The idea behind whitening is that it produces a generic form of regularized estimator. Before whitening, the noise covariance statistics are explicitly included in the kernel as C. After whitening, the kernel form assumes independent and identically distributed noise statistics (the I matrix in the kernel 3rd equation above). Our paper assumed, for simplicity, this iid hypothesis (although I think I remember we mentioned that noise covariance statistics could be accounted for in principle in forming the regularized imaging kernel).
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