You can solve the colors by solving a system of linear equations if you make few assumptions:
- The alpha of the color wheel is constant (i.e. same for all colors in the wheel)
- The beige background extends half way through the color wheel, and is constant for top and bottom halves of the image
- The background in the center is constant for all colors
- The alpha blending equation is the standard $f=c*\alpha+b*(1-\alpha)$, where $f$ = final color, $c$ = color on the wheel, $b$ = background color, $\alpha$ = alpha of the color wheel
If you now pick two colors with the same beige background (e.g. green and yellow), and feed them into the above alpha blending equation you get 4 equations with 4 unknowns, which you can then solve. I.e.
$$f_1=c_1*\alpha+b_1*(1-\alpha)$$
$$f_2=c_1*\alpha+b_2*(1-\alpha)$$
$$f_3=c_2*\alpha+b_1*(1-\alpha)$$
$$f_4=c_2*\alpha+b_2*(1-\alpha)$$
where $f_1$ & $f_2$ = brighter & darker green colors respectively, $f_3$ & $f_4$ = brighter & darker yellow colors respectively, $c_1$ & $c_2$ = the unknown green & yellow colors respectively you try to solve, $\alpha$ = alpha you try to solve, $b_1$ = beige background color, $b_2$ = unknown background color in the center.
Then you can repeat this process for all the colors, or because you solved $\alpha$ you can use this information to solve the rest of the colors simply by:
$$c=\frac{f-(b_1*(1-\alpha))}{\alpha}$$
