Algorithm 3.

Seed-refining algorithm (REFINE-SEEDS).

Inputs:

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

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

- $I_{\text{seed}\_\text{maxima}}$ is the binary maxima-seed mask. White regions are centroid locations of the gland objects.

- ${\mathcal{I}}_{\text{lumen}\_\text{clefting}}$ is the lumen-clefting RGB image. Blue objects are lumen and green objects are periacinar retraction clefting on a white background.

Output:

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

Function and variable description:

- PIXEL-IN($I$) returns all the white pixels in the binary image $I$. DILATE($I,d$) and DISTANCE-TRANSFORM($I$) are the same functions as defined in Algorithm 2.

- Given a binary image $I$, $\overline{I}$ defines a binary image with corresponding inverted pixel values. $I_a\wedge I_b$ and $I_a\vee I_b$ are the Boolean operations defining pixel-wise AND and OR operations between $I_a$ and $I_b$, respectively.

- ${\text{pixels}}^{+}$ and ${\text{pixels}}^{-}$ are two sets containing positive and negative charged particles (pixels).

- MAX-ENTROPY-THRESHOLD($\mathcal{I}$) thresholds $\mathcal{I}$ using the entropy-based image thresholding method.47 Output is a binary image.

Parameters:

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

1:  Procedure REFINE-SEEDS ($I_{\text{gland}}^{\text{threshold}}$, $I_{\mathrm{nuclei}}$, $I_{\text{seed}\_\text{maxima}}$, $I_{\text{lumen}\_\text{clefting}}$)

2:   $I_{\mathrm{nuclei}}^{\text{border}}=\mathrm{DILATE}(I_{\mathrm{nuclei}},8)$.

3:   Create binary lumen mask $I_{\text{lumen}}$ from ${\mathcal{I}}_{\text{lumen}\_\text{clefting}}$.    ⊳ $I_{\text{lumen}}$ indicate lumen in white.

4:   Sample some pixels along the edges of lumen objects ($I_{\text{lumen}}$) into ${\text{pixels}}^{+}$.

5:   ${\text{pixels}}^{+}={\text{pixels}}^{+}\cup \mathrm{PIXELS}-\mathrm{IN}(I_{\text{seed}\_\text{maxima}})$.    ⊳ All the maxima seeds are added to ${\text{pixels}}^{+}$.

6:   Extract approximate centroids of nuclei in $I_{\mathrm{nuclei}}$ into ${\text{pixels}}^{-}$.    ⊳ Remove nuclei from the gland regions.

7:   $I_{\mathrm{mask}}=I_{\mathrm{gland}}^{\mathrm{threshold}}\wedge \overline{I_{\mathrm{nuclei}}^{\mathrm{border}}}$.

8:   ${\mathcal{I}}_{\mathrm{field}}=\mathbf{COMPUTE}\mathbf{-}\mathbf{FIELD}(I_{\mathrm{gland}}^{\mathrm{threshold}},{\text{pixels}}^{+},{\text{pixels}}^{-})$.

9:   ${\mathcal{I}}_{\mathrm{field}\_\mathrm{map}}={\mathcal{I}}_{\mathrm{field}}\times \mathrm{DISTANCE}-\mathrm{TRANSFORM}(I_{\mathrm{mask}})$.    ⊳ Pixel-wise multiplication of two real valued images.

10:   Linearly rescale ${\mathcal{I}}_{\text{field}\_\text{map}}$ to values within [0, 255] for further numerical operations.

11:   $I_{\text{field}\_\text{map}}^{\text{thershold}}=\mathrm{MAX}-\text{ENTROPY}-\text{THRESHOLD}({\mathcal{I}}_{\text{field}\_\text{map}})$.

12:   Fill holes in $I_{\text{field}\_\text{map}}^{\text{threshold}}$ image.

13:   $I_{\text{seed}}=I_{\text{seed}\_\text{maxima}}\vee (I_{\text{field}\_\text{map}}^{\text{threshold}}\vee I_{\text{lumen}})$.    ⊳ Unify all the seeds.

14:   return seed mask $I_{\mathrm{seed}}$.

15: end procedure