Look at how the quantities in the CIE standard that depend on field size vary with
field size, particularly for field sizes greater than 10-degrees, which
is the outer limit of what the standard sanctions.

My conclusions are

  1. that since we know macular pigment is declining towards zero with
    field size (see refs below) and the CIE expoential formula has this
    property, it is resonable to use the CIE formula for macular pigment
    density for field sizes greater than 10-degrees.
  2. the CIE formula for pigment optical density asymptotes to a constant
    as one extends past 10-degrees. Whether this is true in the retina or
    not, I am not sure. But as field size gets larger, using this formula
    is going to be no worse than simply using the 10-degree values, since
    they are the same.
  3. one needs to take any estimate of the CMFs for large field sizes
    with a grain of salt. There is going to be variation across the field,
    so any point estimate is likely to be wrong somewhere. The CIE
    formulae, extended using the formulae past their bounds, are a good
    first guess for the mean field properties, but being aware that there
    is variation within the field, as well as individual variation around
    the CIE estimates, is important when considering things like the effect
    of inadvertant stimulation of cones when one tries to isolate
    melanopsin using silent substitution. Note in particular that the CIE
    formula is trying to capture large field color matches where subjects
    are instructed to ignore the center of the field as best they can. In
    a threshold experiment, this might not be how subjects were instructed
    and would in any case be rather hard to do. And if you used annular
    stimuli, you’d be a bit off and might want to think about how to
    estimate the fundamentals from the annulus. Studying the Moreland and
    Alexander paper below in detail might help with thinking on that.

Mooreland & Alexander (1997). Effect of macular pigment on color
matching with field sizes in the 1 deg to 10 deg range. Doc. Opth.
Proc. Ser., 59, 363-368.
Moreland and Alexander make color matches for annuli and for
circular fields, and develop a formula for the equivalent macular
pigment density for color matches of various field sizes. In
these matches, observers are instructed to to ignore the central
Maxwell?s spot, I am pretty sure. M&A say the data are consistent
with the idea that obsevers look near the edge of the field but
not quite at it. Their data are for field sizes of 10 degrees,
and the estimates are fit with exponentials. M&A used these data
together with measurements of macular pigment density by Moreland
& Bhatt (1884) to develop an equivalent (for uniform fields)
macular pigment density to be used in predicting color matches
out to 10 degrees. The formula is an exponential decay.
Something like this made it into the CIE standard, although it
may have been tweaked to make sure the color matching data are
consistent with the 10-deg and 2-deg CMFs. I have not thought
hard about the underlying calculations.

Moreland, J.D. and Bhatt, P. (1984). Retinal distribution ofmacular
pigment. In: Verriest, G. (ed.), Colour Vision Deficiencies VII. Doc.
Ophthalmol. Proc. Ser. 39: 127-132. W. Junk, The Hague.
Uses color-matching data for field sizes out to 18 deg (I think)
to develop estimates of macular pigment density as a function of
eccentricity. Key feature of the data is that density estimates
decline according to an exponential and reach zero at
eccentricity (radius) of about 7 degrees. This paper also
reviews earlier estimates and they all look like decaying

Putnam and Bland (2014). Macular pigment optical density spatial
distribution measured in a subject with oculocutaneous albinism.
Journal of Optometry, 7, 241-245.

See ComputeCIEConeFundamentals, CIEConeFundamentalsTest.

5/25/16 dhb Wrote it.
6/1/16 dhb Polished it up a bit, and added to PTB distribution.

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