血管切片的三维重建

Re construction of Vessel in Three Dimension

本文讨论血管的三维重建问题。我们通过研究 ,证明了以下的定理。 定理 设C(i) 是中轴线和平面Z =i的交点 ,那么存在以C(i) 为中点且端点P1(i) ,P2 (i) 在ω(i) 上的线段 ,并且在P1(i) ,P2 (i) 处ω(i) 的切线相互平行。 根据定理 ,我们找到利用求截面图象边界曲线的平行切线方法找到中轴线和 10 0个截面的交点及管道的直径 5 9.12 38pixel。并用这 10 0个交点的数据拟合出中轴线的方程 : x(t) =- 0 .2 0 780 6 - 0 .6 10 30 3t+0 .2 0 6 45 5t2 - 0 .0 144 935t3+0 .0 0 0 5 17774t4 - 8.3942 4197775 40 47× 10 -6t5+6 .13335 3112 0 35 975× 10 -8t6- 1.6 6 732 182 6 744 480 5× 10 -10 t7 y(t) =15 8.2 11+1.86 5 95t - 0 .2 6 6 798t2 +0 .0 14140 7t3- 0 .0 0 0 32 5 412t4 +3.0 432 75 5 976 80 80 7× 10 -6t5- 9.8991712 746 15 0 6 3× 10 -9t6 z(t) =t 然后我们用中轴线的方程重建了三维血管 ,并求出了重建血管在 40个平面上的截面ω′(i) (30≤i≤ 6 9) ,并与原始截面ω(i) (30≤i≤ 6 9)进行比较 ,截面平均符合率高达 96 .80 2 4%。

The re-construction of vessel in three dimension will be discussed in this essay. We first show the following proposition. Proposition Let C be a curve along which the center of a ball moves, ω (i) a section sliced from the original vessel by the plane Z=i, and ω (i) the boundary of ω (i) (0≤i≤99). Let C (i) be the intersection point of C with the plane Z=i(0≤i≤99), then there are a line segment with two end points P (i) 1,P (i) 2, (P (i) j∈ω (i) ,0≤i≤99, 1≤j≤2) such thatC (i) is the middle point of the line segment P (i) 1 P (i) 2 and two parallel tangent lines which touch ω (i) only at P (i) 1, P (i) 2respectively (0≤i≤99). On the basis of the proposition shown above,C (i) (0≤i≤99) and the diameter of the vessel D=59.1238 pixel have been found. The equation of C is simulated by coordinate data of 100 intersection points: x(t)=-0.207806-0.610303t+0.206455t 2-0.0144935t 3 +0.000517774t 4-8.394241977754047×10 -6 t 5 +6.133353112035975×10 -8 t 6-1.6673218267444805×10 -10 t 7 y(t)=158.211+1.86595t-0.266798t 2+0.0141407t 3 -0.000325412t 4+3.043275597680807×10 -6 t 5 -9.899171274615063×10 -9 t 6 z(t)=t On the ground of that equation, the simulated vessel in three dimension is figured out, and at the same time, the 40 sections sliced from the simulated vessel are also made out, matching the original vessel at the average rate as much as 96.8024%.