Accelerated motion corrected three‐dimensional abdominal MRI using total variation regularized SENSE reconstruction

Purpose Develop a nonrigid motion corrected reconstruction for highly accelerated free‐breathing three‐dimensional (3D) abdominal images without external sensors or additional scans. Methods The proposed method accelerates the acquisition by undersampling and performs motion correction directly in the reconstruction using a general matrix description of the acquisition. Data are acquired using a self‐gated 3D golden radial phase encoding trajectory, enabling a two stage reconstruction to estimate and then correct motion of the same data. In the first stage total variation regularized iterative SENSE is used to reconstruct highly undersampled respiratory resolved images. A nonrigid registration of these images is performed to estimate the complex motion in the abdomen. In the second stage, the estimated motion fields are incorporated in a general matrix reconstruction, which uses total variation regularization and incorporates k‐space data from multiple respiratory positions. The proposed approach was tested on nine healthy volunteers and compared against a standard gated reconstruction using measures of liver sharpness, gradient entropy, visual assessment of image sharpness and overall image quality by two experts. Results The proposed method achieves similar quality to the gated reconstruction with nonsignificant differences for liver sharpness (1.18 and 1.00, respectively), gradient entropy (1.00 and 1.00), visual score of image sharpness (2.22 and 2.44), and visual rank of image quality (3.33 and 3.39). An average reduction of the acquisition time from 102 s to 39 s could be achieved with the proposed method. Conclusion In vivo results demonstrate the feasibility of the proposed method showing similar image quality to the standard gated reconstruction while using data corresponding to a significantly reduced acquisition time. Magn Reson Med, 2015. © 2015 The Authors. Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Magn Reson Med 75:1484–1498, 2016. © 2015 The Authors. Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance.


INTRODUCTION
Respiratory motion is a major source of artifacts in abdominal imaging, causing ghosting and blurring in the reconstructed image (1). In two dimensions, a simple approach to reduce these artifacts is to acquire the image during breathhold(s). Unfortunately, breathholding is difficult in three dimensions without compromising resolution/scan time; therefore, a three-dimensional (3D) freebreathing acquisition is desired. Free-breathing acquisition is usually performed using respiratory gating techniques (2). Respiratory gating monitors the position of the diaphragm in the superior-inferior (SI) direction using external sensors, navigator echoes or self-gating. External sensors require additional preparation and provide only a relative measurement of diaphragm displacement (3). Navigator echoes do not have these limitations, but can interfere with image acquisition (e.g., disrupting the balanced steady state free precession acquisition) (4). Selfgating techniques (5) extract motion information from the acquired data itself, in general using the central kspace profile. Respiratory gating only accepts data within a small gating window, minimizing motion artifacts at the expense of additional scan time. Moreover, in subjects with highly irregular breathing patterns, drift in respiratory motion can lead to scan termination due to low scan efficiency. Respiratory gating monitors SI translational motion, although it is known that abdominal acquisitions are corrupted by large nonrigid components of motion (6)(7)(8).
In cardiac imaging, several approaches have been proposed for motion compensation. In (9), SI motion is estimated by means of self-navigation, enabling the estimation of data weighting factors from the distance to a reference respiratory phase. This information is then used in a weighted iterative reconstruction with total variation regularization to reduce respiratory artifacts. More complex 3D affine motion has been estimated from 3D low resolution image navigators and used to correct the actual acquisition in image space (10). An alternative approach (11) estimates 3D affine motion from respiratory resolved images and uses it to correct the acquired k-space before reconstructing the image. In Prieto et al (12), motion is estimated similarly, but is then corrected directly in the Cartesian reconstruction using the General Matrix Description (GMD) (13). Additional solutions have been proposed to correct the more complex nonrigid abdominal motion. A pixel-by-pixel translation correction has been used to correct nonrigid motion (14) using local autofocus (15). This technique requires the acquisition of additional navigator echoes as well as triggering and gating of a portion of the acquired data. A self-gated motion corrected approach is proposed in Buerger et al (16), where nonrigid motion is estimated from undersampled respiratory resolved images. A motion compensated image is then obtained by warping all undersampled images to a common respiratory position. However, this image-based approach has two major limitations: (i) aliasing artifacts of each undersampled motion state are warped to the common respiratory position and may persist in the final image, and (ii) the averaging of multiple motion states may lead to blurring in the final image. The latter effect has been shown for abdominal and cardiac images in (17,18) when compared with the GMD approach. GMD corrects motion directly during the reconstruction process; although a previous estimation of the motion is required. This motion has been estimated from lowresolution training acquisitions (19) or from a computationally expensive coupled motion reconstruction and motion estimation problem based on external sensors (20). These approaches have been demonstrated for fully and over-sampled acquisitions only. Here, we propose a highly accelerated GMD-based method to correct nonrigid motion in undersampled 3D isotropic abdominal images.
The motion estimation step of the proposed approach is an extension of the one used in (16) to account for undersampled acquisitions. Data are acquired under free breathing with a self-gated goldenradial phase encoding (G-RPE) trajectory (21,22), enabling the reconstruction of highly undersampled images at various motion states (respiratory bins). Unlike Buerger et al (16), where iterative SENSE reconstruction (23) was sufficient to independently reconstruct the bins, here we propose to reconstruct all motion states simultaneously using a spatial and temporal total variation regularized iterative SENSE (TV-SENSE) approach, allowing reliable motion estimation from bins with higher undersampling factors than those reported in (16). Additionally, in the proposed method motion compensation is performed directly in the reconstruction, incorporating parallel imaging information and spatial total variation regularization (TV-GMD). The proposed approach was tested on nine healthy volunteers and compared against a standard gated reconstruction.

