Fast, Iterative Image Reconstruction for MRI in the Presence of Field Inhomogeneities

In magnetic resonance imaging, magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast Fourier transform (FFT) methods. Several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities, but these methods assume that the field map that characterizes the off-resonance frequencies is spatially smooth. Recently, iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off-resonance effects. However, straightforward implementations of such iterative methods suffer from inconveniently long computation times. This paper describes a tool for accelerating iterative reconstruction of field-corrected MR images: a novel time-segmented approximation to the MR signal equation. We use a min-max formulation to derive the temporal interpolator. Speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast Fourier transform with normalized root mean squared approximation errors of 0.07%. The proposed method provides fast, accurate, field-corrected image reconstruction even when the field map is not smooth.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86010/1/Fessler69.pd

Arbitrary shape surface Fresnel diffraction

Fresnel diffraction calculation on an arbitrary shape surface is proposed. This method is capable of calculating Fresnel diffraction from a source surface with an arbitrary shape to a planar destination surface. Although such calculation can be readily calculated by the direct integral of a diffraction calculation, the calculation cost is proportional to $O(N^2)$ in one dimensional or $O(N^4)$ in two dimensional cases, where $N$ is the number of sampling points. However, the calculation cost of the proposed method is $O(N \log N)$ in one dimensional or $O(N^2 \log N)$ in two dimensional cases using non-uniform fast Fourier transform

A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a central component to computing and analyzing the angular power spectrum of the massive CMB data sets. The method uses a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky's fast spherical harmonic transform (Slevinsky, 2019). For a HEALPix grid with $N$ pixels (points), the computational complexity of the method is $\mathcal{O}(N\log^2 N)$, with an initial set-up cost of $\mathcal{O}(N^{3/2}\log N)$. This compares favorably with $\mathcal{O}(N^{3/2})$ runtime complexity of the current methods available in the HEALPix software when multiple maps need to be analyzed at the same time. Using numerical experiments, we demonstrate that the new method also appears to provide better accuracy over the entire angular power spectrum of synthetic data when compared to the current methods, with a convergence rate at least two times higher