Voronoi tesselation v1
Adalberto Merighi2022
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This protocol describes how to perform Voronoi tesselation analysis of cerebellar images. It can be used for any biological images to study cellular sociology and is based on a model of parametrization and quantitation of cellular population topographies developed by Marcelpoil and Usson (1992). It is advantageous to analyze cellular migration and dispersion in longitudinal studies.
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It is well accepted that cells in the tissue can be regarded as tiles tessellating space. A number of approaches were developed to find an appropriate mathematical description of such cell tiling. A particularly useful approach is the so called Voronoi tessellation, built from centers of mass of the cell nuclei (CMVT), which is commonly used for estimating the morphology of cells in epithelial tissues. However, a study providing a statistically sound analysis of this method's accuracy is not available in the literature. We addressed this issue here by comparing a number of morphological measures of the cells, including area, perimeter, and elongation obtained from such a tessellation with identical measures extracted from direct imaging acquired by staining the cell membranes. After analyzing the shapes of 15,000 MDCK II epithelial cells under several conditions, we find that CMVT reasonably well reproduces many of the morphological properties of the tissue with an error that is between 10 and 15%. Moreover, cross-correlations between different morphological measures are reproduced qualitatively correctly by this method. However, all of the properties including the cell perimeters, number of neighbors, and anisotropy measures often suffer from systematic or size dependent errors. These discrepancies originate from the polygonal nature of the tessellation which sets the limits of the applicability of CMVT.
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Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi Polygons; Delaunay Triangulations; Perpendicular Bisectors; Circumcenters; Convex Hulls; Minimal Spanning Trees; Ulam Trees; Pitteway Violations; Circularity; Clark-Evans spatial statistics; Variance to Mean Ratios; Gabriel Graphs; and, Minimal Spanning Trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s Law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of e...
Voronoi tesselation
Adalberto Merighi1, Laura Lossi1
1Department
Adalber
Merighi
OCT 07, 2022
of Veterinary Sciences - University of Turin, Italy
Adalberto Merighi
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DOI:
dx.doi.org/10.17504/protocols.io.y
xmvmnrx6g3p/v1
Protocol Citation: Adalberto
Merighi, Laura Lossi 2022.
Voronoi tesselation. protocols.io
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ABSTRACT
This protocol describes how to perform Voronoi tesselation analysis of cerebellar images.
It can be used for any biological images to study cellular sociology and is based on a
MANUSCRIPT CITATION:
Marcelpoil, R.; Usson, Y. (1992)
Methods for the study of cellular
sociology: Voronoi diagrams and
parametrization of the spatial
relationships. Journal of
Theoretical Biology 154, 359-369.
License: This is an open
access protocol distributed under
the terms of the Creative
Commons Attribution License,
which permits unrestricted use,
distribution, and reproduction in
any medium, provided the original
author and source are credited
model of parametrization and quantitation of
cellular population topographies developed by Marcelpoil and Usson (1992). It is
advantageous to analyze cellular migration and dispersion in longitudinal studies.
GUIDELINES
N/A
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Protocol status: Working
We use this protocol and it's
working
MATERIALS
Created: Jul 24, 2022
Last Modified: Oct 07, 2022
Software
PROTOCOL integer ID: 67460
Voronoi Diagram Generator
Keywords: Brain slices, Cellular
sociology, Cell migration, Cell
dispersion
Frederik Brasz
NAME
DEVELOPER
Software
NAME
FIJI (Image J)
DEVELOPER
NIH
Software
NAME
Microsoft
OS
Windows 10
DEVELOPER
Microsoft
SAFETY WARNINGS
None
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BEFORE START INSTRUCTIONS
Be sure to familiarize yourself with the theory of Voronoi tessellation
Image processing with the Voronoi generator
1
Open the interactive Voronoi diagram (Thiessen polygon) generator (https://cfbrasz.github.io/Voronoi.html).
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The aspect of the Voronoi diagram generator mask
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2
Upload the image to be analyzed as indicated in the gure above. To do so your image (size must be
900x900 pixels and preferably saved as a PNG le) has to be uploaded to the internet rst so that it is
possible to copy and paste its URL in the Voronoi generator. After uploading the generator displays the
image in its working space as shown in the gure below.
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The image to be analyzed is uploaded to the Voronoi generator
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3
Using the mouse click above the center of each cell to generate the Voronoi polygons. In the end, you will
obtain the image shown below. Save the image on your computer (right-click on the image and choose
"save" from the drop-down menu).
Voronoi tesselation of the uploaded image
4
Choose "Visualization Normal" from the Visualization mode drop-down menu of the generator. The
tesselation appears as shown in the image below. Again save the image on your computer (right-click on the
image and choose "save" from the drop-down menu).
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Voronoi polygons in the Normal visualization mode (Elaboration 1)
5
Select "Hide sites" from the Options menu of the generator. The tesselation appears as shown in the image
below (black dots corresponding to cell centers disappear). Again save the image on your computer (rightclick on the image and choose "save" from the drop-down menu).
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Voronoi tesselation with cell sites hidden (Elaboration 2)
Elimination of the marginal polygons with Photoshop
6
Due to the properties of the Voronoi partition, some polygons of the paving are not statistically
representative of the set of polygons - see Marcelpoil, R.; Usson, Y. (1992) Methods for the study of cellular
sociology: Voronoi diagrams and parametrization of the spatial relationships. Journal of Theoretical Biology
154, 359-369.
