Redistributing Color in a RGB Image According to a Gaussian Distribution
After you equalize the histogram you can think of your data as a stream of variables $ {X}_{i} $ where $ X \sim U \left[ 0, 1 \right] $.  

Now all you need is to transform samples of Uniform Random Variable into Gaussian Variable.
You should do that by applying the Inverse CDF of Gaussian Distribution.
Basically applying the Inverse Transform Sampling method.
For Normal Distribution easy choice would be the Box Muller Transform.

After you apply this, you can just multiply each RGB pixel in the Cholesky Factorization of $ \Sigma $ to get exactly what you need.