Algorithm 5.

Three level EGVD algorithm (THREE-LEVEL-EGVD).

Inputs:

- $I_{\text{seed}}$ is the binary seed mask. White regions are seed objects on a black background.

- ${\mathcal{I}}_{\text{gland}\_\text{map}}$ is the grayscale gland image. Grayscale regions are glands on a black background.

- $I_{\mathrm{nuclei}}$ is the binary nuclei mask. White regions are nuclei on a black background.

Output:

- $I_{\text{gland}}^{\text{result}}$ is the binary gland mask. White regions depict gland on a black background.

Function and variable description:

- DILATE($I,d$) conducts “$d$” dilation operations on binary image $I$ with a $3\times 3$ pixels dilation window. Output image is also binary.

- EGVD($I_s,{\mathcal{I}}_f$) is the image segmentation algorithm described by Yu et al.50 where $I_s$ is the seed image and ${\mathcal{I}}_f$ is the grayscale image. It returns the binary segmentation result.

- $I_a\vee I_b$ is a Boolean operation defining the pixel-wise OR operation between $I_a$ and $I_b$.

Parameters:

- Select number of dilation operations $d=8$. This value for $d$ was optimized empirically.

1:  Procedure THREE-LEVEL-EGVD($I_{\text{seed}}$, ${\mathcal{I}}_{\text{gland}\_\text{map}}$, $I_{\mathrm{nuclei}}$)

2:    $I_{\mathrm{nuclei}}^{\text{border}}=\mathrm{DILATE}(I_{\mathrm{nuclei}},8)$.    ⊳ $I_{\mathrm{nuclei}}^{\text{border}}$ indicates nuclei border pixels in white.

3:    Put all white pixels indicated by $I_{\mathrm{nuclei}}^{\text{border}}$ in ${\mathcal{I}}_{\text{gland}\_\text{map}}$ as black to compute ${\mathcal{I}}_{\text{gland}\_\text{temp}}$.

4:    $I_{\mathrm{EGVD}-\mathrm{output}1}=\mathrm{EGVD}(I_{\text{seed}},{\mathcal{I}}_{\text{gland}\_\text{temp}})$.

5:    Remove small objects and noise in $I_{\mathrm{EGVD}-\mathrm{output}1}$ and then save it to $I_{\mathrm{EGVD}-\mathrm{output}1}^{\mathrm{denoised}}$.

6:    $I_{\text{seed}2}=I_{\mathrm{EGVD}-\text{output}1}^{\text{denoised}}\vee I_{\text{seed}}$.

7:    $I_{\mathrm{EGVD}-\text{output}2}=\mathrm{EGVD}(I_{\text{seed}2},{\mathcal{I}}_{\text{gland}\_\text{map}})$.

8:    Remove small objects and noise in $I_{\mathrm{EGVD}-\text{output}2}$ and then save it to $I_{\mathrm{EGVD}-\text{output}2}^{\text{denoised}}$.

9:    Erode $I_{\mathrm{EGVD}-\text{output}2}^{\text{denoised}}$ to compute $I_{\text{seed}3}$.

10:   $I_{\mathrm{EGVD}-\text{output}3}=\mathrm{EGVD}(I_{\text{seed}3},{\mathcal{I}}_{\text{gland}\_\text{map}})$.

11:   Remove small objects and noise in $I_{\mathrm{EGVD}-\text{output}3}$ and then save it to $I_{\mathrm{gland}}^{\mathrm{result}}$.

12:   return binary gland mask $I_{\mathrm{gland}}^{\mathrm{result}}$.

13: end procedure