degree of resulted fitting \param coeffs, Pointer to double, preallocated memory for
00156      * coefficients of local fittings, should have at least 3*(degree+1) doubles.
00157      */
00158     ErrorCode polyfit3d_walf_curve_vertex( const EntityHandle vid,
00159                                            const bool interp,
00160                                            int degree,
00161                                            int minpnts,
00162                                            const bool safeguard,
00163                                            const int ncoords,
00164                                            double* coords,
00165                                            int* degree_out,
00166                                            const int ncoeffs,
00167                                            double* coeffs );
00168 
00169     //! \brief Perform high order projection of points in an element, using estimated geometry by
00170     //! reconstruction class
00171     /** Given an element on the input mesh, and new points in this element, represented as natural
00172      * coordinates in element, estimate their position in surface. This is done by weighted
00173      * averaging of local fittings: for each vertex of this elment, a fitting has been computed and
00174      * the new points could be projected by this fitting. The final result of projection is the
00175      * weighted average of these projections, weights are chosen as the barycentric coordinates of
00176      * the point in this element. The result will be returned to the user preallocated memory \param
00177      * elem EntityHandle, the element on which to perform high order projection. \param nvpe
00178      * Integer, number of nodes of this element, triangle is 3, quad is four. \param npts2fit
00179      * Integer, number of points lying in elem to be projected. \param naturalcoords2fit Pointer to
00180      * array of doubles, size=nvpe*npts2fit, natural coordinates in elem of points to be projected.
00181      * \param newcoords Pointer to array of doubles, preallocated by user, size=3*npts2fit,
00182      * estimated positions of input points.
00183      */
00184     ErrorCode hiproj_walf_in_element( EntityHandle elem,
00185                                       const int nvpe,
00186                                       const int npts2fit,
00187                                       const double* naturalcoords2fit,
00188                                       double* newcoords );
00189 
00190     //! \brief Perform high order projection of points around a vertex, using estimated geometry by
00191     //! reconstruction class
00192     /** Given an vertex on the input mesh, and new points around this vertex, estimate their
00193      * position in surface. This is done by first projecting input points onto the local uv-plane
00194      * around this vertex and use the precomputed local fitting to estimate the ideal position of
00195      * input points. The result will be returned to the user preallocated memory \param vid
00196      * EntityHandle, the vertex around which to perform high order projection. \param npts2fit
00197      * Integer, number of points lying around vid to be fitted. \param coords2fit Pointer to array
00198      * of doubles, size=3*npts2fit, current coordinates of points to be projected. \param newcoords
00199      * Pointer to array of doubles, preallocated by user, size=3*npts2fit, estimated positions of
00200      * input points.
00201      */
00202     ErrorCode hiproj_walf_around_vertex( EntityHandle vid,
00203                                          const int npts2fit,
00204                                          const double* coords2fit,
00205                                          double* hiproj_new );
00206 
00207     //! \brief Perform high order projection of points around a center vertex, assume geometry is
00208     //! surface
00209     /** Given a vertex position and the local fitting parameter around this vertex, estimate the
00210      * ideal position of input position according to the local fitting. This is done by first
00211      * projecting input points onto the local uv-plane around this vertex and use the given fitting
00212      * to estimate the ideal position of input points. The result will be returned to user
00213      * preallocated memory \param local_origin Pointer to 3 doubles, coordinates of the center
00214      * vertex \param local_coords Pointer to 9 doubles, global coordinates of directions of local
00215      * uvw coordinates axis at center vertex \param local_deg Integer, order of local polynomial
00216      * fitting \param local_coeffs Pointer to array of doubles, size=(local_deg+2)(local_deg+1)/2,
00217      * coefficients of local polynomial fittings, in monomial basis \param interp Boolean,
00218      * true=local fitting is interpolation, false=local fitting is least square fitting \param
00219      * npts2fit Integer, number of points to be estimated, around the center vertices \param
00220      * coords2fit Pointer to array of doubles, size=3*npts2fit, current coordinates of points to be
00221      * estimated \param hiproj_new Pointer to array of doubles, size=3*npts2fit, memory preallocated
00222      * by user to store the fitting/estimated positions of input points.
