Conversion between RGB and RYB color spaces

Question

I am currently trying to convert colours between RGB (red, green, blue) colour space and RYB (red, yellow, blue) colour space and back again.

Based on the details in the following paper, I am able to convert from RYB to RGB using trilinear interpolation - where the parametric weightings (s,t,u) are the RYB colors, and the vertices of the cube are 3d points in RGB space.

Paint Inspired Color Mixing and Compositing for Visualisation - Gossett and Chen - Section 2.1 - Realization Details

My difficulties are in reversing the conversion process.

A second paper references the use of this technique and also indicates that the reverse conversion was achieved using Newton's Method. But provides no further details. This would probably indicate root finding in solving the trilinear interpolation equations.

On the Transfer of Painting Style to Photographic Images through Attention to Colour Contrast - Xiaoyan Zhang; Constable, M.; Ying He;

Before I expand on this question with the equations, has anybody seen or solved this in a language such as Java/C/C++/C# ?

My current approach is to take the forward equations of the trilinear interpolation (RYB to RGB), expand and rearrange to provide 3 simultaneous equations for 3 unknowns (the parametric weightings: s,t and u) then work out how to find the roots using the Newton-Raphson method. Am I going about this in the right way?

I will edit this question shortly to provide the equations.

Many thanks for your time,

Ben

Answer 1

Here is a category on UIColor that does the same thing, returning elements between RGB, RYB, and CMYK. Further, you can mix any number of colors in the respective color space (they mix differently, of course, depending).

https://github.com/ddelruss/UIColor-Mixing

Enjoy,

Damien

Answer 2

I managed to solve it in the end.

Take the equations for a trilinear interpolation: wikipedia

Substitute the first equations into the last, the expand and collect the coefficients for: Xd, Yd, Zd, XdYd, XdZd, YdZd, ZdYdZd and the constant.

Then find the partial differentiation of the equation in each of the 3 dimensions each in respect to Xd, Yd and Zd. Use these new equations to populate the (3x3) Jacobian matrix and then use Newton's method to solve in software.

Newton-Raphson Method

Answer 3

I found this JavaScript implementation of RYB->RGB conversion based on cubic splines. Here is my Lua port (all values lie in the interval 0-1):

local ryb2rgb = function( R, Y, B ) 
  local R, Y, B = R*R*(3-R-R), Y*Y*(3-Y-Y), B*B*(3-B-B)
  return 1.0 + B * ( R * (0.337 + Y * -0.137) + (-0.837 + Y * -0.163) ),
    1.0 + B * ( -0.627 + Y * 0.287) + R * (-1.0 + Y * (0.5 + B * -0.693) - B * (-0.627) ),
    1.0 + B * (-0.4 + Y * 0.6) - Y + R * ( -1.0 + B * (0.9 + Y * -1.1) + Y )
end