September 2016
Volume 57, Issue 12
Open Access
ARVO Annual Meeting Abstract  |   September 2016
Pitfalls of Voronoi Tessellation (VT) in Evaluating Cone Packing
Author Affiliations & Notes
  • James D Akula
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
    Ophthalmology, Harvard Medical School, Boston, Massachusetts, United States
  • Robert J Munro
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
  • Tara L Favazza
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
  • Emily A Swanson
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
  • Anne Moskowitz
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
    Ophthalmology, Harvard Medical School, Boston, Massachusetts, United States
  • Ronald M Hansen
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
    Ophthalmology, Harvard Medical School, Boston, Massachusetts, United States
  • Anne B Fulton
    Ophthalmology, Boston Children's Hospital, Boston, Massachusetts, United States
    Ophthalmology, Harvard Medical School, Boston, Massachusetts, United States
  • Footnotes
    Commercial Relationships   James Akula, None; Robert Munro, None; Tara Favazza, None; Emily Swanson, None; Anne Moskowitz, None; Ronald Hansen, None; Anne Fulton, None
  • Footnotes
    Support  DoD Award MR130362; NIH R01-EY010597; The Massachusetts Lions Eye Research Fund, Inc.
Investigative Ophthalmology & Visual Science September 2016, Vol.57, 70. doi:
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    • Get Citation

      James D Akula, Robert J Munro, Tara L Favazza, Emily A Swanson, Anne Moskowitz, Ronald M Hansen, Anne B Fulton; Pitfalls of Voronoi Tessellation (VT) in Evaluating Cone Packing. Invest. Ophthalmol. Vis. Sci. 2016;57(12):70.

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      © ARVO (1962-2015); The Authors (2016-present)

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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 (ST) 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 ST agreed well with the underlying pattern except at high disorder; SD ST increased monotonically but nonlinearly with disorder. Mean ST 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 ST 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 ST. ST, but not S, identified the ‘true’ square packing pattern.

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

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