PCL-surface/on_nurbs模块分析

文章目录

1. B样条曲线
2. B样条曲面
3. 多个B样条曲面的全局优化

1. B样条曲线

基于点距离最小化的三维B样条曲线拟合：
fitting_curve_pdm.h
fitting_curve_pdm.cpp

主要函数：

1)求点xixixi在节点向量的哪个区间：

        /** \brief Find the element in which the parameter xi lies.* \param[in] xi value in parameter domain of the B-Spline curve.* \param[in] elements the vector of elements of the curve.* \return index of the element with respect to elements. */static intfindElement (double xi, const std::vector<double> &elements);

2)通过每两个节点间插入中间节点修正曲线：

/** \brief Refines curve by inserting a knot in the middle of each element. */voidrefine ();

3)根据约束条件，给定系统方程组：

/** \brief Assemble the system of equations for fitting* - for large point-clouds this is time consuming.* - should be done once before refinement to initialize the starting points for point inversion. */voidassemble (const Parameter &parameter);

4)求解系统方程组，可以选择使用Eigen或UmfPack：

/** \brief Solve system of equations using Eigen or UmfPack (can be defined in on_nurbs.cmake),* and updates B-Spline curve if a solution can be obtained. */voidsolve (double damp = 1.0);

5)更新曲线：

/** \brief Update curve according to the current system of equations.* \param[in] damp damping factor from one iteration to the other. */voidupdateCurve (double damp);

6)使用边界点构成的圆或者PCA初始化曲线：

/** \brief Initialize a closed B-Spline curve using the bounding circle of the point-cloud.* \param[in] order polynomial order of the curve.* \param[in] data 3D point-cloud* \return B-Spline curve. */static ON_NurbsCurveinitNurbsCurve2D (int order, const vector_vec2d &data);/** \brief Initialize a closed B-Spline curve using the eigenvalues as elliptic parameters (z=0).* \param[in] order polynomial order of the curve.* \param[in] data 3D point-cloud* \return B-Spline curve. */static ON_NurbsCurveinitNurbsCurvePCA (int order, const vector_vec3d &data, int ncps = 0, double rf = 1.0);

7)点反演，给定一个点pt，使用牛顿法、L2范数，求取曲线上最近的点：

 /** \brief Inverse mapping / point inversion: Given a point pt, this function finds the closest* point on the B-Spline curve using Newtons method and point-distance (L2-, Euclidean norm).* \param[in] nurbs the B-Spline curve.* \param[in] pt the point to which the closest point on the curve will be computed.* \param[in] hint the starting point in parametric domain (warning: may lead to convergence at local minima).* \param[in] error the distance between the point pt and p after convergence.* \param[in] p closest point on curve.* \param[in] t the tangent vector at point p.* \param[in] maxSteps maximum number of iterations.* \param[in] accuracy convergence criteria: if error is lower than accuracy the function returns* \return closest point on curve in parametric domain.*/static doubleinverseMapping (const ON_NurbsCurve &nurbs, const Eigen::Vector3d &pt, const double &hint, double &error,Eigen::Vector3d &p, Eigen::Vector3d &t, int maxSteps = 100, double accuracy = 1e-6,bool quiet = true);

总结：

PCL中B样条曲线拟合的思路是，对初始化的曲线，通过不断插入中间节点来修正曲线，每次求解过程通过点线距离约束，构造矩阵方程组，求解得到控制点，不断迭代修正，对于稠密点云，求解线性方程组的过程较慢，导致拟合速度较慢，使用UmfPack可加快求解速度。

2. B样条曲面

主要用到pdm、sdm两种距离度量：
fitting_surface_pdm.h、fitting_surface_pdm.cpp

fitting_surface_sdm.h、fitting_surface_sdm.cpp

主要函数：
1)得到某个维度上的节点向量：

/** \brief Get the elements of a B-Spline surface.*/static std::vector<double>getElementVector (const ON_NurbsSurface &nurbs, int dim);

2)初始化曲面，可以使用4个角点、PCA或者PCA+边界点：

/** \brief Initializing a B-Spline surface using 4 corners */static ON_NurbsSurfaceinitNurbs4Corners (int order, ON_3dPoint ll, ON_3dPoint lr, ON_3dPoint ur, ON_3dPoint ul);/** \brief Initializing a B-Spline surface using principal-component-analysis and eigen values */static ON_NurbsSurfaceinitNurbsPCA (int order, NurbsDataSurface *data, Eigen::Vector3d z = Eigen::Vector3d (0.0, 0.0, 1.0));/** \brief Initializing a B-Spline surface using principal-component-analysis and bounding box of points */static ON_NurbsSurfaceinitNurbsPCABoundingBox (int order, NurbsDataSurface *data, Eigen::Vector3d z = Eigen::Vector3d (0.0, 0.0, 1.0));

