Reducing the sampling burden of Fourier sensing with a non-rectangular field-of-view

Nicholas Dwork; Erin K. Englund; Alex J. Barker

doi:10.3934/ammc.2025006

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June 2025, Volume 4, 1-16. Doi: 10.3934/ammc.2025006

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Reducing the sampling burden of Fourier sensing with a non-rectangular field-of-view

1.
Department of Biomedical Informatics, University of Colorado — Anschutz Medical Center, USA
2.
Department of Radiology, University of Colorado — Anschutz Medical Center, USA

^*Corresponding author: Nicholas Dwork
^*Corresponding author: Nicholas Dwork

Received: November 2024

Revised: March 2025

Published: June 2025

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Abstract

Fourier sensing devices collect data in the frequency domain. For example, Magnetic Resonance Imaging (MRI) is a Fourier sensing modality where the user has the freedom to choose which spatial frequencies are sampled. With Fourier sensing, it is commonly the case that the field-of-view (FOV), the area of space to be imaged, is known prior to reconstruction. Most commonly, reconstruction algorithms have focused on FOVs with simple geometries. This leads to sampling patterns that are more burdensome (with more samples) than necessary. Due to the reduced area of imaging possible with an arbitrary (e.g., non-rectangular and non-convex) FOV, the number of samples required for a high-quality image is reduced. However, when an arbitrary FOV has been considered, the reconstruction algorithm has been computationally expensive. In this manuscript, we present a method to reduce the sampling burden for an arbitrary FOV with an accompanying direct (non-iterative) and computationally efficient reconstruction algorithm. We also present a method to decrease the computational cost of the iterative POCSENSE algorithm used with a non-rectangular FOV. We present results using MRI data of an ankle, a pineapple, and a brain.

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Mathematics Subject Classification: Primary: 94A12; Secondary: 41A05.

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Figure 1. An axial slice of a brain: a) the yellow rectangle and the cyan contour represent a rectangular and non-rectangular FOV, respectively, and b) a depiction of the reduced area in the non-rectangular FOV

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Figure 2. A depiction of a non-rectangular FOV and its sampling pattern. The dark blue contour in (a) shows the boundary of the non-rectangular FOV, and (b) shows the full sampling pattern associated with a rectangular FOV

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Figure 3. A depiction of how a non-rectangular FOV is separated into components: a) shows an aliased version non-rectangular FOV of Fig. 2a where the dashed lines separate the outer region of (b) and the inner region of (c)

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Figure 4. Sampling patterns related to the FOV: a) the full sampling pattern based on the rectangular FOV, b) the full sampling pattern for the inner region, and c) the proposed sampling pattern for the non-rectangular FOV

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Figure 5. The shape of the largest connected object that can be reconstructed accurately with the method described. Note that the two lengths labeled $ w $ must be equal; however, the lengths labeled $ h_1 $ and $ h_2 $ need not be

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Figure 6. Subfigure (a) shows a sagittal slice of an ankle and foot; (b) shows a corresponding non-rectangular FOV that does not include the upper-right quadrant, and (c) shows the corresponding sampling pattern for the non-rectangular FOV of (b). The sampling burden of the reduced sampling pattern is $ 75\% $

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Figure 7. Magnitude image reconstructions of a sagittal slice of an ankle and foot: a) shows the fully-sampled reconstruction, b) shows the zero-filled reconstruction with the non-rectangular FOV sampling pattern of Fig. 6b with a sampling burden of $ 75\% $, c) shows the reconstruction using Alg. 1, c) the reconstruction using only the even columns of data, d) the reconstruction after subtracting away the Fourier values of the inner region, and e) the difference between (a) and (c)

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Figure 8. Image reconstructions for six coils of a dedicated ankle array - a) Magnitude images for each coil, and b) the non-rectangular FOV determined using the intensity images of (a)

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Figure 9. Magnitude image reconstruction of the ankle using the direct reconstruction methods and the sampling pattern determined according to the individual coil FOVs shown in Fig. 8b with a sampling burden of $ 63\% $. The difference image shows the type of the errors that arise from this method; these errors are due to the fact that individual coils sense the whole subject imaged

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Figure 10. Magnitude image reconstructions of an axial slice of a pineapple from data collected with a single coil: left) the fully sampled reconstruction, center) the direct reconstruction from the undersampled pattern with a burden of $ 87.5\% $, and right) the scaled magnitude of the difference

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Figure 11. Model-based magnitude image reconstructions of an axial slice of a pineapple with sampling burdens of (from left to right) $ 100\% $, $ 87.5\% $, $ 68.8\% $, and $ 43.8\% $

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Figure 12. Magnitude image reconstructions using the direct method for four axial slices of a brain using data collected with an 8 coil array. The sampling pattern of the non-rectangular FOV had a burden of 95%

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