Symmetry orbits and curvature maps

The cornea is the main refracting surface of the human eye. Its front (anterior) surface is approximately 1.3 cm^2 with an average radius of curvature of about 7.8mm.  Typical computer algorithms for corneal curvature measurement (known as keratometry) are based on projecting a pattern of concentric rings of light onto the anterior surface of the cornea and numerically determining the relative separation between the images of these reflected rings of light. This models the  anterior surface of the cornea as a highly polished spherical mirror.  By sampling the curvature at specific circularly equidistant points, a numerical model for the surface curvature may be obtained.

The corneal curvature, (k), and its refractive index, (n), contribute to determining the surface's ptical refractive power k(n-n'), where n' is the reference refractive index of the air. Most of the light refraction takes place at the surface of the cornea, which has refractive index n=1.3376.

Light then passes through the aqueous humor (n=1.336, close to the refractive index of  water) to the lens (n=1.386 to 1.406, where it is refracted further) and through the vitreous humor to the retina at the back of the eye.

The standard unit of refractive optical power is the diopter (D) and is defined to the inverse of the radius of curvature, or approximately equal to the inverse of the focal length of the refracting element. One diopter equals one inverse meter (m^{-1}). For example, using the standard keratometric index n=1.3375, a cornea with a curvature of 7.50mm at 0 degrees has  power (1/0.0075)x0.3375=45D, whereas if the curvature at 90 degrees is 7.25mm, the power is 46.50D.

The difference between these steep (maximum) and the flat (minimum) curvatures, as illustrated in the diagram above (From W.F. Long), is the amount of regular astigmatism present in the optics of the eye, which interferes with a sharp formation of the image on the retina. In the present example, the regular astigmatism is 1.50D. 

The graph below shows a typical curvature contour near the apex of the cornea:

In analogy with the D_2 symmetry orbits derived  for the  visual field data we can evaluate the D_2 orbit

C, vC, hC, vhC

for the curvature contours by applying the corresponding planar reflections {v, h} and double reflection {vh} to the support of the contour. The following graph shows the contour for the difference

C-hC 

shifted by 7D (red contour) to allow its polar plotting relative to that reference value (black contour).

Recalling the multiplication table for D_2 we now know that

h(C-hC) = hC - h^2 C = hC - C = - (C - hC)

so that the contour is horizontally anti-symmetric. 

Here is the 7D-shifted contour for C - vC:

showing its vertical anti-symmetry, and here

is the contour for C - hvC, showing its central (double-reflection) anti-symmetry.

Again as introduced earlier in the study of the  visual field , we seek to describe the orbits invariants for D_2, namely:

I1 = C + vC + hC + hvC,

Iv = C + vC - hC - hvC, 

Ih = C - vC + hC - hvC,

Ihv = C - vC - hC + hvC. 

Here is the contour for the full symmetric invariant I1,

which is vertically, horizontally and centrally (double-reflection) symmetric. Here is the contour for the invariant Ih,

which is horizontally symmetric, vertically and centrally anti-symmetric.  Here

is the contour for Iv, which is vertically symmetric, horizontally and centrally anti-symmetric. And here

is the contour for Ihv, which is centrally symmetric, and vertically and horizontally anti-symmetric. These are then the invariant summaries of a single contour under the D_2 orbit applied to its support. Clearly, its support allows for many more refined groups of symmetries. We will, in the future, revisit these contours after learning the general method for determining the orbit invariants. 

First revised 06/20/2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetry orbits and space orientation

Here is a simple illustration connecting the notions of symmetry orbit  and  space orientation.  Imagine the following image pasted with its center at the crossing of the equator and the prime meridian of a transparent glass globe, matching its standard cardinal directions. The image is one-sided except in its central area, where it is two-sided.

An observer is allowed to survey the image by walking full-circle around the equator and full-circle around the prime meridian, so that what is seen on the left, right, up, down, front, and back sides of the image relative to the observer's position can be annotated. The globe is stationary relative to the surveyor. What is annotated is the number of disks and circles as the observer approaches the image from the cardinal positions. Here 

is what is the view from the other side of the globe, or the back side of the image, and here

is what the observer reported.

