The use of optical tracking systems (OTS) has been demonstrated to be advantageous for prospective motion correction in MRI [1-2]. In this technique, the position of a moving rigid body is tracked by an optical device and the coordinates of the imaging volume are updated in real-time. However, accurate position updates require a precise cross-calibration between the coordinate systems of the MR scanner and the optical device. Current calibration relies on an image-based registration and is prone to non-linear gradient imperfections [1], which manifest themselves as distortions, leading to errors in the image registration as shown in Figure 1. Methods based on the analysis of 3D radial k-space were previously shown to detect rigid body motion from changes of the topology in the k-space [3]. Therefore, our main goal was to study the feasibility of 3D radial acquisitions for rotation and translation detection. We investigated the application of a 3D radial sequence for the detection of rotation parameters during the OTS calibration and compared the acquired position information with the OTS data.

Experiments were conducted on a 3T MRI system (Siemens Magnetom Prisma). Figure 2 shows a custom made spherical phantom used for calibration [4]. It was 3D-printed and had randomized unique for all directions grid structures. 3D radial acquisition with 16384 spokes and 256 readout samples was implemented for motion detection. The spokes were distributed uniformly along the polar axis and the azimuthal increment was given by the Golden angle [5], leading to a multishot spiral sampling pattern on a spherical surface in k-space (Fig. 3A). The phantom was scanned using the sequence after a series of several rotations. In parallel, its motion was recorded by the Moiré phase tracking system (MPT, Metria Innovation Inc., U.S.A)[6], which comprised a camera and a retro-reflective tracking marker. The MPT camera was installed in the magnet bore and tracked motion of a marker affixed to the phantom (Fig. 2A).

Due to a grid-like structure inside the phantom, projections through the center of the acquired k-space yielded a very high signal at certain angles. As can be seen from typical data displayed on a sphere (Figures 3B-D), the surface topology in k-space was represented by 6 peaks. In order to measure rotation parameters, an algorithm was developed and based on this topology. First, the peaks centers $$$k_{i, ref}=[kx_{i, ref}, ky_{i, ref}, kz_{i, ref}]$$$ and $$$k_{j, rot}=[kx_{i, rot}, ky_{i, rot}, kz_{i, rot}] $$$ were determined for datasets before and after the rotation. Then, the phantom’s profiles were found by Fourier transformation of the signals along the spokes at the peaks locations. In order to find the correspondence between the peaks, the correlation coefficients were calculated between the profiles in the reference and rotated data. If the profiles for a peak pair had a correlation higher than 99%, the peaks were said to transform into each other during the rotation. Otherwise, the peaks were considered as non-related. After sorting the peaks transformations $$$ i\rightarrow j$$$, the rotation matrix $$$R$$$ was calculated from the matrix $$$H_{i,j}=\sum_{i,j}k_{i} \cdot k_{j}^{T}$$$ (where ”T” denotes “transpose”) according to $$$R=V \cdot U^{T}$$$. Here, $$$[U,T,V]=svd(H)$$$ is singular value decomposition and rotation angles were determined from the matrix $$$R$$$. Profile shifts were taken into consideration during calculations of the correlation coefficients by a simultaneous translation search, which was aimed at detecting the optimal shifts between profiles by maximizing their correlation

After the peaks’ centers of mass were measured (Figure 3E), the correlation coefficients between the phantom’s profiles through the detected positions were calculated. The correlation was normally higher than 98% for the peaks, which meant a correspondence to each other (Fig. 4A), whereas it was on the order of 85% for the non-related peaks (Fig. 4B). Figures 4C and D display correlation between the profiles, when the phantom underwent a rotation combined with translation. The regularization of the translation improved significantly peak detection. The resulting rotation angles measured in radial acquisition and from the MPT camera are presented in Figure 5. Both optical tracking and radial acquisition are in a good agreement and thus demonstrate a high precision for the developed algorithm.

Due to unique intrinsic structure along radial directions, the described phantom is a promising calibration tool for MPT. Furthermore, 3D radial acquisition is straightforward for detection of rotational motion of such an object. In this manner, the presented method is precise and robust for rotation calibration. As a next step, we would like to implement spherical shell sampling followed by the selected projections for faster motion detection.