METHODS
The proposed framework can be divided into five steps (Figs. 1b,c). In the first step, data are acquired with a self-gated 3D golden-radial phase encoding (G-RPE) trajectory during free breathing. These data are sequentially assigned into different motion states, yielding a set of highly undersampled k-spaces K b at the completion of the acquisition (step 2). Each bin is reconstructed with TV-SENSE (step 3), producing undersampled respiratory resolved images I b . In the next step, nonrigid motion is estimated using image registration of I b . Finally, the estimated motion is used to reconstruct a motion corrected image using TV-GMD.
Image Acquisition G-RPE (21,22) combines a regularly sampled Cartesian readout (k x ) in SI direction with a radial phase encoding order in the k y -k z plane, where radial profiles are separated by the golden angle h G ¼ 111.25 (Figure 1a, top). A G-RPE acquisition can be undersampled radially by skipping readouts within each radial profile and angularly by acquiring fewer radial profiles (Figure 1a, bottom). This trajectory provides a quasi-uniform k-space distribution regardless of the number of profiles used (24), ensuring an optimal distribution of profiles in k-space for different length of data acquisition. Additionally, the central k-space readout (red line in Figure 1b) is acquired every N Â TR seconds where is N the number of readouts within a radial profile and TR is the repetition time, allowing for respiratory self-navigation. If the acquisition time for the radial profile (N Â TR) is fast enough, respiratory motion within the radial profile can be considered negligible and the central k-space readout can be used to navigate the entire radial profile.

Data Binning
We identify the respiratory position of each radial profile by inferring the diaphragm displacement in the (SI) readout direction as described in Buerger et al (25). This selfnavigated respiratory signal is used to group the acquired data into different respiratory positions. During this process, referred as data binning, the radial profiles in each bin will not be equally spaced by the golden angle as they depend on the breathing cycle (16). Therefore a quasi-uniform k-space distribution is not guaranteed for any bin. k-Space gaps may generate artifacts in I b and affect subsequent motion estimation. Thus, it is important to ensure that each bin has adequate coverage of kspace for reliable motion estimation. The artifact level in I b is predicted using the maximum angle between two radial profiles (a), similar to (25). To guarantee that the image quality of I b is sufficient for reliable motion estimation we perform an adaptive binning that forces each bin to have a b (maximum angle between two radial profiles in bin b) smaller than a predetermined a max (determined empirically). To minimize intra-bin motion, only bins with a bin window (w b ) smaller than a maximum bin window (w max ) are accepted.
Here, we perform an adaptive binning (Figs. 1d,e) with constraints to ensure reliable motion estimation. This process is controlled by 4 parameters: maximum overall undersampling factor (R max ), the maximum angle (a max ), the maximum bin window size (w max ) and the minimum gating efficiency (GE min ). Gating efficiency (GE) is given by:

