Motion Aware MR Imaging via Spatial Core Correspondence

Abstract

Motion awareness in MR imaging is essential when it comes to long acquisition times. For volumetric high-resolution or temporal resolved images, sporadic subject movements or respiration induced organ motion has to be considered in order to reduce motion artifacts. We present a novel MR imaging sequence and an associated retrospective reconstruction method incorporating motion via spatial correspondence of the k-space center. The sequence alternatingly samples k-space patches located in the center and in peripheral higher frequency regions. Each patch is transformed into the spatial domain in order to normalize for spatial transformations rigidly as well as non-rigidly. The k-space is reconstructed from the spatially aligned patches where the alignment is derived using image registration of the center patches. Our proposed method assumes neither periodic motion nor requires any binning of motion states to properly compensate for movements during acquisition. As we directly acquire volumes, 2D slice stacking is avoided. We tested our method for brain imaging with sporadic head motion and for chest imaging where a volunteer has been scanned under free breathing. In both cases, we demonstrate high-quality 3D reconstructions.

1 Introduction

The presence of subject motion during magnetic resonance (MR) imaging is a critical factor particularly for brain, arm and leg imaging. When not explicitly considered, it can results in motion artifacts which render the image acquisition useless. Especially for uncooperative subjects such as pediatric, stroke or Parkinson patients a lot of time is wasted in the clinic with re-scanning. Furthermore, organ movements induced by natural respiration are challenging to handle in chest imaging, as e.g. lung patients cannot be asked for breath holding. Hence, when it comes to longer acquisition times, for volumetric high-resolution or dynamic 3D imaging, motion needs to be taken into account.

Fig. 1.

Sampling pattern for the first and second phase encoding dimension in the k-space: 128$\times$128 px and 128$\times$96 px with a shift-factor of $b=16$ and $b=1$ respectively, at time points $t=50$ and $t=150$. Patches at higher time points are visualized brighter. The center patch is colorized in yellow.

In this paper, we present a novel motion aware MR imaging method which can deal with spatial movements, including rigid and non-rigid ones. The key idea is that the k-space is sampled patch-wise such that each patch can be transformed into the spatial domain. Specifically, patches in the k-space center and patches in higher frequency regions are sampled alternatingly. A high frequency patch can thus be corrected for the relative motion between its simultaneously acquired center patch and a reference one. The spatial transformation between two center patches is obtained using image registration. The k-space at a given time-point is finally reconstructed using the motion corrected patches by accumulating their frequency representation. We demonstrate a high-quality 3D reconstruction of a brain image acquired under sporadic head movements. In addition, we are able to reconstruct a chest image acquiring under free breathing.

Motion correction in MR imaging has been widely studied in the past decades. For a comprehensive review about motion correction methods we refer to [13] and for a focus on prospective methods to [5]. Prospective methods correct the pulse sequence according to the subject motion during acquisition. They generally require additional equipment such as optical tracking systems or active markers and are therefore difficult to implement in the clinic. Methods which rely on k-space or image navigators [12] require unused time in the sequence to acquire accurate motion information and have real-time constraints which restricts the computational budget. Retrospective approaches try to invert the motion affected changes in the acquired data after the acquisition has ended. Most retrospective as well as prospective methods are limited to rigid body motion with up to 6 degrees of freedom. Non-rigid motion remains a major challenge. Approaches which consider non-rigid motion are based on gating/triggering [10], binning of motion states [4], are limited to 2D [1, 7] or perform a piece-wise rigid approximation of the non-rigid motion [1, 4]. In present 4D approaches, partial image data is acquired over several periodic motion cycles and retrospectively sorted using surrogate signals or image similarity measures in order to reconstruct coherent volumes [11]. It is, however, difficult to validate that the reconstructed partial data yields a valid 3D volume.

