Optimize 4 input values that min. abs. val. & var. of 3 output values against their expected output values

[isotope493]

Overview:

I have a color image scanner where the only settings that can be adjusted are: gamma ([MATH]g[/MATH]); saturation ([MATH]s[/MATH]); brightness [MATH](b[/MATH]); contrast ([MATH]c[/MATH]). I have input samples of known color properties; namely, red ([MATH]t[/MATH]); green ([MATH]u[/MATH]); and blue ([MATH]v[/MATH]) color components. Further, after the samples are scanned, I can measure how the scanner interpreted the red ([MATH]x[/MATH]), green ([MATH]y[/MATH]), and blue ([MATH]z[/MATH]) color components of each sample. In short, I know the difference between the expected color component and actual color component.

Problem:

I need to setup an equation or series of equations to determine the optimal [MATH]g,s,b,c[/MATH] values such that I minimize the absolute value of [MATH]|x-t|[/MATH], [MATH]|y-u|[/MATH], and [MATH]|z-v|[/MATH] both individually and as a group. I need to determine the best mathematical approach and how to setup a solution. Further, there is unlikely a combination of [MATH]g,s,b,c[/MATH] that will perfectly result in [MATH]|x-t|[/MATH], [MATH]|y-u|[/MATH], and [MATH]|z-v|[/MATH] all being zero; so, some sort of fuzzy or approximate solution is needed to get as close as possible to ideal values of [MATH]g,s,b,c[/MATH].

My knowledge:

I have studied calculus, linear algebra, and basic optimization modeling (i.e. Excel's solver function). However, I have not done serious math in at least 5 years.

Possible approaches I have contemplated:

1. My instinct is that this problem calls for a series of linear equations to be solved but I am not sure if this is the correct approach or workable. If this is the best, I am unsure how I would approach setting up this problem.
2. I could probably setup a brute force solution using Excel's solver function to guess [MATH]g,s,b,c[/MATH] values within predefined ranges that result in (Excel formula defined) outputs of [MATH]|x-t|[/MATH], [MATH]|y-u|[/MATH], and [MATH]|z-v|[/MATH] that have been measured through known samples and seek to minimize the value of some calculated number incorporating the sum of absolute values of [MATH]|x-t|[/MATH], [MATH]|y-u|[/MATH], and [MATH]|z-v|[/MATH] and the difference between each group's absolute value of the difference between measured and anticipated color component and the standard deviation of all three component groups.

In closing:

Thank you for you help in determining and setting up a solution to help me solve this problem.