Symmetries orbits and the visual field

The matrix

describes the output of an automated perimetry test used in the assessment of the visual field. The entries represent the sensitivity of the retina in detecting the light stimulus and is expressed in decibels units, with a maximal possible reading of 50db. A 50db target is the dimmest target the instrument can project. The smallest the reading the least the sensitivity at that retinal location given by reference to the rows and columns of the matrix. The matrix then describes a function

F( x, y )

defined over a symmetric array of 10x10 points. Here is a graphical display of F:

This says that we can explore the visual field data by letting certain symmetries act in the array where the field is defined. That is, we can study the transformed fields

F,  vF,  hF,  hvF 

forming the symmetry orbit of  F acted by the dihedral group

 D_2 = {1, v, h, o},

where the line reflections are with respect to rows and columns of F. It is opportune to remark that F  plays the role of coloring the underlying framework, in analogy with the colorings of flags and triangles introduced earlier. Here are the results:

  

• Horizontal reflection:

• Vertical reflection

• Double reflection:

The following fields, shown below, are obtained from the symmetry orbit as follows:

• I1 = F + vF + hF + hvF 
• Iv = F + vF - hF - hvF
• Ih = F - vF + hF - hvF
• Io = F - vF - hF + hvF 

They have some remarkable properties, and are referred to as orbit invariants: in the sense that each one of these matrices, indicated here as I,  is such that

wI = +/- I

for all w in  D_2 = { 1, v, h, o }.   Here they are, for your verification.

• The invariant field I1:

• The invariant field Iv:

• The invariant field Ih:

• The invariant field Io (double reflection):

Orbital invariants are important tools in the analysis of data associated with symmetry orbits, and we will often return to that notion in future postings.

Last revised  05/15/2011

These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)