Holographic Coding For Magnetic Resonance Imaging Using Second Order Spherical Harmonics
Open Access
- Author:
- Consevage, Steven
- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 05, 2021
- Committee Members:
- Nitin Samarth, Program Head/Chair
Dezhe Jin, Major Field Member
Nanyin Zhang, Outside Unit Member
Thomas Neuberger, Outside Field Member
Srinivas Tadigadapa, Special Member
Steven Schiff, Chair & Dissertation Advisor - Keywords:
- MRI
Holography
Compressed Sensing
Low-Field MRI
Diffractive Imaging - Abstract:
- Fourier encoding is nearly ubiquitous for spatial magnetization localization in Magnetic Resonance Imaging (MRI). The encoding is conducted by applying gradient magnetic fields which vary linearly along the principal axes in a Euclidean volume. This varies the local Larmor precession frequencies such that the spectral Faraday signal is the imaged object's band limited spatial Fourier transform. MRI Fourier coding can be likened to coherent scattering measurements in the Fraunhofer regime where the 'detector plane' resides in the very far-field region and the measured incident wavefronts are planar, lacking curvature. Image recovery by inverse Fourier Transform is then readily analogous to Fourier transform via lensing. It follows then that MRI could be conducted in the near field region given the proper spatial encoding. This was first done by Ito \emph{et al.} in \cite{Ito-2002} and the references therein. There, an approximate Fresnel phase was encoded with non-linear spatial gradients and images were recovered through numerical refocusing in what was coined Diffractive MRI. The recoding of the Fresnel encoded signal is effectively a hologram, of the type encountered in acoustic or optical holograpahy. Critically, holographically recorded and reconstructed 'images' are known to retain some three dimensional properties of the source, exhibiting parallax and depth of focus. This work, expands on the concept of using Fresnel encoding of the MRI phase both theoretically and with improved methodologies for potential clinical application. I formulate the MRI signal as a problem of inverse scattering source recovery, allowing the use of diffractive imaging concepts, such as the Ewald sphere, to characterize the holographically measured object Fourier space support. I also demonstrate that generation of the Fresnel phase needed to emulate near-field imaging can be well approximated with the use of dynamic encoding gradients from the set of $l = 0,1,2$ spherical harmonic solutions to Maxwell's vacuum field equations. This is an appreciable advancement over customized gradient arrangements which should allow for less engineering difficulties, clinically relevant imaging volumes, and arbitrary viewing angles. The principal appeal of holographic coding is in the pursuit of a dimensionally compressive imaging method. Since MRI measurements are time gated, reductions in the measurement space represent potentially large improvements in measurement speed and efficiency. The near field-like measurements of holographic MRI captures the curvature of Ewald sphere subsurfaces of the imaged object's Fourier spectrum. The holographic MRI measurement consequently records axial spatial information from measurements along only the transverse axes, resulting in a form of sub-dimensional image acquisition. While the recorded Ewald sub-spheres are surfaces embedded in the object's Fourier volume, tomographic reconstruction of the object's scattering potential is ill-posed under classical Nyquist sampling theorem. Compressive Sensing (CS) regularization is then needed for source recovery. Within this work is a full description of a novel method for Holographic Coding for MRI. A detailed derivation from first principles is provided which couches the MRI measurement process as a more general optical scattering problem. It is shown that the Fresnel kernel used in \cite{Ito-2002} may be successfully encoded with spherical harmonic spatial encoding coils which then allow for arbitrary viewing angles. Methods for adapting spin-echo sequences are shown and the effects of and corrections for magnetic field aberrations are explored. Of principal importance, compressive sensing regularization techniques are shown to condition tomographic reconstruction from simulated holographic MRI data. Finally, application of the Ewald sphere concept to holographic MRI provides a means to translate between Fourier and holographic MRI, particularly for the purpose of volumetric compressive sensing measurements.