Phase-Pole-Free Images and Smooth Coil Sensitivity Maps by Regularized Nonlinear Inversion

TL;DR

This paper addresses phase singularities arising from phase ambiguity in MR image reconstructions with auto-calibrated coil sensitivities by introducing a curl-based phase-pole detection scheme applied to coil sensitivity maps. The authors integrate a phase-pole correction into the NLINV framework via a global optimization step that multiplies the image by a phase vortex and the coil maps by the conjugate pole, iteratively refining this correction within IRGNM. They demonstrate reliable removal of phase poles in both Cartesian brain MPRAGE and interactive radial real-time cardiac MRI, achieving accurate coil sensitivity estimation even from very small AC regions (as small as 7×7). The approach incurs minimal computational overhead (<10%) and enables robust NLINV reconstructions, offering a practical, real-time-capable solution for phase-singularity-free imaging and smooth coil sensitivity maps in challenging MRI applications.

Abstract

Purpose: Phase singularities are a common problem in image reconstruction with auto-calibrated sensitivities due to an inherent ambiguity of the estimation problem. The purpose of this work is to develop a method for detecting and correcting phase poles in non-linear inverse (NLINV) reconstruction of MR images and coil sensitivity maps. Methods: Phase poles are detected in individual coil sensitivity maps by computing the curl in each pixel. A weighted average of the curl in each coil is computed to detect phase poles. Phase pole detection and correction is then integrated into the iteratively regularized Gauss-Newton method of the NLINV algorithm, which then avoid phase singularities in the reconstructed images. The method is evaluated for reconstruction of accelerated Cartesian MPRAGE data of the brain and interactive radial real-time MRI of the human heart. Results: Phase poles are reliably removed in NLINV reconstructions for both applications. NLINV with phase pole correction can reliably and efficiently estimate coil sensitivity profiles free from singularities even from very small ($7\times7$) auto-calibration (AC) regions. Conclusion: NLINV emerges as an efficient and reliable tool for image reconstruction and coil sensitivity estimation in challenging MRI applications.

Figures (8)

• Figure 1: Example for a phase singularity in the coil sensitivity profile. The curves $\mathcal{C}_1$ and $\mathcal{C}_2$ enclose the same phase singularity such that their image $c(\mathcal{C}_1)$ and $c(\mathcal{C}_2)$ wind once around the origin in the complex plane. The curve $\mathcal{C}_3$ does not enclose the singularity and therefore has a winding number of zero.
• Figure 2: Image-based (A) and coil-based-phase (B) phase pole detection. The image-based detection is not robust against noise leading to a random correction function. In the coil-based detection, multiple poles are detected. Those corresponding to true singularities in the sensitivities are marked by white circles and the corresponding poles are also contained in the coil-images. After taking the weighted average, only the false pole which is present in all coils and in the image is detected, leading to a single factor in the correction function. In this example, we do not show the thresholding step in (4) as it has no visible effect. Further, we skip the morphological closing in step (5), as it would connect all noise poles in the image-based detection. We refer to Supporting Figure \ref{['supfig:detection_real']} for an example with real data.
• Figure 3: NLINV reconstructions (A) and the first two coil sensitivities (B) with and without phase pole correction. Insets in (A) show the complex image before normalization (\ref{['eq:normalize_rss']}). Coil sensitivities are shown once as complex maps (first row) and once phase only (second row) to highlight the phase poles which are located in the magnitude regions of the coils. After eight iterations the phase pole is corrected visible by the changed phase in the image and coils. After that the magnitude normalizes over the remaining iterations. In the final reconstruction, the phase pole in the non-corrected reconstruction leads to a black hole marked by the red arrow. Orange circles mark the position of the phase pole in the coil sensitivities. The white circles mark the position of a true phase pole only present in the 2nd coil sensitivity map. Phase is color-coded as in Figure 1.
• Figure 4: Evaluation of coils estimated with ESPIRiT (A), NLINV (B), and NLINV with phase pole correction (C). The first column shows the root-sum-of-squares difference between the coil images and their point-wise projection to the space spanned by the coil sensitivities. The second column shows reconstructed images using the respective coils in an $\ell_1$-Wavelet reconstruction. The third column shows the difference of reconstructions to the fully sampled reference. Rows correspond to different sizes of full-sampled AC regions. The inlaid images show the first estimated virtual coil to visualize the position of the phase pole (phase is color-coded as in Figure 1). NLINV provides good coil sensitivities starting from $7\times7$ AC regions, while phase pole correction removes remaining artifacts in this example. ESPIRiT requires larger AC regions to achieve similar results.
• Figure 5: Comparison of different coil sensitivity estimation methods in terms of estimation time, projection error, and reconstruction error. In the top row (A-D), coil sensitivities (phase is color-coded as in Figure 1) are estimated on the full resolution grid. ESPIRiT and PISCO are similarly fast, while NLINV is significantly slower. However, with phase pole correction, NLINV provides the best coil sensitivities in terms of the projection test, leading to the lowest difference of the reconstruction to the reference. Phase poles are marked by orange arrows and corresponding aliasing artifacts by green arrows. The bottom row (E-G) shows results for coil sensitivities estimated on a low resolution grid by PISCO and NLINV. Due to interpolation of a non-smooth phase, PISCO introduces an artifact (red arrow) in the coils, which is visible in the projection test and leads to an artifact (dark spot visible in the inset) in the reconstruction.