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

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Paper

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The authors' code

MATLAB · 68 lines · 2 KB · MIT

function C = C_matrix_2D(x, varargin)
% Function that calculates the C matrix.
%
% Input parameters:
% --x: N1 x N2 x Nc Nyquist-sampled k-space data, where N1 and
% N2 are the data dimensions, and Nc is the number of
% channels in the array.
%
% --tau: Parameter (in Nyquist units) that determines the size
% of the k-space kernel. For a rectangular kernel, the
% size corresponds to (2*tau+1) x (2*tau+1). For an
% ellipsoidal kernel, it corresponds to the radius of the
% associated neighborhood. Default: 3.
%
% --kernel_shape: Binary variable. 0 = rectangular kernel, 1 = ellipsoidal
% kernel. Default: 1.
p = inputParser;
p.addRequired('x', @(x) isnumeric(x) && ndims(x) == 3);
p.addParameter('tau', 3, @(x) isnumeric(x) && isscalar(x));
p.addParameter('kernel_shape', 1, @(x) isnumeric(x) && isscalar(x) && (x == 0 || x == 1));
if isempty(varargin)
parse(p, x);
else
parse(p, x, varargin{:});
end
[N1, N2, Nc] = size(p.Results.x);
x = reshape(x, N1 * N2, Nc);
[in1, in2] = meshgrid(-p.Results.tau:p.Results.tau, -p.Results.tau:p.Results.tau);
if p.Results.kernel_shape == 1
i = find(in1.^2 + in2.^2 <= p.Results.tau^2);
else
i = (1:numel(in1));
end
in1 = in1(i(:));
in2 = in2(i(:));
patchSize = numel(i);
i_centers = p.Results.tau + 1 + utils.even_pisco(N1):N1 - p.Results.tau;
j_centers = p.Results.tau + 1 + utils.even_pisco(N2):N2 - p.Results.tau;
[I_centers, J_centers] = meshgrid(i_centers, j_centers);
centers = [I_centers(:), J_centers(:)];
numCenters = size(centers, 1);
in1_row = in1(:)';
in2_row = in2(:)';
I_all = centers(:,1) + in1_row;
J_all = centers(:,2) + in2_row;
ind_all = sub2ind([N1, N2], I_all, J_all);
x_selected = x(ind_all, :);
x_patches = reshape(x_selected, numCenters, patchSize, Nc);
C = reshape(x_patches, numCenters, patchSize * Nc);
end

C_matrix_2D.m at commit 986a00b, under MIT · at the source

Overview

Authors: Moritz Blumenthal¹, Martin Uecker^1,2,3,4

ORCID iDs: Moritz Blumenthal, Martin Uecker

Institutions: Graz University of Technology (Austria); Universitätsmedizin Göttingen (Germany); BioTechMed-Graz (Austria); German Centre for Cardiovascular Research (Germany); University of Göttingen (Germany)

Journal: Magnetic resonance in medicine, volume 96, issue 1, pages 134-145

Dates: received 8 August 2025; accepted 23 February 2026; published online 12 March 2026; in print July 2026

Type: Research article · Language: English

License: CC BY

Identifiers: DOI 10.1002/mrm.70333 · PMID 41820227 · PMCID PMC13156459 · OpenAlex W4413113798

Open access: green, a free copy (OpenAlex)

Status: code verified

Categories: structural MRI / diffusion (modality), human (organism)

Methods: Spectral & time-frequency

Keywords: image reconstruction, MRI, non‐linear inverse problems, parallel imaging, phase singularity

MeSH: Heart*, Image Processing, Computer-Assisted*, Magnetic Resonance Imaging*, Algorithms, Brain, Humans, Image Enhancement, Nonlinear Dynamics, Reproducibility of Results, Sensitivity and Specificity (* major topic)

Topic: Advanced MRI Techniques and Applications (Radiology, Nuclear Medicine and Imaging, Medicine), according to OpenAlex

Funding: Deutsches Zentrum für Herz-Kreislaufforschung (81Z0300115); Austrian Science Fund FWF (10.55776/F100800); Deutsche Forschungsgemeinschaft (432680300-SFB 1456, 432680300—SFB 1456); NIH HHS (U24EB029240); National Institutes of Health (U24EB029240)

Citations: cited by 4 papers (Europe PMC); 75 references in the paper

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. In addition, we demonstrate the algorithm can also remove phase poles in ESPIRiT coil sensitivity maps. Phase pole correction is evaluated for reconstruction of accelerated Cartesian MPRAGE data of the brain and interactive radial real‐time MRI of the human heart.