GE ¼
Accepted Profiles Total Profiles Acquired [1] Respiratory bins are chosen such that within each bin, a b < a max (to limit reconstruction artifacts), and w b < w max (to limit intra-bin motion). To limit undersampling artifacts in the final reconstruction of the complete dataset, a minimum amount of data to respect R max is enforced. A potential bin is first generated at the end-exhale position with a bin window size equal to the image resolution. The bin window is increased until it has enough data to comply with the a max constraint (Fig.  1d). If the bin window size exceeds w max the bin is discarded to minimize intra bin motion, otherwise it is accepted. A sliding window is used to initialize a potential bin in the next respiratory position and the process in Figure 1d is repeated.
Once the considered data thus far have been binned, the algorithm tests if the selected dataset satisfies the minimum gating efficiency. If the current gating efficiency is less than GE min (Fig. 1e), then an additional radial profile is considered and the binning process is Cartesian readouts (k x direction) are acquired with a radial order in the phase encoding plane (k y -k z ). Consecutive radial profiles (numbered 1, 2, and 3) are separated by the golden angle h G ¼ 111.25 . a: Bottom: Diagram of GRPE undersampling in the phase encoding plane. Angular undersampling is achieved by skipping complete radial profiles (full gray radial profiles), while radial undersampling is performed along each radial profile by uniformly skipping readouts (gray readouts along radial profiles 1, 2, and 3). b,c: Diagram of the proposed approach in five steps. The acquisition is comprised of two parts: step 1. Image is acquired with 3D G-RPE; step 2. The central k-space line yields a respiratory signal which is used to bin data. The reconstruction is divided into 3 parts: step 3. Binned datasets are simultaneously reconstructed with TV-SENSE; step 4. Motion is estimated by image registration (LREG) of the binned reconstructions; step 5. The estimated motion is used in the motion compensated reconstruction. Only the acquired data are needed to produce a motion compensated image. Adaptive data binning is performed in two steps: d: Each bin is initialized with a window size w b . If a b is larger than a max , w b is increased, providing additional data for this bin. This process is repeated until a max is respected. If the final w b is larger than w max , the bin is discarded; otherwise it is accepted. e: Insufficient data cause a significant fraction of bins to be discarded, leading to a poor gating efficiency (GE). If GE is smaller than GE min , an additional radial profile is considered and the full dataset (including the newly considered radial profile) is rebinned. This process terminates when the GE min , a max and w max constraints are simultaneously respected.
restarted (back to Figure 1d). Data binning terminates once enough motion states are adequately populated (a b < a max , w b < w max ) and the complete dataset satisfies the condition GE > GE min .

Reconstruction of Respiratory Bins
The undersampled bins are simultaneously reconstructed with total variation regularized iterative SENSE (TV-SENSE) (26,27) given by: where K b is the acquired data at each bin b, E is the encoding matrix including coil sensitivities, gridding and Fourier transformation, I b the binned images to be reconstructed, TV s represents the 3D spatial total variation (intra-bin) and TV t the temporal total variation in the respiratory direction (inter-bin). The l 2 -norm is the data consistency term. TV s is the sum of absolute differences in each bin; TV t is the sum of absolute differences between adjacent bins. Both of these transforms are l 1norms operating in image space, defined as: where r s and r t represent the 3D spatial gradient and the 1D temporal gradient, respectively. The parameters k s and k t define the balance between total variation regularization and data consistency. The TV s term removes noise-like aliasing in the image and sharpens the main edges in the image, e.g., the diaphragm-lung boundary, allowing better image registration. Large k s values enhance these effects at the expense of additional blurring (due to over-smoothing). While blurring limits the accuracy of subsequent image registration, aliasing introduces errors in I b that can make the registration fail altogether. Hence, we use a large value for k s to reconstruct artifact-free respiratory bins and allow reliable motion estimation. The TV t removes additional aliasing in areas where there is little motion, like the central part of the liver. Large k t values will introduce blurring in regions with large motion, e.g., diaphragmlung boundary. Thus, we use small k t values to distinguish motion states. Here, the TV t term amounts to a small regularization compared with TV s , yielding minor improvements in the reconstructed image. The values for k s and k t were determined empirically by inspecting the output reconstructions of four training datasets covering the expected range of undersampling factors and respiratory amplitudes and choosing values that removed the most background aliasing without significantly blurring the image.
To accelerate bin reconstruction, a data consistency reconstruction (i.e. k s and k t set to zero) is used as the starting estimate for TV-SENSE. This approach accelerates convergence and delivers a good approximation of the reconstructed image. These reconstructions are preconditioned (23) by intensity correction (from coil sensitivities) and a Voronoi based density compensation function (28), as our data acquisition (G-RPE) is non-Cartesian.

