A New k-Space Model for Non-Cartesian Fourier Imaging

TL;DR

The paper identifies fundamental limitations of the conventional image-domain voxel-based model for non-Cartesian Fourier imaging, including k-space periodicity, limited representation capacity, and structured artifacts. It introduces a dual Fourier-domain model with localized, nonperiodic k-space basis functions $\Psi(k)$, enabling sparse forward operators, center-weighted subspace energy, and faster convergence. The approach is demonstrated through 1D/2D MRI reconstructions, showing reduced artifacts and substantial speedups in both single- and multi-channel settings (LORAKS and SENSE+TV), with oversampling parameter $\rho$ and B-spline degree $P$ governing performance. The results suggest broad applicability to non-Cartesian MRI and potentially other Fourier-imaging modalities, highlighting practical benefits in reconstruction quality and efficiency.

Abstract

For the past several decades, it has been popular to reconstruct Fourier imaging data using model-based approaches that can easily incorporate physical constraints and advanced regularization/machine learning priors. The most common modeling approach is to represent the continuous image as a linear combination of shifted "voxel" basis functions. Although well-studied and widely-deployed, this voxel-based model is associated with longstanding limitations, including high computational costs, slow convergence, and a propensity for artifacts. In this work, we reexamine this model from a fresh perspective, identifying new issues that may have been previously overlooked (including undesirable approximation, periodicity, and nullspace characteristics). Our insights motivate us to propose a new model that is more resilient to the limitations (old and new) of the previous approach. Specifically, the new model is based on a Fourier-domain basis expansion rather than the standard image-domain voxel-based approach. Illustrative results, which are presented in the context of non-Cartesian MRI reconstruction, demonstrate that the new model enables improved image quality (reduced artifacts) and/or reduced computational complexity (faster computations and improved convergence).

Figures (11)

• Figure 1: (top) Examples of k-space basis functions associated with Eq. \ref{['eq:img-model']}, corresponding to different shifts of $\xi_N^{(\Delta x)}(k)$. While the functions with small shifts near the center of k-space (green) have unremarkable characteristics, those with large shifts (orange) wrap around from one side of k-space to the other. This can have undesirable consequences for image reconstruction, where (bottom) a simple minimum-norm least squares reconstruction (blue, obtained using $\hat{\mathbf{b}} = \mathbf{A}^\dagger \mathbf{d}$) of a single off-grid k-space sample (red) on one side of k-space results in signal leaking to the opposite side. (Magnitude plots are shown, each curve has generalized linear phase).
• Figure 2: Plot of $\mathcal{E}(x_0)$, the best-case relative approximation error between the ideal signal arising from spatial location $x_0$ and the k-space signal model associated with Eq. \ref{['eq:img-model']}. Voxel locations are marked with black circles.
• Figure 3: Structured k-space artifacts produced using the voxel-based model, for radial glover1992, spiral meyer1992, rosette noll1997, and bunched phase encoding (BPE) moriguchi2006 trajectories. (left) k-space trajectories. (middle) Reconstructed k-space using Eq. \ref{['eq:rec']}. (right) Projection of the reconstruction onto the near-nullspace of $\mathbf{A}$.
• Figure 4: (top) Examples of different shifts of a compactly-supported nonperiodic k-space basis function $\Psi(k)$ (a third-degree B-spline in this case unser1999) associated with Eq. \ref{['eq:kspace-model']}. The wraparound seen in Fig. \ref{['fig:periodicity']} is not present. In addition, (bottom) we no longer observe signal leakage from one side of k-space to the other when performing minimum-norm least squares reconstruction (blue, obtained using $\hat{\mathbf{b}} = \mathbf{A}^\dagger \mathbf{d}$) of an off-grid k-space sample (red).
• Figure 5: Plots of $\mathcal{E}(x_0)$ for different models associated with Eqs. \ref{['eq:img-model']} and \ref{['eq:kspace-model']}.