Why are you trying to match 3 values with a linear relation?

Consider only the red channel. You have 3 reference values (taken from your original photo): Rw for the white color, Rg for the gray and Rb for the black.

You want to find a function mapping Rb to 0, Rg to 128 and Rw to 255. With 3 references, you can't use a linear function if you want a perfect match, you have to use a polynomial function with a degree at least of N-1, N being the number of references to match.

The grey reference (here 128) is related to an 18% grey card (more info: Why is 18% grey considered to be in the middle for photography?). It may be different for your grey.

The function ax² + bx + c = v (1) is of degree 2, exactly what you want for a perfect match ("fitting") with 3 values. Now, you want to find a, b and c such as (1) is verified for Rb, Rg and Rw. In other words, you have 3 equations:

• a (Rb)² + b (Rb) + c = 0
• a (Rg)² + b (Rg) + c = 128
• a (Rw)² + b (Rw) + c = 255

Solving this equation is easy (but not related to photography).

Once you have found a, b and c you can map any red value to a "calibrated" red. You may have negative values or values greater than 255, be careful.

An example maybe?

If Rb=10, Rg=100 and Rw=254, solving for (1) will yield a=-0.002449, b=1.6916 and c=-16.671 (need help ? Wolfram Alpha) so (1) <=> f(x)=-0.002449*x²+1.6916*x-16.671.

On the absciss (x-axis) lies your original values, on the ordinate (y-axis) the "calibrated" ones.

As you can see, all original values bellow 10 are mapped to negative values and should be set to 0... and values beyond 254 are mapped to values greater than 255. In fact, the [0,255] interval is mapped to [f(0),f(255)] = [-16.671,255.44].

If you don't want to clip your values (=losing information), you should use a new function to map [-16.671,255.44] to [0,255]. This function can be linear, such as ax+b.

You can replace (1) with any polynomial function of degree 2, you will get differents results.

If you really want to use a linear relation, say ax+b=v, you will need to find a and b minimizing errors between expected values (0, 128 and 255) and "calibrated" ones.

For more information on Polynomial Interpolation, take a look at https://math.stackexchange.com/questions/7125/polynomial-fitting-how-to-fit-and-what-is-polynomial-fitting or in Wikipedia (https://en.wikipedia.org/wiki/Polynomial_interpolation and https://en.wikipedia.org/wiki/Curve_fitting)