A Unitary Transform of an i.i.d. Normal Random Variable

Consider an independent identically distributed Normal random variable $\mathbf{x}$ with mean $\mu \in \mathbb{C}^C$ and covariance $\Sigma \in \mathbb{C}^{C \times C}$ . Let $\mathbf{x}$ have $N$ vector-valued pixels, each with $C$ channels, i.e. $\mathbf{x} \in \mathbb{R}^{NC}$ and

\mathbf{x}[i] \sim \mathcal{N}(\mu, ~ \Sigma) \quad \forall \quad 1 \leq i \leq N.

Further, let $\mathbf{x}$ have the channel-wise vectorization $\mathbf{x} = [\mathbf{x}_1^T, \, \mathbf{x}_2^T, \, \dots, \, \mathbf{x}_C^T]^T$ . We can then write this via Kronecker product notation as

\mathbf{x} \sim \mathcal{N}(\mu \otimes \mathbf{1}_N, ~ \Sigma \otimes I_N).

Now, consider a linear transformation $\mathbf{y} = A\mathbf{x}$ . We know from linearity of expectation that $\mathbb{E}[A\mathbf{x}] = A\mathbb{E}[\mathbf{x}] = A(\mu \otimes \mathbf{1}_N)$ and that $\mathrm{Cov}(A\mathbf{x}) = A\mathrm{Cov}(\mathbf{x})A^H$ .

Further, consider a channel-wise unitary matrix, i.e. $A = (I_C \otimes Q)$ where $Q \in \mathbb{C}^{N\times N}$ is unitary. The Kronecker notation here means that $A$ is a block diagonal matrix with $C$ blocks of $Q$ on its diagonal. So, the mean and covariance of our transformed variable are

\mathbb{E}[A\mathbf{x}] = (I_C \otimes Q)(\mu \otimes \mathbf{1}_N) \qquad \mathrm{Cov}(A\mathbf{x}) = (I_C \otimes Q)(\Sigma \otimes I_N)(I_C \otimes Q)^H

The mixed-product property of Kronecker product tells us that, for sensible matrix sizes, $(A \otimes B)(C \otimes D) = (AC \otimes BD)$ . Thus,

\mathbb{E}[A\mathbf{x}] = (\mu \otimes Q\mathbf{1}_N) \qquad \mathrm{Cov}(A\mathbf{x}) = (\Sigma \otimes I_N),

as $QQ^H = I_N$ . For zero-mean Normally distributed noise, this results in the channel-wise unitary transform of said noise having the exact same distribution as before the transformation.

Parallel MRI

This is a particularly useful result in MRI, where the measurement domain is a $C$ -channel (coil) Fourier domain contaminated by zero-mean additive white Gaussian noise (AWGN), i.e. we model measurements ( $\mathbf{y} \in \mathbb{C}^{NC}$ ) of image ( $\mathbf{x} \in \mathbb{C}^N$ ) with coil sensitivity operator $S \in \mathbb{C}^{NC \times N}$ via

\mathbf{y} = (I_C \otimes F_N)S\mathbf{x} + v, \quad v \sim \mathcal{N}(0, ~ (\Sigma \otimes I_N)).

where $F_N$ is the N-dimensional DFT matrix. Equivalently we can write $\mathbf{y} \sim \mathcal{N}((I_C \otimes F_N)S\mathbf{x}, ~ (\Sigma \otimes I_N))$ . As the DFT matrix is unitary, we know from the above discussion that the (multi-coil) image domain signal is contaminated with noise from the same distribution,

(I_C \otimes F_N^H)\mathbf{y} \sim \mathcal{N}(S\mathbf{x}, ~ (\Sigma \otimes I_N)).