Our contributions are the following: our method is able to correct for non-rigid motion, which is derived from low-frequency representations of the imaged object, without the use of rigid approximation techniques. At each time-point a full 3D volume can be reconstructed. The specific k-space sampling pattern of our sequence allows for a gradual increase of the image quality. For our method, neither periodic motion has to be assumed nor binning of motion states or gating is required. As we acquire volumes, 2D slice stacking is avoided.

2 Method

For large 3D volumes, conventional k-space sampling strategies often exceed the time frame given for moving structure. Our idea is therefore, to split the k-space into smaller patches which are sampled at a high-temporal rate where motion artifacts are not dominant. Patches in the k-space center and peripheral patches in higher frequency regions are sampled alternatingly. The center patches, transformed into the spatial domain, are low-frequency representations of the imaged object and are used to estimate relative spatial transformations. Due to the patch-wise sampling not only the center but also high-frequencies can be transformed into the spatial domain where they are rigidly and non-rigidly aligned with the transformation found using the center patches.

2.1 Sequence

For this work, a product sequence for a radio-frequency spoiled gradient echo acquisition with short repetition time and low flip angles [2] was modified in order to alter the sampling of k-space. The full resolution in the readout direction is acquired with a short repetition time. In the first and second phase encoding dimensions circular patches are pseudo-randomly sampled.

Undersampling of the k-space. The patch-wise undersampling should cover the k-space with a variable density which is higher in low-frequency regions to improve the fidelity of the reconstructed images. Furthermore, it should gradually increase the coverage of the k-space such that at any time the acquisition can be stopped while the k-space coverage is maximized. At each time point where a patch is sampled therefore, an image can be flexibly reconstructed using patches of an arbitrary time interval. To this end, frequencies which are sampled multiple times should be averaged which results in a gradual increase of the image quality. To integrate these features into our sequence, the peripheral patch centers are derived by a variant of the radial CIRCUS method [3], a low discrepancy additive recurrence sequence for pseudo-randomly sample the k-space.

We define the sampling sequence in the first two phase encoding dimensions as follows: Let $N_\text {min}$ be the number of k-space points of the smaller of the two dimensions. With patch radii $r_p$ we define the patch centers $k_c\in \mathrm{I\!R}^2$ as

(1)

where $\alpha =\frac{1+\sqrt{5}}{2}$ is the golden angle ratio, $K=1000$ the number of points along a perimeter, $r = \frac{t\cdot N_\text {min}}{2\delta }$ is the radius with steps $\delta =\frac{2N_\text {min}}{2r_p+1}-1$, T the number of time points $t\in \{0,1,2,\dots ,T\}$ and $s:=b\cdot r$ where $b\in \mathrm{I\!N}_+$ is a shift parameter. If the number of points in the two dimensions is not equal, the radius r is adjusted accordingly. We optimize b such that the k-space coverage $A_T$ with T time points is maximized

$$\begin{aligned} \mathop {\text {arg max}}\limits _{b\le 100} \sum _{T\in \{10,20,\dots , 300\}}\log {A_T}. \end{aligned}$$

(2)

In Fig. 1, examples of the patch sampling sequence are visualized with a center patch radius of 6 px and peripheral patch radius of 5 px.

2.2 Reconstruction

For each time point t, a part of the k-space $P_t\subset \mathbb {C}^3$ is sampled consisting of a patch in the center (core) $C_t$ and a high-frequency peripheral patch $H_t$. The relative spatial motion $u_t:\mathcal {X}\rightarrow \mathrm{I\!R}^3$ on the image domain $\mathcal {X}\subset \mathrm{I\!R}^3$ between a reference patch $P_r$ and any other patch $P_t$ is derived via correspondence estimation between the reference core $C_r$ and $C_t$ in the spatial domain:

$$\begin{aligned} u_t := \mathop {\text {arg min}}\limits _u~\mathcal {D}\big (\mathcal {F}^{-1}(C_r)\circ u, \mathcal {F}^{-1}(C_t)\big ), \end{aligned}$$

(3)

where $\mathcal {D}$ is a dissimilarity measure, $\mathcal {F}^{-1}$ is the inverse Fourier transform and $\circ$ the function composition. For the reconstruction of an image $I_r$ at the reference time point we distinguish between rigid and non-rigid transformations $u_t$.