Motion Estimation
A 3D nonrigid respiratory motion model is obtained from image registration of reconstructed bin images to a common respiratory position. This is done using the LREG tool (29), which performs an intensity based registration through a succession of local affine registrations, starting from a global affine down to small image blocks. The resulting deformation field is thus nonrigid.

Image Reconstruction
The final undersampled image is reconstructed using a spatial total variation regularized GMD approach, based on the formalism introduced in Batchelor et al (13). The GMD approach is described by: where I is the ideal (motionless) image, U b are the spatial transformations for bin b obtained with LREG, S c are the coil sensitivities for coil c, F is the forward Fourier transform, A b is a logical matrix that selects the k-space lines acquired for bin b, and K is the acquired multichannel k-space data. Equation [5] is in the reference frame of the coils, where they are assumed to remain static. I may be obtained by inverting the encoding operator E. Practical inversion of E is obtained with iterative methods such as the conjugate gradient (CG). The CG requires a symmetric positive-definite matrix, thus we apply the method to the equivalent normal equation: where E H is the Hermitian transpose of E. The proposed approach uses undersampled data, which results in noise amplification in the reconstructed motion corrected images due to poor conditioning of the inversion. This happens for two reasons: first, if the object moves outside the field-of-view (FOV) then information is permanently lost; second, correcting for motion within the FOV may cause k-space inconsistencies, causing some samples to overlap and creating k-space gaps. This effectively increases the undersampling factor (R), which contributes to making Eq. [6] undetermined (30). Furthermore, in practice the motion is not known exactly and errors in motion estimation can further contribute to the poor conditioning of this problem. It has been shown how multiple channels improve the condition of the motion reconstruction problem for fully sampled images (30). To reduce the effect of poor conditioning in undersampled datasets we include spatial total variation regularization TV s in the GMD reconstruction (TV-GMD), given by: This is solved with a nonlinear conjugate gradient reconstruction using the solution to the nonregularized approach as a starting estimate, similar to the previous description of the bin reconstruction. In the TV-GMD reconstruction, the TV s term is used to remove noise-like aliasing with small k s values. The k s for the TV-GMD was determined empirically by choosing a value that reduces the remaining artifacts without significantly affecting image sharpness.

Data Acquisition
Nine healthy volunteers were scanned under freebreathing on a 1.5 Tesla (T) clinical scanner (Philips Achieva, Philips Healthcare, Best, The Netherlands) using a 32-channel cardiac coil. Written informed consent was obtained according to institutional guidelines. Data were acquired with a balanced steady state free precession sequence with the following parameters: 287 Â 287 Â 287 mm isotropic FOV, 1.75 Â 1.75 Â 1.75 mm isotropic resolution, (TR/TE) ¼ 3/1.43 ms, flip angle 30 . During acquisition, each radial profile was uniformly undersampled in the radial direction by a factor of 2. For the resulting reconstructed matrix size of 164 Â 164 Â 164, we need 256 radial profiles to approximately fulfil the Nyquist criterion in the angular direction, assuming a quasi-uniform profile distribution. During the scan, 820 radial profiles were acquired to guarantee all volunteers provided enough data for different reconstruction approaches, including the self-gated reconstruction which has a low scan efficiency.

Simulations
The proposed approach estimates motion from highly undersampled respiratory resolved bin reconstructions (I b ). A pseudo-random distribution of radial profiles is sampled at each respiratory bin, depending on the breathing pattern. This may result in large k-space gaps that can introduce artifacts in the reconstructed images. Specifically, k-space gaps affect the image point spread function by broadening the main lobe/increasing side lobe amplitude, which results in blurring/aliasing artifacts (24). Blurring and residual artifacts may hinder subsequent image registration, producing inaccurate motion fields. A simulation was conducted in one representative volunteer to study the relationship between the maximum angle between two radial profiles (a max ) and the accuracy of motion estimation. An end-inhale image (F) was reconstructed with 25 different numbers of radial profiles (p), from fully sampled (256 profiles) to highly undersampled (26 profiles). The obtained undersampled images (F p ) were registered to a reference fully sampled end-exhale image (F r ). The obtained motion fields were then compared with the ground-truth: the registration between F r and F. We measure the average displacement error in the motion field e(a) similar to (25): where U a(Kp) is the motion field between F p and F r , U Fr is the motion field between F and F r , and the sum is taken over all the voxels (N being the total number of voxels in the image).

