Suppose that we have a color image $C_1$ where every pixel has a color given by $(R,G,B)=)$ where $K$ is a random variable that has a uniform distribution between $0$ and $1.$ Thus each pixel in $C_1$ has a blue hue and an intensity that varies with $K.$ Assume that the intensity levels in the images vary continuously $($i$.$e$.$ are not constrained to be integers$).$

a$)$ Plot the gray level histogram for the blue band of the image.

b$)$ Suppose that we apply histogram equalization to each of the three bands of $C_1$ individually to generate the transformed bands $R^{\text{'}}(x,y),G^{\text{'}}(x,y),$ and $.$ Assume that the maximum gray level for each band is $255.$ The three transformed bands can be combined to form a tranformed color image $C_2$ where each pixel in $C_2$ is described by a color vector $(R,G,B).$ What color vectors $(R,G,B)$ will occur in $C_2?$ Explain your answer.

c$)$ Suppose that we have a color image $C_3$ for which the red, green, and blue band histograms are identical to those for $C_1.$ Will images $C_1$ and $C_3$ contain the same set of color vectors $(R,G,B)?$ Explain your answer.