Visual Perception of Symmetry - II

Let us now revisit the  flag experiment with the language we have introduced up to now. The experiment consisted of ranking the flags within each row in the figure below:

Each row, as we now recognize, is a symmetry orbit generated by

D_2={1,h,o,v}.

 The flags in columns B,C, and D are generated from the flag in column A by applying a horizontal reflection, a  double reflection, and a vertical reflection respectively.

The rankings were purely subjective, as there were no other directions given at the time of the   flag experiment.

The following table summarizes the frequency counts for the number of occasions in which the first choice is the flag with the row label and the second choice is the flag with the column label in that table. For example, in 14 occasions, flag B was the first choice and flag C was the second choice. I will refer to these frequency counts as transition counts, and write BC=14 to indicate the transition count just described.

With this is mind, a direct inspection of the flags in any one of the orbits shows that the transitions

• BC, CB, AD, DA are associated with vertical reflections,
• AB, BA, CD, DC are associated with  horizontal reflections,
• AC, CA, BD, DB are associated with  double reflections.

The posterior probability densities associated with these symmetry transitions of first-to-second preference, based on the joint frequency counts above, are shown in Figure 1. It is evident that the most likely pairing of first-to-second preferences is associated with those flags that are vertical reflection of each other.  

Figure 1.

Moreover, the vertical transitions form a symmetry orbit, as illustrated below:

For example, starting in (A,D) you move to (D,A) by a vertical reflection on each of the component flags; move to (B,C) by a horizontal reflection and to (C,B) by a double reflection.  The transitions were arranged in the diagram in a way that they are related by their relative positions in the square. The fellow diagram shows the corresponding transition counts, obtained from the summary data table at the top of the page.

These vertical transition data are, therefore, indexed by an orbit of  D_2. However, the assignment of data to the labels in D_2 is arbitrary, in the following way:

If we start the orbit in (A,D) then

• x_1 = 12;
• x_v = 16;
• x_o = 11;
• x_h = 14.

However, if we start the orbit in (B,C), then:

• x_1 = 14;
• x_v = 11;
• x_o = 16;
• x_h = 12.

Clearly, since each D_2 orbit has 4 points, there are 4 possible starting points and hence four relabelings of the data.

Similarly, the horizontal transition counts are labeled by a symmetry orbit:

And finally, the orbit for the the double reflection counts:

These transitions orbits, together, are illustrated here,

in relation to the joint frequency counts table shown at the top of this page.

The question to be investigated is this:

What are the summaries of the frequency counts that do NOT depend on a particular relabeling?

We have alluded to the answer, the orbit invariants for that specific orbit, here.  In all that is coming, the aim is exploring ways of systematically determining the invariants for most types of elementary orbits. Stay tuned.

Assignments:
[1] Following the discussion of the vertical transitions orbit, write the data assigned to the orbit with (A,D) as its starting point in the form of the symbolic sum

X=12*1 + 16*v + 11*o + 14*h. 

Then, referring to the multiplication table of D_2, evaluate the left multiplications

• h*X 
• v*X
• o*X

and verify that all regular relabellings of a D_2 orbit can be obtained that way.

Posted: 05/14/2011.
Last revised: 01/28/2015.
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

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)