Sensitivity to motion remains a major limitation to the clinical utility of MRI. Several acquisition trajectories have been developed to enable self-navigation including translation and rotation.^1,2 Self-navigating Cartesian trajectory (Butterfly) can provide coil-by-coil estimates of local linear translation without lengthening scan time or attaching external sensors.³ However, direct rotation estimation is currently not available for Butterfly. In this work, we propose to combine translational motion estimates with the geometry of the differing coil sensitivities to estimate global translation and rotation. These estimates can then be used to retrospectively correct for motion.

Simulation of known motion was used to design our methods. To accurately model translation and rotation the sum of squares image from motion free data was translated and rotated, and then multiplied by the stationary sensitivity maps. Sensitivity maps were estimated from the image data.⁴ Butterfly navigator was simulated by extracting the appropriate points in kspace from the rotated and translated multi-coil image (Fig. 1). Our work here focuses on 2D slices.

Extracting Rotation with Known Translation: Assuming that rotation occurs without translation or with known translation we can estimate rotation of small angles using the simple geometrical relationship $$$s={\theta}r$$$, where s is arc length,θ the angle of rotation in radians, and r the radius. For each coil we take s to be the magnitude of estimated translation and r to be the distance from the center of rotation. θ is then chosen by the best linear approximation of r to s. To determine the appropriate reference point to compute the distance r, two different approaches are investigated:

1. Coil Center: For each coil, the coil center is estimated to be a linear combination of the point of highest sensitivity and the centroid of sensitivity.

2. Virtual Coils: Virtual coils, formed as weighted sums of the array coils⁵ allow enhanced localization of translation estimates. The weights are chosen for the sensitivities to best fit an elliptical region of interest (ROI) smoothed with a Hanning window. Virtual navigators are generated by summing array coil navigators with the same weights. Local translation estimates are then extracted from each virtual navigator as usual and the coil center is set to the center of the elliptical ROIs.

Extracting Unknown Rotation and Translation: We can also simultaneously extract unknown rotation and unknown translation. We take the initial locations of the points to be the coil centers and the new locations to be the coil centers plus the estimated translations and find the best mapping in the least squares sense.⁶ This method can be applied to either the array coils or the virtual coils.

A major challenge for using the array coils directly is choosing the correct location for coil center. The points of maximum sensitivity seem to be too far from the center of the image while centroids appear to be too close. For the appropriate choice of a linear combination of these coils we obtain high accuracy for small angles of rotation, though for very small angles we underestimate motion. (Fig. 2) Virtual coils resolve this difficulty by providing a ready and accurate choice for the coil center.

We observe a significant correlation between the virtual coil center locations that best predict translation in the x direction, y direction, and in rotation (Fig. 3). This suggests that virtual coils centered at certain locations provide more accurate estimates of local linear translation than others. We also observe that regions of highest accuracy are those near the top, bottom, left, and right edge of the head. We hypothesize this is because the motion points closer to the edge of the head more closely approximates translation for rotation about the center. We further hypothesize these edge regions have higher accuracy than diagonal edges because their motion is concentrated solely in the x or y direction which are the directions of our navigators. We propose a motion corrupted image can be rotated and translated in simulation to determine the optimal virtual coils for that image, before using virtual coils to correct for the motion.

The technique for extracting rotation and translation simultaneously proved highly effective when coupled with virtual coils (Fig. 4). The algorithm used for extracting rotation and translation simultaneously natively accepts a 3D set of points. Generalization of these techniques to 3D should prove a promising and ready direction for future work.