00223      */
00224     void walf3d_surf_vertex_eval( const double* local_origin,
00225                                   const double* local_coords,
00226                                   const int local_deg,
00227                                   const double* local_coeffs,
00228                                   const bool interp,
00229                                   const int npts2fit,
00230                                   const double* coords2fit,
00231                                   double* hiproj_new );
00232 
00233     //! \brief Perform high order projection of points around a center vertex, assume geometry is
00234     //! curve
00235     /** Given a vertex position and the local one-parameter fittings parameter around this vertex,
00236      * estimate the ideal position of input position according to the local fittings. This is done
00237      * by first projecting input points onto the local u-direction at this vertex and then use the
00238      * value u as parameter for the three fittings, one for each coordinates axis of xyz. The result
00239      * will be returned to user preallocated memory \param local_origin Pointer to 3 doubles,
00240      * coordinates of the center vertex \param local_coords Pointer to 3 doubles, global coordinates
00241      * of direction of local u coordinate axis at center vertex \param local_deg Integer, order of
00242      * local polynomial fitting \param local_coeffs Pointer to array of doubles,
00243      * size=3*(local_deg+1), coefficients of three local polynomial fittings, in monomial basis. For
00244      * each fitting, local_deg+1 parameters. \param interp Boolean, true=local fitting is
00245      * interpolation, false=local fitting is least square fitting \param npts2fit Integer, number of
00246      * points to be estimated, around the center vertices \param coords2fit Pointer to array of
00247      * doubles, size=3*npts2fit, current coordinates of points to be estimated \param hiproj_new
00248      * Pointer to array of doubles, size=3*npts2fit, memory preallocated by user to store the
00249      * fitting/estimated positions of input points.
00250      */
00251     void walf3d_curve_vertex_eval( const double* local_origin,
00252                                    const double* local_coords,
00253                                    const int local_deg,
00254                                    const double* local_coeffs,
00255                                    const bool interp,
00256                                    const int npts2fit,
00257                                    const double* coords2fit,
00258                                    double* hiproj_new );
00259 
00260     //! \brief Get interally stored fitting results
00261     /** Get fittings results of a vertex, stored internally, results will be writtend to user
00262      * provided memory \param vid EntityHandle, a vertex in _verts2rec \param geomtype GEOMTYPE, one
00263      * of HISURFACE,HI3DCURVE,HI2DCURVE \param coords vector, global coordinates of local uvw
00264      * coordinate system axis directions will be appended to the end of coords \param degree_out
00265      * Reference to Integer, order of polynomial fittings for vid \param coeffs vector, coefficients
00266      * of local polynomial fittings in monomial basis will be appended to the end of coeffs \param
00267      * interp Reference to Boolean, true =  interpolation
00268      */
00269     bool get_fittings_data( EntityHandle vid,
00270                             GEOMTYPE& geomtype,
00271                             std::vector< double >& coords,
00272                             int& degree_out,
00273                             std::vector< double >& coeffs,
00274                             bool& interp );
00275 
00276     // Helper function: estimate require number of ghost layers in parallel setting
00277     static int estimate_num_ghost_layers( int degree, bool interp = false )
00278     {
00279         return 1 + ( interp ? ( ( degree + 1 ) >> 1 ) + ( ( degree + 1 ) & 1 )
00280                             : ( ( degree + 2 ) >> 1 ) + ( ( degree + 2 ) & 1 ) );
00281     };
00282 
00283   protected:
00284     Core* mbImpl;
00285     ParallelComm* pcomm;
00286     HalfFacetRep* ahf;
00287     // prevent copying
00288     HiReconstruction( const HiReconstruction& source );
00289     HiReconstruction& operator=( const HiReconstruction& right );
00290 
00291     // mesh on which to perform reconstruction
00292     EntityHandle _mesh2rec;
00293     //_verts2rec all locally hosted vertices, in parallel might be different from _invert which is
