Physics:Adaptive coil combination

Short description: MRI technique

Adaptive coil combination is a method used in Magnetic Resonance Imaging (MRI) to merge signals from multiple receiver coil elements into a single image. A weighted sum of the individual coil images is performed with a different weighting vector $\mathbf{𝐦}$ for each pixel. Each vector $\mathbf{𝐦}$ maximizes the signal-to-noise ratio (SNR) of a region of interest (ROI) around the pixel. $\mathbf{𝐦}$ is calculated using the following equations derived by David O. Walsh:

$$\begin{array}{rl}& R_N=\sum _{(x,y)\in ROI}\mathbf{𝐧}(x,y){\mathbf{𝐧}}^H(x,y)\\ & R_S=\sum _{(x,y)\in ROI}\mathbf{𝐜}(x,y){\mathbf{𝐜}}^H(x,y)\\ & \mathbf{𝐦}={\text{v}}_{max}(R_N^{-1}{R}_S)\end{array}$$

For a system with $l$ coils, $\mathbf{𝐧}\in {\mathbb{ℂ}}^l$ and is a column vector of the noise of each coil at location x,y. This can be obtained by capturing images without a subject, or if noise is assumed to be uncorrelated white, $R_N$ becomes identity. $H$ is the conjugate transpose. $\mathbf{𝐜}\in {\mathbb{ℂ}}^l$ and is the measured value of signal + noise at location x,y. ${\text{v}}_{max}$ denotes the largest eigenvector of $R_N^{-1}{R}_S$. $R_S$ is an estimate of the signal correlation matrix, which works in practice because signal is fairly constant over a small ROI, but thermal noise is white in the image domain so spatial averaging reduces noise-induced bias. The vectors $\text{m}$ can be concatenated into a coil sensitivity map and used for techniques like parallel imaging.

Derivation

The following derivation was first published by Walsh. We wish to find a vector $\mathbf{𝐦}\in {\mathbb{ℂ}}^l$ that maximizes SNR over an ROI with $p$ pixels and $l$ coils. If we put the measured signal in our ROI into a matrix $S\in {\mathbb{ℂ}}^{l\times p}$, and measured noise into a matrix $N\in {\mathbb{ℂ}}^{l\times p}$ we can write the SNR as:

$$\begin{array}{rl}& Powe{r}_{signal}=\frac{|{\mathbf{𝐦}}^{H}S{|}^2}{l}=\frac{{\mathbf{𝐦}}^{H}S{S}^H\mathbf{𝐦}}{l}\\ & Powe{r}_{noise}=\frac{|{\mathbf{𝐦}}^{H}N{|}^2}{l}=\frac{{\mathbf{𝐦}}^{H}N{N}^H\mathbf{𝐦}}{l}\\ & \frac{Powe{r}_{signal}}{Powe{r}_{noise}}=\frac{{\mathbf{𝐦}}^{H}S{S}^H\mathbf{𝐦}}{{\mathbf{𝐦}}^{H}N{N}^H\mathbf{𝐦}}=\frac{{\mathbf{𝐦}}^H{R}_S\mathbf{𝐦}}{{\mathbf{𝐦}}^H{R}_N\mathbf{𝐦}}\\ \end{array}$$

Because $R_S$ and $R_N$ are Hermitian, we can perform a simultaneous diagonalization with a new matrix $P$ by requiring:

$$\begin{array}{rl}& P^H{R}_{N}P=I\\ & P^H{R}_{S}P=D\end{array}$$

where $I$ is identity and $D$ is diagonal. By multiplying the two equations we get:

$$\begin{array}{rl}& P^H{R}_{S}P=P^H{R}_{N}PD\\ & R_{S}P=R_{N}PD\\ & R_N^{-1}{R}_{S}P=PD\end{array}$$

It can be seen that $P$ and $D$ are the eigenvector and eigenvalue matrices respectively of $R_N^{-1}{R}_S$. Performing a change of basis with $\mathbf{𝐪}=P^{-1}\mathbf{𝐦}$ results in:

$\frac{Powe{r}_{signal}}{Powe{r}_{noise}}=\frac{{\mathbf{𝐪}}^H{P}^H{R}_{S}P\mathbf{𝐪}}{{\mathbf{𝐪}}^H{P}^H{R}_{N}P\mathbf{𝐪}}=\frac{{\mathbf{𝐪}}^{H}D\mathbf{𝐪}}{{\mathbf{𝐪}}^H\mathbf{𝐪}}$

This is the Rayleigh quotient and so the maximum value of $\mathbf{𝐪}$ corresponds to the maximum eigenvector of D, which is ${\text{q}}_{max}=[1,0,\cdots ,0]$ when D is sorted by descending order. Therefore ${\text{m}}_{max}=P{\text{q}}_{max}={\text{v}}_{max}(R_N^{-1}{R}_S)$.