Automatic Spatial Calibration of Ultra-Low-Field MRI for High-Accuracy Hybrid MEG--MRI

With a hybrid MEG--MRI device that uses the same sensors for both modalities, the co-registration of MRI and MEG data can be replaced by an automatic calibration step. Based on the highly accurate signal model of ultra-low-field (ULF) MRI, we introduce a calibration method that eliminates the error sources of traditional co-registration. The signal model includes complex sensitivity profiles of the superconducting pickup coils. In ULF MRI, the profiles are independent of the sample and therefore well-defined. In the most basic form, the spatial information of the profiles, captured in parallel ULF-MR acquisitions, is used to find the exact coordinate transformation required. We assessed our calibration method by simulations assuming a helmet-shaped pickup-coil-array geometry. Using a carefully constructed objective function and sufficient approximations, even with low-SNR images, sub-voxel and sub-millimeter calibration accuracy was achieved. After the calibration, distortion-free MRI and high spatial accuracy for MEG source localization can be achieved. For an accurate sensor-array geometry, the co-registration and associated errors are eliminated, and the positional error can be reduced to a negligible level.Comment: 11 pages, 8 figures. This work is part of the BREAKBEN project and has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 68686

Application of the Fractional Fourier Transform to Image Reconstruction in MRI

The classic paradigm for MRI requires a homogeneous B 0 field in combination with linear encoding gradients. Distortions are produced when the B 0 is not homogeneous, and several postprocessing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for shortbore magnets and higher B 0 fields. Nonlinear magnetic components can also arise from concomitant fields, particularly in low-field imaging, or intentionally used for nonlinear encoding. In any of these situations, the second-order component is key, because it constitutes the first step to approximate higher-order fields. We propose to use the fractional Fourier transform for analyzing and reconstructing the object's magnetization under the presence of quadratic fields. The fractional fourier transform provides a precise theoretical framework for this. We show how it can be used for reconstruction and for gaining a better understanding of the quadratic field-induced distortions, including examples of reconstruction for simulated and in vivo data. The standard paradigm for MRI requires a strong magnetic field with uniform intensity and time-varying linear encoding gradients across the entire field of view. However, deviations in the main field are common as uniform fields are physically difficult to achieve and also because of off-resonance effects from susceptibility changes. Such frequency variations introduce an accumulating phase over time, which cannot be demodulated easily as it varies spatially. This problem is worse for stronger fields and for sequences with long acquisition times. Great efforts are put in building the system with the highest possible homogeneity, for example, with passive or active shimming which partially correct first-and second-order field variations. Additionally, there has been an increasing interest in spatial encoding by nonhomogeneous, nonbijective spatial encoding magnetic fields (SEMs; Ref. 1). Starting from a general nonlinear field concept, novel techniques including PatLoc (1), O-Space (2), Null Space (3), and PhaseScrambled imaging (4-6) have all introduced the use of second-order SEMs as the first and simplest approach in simulations, custom-built hardware and experiments (7-12). One of the challenges of these approaches is to have an appropriate reconstruction technique. Higher-order fields also appear as a natural component of linear gradients. These concomitant fields can be well approximated by quadratic functions Several image reconstruction methods have been proposed to correct distortions, or to reconstruct an image produced by nonhomogeneous fields, being an active field of research (19-28). There is a well-known theoretical background for the linear correction approaches, in which an exact analytical solution is provided The fractional Fourier transform (FrFT) is a generalization of the standard Fourier transform (FT) by means of the continuous fractional order a, which covers densely the entire transition between image (or time) domain (a = 0) and the Fourier domain (a = 1; Ref. 32). The FrFT is a special case of the linear canonical transform and can be defined in several different ways leading to different physical interpretations and thus, it has become useful in many applications It is of general knowledge that the magnetization of an object under a linear magnetic field can be related to the FT of the MR signal due to the mathematical equivalence of the signal with the FT kernel. Similarly, the magnetization of an object under a quadratic field can be related to its FrFT. The kernel of the integral definition of the FrFT presents a resemblance with the MR signal. This fact, along wit

Gaussian process regression can turn non-uniform and undersampled diffusion MRI data into diffusion spectrum imaging

We propose to use Gaussian process regression to accurately estimate the diffusion MRI signal at arbitrary locations in q-space. By estimating the signal on a grid, we can do synthetic diffusion spectrum imaging: reconstructing the ensemble averaged propagator (EAP) by an inverse Fourier transform. We also propose an alternative reconstruction method guaranteeing a nonnegative EAP that integrates to unity. The reconstruction is validated on data simulated from two Gaussians at various crossing angles. Moreover, we demonstrate on non-uniformly sampled in vivo data that the method is far superior to linear interpolation, and allows a drastic undersampling of the data with only a minor loss of accuracy. We envision the method as a potential replacement for standard diffusion spectrum imaging, in particular when acquistion time is limited.Comment: 5 page

Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie

Sub-dekahertz ultraviolet spectroscopy of 199Hg+

Using a laser that is frequency-locked to a Fabry-Perot etalon of high finesse and stability, we probe the 5d10 6s 2S_1/2 (F=0) - 5d9 6s 2D_5/2 (F=2) Delta-m_F = 0 electric-quadrupole transition of a single laser-cooled 199Hg+ ion stored in a cryogenic radio-frequency ion trap. We observe Fourier-transform limited linewidths as narrow as 6.7 Hz at 282 nm (1.06 X 10^15 Hz), yielding a line Q = 1.6 X 10^14. We perform a preliminary measurement of the 5d9 6s2 2D_5/2 electric-quadrupole shift due to interaction with the static fields of the trap, and discuss the implications for future trapped-ion optical frequency standards.Comment: 4 pages, 4 figures, submitted for publicatio

High efficiency, low distortion 3D diffusion tensor imaging with variable density spiral fast spin echoes (3D DW VDS RARE)

We present an acquisition and reconstruction method designed to acquire high resolution 3D fast spin echo diffusion tensor images while mitigating the major sources of artifacts in DTI-field distortions, eddy currents and motion. The resulting images, being 3D, are of high SNR, and being fast spin echoes, exhibit greatly reduced field distortions. This sequence utilizes variable density spiral acquisition gradients, which allow for the implementation of a self-navigation scheme by which both eddy current and motion artifacts are removed. The result is that high resolution 3D DTI images are produced without the need for eddy current compensating gradients or B_0 field correction. In addition, a novel method for fast and accurate reconstruction of the non-Cartesian data is employed. Results are demonstrated in the brains of normal human volunteers