义齿模型三维重建算法的研究
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摘要
逆向工程在生物医学领域不断地拓展和应用，CAD技术在口腔医学中得到了迅速的发展。利用CAD系统制作的义齿精度高、制作周期短备受医生和患者的青睐，是口腔医学发展的趋势。虽然在国外已经设计出义齿CAD系统，但现有的义齿CAD系统对三维重建的数据要求很高而且价格昂贵，限制了义齿CAD系统的广泛应用。本文以义齿的三维点云数据作为研究对象，系统研究了只含有位置信息的义齿点云数据在三维重建技术中预处理、曲面重建以及重建后处理的基础理论和方法。
针对牙齿数据只含有位置信息缺少拓扑关系的问题，提出一种基于Morton序列的动态连续k邻近查询算法。首先将三维数据进行空间划分并存储到八叉树结构中，利用Morton序列为每个点云数据生成相应的Morton编码，即将三维数据转换为一维数据；然后采用基于欧氏距离的三维点云并行快速排序算法使点云数据的Morton编码有序，并存储在一维线性链表中；最后通过访问一维线性链表中Morton编码直接定位到相应的点云数据，避免了在子空间的搜索甚至到邻近子空间搜索的复杂性。结果证明了此算法优越性和准确性。
针对现有曲面重建方法依赖法向量的问题，提出了义齿点云数据法向量模糊估值算法。利用模糊逻辑系统将义齿点云数据进行分类，根据点云的疏密程度和曲率变化将点云分为均匀光滑（牙冠）部分、薄片特征（牙尖牙棱）部分和尖锐特征（臼齿咬合面）部分。义齿点云分布均匀的区域采用PCA算法；含有薄片特征的区域采用增加检测器的算法；含有尖锐特征的区域采用增加附加点的算法。模糊逻辑系统的输出确定牙齿点云数据应采取相应的法向量估算方法。后两种算法比较复杂但估值准确，由于义齿模型中具有尖锐特征和薄片特征的区域较少所以应用这两种算法的次数很少，而义齿点云中分布均匀的区域占多数，使用PCA算法的次数最多，实验结果表明义齿点云的法向量整体估算效率高且估值准确。
针对隐式曲面重建对海量点云数量的限制，提出渐近式义齿曲面重建算法。通过对子立方体空间中心点确定偏移点，改进了增加偏移点的方式，减少了计算径向基函数所需偏移点的数量；在此基础上采用Schur补计算隐函数系数的方法，将系数矩阵划为四个子矩阵，只有其中一个子矩阵在曲面拟合过程中需要更新计算，从而计算量被减少了一个数量级；在八叉树细化的子立方体空间给出渐近式拟合局部隐式曲面算法。通过对牙颌模型和义齿模型的曲面拟合，验证了算法的可行性，同时体现出算法在执行时间上的优势。
针对义齿曲面尖锐特征变化不规律孔洞修补效果差的问题，提出尖锐度滤波器的概念，通过对义齿孔洞网格顶点尖锐度的计算，将顶点分为两类，对于尖锐度小的顶点直接使用隐式曲面重建形成初始修补模型即可；对于尖锐度大的顶点（如牙尖牙窝的位置），提出曲面特征恢复算法，将孔洞周围的尖锐特征延伸到孔洞修补曲面上，恢复尖锐程度根据牙齿局部咬合面积的参数确定，从而恢复了曲面的尖锐特征；最后研究了义齿曲面光顺方法，将光顺问题简化为求解耦合非线性常微分方程和使用显式方案进行曲面拟合的推导问题，理论分析和算法验证的结果均表明该修补算法和光顺处理过程简单实用，有效地修补义齿的孔洞，恢复了模型原有特征。
Reverse engineering is developed and applied in the field of biomedicalscience constantly. CAD technology is used to oral medicine quickly. With theadvantages of high precision of denture and short time of production cycle, etc, itwins good graces from doctors and clients and will be the inevitable trend of oralmedicine. Although the denture CAD system has been designed overseas, itrequests the point data of denture too strictly and expensively to be appliedwidely. The three dimensional point data of denture is regarded as research objectin the paper. The basic theories and methods are researched on the denture pointdata with only position information, namely pretreatment, surface reconstructionand post-processing reconstruction in three dimensional reconstructiontechniques.
According to the denture data which have only position information and lacktopological relations, this paper proposes a dynamic k-nearest neighbor queryalgorithm based on Morton order. Three dimensional data are divided in spacefirstly and stored in Octree structure. Morton code for every point is generated byMorton order. Three dimensional data can be changed to one dimensional data.And then a parallel fast sort algorithm based on Euclidean distance is adopted tomake the Morton code in order and stored in one dimensional linear chain.Finally, Morton code can be located corresponding point clouds directly byvisiting the linear chain, avoiding the complexity to search in subspace even toneighbor subspace. The experimental results show that the proposed algorithm issuperior and accutate.
