Voronoi tesselation

Voronoi-tesselation is the process of subdividing an area accoding to a Voronoi-diagram. In bio-imaging this method can be used to estimate where cells touch each other in case imaging data does not allwo this, e.g. if no membrane marker was used.

from skimage.io import imread
import pyclesperanto_prototype as cle

To demonstrate tesselation, we first load an image showing some objects.

image = imread("../../data/blobs.tif")
cle.asarray(image)

cle._ image shape (254, 256) dtype float32 size 254.0 kB min 8.0 max 248.0

We also segment the nuclei in this image and label them.

#background_subtracted = cle.top_hat_sphere(image, radius_x=50, radius_y=50)
binary = cle.threshold_otsu(image)
labels = cle.connected_components_labeling_box(binary)
labels

cle._ image shape (254, 256) dtype uint32 size 254.0 kB min 0.0 max 64.0

By dilating the labels, we fill up the space between them. We could do this in a loop:

for radius in range(0, 10, 2):
    dilated_labels = cle.dilate_labels(labels, radius=radius)
    
    print("Radius:", radius)
    cle.imshow(dilated_labels, labels=True)

Radius: 0

Radius: 2

Radius: 4

Radius: 6

Radius: 8

We can also do this with a single function call and fill up background intensity until its gone. This process is called Voronoi-Tesselation. The result is a Voronoi-label-image. You can also see it as a discrete Voronoi-diagram.

voronoi_label_image = cle.voronoi_labeling(labels)
voronoi_label_image

cle._ image shape (254, 256) dtype uint32 size 254.0 kB min 1.0 max 64.0

An common alternative visualization is the corresponding label-edge-image.

label_edges = cle.detect_label_edges(voronoi_label_image)
label_edges

cle._ image shape (254, 256) dtype uint8 size 63.5 kB min 0.0 max 1.0

Voronoi and Delauney diagrams

There is a close relationship with the Delauney triangulation. Voronoi and Delauney diagrams are dual graphs. In the continuous case, you can compute the one from the other and vice-versa. In the discrete case, one can determine an estimation of the Delauney graph from a Voronoi-label-image by determining the label’s centroids and connecting them:

touching_neighbor_mesh = cle.draw_mesh_between_touching_labels(voronoi_label_image)
touching_neighbor_mesh

cle._ image shape (254, 256) dtype float32 size 254.0 kB min 0.0 max 1.0

By visualizing the label-edge-image and neighbor-mesh in the same image, their relationship becomes more obvious.

sum_image = cle.maximum_images(touching_neighbor_mesh * 2, label_edges)
cle.imshow(sum_image, labels=True)

Exercise

Visualize the neighbor-mesh and the label-edges of this label_image together as shown above:

label_image = cle.artificial_tissue_2d(width=200, height=100)
label_image

cle._ image shape (100, 200) dtype uint32 size 78.1 kB min 1.0 max 60.0