Munsell color space, CIE, and the mysterious 1.0257

Question

This is an arcane question, but here's hoping.

It has been established empirically, that the luminance Y in the CIE color spaces is a function of the Munsell value V. It does not depend on hue or chroma. The function can be approximated by a 5th degree polynomial (Eq. 2 of ASTM Standard D 1535-08.)

double munV_to_cieY(const double V) {
    // V is in [0,10] 
    // Y is in [0,100] 
    double Y = V*(1.1914+V*(-0.22533+V*(0.23352+V*(-0.020484+V*0.00081939))));
    return Y; 
}

It is the case that in the Munsell renotation data, which associates HVC triples with xyY triples, the Y in xyY is effectively multiplied by a mysterious 1.0257. I have verified that that is the case. I read somewhere, years ago, that the mystery number is due to some kind of mistake, but that is all I remember.

What was the mistake?

I want to know because I have a program that maps sRGB pixels to Munsell. Should I only use the 1.0257 when processing the renotation training data? Or should I also use it in the program per se?

Can someone tell me what's up with that 1.0257, or suggest a way to find out?

EDIT: I inverted the function, using a cubic spline. When applied to the renotation data that I have, here's what I get

V   Y       V'     
1   1.21    1.022   
2   3.126   2.031   
3   6.55    3.037   
4   12.00   4.046   
5   19.77   5.056   
6   30.03   6.064  
7   43.06   7.076   
8   59.1    8.085   
9   78.66   9.092   

I am trying to wrap my head around that.

Answer 1

Short answer is, the renotation Y-data are too high by a factor of 1.0257, but the quintic formula listed in the question is correct.

(The discrepancy is apparently related to the reflectance of 0.975 of magnesium dioxide, a white substance which one presumes was used somehow when analysing the color samples.)

A predecessor to the quintic above was fitted to the uncorrected renotation data. The ASTM standard contains a note to the effect that the correct quintic was obtained from the previous one by dividing the original by 1.0257 (I.e. multiplying each coefficient by 0.975.)

Answer 2

In regard to Eq.2, ASTM D1535-08e1 states that:

The coefficients of this equation are obtained from the 1943 equation by multiplying each coefficient by 0.975, the reflectance factor of magnesium oxide with respect to the perfect reflecting diffuser, and rounding to five digits of precision.

The Munsell Renotation data commonly available through the all.dat, experimental.dat and real.dat files features CIE xyY values that are scaled by a 1 / 0.975 ±= 1.0257 factor because the reference white used for measurement was magnesium oxide.

Magnesium oxide reflectance relative to the perfect reflecting diffuser is 97.5%, thus Munsell Value (V) = 10 converts to Luminance (Y) = 100/97.5 ≈ 102.57 if magnesium oxide is the reference white.

If you are performing conversions using the Munsell Specification, e.g. 2.5R 9/2, using ASTM D1535-08e1 practise, you do not have to scale the computed Luminance (Y) because the adopted reference white is the perfect reflecting diffuser. However, if you are directly using the Munsell Renotation CIE xyY data, you will have to scale it first accordingly to the ratio between magnesium oxide and your reference white.