The arrows indicate the front / back side viewing of the image. For example, the observer approaching the image from the East would have seen 16 circles in the front central area, 16 disks up, 25 down, 9 on the left, and 4 on the right side of the image. When at the other side of the globe the observer would have seen 4 circles in back central area, 25 disks up, 16 down, 9 on the left, and 4 on the right side of the image, when again approaching the image from the East.

It turns out that the observer's  path has enough information to map it to a path along a symmetry orbit of the square. That is, a dihedral D_4 orbit. Each up/down, left/right, front/back observed data vector (U,D,L,R,F,B) with the reported number of disks and circles is then indexed by a point in that orbit. 

This will be (somewhat) more evident if we now hold the observer stationary and do a two-step mechanism (the dihedral trick):

• Rotate (say clockwise) the globe along the central axis through the equator-prime meridian intersection (call it the x-axis) by 90, 180, 270, 360 degrees, thus rotating the image around its center;
• Rotate the globe (either direction) 180 deg along the N-S (z) axis, then repeat the four-fold rotations described above. The projections of the 8 transformations onto the z-y plane reproduces the planar (D_4) rotations and reversals.  

The standard question, then, is: what are the summaries of the reported data that remain invariant under the D_4 relabeling? In the present context:  What are the summaries of the data that do not depend on the up/down, left/right, in front/back relabeling? Shortly: What are are the orbit invariants? Or yet: What are the invariants that resolve the arbitrariness in the labels? The corresponding questions formulated for D_2 were discussed in the context of  visual field data. 

We do not have all the tools to develop the complete set of invariants for D_4 yet. However, as we shall see coming along these postings,  the D_4 orbits have exactly 5 invariant summaries. In the present posting we will just enunciate 2 of them. Here they are:

The orbit invariant on the top combines within rotation variation and within-reversals variation, whereas the other one compares rotations with reversals, briefly stating it. Both invariants define one-dimensional subspaces for the data. The first invariant thus resolve the arbitrariness in the left-right, up-down orientation. The relabeling of the planar orientation has the effect of at most changing the sign of the summary (+/- 56). The second invariant resolves the arbitrariness in the front-back orientation. Again, the summary (+/- 48) stays in a one-dimensional subspace. 

These two invariant (subspaces) account for two of the eight dimensions afforded by D_4. As we move along the remaining invariants will be introduced. When all invariants are available, the inverse problem of recovering the original data along the orbit can be effected.  

First revised 06/08/2011
Text with this color was revised in  06/09/2011

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

Coloring R.A. Fisher

Contemporary statistics owns several chapters of its history to Sir Ronald A. Fisher. In one of his papers published in 1942 in the Annals of Eugenics, Fisher discusses the concept of confounding in factorial experiments making use of  a very useful group of symmetries, illustrated in this page to highlight again the notion of experimental results indexed by a symmetry orbit. At a later page we will return to discuss Fisher's ideas.  

Suppose we have three attributes to experiment with by replacing some or all of them into an initial composition. Let's say that these attributes are represented, or labelled,  by the primary colors R (Red), G (Green), and B (Blue).  Here

are the resulting (additive) color labels if we started with

(R,G,B) = (0,0,0)

indicating no red, green, or blue, and ended up with

(R,G,B) = (1,1,1)

mixing all three colors in the coloring of the squares, following the sequence

(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1).

In the sequence, between Black and White,  we obtained, respectively, the colors

 Red, Blue, Green, Yellow, Magenta, Cyan.

In Fisher's theory, these experimental conditions would have been represented in terms of the set

S = {{ }, {R}, {G}, {B}, {R,G}, {R,B}, {G,B}, {R,G,B}},

of all subsets of  {R,G,B}. Fisher also observed that the subsets have a  composition rule defined by including the elements that are not common and excluding the elements that are common, so that, for example,

{R}+{G} = {R,G},  {R}+{R,G,B} = {G,B}.   