Results: For both applications, phase pole correction can be used for estimation of coil sensitivity profiles free from singularities. For NLINV, we demonstrate reliable and efficient estimation even from very small (7×7) auto‐calibration (AC) regions.

Conclusion: With the proposed method, phase poles can be reliably removed in reconstructions and coil sensitivities.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repositories

Its files are read in the Code ↔ Paper reader above.

ralobos/PISCO

License: MIT

State: the link answers, verified on 30 September 2026

Evidence: files inventoried

Commit: 986a00bc1613fa9258193e3ed2e10a571073146b, 2 September 2025

Languages: MATLAB (21)

Size: 28 files, 21 scripts

Software Heritage: not archived

Found in: the text, “ENDNOTES”

Holds: README, license file

Not found: CITATION.cff, environment file, tests, continuous integration, documentation

Availability: 1 check, the latest on 30 September 2026: the link answers

30 September 2026: the link answers

23 files

+utils/C_matrix_2D.m, MATLAB, 68 lines
+utils/C_matrix_3D.m, MATLAB, 70 lines
+utils/ChC_FFT_convolutions_2D.m, MATLAB, 74 lines
+utils/ChC_FFT_convolutions_3D.m, MATLAB, 85 lines
+utils/G_matrices_2D.m, MATLAB, 136 lines
+utils/G_matrices_3D.m, MATLAB, 152 lines
+utils/even_pisco.m, MATLAB, 3 lines
+utils/fft3.m, MATLAB, 18 lines
+utils/ft3.m, MATLAB, 15 lines
+utils/ifft3.m, MATLAB, 14 lines
+utils/ifft3_zp.m, MATLAB, 15 lines
+utils/ift3.m, MATLAB, 15 lines
+utils/mdisp.m, MATLAB, 21 lines
+utils/nullspace_vectors_C_matrix_2D.m, MATLAB, 161 lines
+utils/nullspace_vectors_C_matrix_3D.m, MATLAB, 161 lines
+utils/nullspace_vectors_G_matrix_2D.m, MATLAB, 349 lines
+utils/nullspace_vectors_G_matrix_3D.m, MATLAB, 388 lines
+utils/vect.m, MATLAB, 3 lines
PISCO_sensitivity_map_estimation.m, MATLAB, 354 lines
example_2D.m, MATLAB, 213 lines
example_3D.m, MATLAB, 240 lines
LICENSE, License, 22 lines
readme.md, Text, 57 lines

At the source: github.com/ralobos/PISCO

gitlab.tugraz.at/ibi/mrirecon/papers/phase-pole

License: CC-BY-4.0

State: the link answers, verified on 30 September 2026

Evidence: files inventoried

Commit: 7d8e5fb2b9adcae19fc0100e8e5098c8f166694d, 1 February 2026

Languages: Shell (25), MATLAB (2), Python (1)

Size: 82 files, 28 scripts

Software Heritage: not archived

Found in: “Data Availability Statement”

Holds: README, license file

Not found: CITATION.cff, environment file, tests, continuous integration, documentation

Tools: Parallel Computing Toolbox (1 file), Matplotlib (1 file), NumPy (1 file), SciPy (1 file)

Availability: 1 check, the latest on 30 September 2026: the link answers

30 September 2026: the link answers

30 files

01_data_brain/run.sh, Shell, 57 lines
01_data_brain/run_loop.sh, Shell, 47 lines
02_data_realtime/run.sh, Shell, 31 lines
figure_01_phase_singularity/fig.py, Python, 106 lines
figure_01_phase_singularity/fig.sh, Shell, 13 lines
figure_01_phase_singularity/run.sh, Shell, 33 lines
figure_02_detection/fig.sh, Shell, 39 lines
figure_02_detection/run.sh, Shell, 42 lines
figure_03_iterations/fig.sh, Shell, 28 lines
figure_03_iterations/run.sh, Shell, 51 lines
figure_04_coil_projection/fig.sh, Shell, 32 lines
figure_04_coil_projection/run.sh, Shell, 67 lines
figure_05_speed/PISCO_bart.m, MATLAB, 78 lines
figure_05_speed/PISCO_bart_loop.m, MATLAB, 135 lines
figure_05_speed/fig.sh, Shell, 24 lines
figure_05_speed/run.sh, Shell, 74 lines
figure_05_speed/run_loop.sh, Shell, 71 lines
figure_06_espirit/fig.sh, Shell, 40 lines
figure_06_espirit/run.sh, Shell, 33 lines
figure_07_realtime/fig.sh, Shell, 21 lines
figure_07_realtime/run.sh, Shell, 12 lines
run_all.sh, Shell, 34 lines
sup_figure_01_detection_real/fig.sh, Shell, 62 lines
sup_figure_01_detection_real/run.sh, Shell, 55 lines
sup_figure_03_3D_phase_pole_espirit/fig.sh, Shell, 19 lines
sup_figure_03_3D_phase_pole_espirit/run.sh, Shell, 29 lines
sup_video_01/fig.sh, Shell, 14 lines
sup_video_01/run.sh, Shell, 14 lines
LICENSE, License, 395 lines
README.md, Text, 39 lines

