**6. Candidate selection with shape filter**

The tumor candidates, which were detected in the previous section, include many false positives because the detection used only intensity information. Therefore, in this step, we perform a selection process using a shape filter. We define the following five evaluation criteria: (1) size, (2) shape of each slice, (3) regional variation among the slices, (4) location in the image, and (5) numbers of connectivity among the slices.

In this study, we assume that the shape of a tumor is approximately spherical. Therefore, for the second criteria, we use a shape filter as shown in Fig. 5. For each tumor candidate, we first find a center of gravity, and then calculate distances from it to the 12 points which are on the edge of tumor candidate as shown in Fig.5. 4 points are cross points with the bounding box (line of a rectangle), which are shown in Fig.5 as diamond points. 8 points are boundary points sampled at intervals of 45 degree, which are shown in Fig.5 as small circle points. The ratio of maximum distance and minimum distance is used as a measure of shape. The ratio is a value larger than or equal to 1. If the candidate's shape is like a circle, the ratio will be 1. Since the tumor is considered having a spherical shape, the candidates are rejected if their ratio is larger than a pre-defined threshold (in our research, the threshold is set as 4).

Fig. 5. Illustration of a shape filter

Though most false positive candidates can be rejected by the use of above 5 criterions, some tumor points will also be rejected. In order to recover the rejected true tumors, we use a new criterion, which is shown in Fig. 6 to check the rejected candidates. For each rejected tumor candidate, we first generate an edge image, and then superimpose it with two circles *L*in and *L*out having radius *r*in and *r*out, respectively, and centers corresponding to the center of the tumor candidate's 2D ROI. Their parameters are defined in Eq. 4.

Fig. 6. Illustration of a shape filter for recheck

$$\begin{cases} r\_{\text{out}} = 4 + L \land 2, \quad r\_{\text{in}} = r\_{\text{out}} / 2, & \text{if } (L \ge 10) \\ r\_{\text{out}} = 2 + L \land 2, \quad r\_{\text{in}} = r\_{\text{out}} / 2 - 1, & \text{if } (L < 10) \end{cases} \tag{4}$$

Here, *L* is the longer side length of 2D ROI. If the shape of a tumor candidate is approximately spherical, then a major portion of the tumor region is bounded by the circle *L*in and the edge of the tumor is between the circles *L*in and *L*out.
