1) The traditional GrowCut

GrowCut algorithm is an interactive segmentation method and solves pixel labeling task based on cellular automaton.

A cellular automaton (CA) is defined as a triplet $\mathsf{\text{A}}=\left(\mathsf{\text{S}},\mathsf{\text{N}},\delta \right)$, where S is a set of non-empty state, N is the neighborhood system and $\delta :{\mathsf{\text{S}}}^{\mathsf{\text{N}}}\to \mathsf{\text{S}}$ defines the state transition rule of cells at time t+1 based on the states of neighbor cells at time t. The Moore von (8-connected) and Neumann (4-connected) neighborhoods are two commonly used neighbor systems. The state of each cell is also a tri-plet ${\mathsf{\text{S}}}_{\mathsf{\text{p}}}=\left({\mathsf{\text{l}}}_{\mathsf{\text{p}}},{\theta }_{\mathsf{\text{p}}},{\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{p}}}\right)$, where ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}$ is the label of this cell, ${\theta }_{\mathsf{\text{p}}}$ is the strength of this cell which ranges from 0 to 1, and ${\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{p}}}$ is the feature vector that its value is image intensity.

where ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}$ is the intensity value of each pixel. As the segmentation algorithm proceeds, all pixels in this image are assigned to one of possible labels.

Before starting the segmentation, user should input an initial label matrix manually. The label matrix has a same size with the original image. In the label matrix, there are two kinds of marked points, one is foreground seed point whose label is ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}=1$, and the other is background seed point whose label is ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}=-1$. The original strength of these two kinds of marked points is ${\theta }_{\mathsf{\text{p}}}=1$. Apart from these two kinds of marked points, the label of the remainder of points is ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}=0$. After all the initial operations have been done, the iteration segmentation runs until the label matrix does not change. Finally, the label value of object region is 1 and the label value of background is -1. The iterative process of labels ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}$ and strength ${\theta }_{\mathsf{\text{p}}}$ at time t+1 is shown as follows,

      State transition of CA

      // For each cell...

        for $\forall \mathsf{\text{p}}\in \mathsf{\text{P}}$

      // Copy the previous state

        ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}^{\mathsf{\text{t}}+1}={\mathsf{\text{l}}}_{\mathsf{\text{p}}}^{\mathsf{\text{t}}}$;

        ${\theta }_{\mathsf{\text{p}}}^{\mathsf{\text{t}}+1}={\theta }_{\mathsf{\text{p}}}^{\mathsf{\text{t}}}$;

        // Neighbors try to attack current cell

    for $\forall \mathsf{\text{q}}\in \mathsf{\text{N}}\left(\mathsf{\text{p}}\right)$

            if $\mathsf{\text{g}}\left(\left|\right|{\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{p}}}-{\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{q}}}|{|}_2\right)\cdot {\theta }_{\mathsf{\text{q}}}^{\mathsf{\text{t}}}>{\theta }_{\mathsf{\text{p}}}^{\mathsf{\text{t}}}$

            ${\mathsf{\text{l}}}_{\mathsf{\text{p}}}^{\mathsf{\text{t}}+1}={\mathsf{\text{l}}}_{\mathsf{\text{q}}}^{\mathsf{\text{t}}}$;

            ${\theta }_{\mathsf{\text{p}}}^{\mathsf{\text{t}}+1}=\mathsf{\text{g}}\left(\mathsf{\text{||}}{\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{p}}}-{\overrightarrow{\mathsf{\text{C}}}}_{\mathsf{\text{q}}}||{}_2\right)\cdot {\theta }_{\mathsf{\text{q}}}^{\mathsf{\text{t}}}$;

        end if

    end for

end for

2) The Improved GrowCut

Although GrowCut algorithm is simple and precise, it is an interactive segmentation algorithm. For some complex images, just once interactive operation cannot achieve satisfactory results, and both foreground seed points and background seed points also should be selected carefully. In order to avoid multi-interaction, we propose an improved GrowCut algorithm which can generate seed labels automatically.

In this subsection, the rough segmentation results are continuous in space series and about half of them do not need refined segmentation. Therefore, the initial label matrix can be generated automatically using the result of rough segmentation (denoted as "a seed template image"). The seed template image must meet two conditions. One is that it does not need refined segmentation, and the other is that it is adjacent to the slice which needs refined segmentation. The process of generating both foreground and background seed points is described in Figure 6. Figure 6(a) is an original kidney CT image and Figure 6(d) is the contour of the kidney. Due to the spatial continuity of CT images, Figure 6(d) can be used to generate the initial label matrix for the segmentation of next slice. There are four steps to generate the initial label matrix automatically.

Firstly, we should get the edge points of kidney contour.

Secondly, calculate the altitude difference between two edge points. Figure 6(b) and 6(c) show the altitude difference (d_y) at point P_m and P_n respectively.

Thirdly, there are three threshold values T_h, T₁, T₂ to control the process of generating seed points. The seed points will be located in the vertical direction if d_y is less than T_h, otherwise they will be located in the horizontal direction. Taking Figure 6(e) as an example, the foreground seed point is located at the bottom of P_m and the background seed point is located at the top of P_m , because d_y is less than T_h. In Figure 6(f), the foreground seed point is located at the right of P_n and the background seed point is located at the left of P_n, because d_y is greater than T_h. All foreground seed points are located inside of the kidney contour and the distance between them and edge points are T₁ pixel. All background seed points are located outside of the kidney contour and the distance between them and edge points are T₂ pixel. In Figure 6(e) and 6(f), the red points are denoted as foreground seed points and the green points are denoted as background seed points.

Finally, according to step 2 and 3, we can get both foreground and background seed points of each edge point

Figure 7 shows the refined segmentation result by IGC algorithm. Figure 7(a) is a seed template image. Figure 7(b) is the seed label image which is generated automatically base on IGC algorithm. The red points are foreground seed points and the green points are background seed points. The value of red point is 1, the value of green point is -1, and the remainders are 0. The final result of IGC is shown in Figure 7(c). More refined segmentation results are shown in Figure 8. They have the same slice number with the rough segmentation results in Figure 5.