Rigid Reconstruction. In the special case of a rigid transformation the motion correction can be directly performed in the k-space while the registration is still performed in the spatial domain. The spatial translation $\varDelta x_t\in \mathrm{I\!R}^3$ corresponds to a phase shift in the k-space and the spatial rotation with the rotation matrix $R_t$ is equivalent to the same rotation in the k-space:

$$\begin{aligned} P_t^{\varDelta x}(k_x) :=~&e^{i2\pi (k_x^T\varDelta x_t)}P_t(k_x)\end{aligned}$$

(4)

$$\begin{aligned} \hat{P}_t(k_x) =~&P_t^{\varDelta x}(R_tk_x), \end{aligned}$$

(5)

where $k_x$ is the k-space coordinate and $\hat{P}_t$ is the rigidly corrected patch of $P_t$. The reconstruction yields

$$\begin{aligned} \hat{I}_r&= \mathcal {F}^{-1}\left( \hat{W}\sum _{t=1}^T\hat{P}_t\right) , \\ \hat{W}(k_x)&= {\left\{ \begin{array}{ll} \frac{1}{\hat{w}_{k_x}} &{} \hat{w}_{k_x}>0\\ 1 &{} \text {otherwise} \end{array}\right. },~~ \hat{w}_{k_x}=\sum _{t=1}^T \hat{W}^t(k_x), ~~ \hat{W}^t(k_x) = {\left\{ \begin{array}{ll} 1 &{} \vert \hat{P}_t(k_x)\vert > 0\\ 0 &{} \text {otherwise}, \end{array}\right. }\nonumber \end{aligned}$$

(6)

where $\hat{W}$ normalizes overlapping patch regions.

Non-Rigid Reconstruction. For non-rigid transformations, the motion correction is performed in the spatial domain. The reconstruction is

$$\begin{aligned} \widetilde{I}_r = \mathcal {F}^{-1}\left( \widetilde{W}\sum _{t=1}^T \mathcal {F}\Big (\mathcal {F}^{-1}\big (P_t\big )\circ u_t\Big )\right) , \end{aligned}$$

(7)

where $\mathcal {F}$ is the forward Fourier transform. In the non-rigid case, it is not clear how to derive an exact normalization $\widetilde{W}$ which would correspond to the normalization $\hat{W}$ in the rigid case. This is because the non-rigid transformed patch ${\widetilde{P}_t=\mathcal {F}\big (\mathcal {F}^{-1}(P_t)\circ u_t\big )}$ contains frequencies not present in $P_t$. To still derive a valid reconstruction of the k-space, we propose to keep only motion corrected points which originally have a non-zero magnitude

$$\begin{aligned} \bar{I}_r&= \mathcal {F}^{-1}\left( W\sum _{t=1}^T W^t\mathcal {F}\Big (\mathcal {F}^{-1}\big (P_t\big )\circ u_t\Big )\right) ,\\ W(k_x)&= {\left\{ \begin{array}{ll} \frac{1}{w_{k_x}} &{} w_{k_x}>0\\ 1 &{} \text {otherwise} \end{array}\right. },~~ w_{k_x}=\sum _{t=1}^T W^t(k_x), \quad W^t(k_x) = {\left\{ \begin{array}{ll} 1 &{} \vert P_t(k_x)\vert > 0\\ 0 &{} \text {otherwise}, \end{array}\right. }\nonumber \end{aligned}$$

(8)

where W normalizes overlapping patches. Upcoming frequencies $\widetilde{P}_t\cap P_t$ which are masked out by $W^t$ are covered by other patches located at the respective k-space region.