The set S together with that composition rule form a group, so that the collection of experimental conditions listed above is then a symmetry orbit of S.   Here are all orbit representations, relative to the initial control condition, illustrated by the color of the square at the leftmost position:

This is also a picture of the multiplication table of S.

The outstanding question, common to all the previous pages in this blog, is that of determining the summaries of the experimental results obtained along a symmetry orbit in a way that they do not depend on the initial condition. Shortly, the question is: What are the symmetry orbit invariants? 

The operation in S is clearly commutative: that is X+Y=Y+X for all X and Y in S. Note the symmetric pattern of the multiplication table shown above. We also say in this case that S is an Abelian group. Also note that every element X in S is an involution, that is: 

 X+X= { }.

These two facts greatly facilitate the determination of the orbit invariants: Because the group is finite and commutative we (will) know that there are as many invariants as the number of elements in the group. The involutions factor S into the product FxFxF of three copies of an additive group F = {0,1} in which 1 is an involution.  

The factorization becomes evident as we rewrite Fisher's formulation of the group operation in terms of 

{ } = (0,0,0), {R} = (1,0,0), {G} = (0,1,0), {B} = (0,0,1),

 {R,G} = (1,1,0), {R,B} = (1,0,1), {G,B} = (0,1,1), {R,G,B} = (1,1,1),

and observe that the original operation in S corresponds to adding any two of its elements using the operation rules from F jointly in each component. As illustrated above,

{R}+{R,G,B} = (1,0,0)+ (1,1,1) = (0,1,1) = {G,B}.

The orbit invariants will result from all (eight) possible product of three copies of the two elementary invariants for F, indicated by u and s,  and given by 

u(0) = 1, u(1) = 1        and     s(0) = 1, s(1) = -1.

The eight invariants are then described by the products

uuu, suu, usu, uus, ssu, sus, uss, sss.  

For example, the evaluation of the product ssu gives:

s(0)s(0)u(0)= +1

s(1)s(0)u(0)=  -1

s(0)s(1)u(0)=  -1

s(0)s(0)u(1)= +1

s(1)s(1)u(0)= +1

s(1)s(0)u(1)=  -1

s(0)s(1)u(1)=  -1

s(1)s(1)u(1)= +1

so that it contrasts the total effects indexed by the experimental labels with colors  

(0,0,0), (0,0,1), (1,1,0), (1,1,1),

with those indexed by the colors

(1,0,0), (0,1,0), (1,0,1), (0,1,1).

This invariant (and all others with the exception of uuu) factors the original group into the two halves

one of which contains the identity and forms a subgroup H and the other that is a coset of H in G. Here

H = { Black, Red, Blue, Cyan}

whereas the coset is,

Green + H = {Green, Cyan, Yellow, White}.

Together, we have

S = {Black + H} + {Green + H}.

Note that  Cyan, Yellow, and White all produce the same coset of H in S. They are representatives of the coset. Here the representatives are involutions, and each one together with Black gives a subgroup of S with  the same structure as the factor  F introduced above. We say that F is a factor subgroup of H in S. Also due to the commutativity in S, the coset space

S/H = {Black + H,  Green + H}

forms a group under the operation borrowed from F = {Black, Green}. It is called the quotient group of S by H.

Fisher's 1942 paper is a classical application of quotient groups with the purpose of finding suitable factors (or fractions of the initial factorial experiments) that are more homogeneous and yet retain the comparisons of primary interest (single factors and two-factor interactions). For example, in the factorization obtained above,

only half of the experimental labels is used, at the cost of using the attribute Magenta, which is a confounding of Green and Red.

This was a longer then usual page! A quick summary is simply that all classical contrasts in factorial experiments are the orbit invariants for the type of groups introduced above. They are of the same nature as those invariants described in the flag preference experiment and also here, and have behind their recipe a common methodology. This is what I am proposing to developed here.

Note:

The collected papers of R.A. Fisher related to statistics, mathematical theory and applications is available here. His collected papers related to genetics, evolution and eugenics are available here.

