Since 2000, the equipment of multiple receiver coils become the standard for MRI scanners, chiefly due to the invention of accelerated data acquisition techniques such as SENSE and GRAPPA.

In this post, I will summarize one method to combine images from different coils when there is no under-sampling. In this case, the goal is to achieve the best signal-noise-ratio (SNR) for the combined image.

The noise correlation function is defined as \begin{align} \Psi_{ij}(\vec r ) = \left< n_i^* n_j\right>, \end{align} where $$n_i(\vec r, t)$$ is the temporal noise for the i’th coil, and $$\left<\cdot\right>\equiv \int_0^T \cdot dt / T$$ is the temporal average. Note $$\Psi$$ is a Hermitian matrix.

We further assume that the complex data from different coils are linearly combined, i.e.,

\begin{align} s =& \sum_i w_i s_i = f(t) \sum_i w_i c_i\\ n =& \sum_i w_i n_i \end{align}

Here $$w_i$$ is the weight to be determined, $$s_i$$ is the noiseless signal from the $$i$$‘th coil, $$c_i$$ is the coil sensitivity function of the $$i$$‘th coil, and $$f(t)$$ is the true signal.

The square of SNR is given by the Rayleigh quotient \begin{align} \lambda \equiv \frac{\left<s^* s\right>}{\left<n^* n\right>} = \frac{w^\dagger F w}{w^\dagger \Psi w} \end{align} where the coil sensitivities $$c_i$$’s are assumed to be time invariant and their correlations are defined from \begin{align} F_{ij} = c_i^* c_j. \end{align}

Without loss of generality, we assume that $$\left<|f|^2\right>=1$$ and $$w^\dagger\Psi w\neq0$$. Note both $$\Psi$$ and $$F$$ are non-negative Hermitian matrices.

To maximize SNR, we take $${\partial \lambda}/{\partial w_i^*} = 0$$, which simplifies to \begin{align} Fw = \lambda \Psi w \end{align}

If $$\Psi$$ is invertible, the optimal weights $$w$$ form the eigenvector of $$\Psi^{-1}F$$, i.e., \begin{align} \Psi^{-1}F w = \lambda w. \end{align} As a result, the maximal SNR is simply the square root of the maximum eigenvalue of this secular equation, i.e., $$\sqrt{\lambda_{\max}}$$.

Note both $$\Psi$$ and $$F$$ have the structure of $$v^\dagger v$$. Thus they are both projection operators. As a result, $$\Psi^{-1}F$$ has only one degree of freedom, and there is only one nonzero eigenvalue, given by $$\lambda_{\max}=c^\dagger \Psi^{-1}c$$.

In practice, it is difficult to obtain $$\Psi$$ and $$c_i$$. For example, one can scan empty space for some time to deduce the noise term $$\Psi$$ and scan uniform objects to get the coil sensitivity term $$c_i$$. A simplifying assumption is that $$\Psi=I$$ and $$c_i=s_i$$. In other words, all coils are independent with each other and share the same amount of noise. In this case \begin{align} \lambda_{\max}=\sum_i|s_i|^2 \end{align} and $$w=s^*$$.