Algorithm 1:

P-Count

Input: A 3D soft lesion probability map, and a threshold θ

Output: Number of lesions

Definition: G = (V, E) denote a graph; f(v) is the intensity value of node v; $\text{lower\_star}(v)=\{(u,v)\in E\mid f(u)<f(v)\}$; $cc(v)$ is the connected component id of node v.

1: $\text{PD}=\varnothing$; Build the proximity graph (6-connectivity) for 3D image;

2: $U=V$ sorted according to f(v); T a sub-graph, which includes all the nodes and edges whose value < t.

3: for v in U do

4:   $t=f(v)$,$T=T+\left\{v\right\}$

5:   for (u, v) in $\text{lower\_star}(v)$ do

5:     Assert $u\in T$

6:     if $cc(u)=cc(v)$ then

6:      Continue

7:     else

7:      $\text{younger\_cc}=\arg {\max }_{w=cc(u),cc(v)}f(w)$

7:      $\text{older\_}cc=\arg {\min }_{w=cc(u),cc(v)}f(w)$

7:      $pers=t-f(younger\text{\_}cc)$

8:      if $\text{pers}<=\theta$ then

9:       for w in $\text{younger\_cc}$ do

9:        $cc(w)=\text{older\_cc}$

10:      end for

11:     end if

11:     $\text{PD}=\text{PD}+(f(\text{younger\_cc}),t)$

12:    end if

13:   end for

14: end for

15: return # of lesions = len(cc).