Comparison with Image Based Motion Compensation
A preliminary study was performed to compare the approach described in Buerger et al (16), hereby referred as image-based motion compensation (IMC) with the proposed TV-GMD. To evaluate the effects of blurring and residual artifacts in motion estimation, bin reconstructions using the a max value, obtained by means of simulations, were performed using iterative SENSE (as in IMC) and with the proposed TV-SENSE. Motion fields between neighboring bins of representative volunteers were obtained by means of LREG. A comparison of motion estimation obtained from undersampled SENSE and TV-SENSE bin reconstructions was made by computing their displacement errors to motion fields obtained from a dataset 3Â oversampled in the angular direction (considered as ground truth). Additionally, all nine datasets were reconstructed with IMC, using the same amount of data as the TV-GMD reconstruction. This resulted in a total of 160 6 37 acquired profiles for both TV-GMD and IMC reconstructions. IMC reconstruction was performed with iterative SENSE (23), taking approximately 1 h. IMC and TV-GMD reconstructions were compared by means of measures of liver sharpness and gradient entropy. TV-GMD reconstruction and the comparison metrics are described in more detail in the following section.

In Vivo Experiments
The same acquired data were used retrospectively to produce a 2Â undersampled non-motion corrected (NMC) reconstruction (2Â undersampling in radial direction, no undersampling in angular direction), a 2Â undersampled 5 mm self-gated reconstruction (2Â undersampling in radial direction, no undersampling in angular direction), and a highly undersampled TV-GMD reconstruction (2Â undersampling in radial direction, 1.8Â undersampling in angular direction, on average). This resulted in a total of 256, 414 6 147 and 160 6 37 acquired profiles for the NMC, gated and TV-GMD reconstructions.
The following binning parameters were used: a max ¼ 13.75 (determined according to the previous section); w max ¼ 5 mm (same value as the gating window); GE min ¼ 80% (set to remove outliers in the respiratory cycle). Furthermore, to avoid remaining undersampling artifacts in the final motion corrected reconstruction a maximum undersampling factor (R max ) of 4 was used. Therefore, the proposed method was set to reconstruct a minimum of 128 radial profiles, equivalent to 2Â angular undersampling and 2Â radial undersampling. This resulted in data being grouped into three to five bins (varying per volunteer), with bin undersampling factors ranging from 5.2Â to 18.9Â (average of 9.5Â). The motion corrected datasets were undersampled from a minimum of 2.1Â to a maximum of 4Â (average of 3.6Â). Table 1 presents values for undersampling factors, gating efficiency, acquisition time, number of acquired profiles, number of profiles per bin, number of bins, and bin window sizes. The respiratory bins were reconstructed with TV-SENSE. The initial l 2 -norm reconstruction ran with 10 iterations, with k s ¼ 0 and k t ¼ 0. This result was used as a starting estimate for the reconstruction with k s ¼ 0.2 and k t ¼ 0.1 (corresponding to 9.6 Â 10 À6 and 4.8 Â 10 À6 , respectively, in terms of the average jj.jj ' norm of the images), three iterations. The reconstructed bins were registered to estimate the motion for the TV-GMD reconstruction. The initial data consistency GMD reconstruction ran with five iterations, k s ¼ 0. The following TV-GMD reconstruction used k s ¼ 0.0001 (corresponding to 4.8 Â 10 À9 , in terms of the average jj.jj ' norm of the images) and