Numerous prospective and retrospective intensity inhomogeneity correction algorithm based on image processing operation have been proposed in the last decades^1,2. In the era of multi-coil acquisition, the electromagnetic reverse method³ has been suggested to calculate sensitivity map. Correlation matrix analysis to extract the intensity bias distributions was also reported⁴. This work implements image intensity correction in ESPIRiT reconstruction, as well as perform intensity correction to the normal images from the normal sum-of-square (SOS) combinations, using the properly spatially variant scaled coil sensitivity maps modified from the maps in the basic ESPIRiT algorithm.

ESPIRiT⁵ is an auto-calibrating parallel imaging algorithm capable of generating sensitivity maps from the under-sampled imaging data, which is used to unfold the aliasing artifacts with SENSE^6,7 reconstruction. However, the original ESPIRiT reconstruction does not perform image intensity corrections to compensate the coil sensitivity distribution induced spatial intensity bias (Figure 1), because although the ESPIRiT derived coil sensitivity maps (eigenvectors) are good for SENSE, multiplications with the arbitrary spatially variant complex scalers made the same maps not useful to the intensity correction purpose.

SENSE provides a method to solve the image $$$m$$$ with the under-sampled k-space data $$$y_{i}$$$ and the sensitivity map $$$S_{i}$$$ for each of the coils in

$$\hspace{6.77cm}y_{i}=PFS_{i}m,\hspace{6.77cm}(1)$$

where $$$P$$$ is the under sampling operator, $$$F$$$ is the Fourier transform operator, and $$$i$$$ is the coil index. If we divide the sensitivity map by a pixel dependent but coil independent value $$$\lambda$$$, the solution will be original $$$m$$$ multiplied by a corresponding value $$$\lambda$$$. If this $$$\lambda$$$ is not properly formulated, the coil sensitivity bias in the images is not corrected, although the folding artifacts in the under-sampled images are resolved, because the relative complex relations (ratios) between multi-receiver coils is retained for all the imaging pixels.

In ESPIRiT, the data space $$$V$$$ of the auto calibration signal (ACS) region is split into two spaces, the signal space $$$V_{\parallel}$$$ and the null space $$$V_{\perp}$$$, through singular value decomposition (SVD)

$$\hspace{6.9cm}A=U{\Sigma}V^{H},\hspace{6.9cm}(2)$$

where $$$A$$$ is the calibration matrix. The signal space projection operator $$$W$$$ is then constructed as

$$\hspace{5.55cm}W=M^{-1}{\Sigma_r}R_{r}^{H}V_{\parallel}V_{\parallel}^{H}R_{r} , \hspace{5.55cm}(3)$$

where operators $$$R_{r}$$$ represents the operation of choosing a block of k-space out of the entire grid around the k-space positions indexed by $$$r$$$, and $$$M$$$ represents $$$ {\Sigma_r}R_{r}^{H}R_{r}$$$. Next, the operator $$$W$$$ is transformed to the image space to solve the pixel independent eigenvalue decomposition, in order to obtain the coil sensitivity map $$$\overrightarrow{s_{q}}$$$,

$$\hspace{6.4cm}F^{-1}WF{\mid}_q=G_{q}, \hspace{6.4cm}(4) $$

$$\hspace{7cm}G_{q}\overrightarrow{s_{q}}=\overrightarrow{s_{q}}. \hspace{7cm}(5)$$

To generate ESPIRiT sensitivity maps, the singular values $$$\Sigma$$$ are discarded (thus the signal space is “whitened”). Hence, the pixel-to-pixel relative intensity variations in the sensitivity maps are discarded, despite the retaining of the relative complex coil intensity relationships. We propose to keep the singular values to define a vector space

$$\hspace{6.95cm}\bar{V}_{\parallel}=\sqrt{\Sigma}{V}_{\parallel}\hspace{6.95cm}(6)$$

to be used to construct operator $$$\bar{W}$$$, and then the eigenvalues of corresponding image space operator $$$\bar{G}_{q}$$$ will not be all theoretically normalized.

$$\hspace{6.85cm}\bar{G}_{q}\overrightarrow{t_{q}}=\lambda_{q}\overrightarrow{{t}_{q}}. \hspace{6.85cm}(7)$$

Finally, we define

$$\hspace{7.0cm}\overrightarrow{\bar{s_{q}}} = \sqrt{\lambda} \overrightarrow{s_{q}} \hspace{7.0cm}(8)$$

as the spatially variant rescaled modified sensitivity maps in the SENSE reconstruction, to solve the arbitrary scaling problem in (1) and enable the image intensity bias corrected ESPIRiT reconstruction. This algorithm can be also used to generate intensity bias correction maps from values $$$\sqrt{\lambda}$$$ of the each pixel. The coil sensitivity variations caused intensity bias of the SOS combined multi-coil images are corrected by dividing the images by the map.

Under-sampled imaging data with volunteers were acquired on an AllTtech Centauri 1.5T system. Fig. 1 is a head imaging example with an 8-channel head coil and a parallel acceleration factor R=1.8. Comparing the results from with the original ESPIRiT (left images), the coil intensity bias is corrected from the proposed modifications (middle images). Fig. 2 shows examples of C-Spine images with a 7-channel head-spine coil without parallel acceleration. Intensity thresholding mask was applied to only perform the intensity bias corrections over the object region to avoid unnecessary noise and residue artifacts amplifications. Feathering out nearing the boundary generates more natural looking results with the gradual fading out of the corrections.