Those polygons are associated with points located on the border of the cell population and have one or
more summits that do not contain total information on their "surround" (marginal polygons). Such summits
are created by points that belong to a half-plane that does not contain this particular summit. Therefore,
every point of the cell population whose associated polygon satis es one of the two following conditions is
not taken into account in the further calculations.
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- The polygon is open (the point belongs to the convex hull),
- At Ieast one of the summits of the polygon is outside the convex hull.
6.1
Elimination of the open polygons is carried out with Photoshop using the Magic wand tool to
select and cancel them from the image above named Elaboration 1, as shown in the image
below.
Elimination of the polygons with an open side, i.e. a side that is part of the border of the
image (Elaboration 3)
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7
Construct the convex hull from the image Elaboration 3. The convex hull is constructed with the Line tool by
drawing segments that join the site points (cell centers) of the eliminated open polygons so that there are
no concavities, as shown in the image below.
Construction of the convex hull
8
Eliminate the polygons intersected by the convex hull and the polygons with open sides using the image
Elaboration 2 (without cell sites) as shown in the image below.
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All marginal polygons are eliminated (Elaboration 4)
9
Cancel the sides of the marginal polygons. Use the Magic wand tool of photoshop followed by the
commands Selection → Expand 2px; Selection → Contract 1px; Cancel; Modify → Stroke (color black)
2px. You should obtain an image in which the area of the marginal polygons is empty as in the gure
below. This is the last elaboration that will be used for the subsequent steps of analysis.
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Last elaboration of Voroni tessalation (Elaboration 5)
Analysis of Voronoi polygons with ImageJ
10
Open the image to be analyzed with ImageJ. Set the appropriate scale with Analyze → Set scale.
Run the following Macro by selecting Plugins → Macros → Run → Voronoi Macro
Voronoi Macro
run("Enhance Contrast...", "saturated=2");
run("8-bit");
run("Find Edges");
//run("Brightness/Contrast...");
setMinAndMax(0, 0);
run("Apply LUT");
run("Set Measurements...", "area perimeter shape limit display redirect=None decimal=6");
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run("Analyze Particles...", "display summarize add in_situ");
The macro enhances image contrast (optional), converts the image into a B&W 8-bit image, nds the edges
of the Voronoi polygons, and optimizes their contrast as in the gure below.
Image of Voronoi polygons after application of the Find Edges command and optimization of contrast
(Elaboration 6)
It then sets up the measurements necessary for the following analysis of polygons: Area, Shape descriptors,
and Perimeter. It also permits the creation of an image (below) with the overlay indication of the individual
polygons that the program has measured (Add to overlay and Display label). It nally sets the number of
Decimal places to 6. When run, the Macro performs the command Analyze Particles.
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Image of the Voronoi polygons after performing the Analyze Particles command. Each polygon is
assigned a progressive number. Note the number 1 at the center of the image (encircled in red). This
corresponds to the rst counted particle that the program considers being the ensemble of the
marginal polygons (highlighted in red in the following image). Note that the red circle is added here for
clarity but the program does not display it at all.
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The image displays in red the particle that the program numbers as particle 1 that must be discarded in
the following analysis.
At the end of the Macro, all computed values are saved in a .csv or a .xls le (according to the version of
ImageJ used) that must be converted into a .xlsx Excel le.
Analysis of data
11
Open the .csv or .xls le generated by ImageJ with Microsoft excel. The le appears as follows
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Screenshot of the .xls le generated by ImageJ (the le is created from the analysis of the image
elaborated as described above and contains 318 lines, i.e. the information about the 317 particles
(polygons) counted by the program.
The le contains the following information: Column A: progressive numbering of the particles (polygons)
counted by ImageJ; Column B: Identi cation of image analyzed; Column C: Area (in µm2 if the Set scale
command has been set properly); Column D: Perimeter (in µm if the Set scale command has been set
properly); Column E: Circularity (or Roundness factor); Columns F-H: Other shape descriptors computed by
ImageJ that are not used in the analysis. Note that line 2 (highlighted in yellow) corresponding to Particle 1
must be deleted (as indicated above).
Save the le as a .xlsx le.
12
Open the .xlsx le in Excel and calculate the following:
- Mean of area, perimeter, and circularity (roundness)
- Standard deviation of area, perimeter, and circularity (roundness)
- Area Disorder (AD)
- Roundness Factor Homogeneity (RFH)
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The mean circularity (roundness) (RFav) is computed directly by the ImageJ program using the following
formula
RF av = 1/N ∑( i = 1)N 4πA(Xi )/(L(Xi )2 )
where A(x) is the area and L(x) is the perimeter of the N polygons generated by the Voronoi generator. RFav
is a pure number (0< RFav≤1).
The AD is calculated as follows:
AD = 1 − (1 + σA /Aa v)− 1
where σA is the area standard deviation, and Aav is the mean area.
The RFH is calculated as follows:
RF H = (1 + σR F /RFa v)− 1
where σRF is the roundness factor standard deviation, and RFav is the mean roundness factor.
Both are pure numbers with values >0 and ≤1.
13
Transfer the values of RFav, AD, and RFH to a new Excel spreadsheet for subsequent statistical analysis.
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