Fig. 2.

Measured rigid head motion using the rigid registration.

3 Results

We demonstrate the reconstruction performance of our method on brain MR imaging under sporadic head motion and on dynamic chest MR imaging under free breathing. The volunteers were scanned at 3 T using a Siemens MAGNETOM Prisma. We distinguish between three variants of our method: Static where no motion correction is performed ($u_t=id$), Rigid where the rigid motion correction is performed directly in the k-space cf. Eq. 6 and Non-rigid where the motion is corrected non-rigidly using Eq. 8. In all acquisitions, the center and peripheral patch radius was set to 6 and 5 pixels respectively yielding 109 and 69 k-space pixels to sample in the first two phase encoding dimensions per time point. The signals from multiple coils are combined in the spatial domain using the absolute sum of squares method [8].

Fig. 3.

(a) Full k-space sampling (no head motion), (b) Static reconstruction (no head motion), (c) Static reconstruction (with head motion), (d) Rigid motion compensated reconstruction (with head motion).

3.1 Brain Experiment Rigid Reconstruction

We asked the volunteer to sporadically move their head during the acquisition of 2000 time points. In Fig. 2, the rotation angle and translation estimated by the rigid registration are plotted. The first two phase encoding dimensions have been set to the traversal plane with a size of 160 $\times$ 144 px while the readout direction was craniocaudal with 160 px. With an echo time of TE = 2.5 ms and a repetition time of TR = 5.9 ms the duration of a time point yields $\sim {1}$ s. In the registration (Eq. 3), the Mattes’ mutual information dissimilarity metric [6] has been used. For a reference scan, we repeated the experiment but with the volunteer holding their head still. In Fig. 3, a sagittal slice of the reconstructed volume from the different reconstruction variants are shown. Comparing the reconstruction of the full k-space and the Static reconstruction of the reference scan Static achieves a better image quality which is the result of the high oversampling. By considering rigid motion, Rigid substantially reduces motion artifacts and generally yields better quality and higher contrast than Static. Some intensity inhomogeneities remain e.g. in the brain stem and mouth regions.

Fig. 4.

(a) Breath-hold reference exhalation scan, (b) Static and (c) Non-rigid reconstruction under free-breathing. For comparison, the green line marks the exhalation level of the diaphragm.

3.2 Chest Experiment Non-rigid Reconstruction

The chest has been acquired under free breathing for 2000 time points. The first two phase encoding dimensions have been set to the sagittal plane with a size of 256 $\times$ 88 px while the readout direction was left-right with 128 px. With an echo time of TE = 1.0 ms and a repetition time of TR = 2.5 ms the duration of a time point yields 445 ms. For the registration, a graph-based method [9] has been used applying the local cross correlation as dissimilarity measure. A reference core in inhalation state was chosen. In its spatial representation, the expected sliding boundaries have been masked and considered in the registration. Figure 4a shows a reference breath-hold scan at exhalation with full k-space sampling. Due to the averaging, Static reconstructs a perceived exhalation state, as this is the most frequent respiratory state in natural breathing (Fig. 4b). However, a lot of details are lost and motion lines are visible especially in the liver. With our method (Fig. 4c), the respiratory state of the reconstruction can be controlled by selecting the reference core. Hence, Non-rigid reconstructs a high quality image in inhalation state, where the motion lines are disappeared.

4 Conclusion

We have presented a new 3D MR imaging method which corrects for rigid and non-rigid subject motion derived from the k-space center. The introduced sequence samples the k-space patch-wise which allows to correct for non-rigid transformations for each patch separately. We demonstrated the feasibility of the method with a brain acquisition under sporadic movements and a chest acquisition under free-breathing. In both cases, we have achieved high-quality reconstructions. We plan to investigate compressed sensing reconstructions which will allow to drastically improve acquisition time by acquiring fewer time points.