【摘要】 为了重构出相邻齿断面由于密度高度一致，造成在CT图像中丢失的边界轮廓线，先在已提取的牙列整体连续外轮廓上，根据切矢方向的变化，确定出相邻齿廓线的交汇点;再分别以两个相邻交汇点为端点，其间的轮廓点为型值点，构建标准B样条曲线作为部分齿廓线;然后，分别求出相邻齿廓线在交汇点处的一阶导矢，进而求出该交汇点处的平均导矢;最后，以两两相对的交汇点为端点，构建Hermit样条插值曲线，该曲线即作为相邻齿间被丢失的轮廓线。用该方法可以成功地对两颗牙齿进行边界区分，进而提取轮廓，以达到对单颗牙齿进行三维重建，在仿真上能满足临床要求，为后续的牙颌仿真工作奠定了基础。
【关键词】 牙齿轮廓;牙列CT图像;轮廓提取;样条曲线;三维重构
Rebuilding Invisible Boundaries between Adjoing
Teeth from CT Images based on Spline CurvesLI Wenyu，WANG Yu，XU Min
(Information Engineering College, Qingdao University，Qingdao 266071, China)
Abstract：For reconstructing the contours, which are invisible in CT images due to high identity in the density value, between adjoining teeth,we presented a simple, and feasible method: first, the intersecting points, which were formed by the adjoining contours being parts of outer continuous contour extracted from a tooth-array cross-section in a CT image, were found out according to the saltation of the tangent vector directions on the points. Second, a B-spline, as the representation of a piece of outer contour, was built with two neighboring intersection-points as the end-points, and the outer-contour points as the interpolating points. Third, the one-order derivatives of two adjoining B-splines at common endpoint were resolved respectively, and then a mean derivative at the point was computed. Finally, cubic Hermit curves, as the replace of invisible contours between adjoining teeth, were built with a pairs of the opposite intersecting point as endpoints.Using this method to determine the edge of the adjoin teeth, and further to reconstruct 3D teeth based on the contours rebuilt with the method mentioned above has proven a satisfying result.
Key words：Tooth contour; CT image of tooth-array; Contour extracting; Spline curves, 3D reconstruction
1 引 言
CT图像是基于组织的密度以灰度形式来成像的。而在同一扫描层面内，紧密排列的牙列断层的牙釉质具有相当一致的高密度值，造成了相邻牙齿接触处轮廓线的完全丢失，无法辨认，最终导致重构的三维牙列模型相邻牙齿呈现互相粘连的情况，见图1。
目前，对于图像的自动分割，已有许多经典的算法，如分水岭算法[1]、活动轮廓算法[2]、水平集方法[3]等，但对于CT图像中牙齿间由于密度极其相似而出现的边界丢失，这些算法则显得无能为力。国外关于牙齿三维建模的文献对于单颗牙齿的三维模型的构建，一般也仅限于对某一颗牙齿的建模[4]或通过与已有的牙齿库进行匹配[5]。
而在基于CT对牙齿建模的过程中，对每一颗牙齿单独的建模及操作是牙齿虚拟仿真中最基本的要求。因此，需要一种算法来确定相邻两颗牙齿之间缺失的齿廓线。
2 丢失齿廓的重建
2.1 整体外齿廓线的绘制
利用edge(I1,'soble')函数自动进行轮廓提取，对图像利用[L,num]=bwlabel(I1,8);函数做分块处理，使得图像只显示当前待处理的块，见图2。
(a) (b)
图2 获得的牙列断面图像的轮廓线
(a)牙列断面图像的轮廓线;(b)本图只需对第三块进行处理
Fig 2 The contour of dentition cross-sectional drawing
(a)the contour of dentition cross-sectional drawingin;
(b)this case just focus on the 3rd part
2.2 求相邻齿廓线的交点
采用半交互式的方法，在图像中选定所求凹点的大致区域，视具体情况自动地对该区域中所有轮廓线上的点进行查找，选出x值和y值符合极值条件的点，则该点就是所求的凹点。得出的凹点为图3中标识的圆点。
3 齿廓线的拟合
3.1 相邻齿廓线的拟合计算
由于被丢失的齿廓线要采用上述两个相对的交点为端点，并依据交点处的坐标和一阶导矢为几何参数所生成的一条三次的Hermit曲线来近似，所以对上述已生成的整体外轮廓采用三次标准B样条曲线拟合。
由以上方法可由已知的Pi—型值点序列得出在B样条中相应的控制点序列。从而构造标准的B样条曲线，其中Vi为控制点。图4为用标准B样条曲线拟合出的牙齿的轮廓线。
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