Sampling Density Compensation using Fast Fourier Deconvolution

TL;DR

This work tackles the slow, iterative computation of density compensation functions (DCFs) in non-Cartesian MRI by introducing a fast, non-iterative approach based on Fast Fourier Deconvolution (FFD). By decomposing the PSF and optimizing a windowed sampling pattern, the method derives an optimal weighted pattern $E^{\star}(\mathbf{k})$ that steers the PSF toward an impulse within the field of view, implemented via $FFD$. Across 2D and 3D trajectories, the proposed method achieves reconstruction-quality metrics comparable to a state-of-the-art iterative baseline while reducing DCF computation time from minutes to around 20 seconds for 3D trajectories. This substantial speedup enables efficient non-Cartesian MRI pipelines and broad practical adoption without sacrificing image fidelity.

Abstract

Density Compensation Function (DCF) is widely used in non-Cartesian MRI reconstruction, either for direct Non-Uniform Fast Fourier Transform (NUFFT) reconstruction or for iterative undersampled reconstruction. Current state-of-the-art methods involve time-consuming tens of iterations, which is one of the main hurdles for widespread application of the highly efficient non-Cartesian MRI. In this paper, we propose an efficient, non-iterative method to calculate DCF for arbitrary non-Cartesian $k$-space trajectories using Fast Fourier Deconvolution. Simulation experiments demonstrate that the proposed method is able to yield DCF for 3D non-Cartesian reconstruction in around 20 seconds, achieving orders of magnitude speed improvement compared to the state-of-the-art method while achieving similar reconstruction quality.

Figures (5)

• Figure 1: The complex-valued digital phantom used for simulation. The top row displays magnitude images of the (A) axial, (B) coronal, and (C) sagittal center slices. The bottom row shows the corresponding phase images for the same slices (D, E, F).
• Figure 2: (A) Min-max parameter search in the 1D random sampling experiment. (B) The adopted $W^\star(\mathbf{x})$.
• Figure 3: DCF calculated using the baseline and the proposed methods for an interleaf of the Yarnball trajectory.
• Figure 4: PSF resulting from the DCF calculated by the baseline method (A-C) and the proposed method (D-F). The figure displays logarithmic-scaled PSF of three cross-sectional planes: $Z=0$ (A,D), $Y=0$ (B,E), and $X=0$ (C,F).
• Figure 5: The digital phantom images of three orthogonal slices reconstructed using the DCF calculated by the baseline method (A-C) and the proposed method (G-I). The difference images between the reconstructed and ground truth images are shown below the reconstruction for each method.