00294     // all the vertices in _mesh2rec, including ghost vertices
00295     Range _verts2rec, _inverts, _inedges, _infaces, _incells;
00296     size_t _nv2rec;  // size of _verts2rec
00297 
00298     int _MAXPNTS, _MINPNTS;
00299     double _MINEPS;
00300 
00301     // in curve mesh, _hasderiv=true means vertex tangent vectors have been computed over _verts2rec
00302     // in surface mesh, _hasderiv=true means vertex normals have been computed over _verts2rec
00303     bool _hasderiv;
00304 
00305     GEOMTYPE _geom;
00306     int _dim;
00307     bool _hasfittings;
00308     bool _initfittings;
00309     std::vector< double > _local_coords;
00310     std::vector< double > _local_fit_coeffs;
00311     std::vector< size_t > _vertID2coeffID;
00312     std::vector< int > _degrees_out;
00313     std::vector< bool > _interps;
00314 
00315     // Estimate stencil size
00316     int estimate_num_rings( int degree, bool interp );
00317 
00318     //! \brief Given a vertex, return the incident elements with dimension elemdim
00319     /** Wrapper of MOAB Core->get_adjacencies and HalfRep->get_up_adjacencies, depends on if USE_AHF
00320      * is defined \param vid EntityHandle of vertex \param elemdim Integer, dimension of elements
00321      * incidented in vid \param adjents vector<EntityHandle>, container which push incident elements
00322      * in
00323      */
00324     ErrorCode vertex_get_incident_elements( const EntityHandle& vid,
00325                                             const int elemdim,
00326                                             std::vector< EntityHandle >& adjents );
00327 
00328     //! \brief Get n-ring neighbor vertices, assuming curve/surface mesh, not volume mesh
00329     /** Given a vertex, find its n-ring neighbor vertices including itself in _mesrh2rec.
00330      * 1-ring neighbor vertices of a vertex are the vertices connected with this vertex with an edge
00331      * n-ring vertices are obtained first get the 1-ring vertices and then get the 1-ring of these
00332      * vertices, and so on \param vid EntityHandle, vertex around which to get n-ring vertices
00333      * \param ring Integer, number of rings
00334      * \param minpnts Integer, number of minimum vertices to obtain, if the input ring could not
00335      * provide enough vertices, i.e. more than minpnts, then expand the number of rings \param ngbvs
00336      * Range, the n-ring vertices of vid, including vid. If too many points found, i.e. more than
00337      * _MAXPNTS, then terminate early.
00338      */
00339     ErrorCode obtain_nring_ngbvs( const EntityHandle vid, int ring, const int minpnts, Range& ngbvs );
00340 
00341     /** Initialize the storage for fitting results over _mesh2rec, curve/surface mesh
00342      * Two options are provided: a) use uniform degree for all vertices b) use customized degrees
00343      * for different vertices After calling of initializing functions, _initfitting is set to be
00344      * true, the fitting result could be stored internally
00345      */
00346     void initialize_surf_geom( const int degree );
00347     void initialize_surf_geom( const size_t npts, const int* degrees );
00348     void initialize_3Dcurve_geom( const int degree );
00349     void initialize_3Dcurve_geom( const size_t npts, const int* degrees );
00350 
00351     /** Save fitting results of a vertex into internal storage
00352      * \param vid EntityHandle, a vertex in _mesh2rec, in _verts2rec
00353      * \param coords Pointer to double array, global coordinates of local uvw coordinate system axis
00354      * directions \param degree_out Integer, order of polynomial fittings for vid \param coeffs
00355      * Pointer to double array, coefficients of local polynomial fittings in monomial basis \param
00356      * interp Boolean, true =  interpolation
00357      */
00358     // ErrorCode set_geom_data_surf(const EntityHandle vid, const double* coords, const double
00359     // degree_out, const double* coeffs, bool interp); ErrorCode set_geom_data_3Dcurve(const
00360     // EntityHandle vid, const double* coords, const double degree_out, const double* coeffs, bool
00361     // interp);
00362 
00363     /** Compute area weighted average vertex normals for given vertex, assuming surface mesh
00364      * For arbitrary polygon mesh, use incident two edges of each incident polygon of this vertex to
00365      * form a triangle, then use these incident "triangles" to compute area weighted average vertex
00366      * normals \param vid EntityHandle, vertex in _mesh2rec, might be ghost vertex \param nrm
