Image Processing Reference
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2 Estimating an ideal resected region using the TDR
Perfused regions are typically estimated by using a Voronoi diagram [ 3 ] . Figure 3 shows how
to compute perfused regions using Voronoi diagram. A Voronoi diagram is a way of dividing
space into a number of regions as shown in Figure 3(a) . A set of points (called seeds) is speci-
ied beforehand and for each seed there will be a corresponding region consisting of all points
closer to that seed than to any other. The corresponding regions are called Voronoi cells. In es-
timating perfused regions, a seed and Voronoi cell in a Voronoi diagram correspond to a point
on a portal vein and a region perfumed by the portal point, respectively, as shown in Figure
3(b) . Note that, in Figure 3(b) , portal-vein points (seeds) exist continuously, whereas seeds in
the original Voronoi diagram are located apart from each other. As a result, the liver region is
divided into small regions of the same number as the portal-vein points.
FIGURE 3 Computing perfused regions using Voronoi diagram.
The TDR of a portal-vein point P is defined as
(1)
where the denominator and enumerator are illustrated in Figure 3(c) . The TDR reflects how
much the portal-vein point feed the tumor and is also affected by the tumor.
Given a tumor location, let us compute an ideal resected region by using the TDR, where the
ideal resected region means a minimum region including all subregions perfused by tumor-
related portal points. The ideal resected region is estimated by following steps ( Figure 4 ) .
FIGURE 4 Computing ideal resected region.
Step 1 : Among the portal-vein points with TDR larger than 0 on a branch, select the point
closest to the main stem as a cut-point of the branch. For example, in Figure 4(a) , P i ( i = 1,

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