Adaptive coil combination

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 ${\displaystyle \mathbf {m} }$ for each pixel. Each vector ${\displaystyle \mathbf {m} }$ maximizes the signal-to-noise ratio (SNR) of a region of interest (ROI) around the pixel. ${\displaystyle \mathbf {m} }$ is calculated using the following equations derived by David O. Walsh: ^{[1] [2]}

$${\displaystyle {\begin{aligned}&R_{N}=\sum _{(x,y)\in {ROI}}\mathbf {n} (x,y)\mathbf {n} ^{H}(x,y)\\&R_{S}=\sum _{(x,y)\in {ROI}}\mathbf {c} (x,y)\mathbf {c} ^{H}(x,y)\\&\mathbf {m} ={\textbf {v}}_{max}(R_{N}^{-1}R_{S})\end{aligned}}}$$

For a system with ${\displaystyle l}$ coils, ${\displaystyle \mathbf {n} \in \mathbb {C} ^{l}}$ 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, ${\displaystyle R_{N}}$ becomes identity. ${\displaystyle H}$ is the conjugate transpose. ${\displaystyle \mathbf {c} \in \mathbb {C} ^{l}}$ and is the measured value of signal + noise at location x,y. ${\displaystyle {\textbf {v}}_{max}}$ denotes the largest eigenvector of ${\displaystyle R_{N}^{-1}R_{S}}$. ${\displaystyle R_{S}}$ 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 ${\displaystyle {\textbf {m}}}$ can be concatenated into a coil sensitivity map and used for techniques like parallel imaging. ^{[3] [4]}

Derivation

The following derivation was first published by Walsh. ^[1] We wish to find a vector ${\displaystyle \mathbf {m} \in \mathbb {C} ^{l}}$ that maximizes SNR over an ROI with ${\displaystyle p}$ pixels and ${\displaystyle l}$ coils. If we put the measured signal in our ROI into a matrix ${\displaystyle S\in \mathbb {C} ^{l\times p}}$, and measured noise into a matrix ${\displaystyle N\in \mathbb {C} ^{l\times p}}$ we can write the SNR as:

$${\displaystyle {\begin{aligned}&Power_{signal}={\frac {|\mathbf {m} ^{H}S|^{2}}{l}}={\frac {\mathbf {m} ^{H}SS^{H}\mathbf {m} }{l}}\\&Power_{noise}={\frac {|\mathbf {m} ^{H}N|^{2}}{l}}={\frac {\mathbf {m} ^{H}NN^{H}\mathbf {m} }{l}}\\&{\frac {Power_{signal}}{Power_{noise}}}={\frac {\mathbf {m} ^{H}SS^{H}\mathbf {m} }{\mathbf {m} ^{H}NN^{H}\mathbf {m} }}={\frac {\mathbf {m} ^{H}R_{S}\mathbf {m} }{\mathbf {m} ^{H}R_{N}\mathbf {m} }}\\\end{aligned}}}$$

Because ${\displaystyle R_{S}}$ and ${\displaystyle R_{N}}$ are Hermitian, we can perform a simultaneous diagonalization with a new matrix ${\displaystyle P}$ by requiring:

$${\displaystyle {\begin{aligned}&P^{H}R_{N}P=I\\&P^{H}R_{S}P=D\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}&P^{H}R_{S}P=P^{H}R_{N}PD\\&R_{S}P=R_{N}PD\\&R_{N}^{-1}R_{S}P=PD\end{aligned}}}$$

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

${\displaystyle {\frac {\mathbf {q} ^{H}P^{H}R_{S}P\mathbf {q} }{\mathbf {q} ^{H}P^{H}R_{N}P\mathbf {q} }}={\frac {\mathbf {q} ^{H}D\mathbf {q} }{\mathbf {q} ^{H}\mathbf {q} }}}$

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

References

1. D. Walsh (2000). "Adaptive Reconstruction of Phased Array MR Imagery". Magn Reson Med. 43 (5): 682–690. doi:10.1002/(sici)1522-2594(200005)43:5<682::aid-mrm10>3.0.co;2-g. PMID   10800033.
2. M. Griswold; D. Walsh (2002). "The use of an adaptive reconstruction for array coil sensitivity mapping and intensity normalization" (PDF). Intl. Soc. Mag. Reson. Med.
3. A. Deshmane (2012). "Parallel MR imaging". J Magn Reson Imaging. 36 (1): 55–72. doi:10.1002/jmri.23639. PMC   4459721 . PMID   22696125.
4. M. Griswold (2006). "Autocalibrated coil sensitivity estimation for parallel imaging". NMR Biomed. 19 (3): 316–324. doi:10.1002/nbm.1048. PMID   16705632.