00367      * Pointer to 3-doubles array, preallocated by user
00368      */
00369     ErrorCode average_vertex_normal( const EntityHandle vid, double* nrm );
00370 
00371     /** Compute weighted average vertex normals for all vertices in _verts2rec, not including ghost
00372      * vertices, results are stored interally in _local_coords
00373      */
00374     ErrorCode compute_average_vertex_normals_surf();
00375 
00376     /** Return the normals of given vertices in a Range, writing to preallocated memory
00377      * If normals have been computed and stored, just access them
00378      * If not, compute on the fly
00379      * \param vertsh Range, EntityHandles of vertices
00380      * \param nrms Pointer of array of doubles, size = 3*vertsh.size()
00381      */
00382     ErrorCode get_normals_surf( const Range& vertsh, double* nrms );
00383 
00384     /** Compute area weighted average vertex tangent vector for given vertex, assuming curve mesh
00385      * Use incident two edges of vertex as estimatation of tangent vectors, weighted by length
00386      * \param vid EntityHandle, vertex in _mesh2rec, might be ghost vertex
00387      * \param tang Pointer to 3-doubles array, preallocated by user
00388      */
00389     ErrorCode average_vertex_tangent( const EntityHandle vid, double* tang );
00390 
00391     /** Compute weighted average vertex tangent vectors for all vertices in _verts2rec, not
00392      * including ghost vertices, results are stored interally in _local_coords
00393      */
00394     ErrorCode compute_average_vertex_tangents_curve();
00395 
00396     /** Return the tangent vectors of given vertices in a Range, writing to preallocated memory
00397      * If tangent vectors have been computed and stored, just access them
00398      * If not, compute on the fly
00399      * \param vertsh Range, EntityHandles of vertices
00400      * \param tangs Pointer of array of doubles, size = 3*vertsh.size()
00401      */
00402     ErrorCode get_tangents_curve( const Range& vertsh, double* tangs );
00403 
00404     //! \brief Compute local coordinates system of a vertex, and perform vertex based polynomial
00405     //! fittings of local height function
00406     /** This function take the first vertex of input as center, and build local uv-plane by
00407      * estimating vertex normals and tangent planes Then other vertices forms vectors starting from
00408      * center and then are projectd onto this uv-plane to form a local height function. Local
00409      * fitting of this local height function is performed in WLS sense, according if interpolation
00410      * required or not. \param nverts Integer, number of vertices of input \param ngbcors Pointer to
00411      * array of doubles, size = 3*nverts, coordinates of input vertices, first will be center \param
00412      * ngbnrms Pointer to array of doubles, size = 3*nverts, vertex normals of input vertices \param
00413      * degree Integer, user specified fitting degree \param interp Boolean, user input,
00414      * interpolation or not \param safeguard Boolean, true = use safeguarded numerical method in
00415      * computing \param ncoords Integer, size of *coords, should be 9, used for check \param coords
00416      * Pointer to array of doubles, preallocated memory for storing the glocal coordinates of local
00417      * uvw axis directions \param ncoeffs Integer, size of *coeffs, should be
00418      * (degree+2)(degree+1)/2, used for check \param coeffs Pointer to array of doubles,
00419      * preallocated memory for storing coefficients of local fittings in monomial basis \param
00420      * degree_out Pointer to integer, order of resulted polynomial of fitting, could be downgraded
00421      * due to numerical issues \param degree_pnt Pointer to integer, polynomial fitting order
00422      * determined by stencil size/number of points \param degree_qr Pointer to integer, polynomial
00423      * fitting order determined by Vandermonde system condition number
00424      */
00425     void polyfit3d_surf_get_coeff( const int nverts,
00426                                    const double* ngbcors,
00427                                    const double* ngbnrms,
00428                                    int degree,
00429                                    const bool interp,
00430                                    const bool safeguard,
00431                                    const int ncoords,
00432                                    double* coords,
00433                                    const int ncoeffs,
00434                                    double* coeffs,
00435                                    int* degree_out,