3)给定一个点pt，使用牛顿法，pdm距离或者sdm距离求解曲面上的最近一点：

      /** \brief Inverse mapping / point inversion: Given a point pt, this function finds the closest* point on the B-Spline surface using Newtons method and point-distance (L2-, Euclidean norm).* \param[in] nurbs the B-Spline surface.* \param[in] pt the point to which the closest point on the surface will be computed.* \param[in] hint the starting point in parametric domain (warning: may lead to convergence at local minima).* \param[in] error the distance between the point pt and p after convergence.* \param[in] p closest point on surface.* \param[in] tu the tangent vector at point p in u-direction.* \param[in] tv the tangent vector at point p in v-direction.* \param[in] maxSteps maximum number of iterations.* \param[in] accuracy convergence criteria: if error is lower than accuracy the function returns* \return closest point on surface in parametric domain.*/static Eigen::Vector2dinverseMapping (const ON_NurbsSurface &nurbs, const Eigen::Vector3d &pt, const Eigen::Vector2d &hint,double &error, Eigen::Vector3d &p, Eigen::Vector3d &tu, Eigen::Vector3d &tv, int maxSteps = 100,double accuracy = 1e-6, bool quiet = true);static Eigen::Vector2dinverseMapping (const ON_NurbsSurface &nurbs, const Eigen::Vector3d &pt, const Eigen::Vector2d &hint,Eigen::Vector3d &p, int maxSteps, double accuracy, bool quiet);

4)给定一个点pt，使用牛顿法，pdm距离或者sdm距离求解曲面边界上的最近一点：

 /** \brief Inverse mapping / point inversion: Given a point pt, this function finds the closest* point on the boundary of the B-Spline surface using Newtons method and point-distance (L2-, Euclidean norm).* \param[in] nurbs the B-Spline surface.* \param[in] pt the point to which the closest point on the surface will be computed.* \param[in] error the distance between the point pt and p after convergence.* \param[in] p closest boundary point on surface.* \param[in] tu the tangent vector at point p in u-direction.* \param[in] tv the tangent vector at point p in v-direction.* \param[in] maxSteps maximum number of iterations.* \param[in] accuracy convergence criteria: if error is lower than accuracy the function returns* \return closest point on surface in parametric domain.*/static Eigen::Vector2dinverseMappingBoundary (const ON_NurbsSurface &nurbs, const Eigen::Vector3d &pt, double &error,Eigen::Vector3d &p, Eigen::Vector3d &tu, Eigen::Vector3d &tv, int maxSteps = 100,double accuracy = 1e-6, bool quiet = true);/** \brief Inverse mapping / point inversion: Given a point pt, this function finds the closest* point on one side of the boundary of the B-Spline surface using Newtons method and* point-distance (L2-, Euclidean norm).* \param[in] nurbs the B-Spline surface.* \param[in] pt the point to which the closest point on the surface will be computed.* \param[in] side the side of the boundary (NORTH, SOUTH, EAST, WEST)* \param[in] hint the starting point in parametric domain (warning: may lead to convergence at local minima).* \param[in] error the distance between the point pt and p after convergence.* \param[in] p closest boundary point on surface.* \param[in] tu the tangent vector at point p in u-direction.* \param[in] tv the tangent vector at point p in v-direction.* \param[in] maxSteps maximum number of iterations.* \param[in] accuracy convergence criteria: if error is lower than accuracy the function returns* \return closest point on surface in parametric domain.*/static Eigen::Vector2dinverseMappingBoundary (const ON_NurbsSurface &nurbs, const Eigen::Vector3d &pt, int side, double hint,double &error, Eigen::Vector3d &p, Eigen::Vector3d &tu, Eigen::Vector3d &tv,int maxSteps = 100, double accuracy = 1e-6, bool quiet = true);

5)根据约束条件，设定待求解的系统方程：

 /** \brief Assemble the system of equations for fitting* - for large point-clouds this is time consuming.* - should be done once before refinement to initialize the starting points for point inversion.*/virtual voidassemble (Parameter param = Parameter ());

6)系统方程的求解，使用Eigen或者UmfPack，UmfPack可以加速求解稀疏矩阵方程组：

      /** \brief Solve system of equations using Eigen or UmfPack (can be defined in on_nurbs.cmake),* and updates B-Spline surface if a solution can be obtained.*/virtual voidsolve (double damp = 1.0);

7)曲面的更新：

/** \brief Update surface according to the current system of equations.* \param[in] damp damping factor from one iteration to the other.*/virtual voidupdateSurf (double damp);

8)通过插入中间节点修正曲面：

/** \brief Refines surface by inserting a knot in the middle of each element.* \param[in] dim dimension of refinement (0,1)*/voidrefine (int dim);static voidrefine (ON_NurbsSurface &nurbs, int dim);

总结：

PCL中B样条曲面拟合的思路是，对初始化的曲面，通过在两个维度上，不断插入中间节点来修正曲面，每次求解过程通过点线距离约束或者切线距离约束，构造矩阵方程组，求解得到控制点，不断迭代修正，对于稠密点云，求解矩阵方程组的过程较慢，导致拟合速度较慢，使用UmfPack可加快求解速度。

3. 多个B样条曲面的全局优化

global_optimization_pdm.h、global_optimization_pdm.cpp

global_optimization_sdm.h、global_optimization_sdm.cpp

在单个方程组中使用点距离或切线距离最小化法，从全局的三维点云中拟合和优化多个B样条曲面.

主要函数：
设置两个B样条曲面的共同边界点

      /** \brief Set common boundary points two NURBS should lie on*  \param[in] boundary vector of boundary points.*  \param[in] nurbs_indices vector of 2 NURBS indices sharing the boundary point. */voidsetCommonBoundary (const vector_vec3d &boundary, const vector_vec2i &nurbs_indices);

其他函数与单个B样条曲面拟合基本相同。

参考
https://github.com/PointCloudLibrary/pcl/tree/master/surface