The existing surface reconstruction methods are excessively dependent on thenormal, so fuzzy normal estimation algorithm is proposed for point clouds ofdenture model. The point clouds are classified by fuzzy logical system accordingto point clouds density and curvature change. The smooth area of crown, thethin-plate one of cusp or teeth edge, and sharp one of molar occlusion surface are divided. PCA algorithm is used to smooth area, the checker one is used to thin-plate area, and the attach point one is used to sharp area. The output of fuzzylogic system determines the tooth point cloud data should adopt thecorresponding normal vector estimation method. The two latter methods arecomplex but accurate. Because the area including thin-plate and sharp feature isless than smooth area in denture, the two latter algorithms are used low frequency.On the contrary, PCA algorithm is used frequently. The experimental resultsshow that normal estimation of denture points is efficient and accurate.
Due to limitation on the number of mass cloud points, a progressive implicitsurface reconstruction is proposed. The selection of offset point is developedbased on the center point of sub-cube to decrease the number of radial basisfunction needed during the denture reconstruction. A simple coefficientcalculation of implicit function is proposed based on Schur compensation, thecoefficient matrix is partitioned into four blocks, and only one of the four blocksneeds to be updated during the surface fitting. In this way, the calculation reducesan order of magnitude for mass point clouds. The local surface reconstructionalgorithm is proposed in Octree sub-cube. The denture model and tooth modelare fitted by this algorithm. The results show that it is feasible and its advantagefor saving time is obviously compared with other methods.
Aiming at the irregular of sharp feature distribution in denture and poor effectof hole-filling, the definition of sharpness-filter is proposed. The vertexes of holeare divided into two classes by computing sharpness of vertex on the mesh ofhole. One class with small sharpness is repaired and the initial model isconformed by implicit surface. Another with large sharpness such as canine ortooth fossa is used by the algorithm proposed in this paper based on implicitsurface. The function of proposed algorithm is to recover the feature of surface,which allows the sharp feature to be extended to the surface patch inside of thehole and the parameter of sharpness is adjusted according to the partial occlusionarea of teeth. At last, this paper proposes a simple surface smoothing method ofdenture model. The method is equivalent to transporting the scalar implicitfunction by solving the system of coupled non-linear ordinary differentialequations (ODE) and the propagation of the surface can be performed by usingthe approximate solution of an explicit scheme. Theoretical analysis and experiment show that the hole-filling algorithm and surface smoothing are simpleand useful for denture, which can repair and fill the hole of denture effectivelyand recover the shape of original model.
针对牙齿数据只含有位置信息缺少拓扑关系的问题，提出一种基于Morton序列的动态连续k邻近查询算法。首先将三维数据进行空间划分并存储到八叉树结构中，利用Morton序列为每个点云数据生成相应的Morton编码，即将三维数据转换为一维数据；然后采用基于欧氏距离的三维点云并行快速排序算法使点云数据的Morton编码有序，并存储在一维线性链表中；最后通过访问一维线性链表中Morton编码直接定位到相应的点云数据，避免了在子空间的搜索甚至到邻近子空间搜索的复杂性。结果证明了此算法优越性和准确性。
针对现有曲面重建方法依赖法向量的问题，提出了义齿点云数据法向量模糊估值算法。利用模糊逻辑系统将义齿点云数据进行分类，根据点云的疏密程度和曲率变化将点云分为均匀光滑（牙冠）部分、薄片特征（牙尖牙棱）部分和尖锐特征（臼齿咬合面）部分。义齿点云分布均匀的区域采用PCA算法；含有薄片特征的区域采用增加检测器的算法；含有尖锐特征的区域采用增加附加点的算法。模糊逻辑系统的输出确定牙齿点云数据应采取相应的法向量估算方法。后两种算法比较复杂但估值准确，由于义齿模型中具有尖锐特征和薄片特征的区域较少所以应用这两种算法的次数很少，而义齿点云中分布均匀的区域占多数，使用PCA算法的次数最多，实验结果表明义齿点云的法向量整体估算效率高且估值准确。
针对隐式曲面重建对海量点云数量的限制，提出渐近式义齿曲面重建算法。通过对子立方体空间中心点确定偏移点，改进了增加偏移点的方式，减少了计算径向基函数所需偏移点的数量；在此基础上采用Schur补计算隐函数系数的方法，将系数矩阵划为四个子矩阵，只有其中一个子矩阵在曲面拟合过程中需要更新计算，从而计算量被减少了一个数量级；在八叉树细化的子立方体空间给出渐近式拟合局部隐式曲面算法。通过对牙颌模型和义齿模型的曲面拟合，验证了算法的可行性，同时体现出算法在执行时间上的优势。
针对义齿曲面尖锐特征变化不规律孔洞修补效果差的问题，提出尖锐度滤波器的概念，通过对义齿孔洞网格顶点尖锐度的计算，将顶点分为两类，对于尖锐度小的顶点直接使用隐式曲面重建形成初始修补模型即可；对于尖锐度大的顶点（如牙尖牙窝的位置），提出曲面特征恢复算法，将孔洞周围的尖锐特征延伸到孔洞修补曲面上，恢复尖锐程度根据牙齿局部咬合面积的参数确定，从而恢复了曲面的尖锐特征；最后研究了义齿曲面光顺方法，将光顺问题简化为求解耦合非线性常微分方程和使用显式方案进行曲面拟合的推导问题，理论分析和算法验证的结果均表明该修补算法和光顺处理过程简单实用，有效地修补义齿的孔洞，恢复了模型原有特征。
