Table 1:

Summary of coil combination methods with spectroscopy and imaging applications.

	Spectroscopy Application	Imaging Application
SOS	${\mathit{I}}_{\mathit{s}\mathit{o}\mathit{s}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f}\right)=\sqrt{\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\left|\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f},{\mathit{c}}_{\mathit{k}}\right)\right|}^{\mathbf{2}}}$	${\mathit{I}}_{\mathit{s}\mathit{o}\mathit{s}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N}\right)=\sqrt{\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\left|\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N},{\mathit{c}}_{\mathit{k}}\right)\right|}^{\mathbf{2}}}$
SVD	U × Σ × VH = SVD(I(xi,yj,f,c)) σ1 = max(Σ) u1 = U·,1 ISVD(xi,yj,f) = σ1u1∅	N/A
RefPeak	$\mathit{S}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)=\frac{\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{f}}_{\mathit{R}\mathit{e}\mathit{f}},{\mathit{c}}_{\mathit{k}}\right)}{{\mathit{I}}_{\mathit{S}\mathit{O}\mathit{S}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{f}}_{\mathit{R}\mathit{e}\mathit{f}}\right)}$ ${\mathit{I}}_{\mathit{R}\mathit{e}\mathit{f}\mathit{P}\mathit{e}\mathit{a}\mathit{k}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f}\right)=\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\mathit{S}}^{*}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f},{\mathit{c}}_{\mathit{k}}\right)$	$\mathit{S}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)=\frac{\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{N}}_{\mathit{R}\mathit{e}\mathit{f}},{\mathit{c}}_{\mathit{k}}\right)}{{\mathit{I}}_{\mathit{R}\mathit{e}{\mathit{f}}_{-}\mathit{s}\mathit{o}\mathit{s}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{N}}_{\mathit{R}\mathit{e}\mathit{f}}\right)}$ ${\mathit{I}}_{\mathit{R}\mathit{e}\mathit{f}\mathit{P}\mathit{e}\mathit{a}\mathit{k}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N}\right)=\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\mathit{S}}^{*}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N},{\mathit{c}}_{\mathit{k}}\right)$
ESPIRiT	$\mathit{S}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)=\mathit{E}\mathit{c}\mathit{a}\mathit{l}\mathit{i}\mathit{b}\left(\mathit{k}\mathit{s}{\mathit{p}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{i}\mathit{b}}\right)$ ${\mathit{I}}_{\mathit{E}\mathit{S}\mathit{P}\mathit{I}\mathit{R}\mathit{i}\mathit{T}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f}\right)=\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\mathit{S}}^{*}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{f},{\mathit{c}}_{\mathit{k}}\right)$	$\mathit{S}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)=\mathit{E}\mathit{c}\mathit{a}\mathit{l}\mathit{i}\mathit{b}\left(\mathit{k}\mathit{s}{\mathit{p}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{i}\mathit{b}}\right)$ ${\mathit{I}}_{\mathit{E}\mathit{S}\mathit{P}\mathit{I}\mathit{R}\mathit{i}\mathit{T}}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N}\right)=\sum _{\mathit{k}}^{{\mathit{K}}_{\mathit{C}\mathit{o}\mathit{i}\mathit{l}}}{\mathit{S}}^{*}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}\right)\mathit{I}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}},\mathit{N},{\mathit{c}}_{\mathit{k}}\right)$

I: original data; x_i: i^thvoxel in one spatial dimension; y_j: j^th voxel in the other spatial dimension; f: spectral points; c_k and K: k^th channel with a total of K channels; N: metabolite in metabolite-specific imaging approach; ∅: arbitrary phase for each voxel; S: sensitivity map; *: conjugate; H: conjugate transpose; f_Ref: reference peak picked from spectroscopic datasets for coil combination; N_Ref: reference metabolite image used for imaging datasets coil combination; ksp_calib: part of k-space data used for ESPIRiT-derived sensitivity map calculation I_SOS, I_SVD, I_RefPeak, I_ESPIRiT: combined spectroscopy or imaging datasets with the SOS, SVD, RefPeak, and ESPIRiT methods.