Spatiotemporal Maps for Dynamic MRI Reconstruction

TL;DR

This paper introduces Spatiotemporal Maps (STMs) to address limitations of traditional PSF models in dynamic MRI by allowing voxel-dependent temporal subspaces, enabling a more parsimonious yet flexible representation of $\rho(\vec{x},t)$ via $\rho(\vec{x},t) \approx \sum_{l=1}^{L(\vec{x})} s_l(\vec{x},t) \, \rho_l(\vec{x})$.The authors establish a theoretical framework with Shift-Invariant Linear Predictability (SILP) in $k-t$ space, derive sufficient conditions under a multiband spectral model for the existence of SILP, and show how SILP leads to STM computation through voxelwise nullspaces of $\mathbf{G}(\vec{x})=\mathbf{H}^H(\vec{x})\mathbf{H}(\vec{x})$.A practical, ACS-driven workflow computes STMs using multiframe FIR filters, an FFT-based approach to obtain $\mathbf{G}(\vec{x})$, and a sketched SVD to efficiently approximate the nullspace basis, with extension to multichannel data via virtual coil combination.Incorporating STMs into a reconstruction framework reduces the dynamic MRI problem to estimating a small set of static spatial components, enabling regularizers and machine-learning priors to be applied to spatial functions; experiments on 2D GI rat data and 3D fMRI show improved representation and functional activation recovery, along with substantial computational savings from the sketched SVD.

Abstract

The partially separable functions (PSF) model is commonly adopted in dynamic MRI reconstruction, as is the underlying signal model in many reconstruction methods including the ones relying on low-rank assumptions. Even though the PSF model offers a parsimonious representation of the dynamic MRI signal in several applications, its representation capabilities tend to decrease in scenarios where voxels present different temporal/spectral characteristics at different spatial locations. In this work we account for this limitation by proposing a new model, called spatiotemporal maps (STMs), that leverages autoregressive properties of (k, t)-space. The STM model decomposes the spatiotemporal MRI signal into a sum of components, each one consisting of a product between a spatial function and a temporal function that depends on the spatial location. The proposed model can be interpreted as an extension of the PSF model whose temporal functions are independent of the spatial location. We show that spatiotemporal maps can be efficiently computed from autocalibration data by using advanced signal processing and randomized linear algebra techniques, enabling STMs to be used as part of many reconstruction frameworks for accelerated dynamic MRI. As proof-of-concept illustrations, we show that STMs can be used to reconstruct both 2D single-channel animal gastrointestinal MRI data and 3D multichannel human functional MRI data.

Figures (8)

• Figure 1: (a) k-space mask used to retrospectively undersample 2D+T dataset (A.1). White indicates the PE lines considered in each time frame. (b) Analogous k-space mask used to retrospectively undersample 3D+T dataset (B). In this case two PE directions are considered.
• Figure 2: (a) Eigenvalue maps representing the last $10$ eigenvalues (after normalizing and sorting in decreasing order) of the matrices $\bm{G}\xspace(\vec{x}\xspace)$ for each spatial location in the FOV using dataset (A.1). (b) First $10$ singular values of the Casorati matrix for dataset (A.1).
• Figure 3: Normalized projection residual versus the number of components of the PSF and STM models. (b) is a zoom in of (a).
• Figure 4: Visualization of the dynamic behavior of two voxels using dataset (A.1). (a) Approximate locations of the two voxels overlaid with an image corresponding to the first time frame. (b) Time evolutions of $\rho(\vec{x}\xspace, t)$ and $g_{t'}(\vec{x}\xspace, t)$ for voxel A (top), and their respective frequency spectra (bottom). The frequency range is shown after normalization in cycles/sec. (c) Analogous results for voxel B.
• Figure 5: (a) Computation time of the sketched SVD versus sketch dimension for dataset (A.1). The median over $50$ realizations is reported in each case. (b) Mean and standard deviations of the NPR obtained in each case.