In this study, we propose to incorporate the self-consistency property to reconstruct InI acquisitions to obtain self-consistent and 1-self-consistent K-InI reconstructions.
Our simulations show that the reconstructed images using self-consistent and
1-self-consistent K-InI methods showing a larger RMSE than K-InI (Figure 4). We specifically observed that the RMSE of 1-self-consistent K-InI diverged after 40 iterative reconstructions, implying that the 1 -norm constraint may not stable.
Particularly, we found strong streak artifacts (Figure 6). We found that using K-InI as the initial value for self-consistent and the 1-self-consistent K-InI reconstruction can get a smaller RMSE than those using a zero vector as the initial value.
To explain the reason of reconstructing images with diverged RMSE with strong streak artifacts over many iterations, it is useful to specifically contrast between self-consistent K-InI and K-InI. Self-consistent K-InI as well as 1-self-consistent K-InI are different from K-InI in both reconstruction kernel g estimation and skipped data interpolation. K-InI has to create different reconstruction kernels, each of which accounts for a specific sampling pattern relating ACC data and the skipped k-space data point to be interpolated. This is usually an ill-posed inverse problem. After estimating the K-InI reconstruction kernels using ACS data, interpolating skipped data points from ACC is straightforward and well-defined. Differently, there is only one reconstruction kernel g for both self-consistent K-InI and the 1 -self-consistent K-InI. Thus estimating g from ACS data is more constrained than K-InI. Given the estimated g , we then have to use calibration consistency constraint to reconstruct the image (Eq. [13]
and Eq. [16]). The process can be very ill-posed, because InI only collects the central
partition of the 3D k-space. The ill-poseness of this calibration consistency is one reason to account for the divergence of the reconstruction as iteration continues.
The estimation of the reconstruction kernel g in this study (Eq.[15]) is an ill-posed inverse problem because of the number of channels in the coil array, the chosen size of the reconstruction kernel, and the size of the ACS data. Thus Tikhonov regularization is needed for reconstruction kernel estimation (Bydder and Jung, 2009;
Qu et al., 2006). Here we used a prior constraint to minimize the power of the reconstruction kernel to obtain the unique estimate of g (Eq. [15]). However, it is possible to transform this kernel estimation into a well-posed inverse problem. This can be done by 1) increase the size of the ACS data, 2) decrease the size of the reconstruction kernel, and 3) using coil compression technique to reduce the number of the coils virtually (Buehrer et al., 2007; Doneva and Bornert, 2008; King et al., 2010;
Zhang et al., 2012). However, these options respectively have the potential issue of 1) using noisy data at the periphery of the k-space and thus reduce the stability of the reconstruction kernel, 2) reducing the degree of freedom of employing more neighboring k-space data to better interpolate the skipped k-space data points, and 3) losing part of the information from all channels of the coil array for data interpolation.
Thus the optimal configuration of the reconstruction kernel needs to be further studied.
Our 1-self-consistent K-InI results corroborate a recent study using the 1-norm minimization to reconstruct highly accelerated fMRI data (Hugger et al., 2011). While our method is based on k-space data interpolation, the work by Hugger et al is an image-domain reconstruction method. However, both methods show that using the
1-norm minimization can improve the spatial resolution. Moreover, Hugger et al found that replacing the wavelet transform by an identity matrix in Eq. [16] generated similar
results. We expect that we may get similar results and slightly improve the computational time by removing the wavelet transform, since the wavelet transform is a very fast process.
Since self-consistent K-InI and 1-self-consistent K-InI methods are iterative reconstruction methods, the computational load is higher than the analytical K-InI reconstruction method. The computation time of the self-consistent K-InI and the
1-self-consistent K-InI for reconstructing a 2D slice using 32-channel coil array data is about 1 second for one iteration. It took about 10 to 20 minutes to reconstruct a 3D volumetric data with 20 iterations. In contrary, K-InI took 30 to 40 s to reconstruct a 3D data set. This computational efficiency is expected to be improved by, for example, the coil compression technique (Buehrer et al., 2007; Doneva and Bornert, 2008; King et al., 2010; Zhang et al., 2012), which can reduce the number of channels in an RF coil array into fewer virtual channels. The self-consistent GRAPPA (SC-GRAPPA) (Ding et al., 2012) has been proposed to combine traditional GRAPPA reconstruction and the self-consistency with a closed form solution. This may be used in self-consistent K-InI to reduce the computational time. Alternatively, using parallel computational architecture, such as GPGPU (Murphy et al., 2012), may further reduce the reconstruction time.
In conclusion, we demonstrate two iterative reconstruction methods using self-consistent property to reconstruct highly accelerated InI data in k-space. The results show improved spatial resolution and higher sensitivity in detecting brain areas in BOLD fMRI experiment. These reconstruction methods may help researchers to better understand the temporal details of hemodynamic response functions.
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