00436                                    int* degree_pnt,
00437                                    int* degree_qr );
00438     //! \brief Form and solve Vandermonde system of bi-variables
00439     void eval_vander_bivar_cmf( const int npts2fit,
00440                                 const double* us,
00441                                 const int ndim,
00442                                 double* bs,
00443                                 int degree,
00444                                 const double* ws,
00445                                 const bool interp,
00446                                 const bool safeguard,
00447                                 int* degree_out,
00448                                 int* degree_pnt,
00449                                 int* degree_qr );
00450 
00451     //! \brief Compute local single variable coordinate system of a vertex, and perform vertex based
00452     //! polynomial fittings of three global coordinates axis
00453     /** This function take the first vertex of input as center, and build local u-line by estimating
00454      * tangent vector Then other vertices form vectors originating from center and vectors then are
00455      * projectd onto this u-plane to form three local height functions, one for each coordinates
00456      * axis. Local fitting of these local height functions are performed in WLS sense, according if
00457      * interpolation required or not. \param nverts Integer, number of vertices of input \param
00458      * ngbcors Pointer to array of doubles, size = 3*nverts, coordinates of input vertices, first
00459      * will be center \param ngbtangs Pointer to array of doubles, size = 3*nverts, vertex tangent
00460      * vectors of input vertices \param degree Integer, user specified fitting degree \param interp
00461      * Boolean, user input, interpolation or not \param safeguard Boolean, true = use safeguarded
00462      * numerical method in computing \param ncoords Integer, size of *coords, should be 3, used for
00463      * check \param coords Pointer to array of doubles, preallocated memory for storing the glocal
00464      * coordinates of local u axis direction \param ncoeffs Integer, size of *coeffs, should be
00465      * 3*(degree+1), used for check \param coeffs Pointer to array of doubles, preallocated memory
00466      * for storing coefficients of local fittings in monomial basis \param degree_out Pointer to
00467      * integer, order of resulted polynomial of fitting, could be downgraded due to numerical issues
00468      */
00469     void polyfit3d_curve_get_coeff( const int nverts,
00470                                     const double* ngbcors,
00471                                     const double* ngbtangs,
00472                                     int degree,
00473                                     const bool interp,
00474                                     const bool safeguard,
00475                                     const int ncoords,
00476                                     double* coords,
00477                                     const int ncoeffs,
00478                                     double* coeffs,
00479                                     int* degree_out );
00480     //! \brief Form and solve Vandermonde system of single-variables
00481     void eval_vander_univar_cmf( const int npts2fit,
00482                                  const double* us,
00483                                  const int ndim,
00484                                  double* bs,
00485                                  int degree,
00486                                  const double* ws,
00487                                  const bool interp,
00488                                  const bool safeguard,
00489                                  int* degree_out );
00490     //! \brief Compute weights for points selected in weighted least square fittigns
00491     int compute_weights( const int nrows,
00492                          const int ncols,
00493                          const double* us,
00494                          const int nngbs,
00495                          const double* ngbnrms,
00496                          const int degree,
00497                          const double toler,
00498                          double* ws );
00499     //! \brief Check the correctness of barycentric coordination, wi>=0 and sum(wi)=1
00500     bool check_barycentric_coords( const int nws, const double* naturalcoords );
00501 };  // class HiReconstruction
00502 }  // namespace moab
00503 #endif
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