Since the camera's gamma correction is based on brightness and not on color, we have used the six greyscale swatches of the Macbeth ColorChecker to find the gamma function. We use the Y values given by the spectroradiometer and the RGB values given by the camera for each swatch.

            We found the gamma function for each of the red, green, and blue values individually. The first step was to establish normalized data for each of the gray swatches. This is the procedure we followed:
 

• Extract an R-value vector, G-value vector, and B-value vector from the series of six greyscale RGB values from the camera.

• Extract a Y-value vector from the series of six greyscale XYZ values from the spectroradiometer.

• Normalize each of these four vectors by dividing by the whitepoint. For each vector, the whitepoint value is the first value of that vector. This value is associated with the white swatch on the Macbeth ColorChecker.

            With the normalized vectors, we were able to find the gamma and inverse gamma functions. The gamma function is evident when we plot the normalized greyscale RGB nonlinear values as a function of the normalized greyscale Y values.

            Since the Y values are assumed to vary linearly with the linear RGB values, this plot gives the relationship between the nonlinear RGB values and the linear RGB values. We fit the curve using a nonlinear curve-fitting function (described in the appendix) and arrived at the following gamma functions:

Rnon = 1.045*Y^.5312
Gnon = 1.0410*Y^.5406
Bnon = 1.0413*Y^.5437

            To find the inverse gamma function, we plotted the Y values as a function of the nonlinear RGB values and performed the same curve-fitting routine. These are the functions we found for converting nonlinear RGB values to linear RGB values:

Rlin = .9521*Rnon^2.150
Glin = .9583*Gnon^2.0750
Blin = .9554*Bnon^2.05

            Note that these functions are designed for normalized values. We have compiled these inverse gamma functions into a MATLAB function, tolinear.m that will convert nonlinear RGB values to linear RGB values for this project.