function R_Inf = SaundersonCorrectionInverse(R_m, K1, K2);
% Purpose Calculate the inverse of the Saunderson correction.
%
% Description This routine implements Equation (26) of [Allen1980]. This reference also
% gives a readable derivation and discussion. The input variables can apply at just
% one wavelength, or can be taken to be apply to the entire visible spectrum.
%
% The Saunderson correction is an adjustment to the Kubelka-Munk model. The Kubelka-
% Munk model deals with the behavior of light after that light has entered a film
% of paint or ink, and assumes that the light s behavior is the same at the air-film
% interface as it is elsewhere in the film. In fact, the behavior at the interface is
% different. The Saunderson correction models behavior at the interface.
%
% The Saunderson correction relates two terms, R_m and R_Inf. R_m is the reflectance
% fraction that a spectrophotometer (in a 0/0 geometry) would measure. The light from a
% spectrophotometer, of course, originates from outside the paint film. R_inf is the
% bulk reflectance, which is the fraction of light that would escape from the
% top of the paint film, if the light originated just inside the paint film, at the top,
% and was directed into the film.
%
% This routine calculates R_Inf, if R_m is already known. The Saunderson correction
% depends on two physical terms, K1 and K2. These terms are often not known in practice,
% and so must be estimated. K1 can be calculated from the Fresnel Equation (Equation (3.1)
% of [Judd1967]) if the refractive indices of the paint film and the surrounding air
% are known. The refractive index of air is very near 1, but the refractive index of the
% paint film is usually not known. K2 is usually chosen to be between 0.4 and 0.6.
%
% References [Allen1980] Eugene Allen, "Colorant Formulation and Shading," Chap. 7 in
% Optical Radiation Measurements, Vol. 2: Color Measurement (eds. Franc Grum
% and C. James Bartleson), Academic Press, 1980.
% [Judd1967] Deane B. Judd and Gunter Wyszecki, Color in Business, Science, and
% Industry, 3rd printing, John Wiley and Sons, 1967.
%
% Syntax R_Inf = SaundersonCorrectionInverse(R_m, K1, K2);
%
% R_m The reflectance, as a ratio between 0 and 1, of a masstone of a paint or
% ink. R_m has the same size as R_Inf, and can be a matrix, scalar or vector. R_m
% is the reflectance value that a spectrophotometer would measure.
%
% R_Inf The bulk reflectance, as a ratio between 0 and 1, of a masstone
% of a paint or ink. R_Inf is the fraction of light, that has penetrated the
% top surface of a paint film, that is eventually reflected back out the top,
% possibly after complicated internal scattering.
%
% K1 The fraction, as a value between 0 and 1, of the light impinging (from
% OUTside the film) on a paint
% or ink, that is reflected back directly from the paint surface, without
% first entering the paint film. The Fresnel equation (Equation (3.1) of
% [Judd1967] gives an expression for K1 in terms of refractive indices.
%
% K2 The fraction, as a value between 0 and 1, of the light impinging (from
% INside the film) on the interface between the film and the surrounding
% air (or other medium), that is reflected back into the film. P. 308
% of [Allen1980] indicates that a theoretical value for K2, assuming
% perfectly diffuse light, is 0.6. In practice, it is not always clear
% how K2 should be determined, and values as low as 0.4 might be used.
%
% Author Paul Centore (July 28, 2013)
% Revised Paul Centore (October 30, 2013)
% --Used elementwise operations consistently in formula
% Revised Paul Centore (January 5, 2015)
% --Minor revisions to documentations
%
% Copyright 2013, 2015 Paul Centore
%
% This file is part of MunsellAndKubelkaMunkToolbox.
%
% MunsellAndKubelkaMunkToolbox is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% MunsellAndKubelkaMunkToolbox is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with MunsellAndKubelkaMunkToolbox. If not, see .
R_Inf = (R_m - K1)./(1 - K1 - K2 + K2.*R_m) ;