Reverse engineering is developed and applied in the field of biomedicalscience constantly. CAD technology is used to oral medicine quickly. With theadvantages of high precision of denture and short time of production cycle, etc, itwins good graces from doctors and clients and will be the inevitable trend of oralmedicine. Although the denture CAD system has been designed overseas, itrequests the point data of denture too strictly and expensively to be appliedwidely. The three dimensional point data of denture is regarded as research objectin the paper. The basic theories and methods are researched on the denture pointdata with only position information, namely pretreatment, surface reconstructionand post-processing reconstruction in three dimensional reconstructiontechniques.
According to the denture data which have only position information and lacktopological relations, this paper proposes a dynamic k-nearest neighbor queryalgorithm based on Morton order. Three dimensional data are divided in spacefirstly and stored in Octree structure. Morton code for every point is generated byMorton order. Three dimensional data can be changed to one dimensional data.And then a parallel fast sort algorithm based on Euclidean distance is adopted tomake the Morton code in order and stored in one dimensional linear chain.Finally, Morton code can be located corresponding point clouds directly byvisiting the linear chain, avoiding the complexity to search in subspace even toneighbor subspace. The experimental results show that the proposed algorithm issuperior and accutate.
The existing surface reconstruction methods are excessively dependent on thenormal, so fuzzy normal estimation algorithm is proposed for point clouds ofdenture model. The point clouds are classified by fuzzy logical system accordingto point clouds density and curvature change. The smooth area of crown, thethin-plate one of cusp or teeth edge, and sharp one of molar occlusion surface are divided. PCA algorithm is used to smooth area, the checker one is used to thin-plate area, and the attach point one is used to sharp area. The output of fuzzylogic system determines the tooth point cloud data should adopt thecorresponding normal vector estimation method. The two latter methods arecomplex but accurate. Because the area including thin-plate and sharp feature isless than smooth area in denture, the two latter algorithms are used low frequency.On the contrary, PCA algorithm is used frequently. The experimental resultsshow that normal estimation of denture points is efficient and accurate.
Due to limitation on the number of mass cloud points, a progressive implicitsurface reconstruction is proposed. The selection of offset point is developedbased on the center point of sub-cube to decrease the number of radial basisfunction needed during the denture reconstruction. A simple coefficientcalculation of implicit function is proposed based on Schur compensation, thecoefficient matrix is partitioned into four blocks, and only one of the four blocksneeds to be updated during the surface fitting. In this way, the calculation reducesan order of magnitude for mass point clouds. The local surface reconstructionalgorithm is proposed in Octree sub-cube. The denture model and tooth modelare fitted by this algorithm. The results show that it is feasible and its advantagefor saving time is obviously compared with other methods.
Aiming at the irregular of sharp feature distribution in denture and poor effectof hole-filling, the definition of sharpness-filter is proposed. The vertexes of holeare divided into two classes by computing sharpness of vertex on the mesh ofhole. One class with small sharpness is repaired and the initial model isconformed by implicit surface. Another with large sharpness such as canine ortooth fossa is used by the algorithm proposed in this paper based on implicitsurface. The function of proposed algorithm is to recover the feature of surface,which allows the sharp feature to be extended to the surface patch inside of thehole and the parameter of sharpness is adjusted according to the partial occlusionarea of teeth. At last, this paper proposes a simple surface smoothing method ofdenture model. The method is equivalent to transporting the scalar implicitfunction by solving the system of coupled non-linear ordinary differentialequations (ODE) and the propagation of the surface can be performed by usingthe approximate solution of an explicit scheme. Theoretical analysis and experiment show that the hole-filling algorithm and surface smoothing are simpleand useful for denture, which can repair and fill the hole of denture effectivelyand recover the shape of original model.
引文
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