Pitfalls of Voronoi Tessellation (VT) in Evaluating Cone Packing

Abstract

Purpose : VT and, to a lesser extent, its dual graph, Delaunay Triangulation (DT), are often applied to adaptive optics (AO) scanning light ophthalmographs (SLOs) of the retina to evaluate the cone mosaic. For instance, with cone centroids as generators, sidedness (S) of ‘Voronoi cells’ is used to determine packing pattern (i.e., hexagonal), and variability in S is used as a measure of disorder. Therefore, we evaluated VT and DT in actual and simulated cone mosaics to identify key properties and pitfalls.

Methods : AO-SLOs displaying central and eccentric cones were obtained from healthy human retinae. Cone centroids were labeled using a semi-automated method. Simulated triangular, square, and hexagonal packing were then generated with no, low (15%), medium (30%) and high (45%) levels of normally-distributed disorder. VT and DT were performed in MATLAB. Inspection of VT in SLO and simulated mosaics indicated that the ‘apparent’ S of a cell was frequently lower than the actual S due to very short sides or very shallow corners. An algorithm that trimmed sides and corners to agree with subjective sidedness (S_T) was developed and applied to the SLO and simulated mosaics. DT was used to objectively determine each centroid’s neighbors and the standard deviation (SD) of mean inter-centroid distance (ICD) was calculated.

Results : Mean S in every image of every subject was very close to 6. SD S differed slightly between SLOs but not systematically with any subjective impression of the mosaic. The triangle, square and hexagon simulations had mean 3, 4, and 6 S but, when any amount of disorder was introduced, mean S always jumped to ~6; SD S did not change monotonically with disorder. Mean S_T agreed well with the underlying pattern except at high disorder; SD S_T increased monotonically but nonlinearly with disorder. Mean S_T was between 5 and 6 in all SLOs. SD ICD increased monotonically and remarkably linearly with increasing disorder.

Conclusions : Given that some disorder is inherent biological systems, mean S will always indicate hexagonal packing. SD S is only a poor measure of disorder, SD S_T is an improvement, and SD ICT is much better. The true packing of the cones is somewhere between pentagonal and hexagonal (like a soccer ball).

This is an abstract that was submitted for the 2016 ARVO Annual Meeting, held in Seattle, Wash., May 1-5, 2016.

 

Left: Voronoi cells color-coded based upon S. Right: Histograms of S and S_T. S_T, but not S, identified the ‘true’ square packing pattern.