At the source: gitlab.tugraz.at/ibi/mrirecon/papers/phase-pole

The paper's code and data availability statement is in the Data section.

Tracing map

Proposed by the machine: these links were found in the paper and verified at the source, without human review. The map will receive a Zenodo DOI once one of the paper's authors has validated it with their ORCID.

What the map holds:

2 repositories of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
49 scripts, each with its path and the digest of its content;
no match between paragraphs and code yet;
neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

Datasets cited

zenodo:16737746, at Zenodo; found in “Data Availability Statement”

Data Availability Statement

In the spirit of reproducible research, the code to reproduce the results of this paper is available at https://gitlab.tugraz.at/ibi/mrirecon/papers/phase‐pole (https://gitlab.tugraz.at/ibi/mrirecon/papers/phase-pole) (Version v0.2). All reconstructions have been performed with BART, available at https://github.com/mrirecon/bart. The data used in this study is available at Zenodo https://doi.org/10.5281/zenodo.16737746.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

The history of this record: each version stored by the harvester or made by a correction of its authors or of the maintainers of its code, and what changed in its facts. The texts of the paper (its abstract, its availability statements) are not part of it; versions that changed only those are not listed.

Version 1, 30 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 2 authors, 5 keywords, 10 MeSH terms, 5 funders, 63 references.

Cite

This paper

Blumenthal, M., & Uecker, M. (2026). Phase-Pole-Free Images and Smooth Coil Sensitivity Maps by Regularized Nonlinear Inversion. Magnetic resonance in medicine, 96(1), 134-145. https://doi.org/10.1002/mrm.70333

BibTeX

@article{blumenthal2026phase, author = {Blumenthal, Moritz and Uecker, Martin}, title = {{Phase-Pole-Free Images and Smooth Coil Sensitivity Maps by Regularized Nonlinear Inversion}}, journal = {Magnetic resonance in medicine}, year = {2026}, month = mar, volume = {96}, number = {1}, pages = {134--145}, publisher = {Wiley}, issn = {0740-3194}, doi = {10.1002/mrm.70333}, url = {https://doi.org/10.1002/mrm.70333}, pmid = {41820227}, pmcid = {PMC13156459} }

RIS

TY - JOUR AU - Blumenthal, Moritz AU - Uecker, Martin TI - Phase-Pole-Free Images and Smooth Coil Sensitivity Maps by Regularized Nonlinear Inversion T2 - Magnetic resonance in medicine J2 - Magn Reson Med PY - 2026 DA - 2026/03/12 VL - 96 IS - 1 SP - 134 EP - 145 SN - 0740-3194 PB - Wiley DO - 10.1002/mrm.70333 UR - https://doi.org/10.1002/mrm.70333 LA - en ER -

CSL-JSON

{ "id": "10.1002/mrm.70333", "type": "article-journal", "title": "Phase-Pole-Free Images and Smooth Coil Sensitivity Maps by Regularized Nonlinear Inversion", "container-title": "Magnetic resonance in medicine", "author": [ { "family": "Blumenthal", "given": "Moritz" }, { "family": "Uecker", "given": "Martin" } ], "container-title-short": "Magn Reson Med", "volume": "96", "issue": "1", "page": "134-145", "DOI": "10.1002/mrm.70333", "PMID": "41820227", "PMCID": "PMC13156459", "ISSN": "0740-3194", "publisher": "Wiley", "URL": "https://doi.org/10.1002/mrm.70333", "language": "en", "issued": { "date-parts": [ [ 2026, 3, 12 ] ] } }

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