five iterations. Additionally, data were reconstructed with GMD (4 iterations) for additional comparison with TV-GMD. It took approximately 1.5 h to reconstruct bins (TV-SENSE) and an additional 1.5 h for the motion compensated reconstruction (TV-GMD) on a 12-core implementation. The gated and nonmotion corrected images were reconstructed with iterative SENSE (23), both taking approximately 5 min to reconstruct. Coil sensitivities were estimated from a reference scan. All images were reconstructed using Matlab (The Math-Works, Natick, MA).
To observe the effect of undersampling in the final motion corrected reconstruction, data from two representative volunteers were reconstructed with the proposed method with undersampling factors of 4Â, 3Â, 2Â, and 1Â, using the same estimated motion as described above.
The proposed TV-GMD approach was compared with the gated reconstruction by means of liver sharpness, gradient entropy, visual image scoring and ranking (31). The liver sharpness and gradient entropy metrics were computed on a set of 10 coronal slices. To compute liver sharpness, twenty-five 1D profiles were manually selected across the liver-lung interface. The sharpness measure for each profile was obtained by the maximum gradient normalized to the maximum intensity, similar to Botnar et al (32). Gradient entropy has previously been used in motion correction as an optimization cost function that favors distinct boundaries and reduced artifacts and was ranked 1 of 24 image quality metrics studied (33). Here we use local gradient entropy as an image quality metric, similar to Vuissoz et al (34). The total gradient entropy of the image is given by the mean of the local gradient entropies. All liver sharpness and gradient entropy values were normalized to the reference gated values. We present the gradient entropy results inverted to allow easier comparison with the other metrics (the higher the metric the better the image). Two experts (a radiologist with 14 years of experience in MRI and an image processing researcher with 18 years of experience in MRI) were asked to "score the sharpness of the main boundaries and features of the images" on a scale of 0 (extreme blurring) to 4 (no blurring) and "rank the overall image quality based on existing artifacts, ghosting and blurring" from 1 (worst) to 4 (best). Statistical significance of gradient entropy and liver sharpness was evaluated using a paired t-test (P-value of 1%); statistical significance of expert sharpness score and quality rank was evaluated using a Wilcoxon signed rank test (Pvalue of 1%).  ¼ 13.75 , and (c) a ¼ 16.25 . For a 13.75 , motion field errors are kept below 1.1 voxels. Table 1 Acquisition and reconstruction information for non-motion corrected (NMC), gated, GMD and TV-GMD reconstructions.
Acquisition and reconstruction NMC Gated GMD TV-GMD  Figure 2 shows the mean voxel error in the registration deformation field decreasing with the increasing number of radial profiles. Angular gaps values (a) resulting for different number of radial profiles are displayed throughout the graph. For values lower than a ¼ 13.75 the mean voxel error in the motion fields remains below 1.1 voxels. For a values larger than 13.75 the error in the motion fields start increasing significantly. Based on this simulation, we chose to set a max ¼ 13.75 . %19Â for bins 1, 2, and 3, respectively. Iterative SENSE reconstruction, as used in Buerger et al (16), shows residual artifacts (Fig. 3b). These artifacts are significantly reduced with a TV-SENSE reconstruction, while only introducing minor blurring (Fig. 3c).

