Conversion between RGB and RYB color spaces
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.
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?
Answer
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.