Last revised 06/01/11

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

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)

Field orbits

The following is a symmetry orbit in which the elements are planar vectors:

To generate the orbit, we may start with any one of its vectors in red, rotating it counterclockwise in steps of 90 degrees, producing the vectors in red; Then we may reflect the original starting vector with respect to the x-axis, and rotated it the same say, thus producing the remaining vectors (black color) in the orbit.  The resulting orbit thus has the symmetries of the square, introduced earlier on  here. Clearly, the orbit would have been the same regardless of its initial condition (generating vector) and direction of rotation.

The orbit above has the symmetry of the square. We will indicate those symmetries with the symbol D_4. We will refer to the symmetries of the regular polygons as dihedral symmetries,  We will write D_3 for the triangle, D_4 for the square and so on, for D_5, D_6, .... We extend the notation to D_2 as well, although in this case it  indicates the symmetries of the rectangle.

These symmetries are the planar line reflections and double reflections (or rotations), or equivalently, the n-fold rotations and rotations preceded or followed by a reflection. These always come in pairs, and referred to as rotations and reversals.

Rotations and reversals have, intrinsically, the same reflexivity found in the naming of left- and right-handed objects. It is perhaps better not to try to say it in words. It gets rather silly very fast.

Field orbits are simple designs to gather experimental results that are then indexed by the points in the orbit.

For example, dihedral symmetry aspects in a visual field can be studied by its response to a dihedral orbit of the square by embedding a set of rotations in the visual field, say:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

and the companion reversals:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

Any summary of the visual field over each dihedral field vector, such as its gradient, average, or extreme values, is then indexed by the dihedral symmetries. The summaries may, of course, be in more than one dimension. Visual fields are usually obtained from fellow eyes, so each orbit may have a fellow orbit obtained from the fellow eye, thus producing a pair of numbers in each point of the dihedral field. 

Similarly, below are the dihedral field orbits of D_5, D_6, and D_10.

The question to be addressed in later postings is that of determining the orbit invariants for those experimental data, their broad interpretations, and plausible methods of statistical inference.

Last revised: May 15th, 2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

The Sloan Fonts classification study

The 10 distinct letters appearing (in Roman font) in the chart below are used in the assessment of visual acuity. The actual charts are set in Sloan fonts and are referred to as  Snellen Charts.

The letters composing each line were selected [1] to balance the between-line overall difficulty in correctly identifying the five letters.

The table below lists the Sloan letters (in Roman font), their estimated probability (p) of being correctly identified by normal-acuity subjects, their corresponding entropy, and their -log contrast sensitivity:

We now refer to the subgroups introduced earlier on here and obtain the follow classification of the fonts according to the (largest) subgroup of symmetries afforded by each font:

• K, R        ------->    { 1 },
• Z, N, S    ------->    { 1, o },
• V             ------->    { 1, v },
• D, C        ------->    { 1, h },
• H, O        -------->  { 1, v, h, o }.

Each font in each class is the symmetry orbit of that subgroup. Shortly, we may say that the font has the symmetry of that subgroup.

Moreover, now we may associate to each subgroup a number, such as the maximum entropy of the fonts with the symmetry of that subgroup. From the above table where the entropies are listed we then obtain:

• K, R        ------->    { 1 }             -----> 0.669,
• Z, N, S    ------->    { 1, o }         -----> 0.693,
• V             ------->    { 1, v }         -----> 0.656,
• D, C        ------->    { 1, h }         -----> 0.687,
• H, O        -------->  { 1, v, h, o }  -----> 0.692.

This is then our first example of subgroups indexed by numerical evaluations. Later on we will remark that this experimental situation can be treated in analogy with the previous case, where the elements of a given group were indexed by numerical evaluations.

=====  Reference Cited =====

[1]  Ferris FL 3rd, Freidlin V, Kassoff A, Green SB, Milton RC. Relative letter and position difficulty on visual acuity  charts from the Early Treatment Diabetic Retinopathy Study. Am J Ophthalmol. 1993 Dec 15;116(6):735-40. 

Last revised on 03/01/2011

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