Comparison with Image Based Motion Compensation
The improvement in motion estimation accuracy with the proposed TV-SENSE in comparison to previously shown iterative SENSE can be seen in Figures 3e-h, for two representative volunteers. Image registration of SENSE reconstructions can produce local errors FIG. 4. Coronal (top), sagittal (middle) and axial (bottom) slices for volunteers 2 and 4 (including zoom-in images, arrows point out some main differences). a: Image based motion correction (IMC) for volunteer 2 with 4Â undersampling at 100% gating efficiency. Residual aliasing may be seen in all slice orientations. Additionally, some blurring is introduced by IMC. b: Proposed TV-GMD for volunteer 2 with 4Â undersampling at 100% gating efficiency. Residual aliasing is reduced and image structures appear sharper when compared with IMC. c: IMC for volunteer 4 with 4Â undersampling at 100% gating efficiency. Residual aliasing is not significant in this volunteer, but the IMC still introduces additional blurring to the image. d: TV-GMD for volunteer 4 with 4Â undersampling at 100% gating efficiency. A sharper reconstruction is obtained with the TV-GMD.
(yellow-red regions in Figures 3f,h) as the LREG algorithm attempts to register aliasing artifacts. TV-SENSE removes most of these artifacts, enabling reliable motion estimation (corresponding blue-green regions in Figures 3e,g).  This aliasing is significantly reduced with the TV-GMD (Fig. 4b). Results for volunteer 4 using IMC (Fig. 4c) present less residual aliasing, but additional blurring is introduced (which can be seen in both volunteers). In contrast, the TV-GMD corrects motion without introducing this additional blurring (Fig. 4d). TV-GMD and IMC presented liver sharpness values of 1.18 and 0.98, respectively, and gradient entropy values of 1.00 and 0.98, both statistically different with a P-value of 1%.
FIG. 6. Coronal (top), sagittal (middle) and axial (bottom) slices for volunteer 2 with maximum respiratory amplitude of 8.2 mm (including zoom-in images, arrows point out some main differences). NMC (non-motion corrected): 2Â undersampled at 100% gating efficiency. Some blurring is visible in image structures and the liver-lung border. Gated: 2Â undersampled at 77% gating efficiency. There is no significant improvement, as the gated reconstruction uses a 5 mm window. GMD: 4Â undersampled at 96% gating efficiency. The high undersampling creates a strong noise-like aliasing. TV-GMD: 4Â undersampled at 96% gating efficiency. The total variation regularization improves GMD undersampled reconstruction at the expense of some minor blurring, despite using only 128 radial profiles.
In Vivo Experiments Figure 5 shows multiple slice orientations for the nonmotion corrected, gated, GMD and TV-GMD reconstructions for volunteer 1. In Figure 5, it can be seen that TV-GMD and gated reconstructions yield images of similar quality, correcting most ghosting and blurring present in the non-motion corrected (NMC). The gated and NMC reconstructions have an undersampling factor of 2Â, while the GMD and TV-GMD resulted in an undersampling of 3.5Â for this volunteer. Remaining artifacts from undersampling and noise amplification can be observed in the GMD reconstruction (Fig. 5). This effect is consistent across all volunteers and increases for higher undersampling factors. Comparison between GMD and TV-GMD highlights how TV regularization improves the conditioning of the reconstruction. Figure 6 shows multiple slice orientations for volunteer 2, where the GMD and TV-GMD reconstructions resulted in an undersampling of 4Â, compared with 2Â for the gated and NMC. The TV-GMD presents a sharper reconstruction than the gated (Fig. 6), due to the fact that the resulting binning windows were smaller (2.76 mm average) than the gating window (5 mm). Note that undersampling artifacts are stronger for this case, but still can be reduced by TV regularization. Reconstructions for the TV-GMD at various undersampling factors in two volunteers are shown in Figures  7a,b. As expected, undersampling artifacts decrease with increasing data used. The proposed method is flexible in the choice of the maximum undersampling factor of the reconstructed image which may be useful for clinical applications that require specific signal-to-noise ratios.
Bar plots of the average acquisition times, undersampling factors, gradient entropies, liver sharpness, sharpness score, and image quality rank over all nine volunteers are shown in Figure 8. When comparing with the gated reconstruction, the TV-GMD reconstruction reduces the average scan time from 101 to 34 s. The gradient entropy indicates the TV-GMD and gated reconstructions have the best image quality (1.00 and 1.00, respectively), significantly better than NMC (0.97) and GMD (0.95). Image rank from visual assessment agreed with gradient entropy, marking TV-GMD (3.33) just below the gated (3.39), significantly better than NMC (1.61) and GMD (1.67). Image sharpness score from visual assessment yielded similar values between the TV-GMD (2.22) and gated (2.44), differentiating them from the NMC (1.17). Liver sharpness measures were not as sensitive as expert sharpness evaluation, but presented a similar behavior, marking TV-GMD as the best (1.18), followed by GMD (1.12) and gated (1.00), significantly better than NMC (0.90).

DISCUSSION
The proposed approach accelerates the acquisition by means of undersampling and 3D nonrigid motion correction, reducing the average scan time by 2.6Â when compared with a gated acquisition. Here, motion is estimated (TV-SENSE) from highly undersampled respiratory resolved images and corrected in the reconstruction (TV-GMD) of the undersampled final data set. Gradient entropy and image quality ranking present similar values for the TV-GMD and gated reconstructions, distinguishing them as superior to the NMC and GMD. The noiselike aliasing present in the GMD reconstruction is responsible for the worse values in gradient entropy. The liver sharpness and expert sharpness score show the TV-GMD removes motion-induced blurring (visible in the NMC) to a similar level as the gated reconstruction. In cases of uniform breathing the TV-GMD achieves sharper FIG. 9. Coronal slices for a nonmotion corrected (NMC) G-RPE 2Â radially undersampled, Cartesian navigator gated (5 mm) 2Â undersampled and TV-GMD G-RPE 2Â radially undersampled for two volunteers. The Cartesian gated and TV-GMD approaches compensate motion to a comparable degree. Residual motion is still present in the Cartesian gated (ghosting, yellow dotted arrows) and TV-GMD (minor blurring, red full arrows).
reconstructions than the gated as it uses binning win-