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)