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. 2014 Jul 17;73(1):82–101. doi: 10.1002/mrm.25358

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© 2014 The Authors. Magnetic Resonance in Medicine Published by Wiley Periodicals, Inc. on behalf of International Society of Medicine in Resonance

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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The ill-posedness of the dipole inverse problem. The unit dipole field in sagittal section (i) and its surface rendered contour (ii). (iii) The zero cone surfaces of the dipole kernel in k-space. (iv) Field map derived at signal-to-noise ratio (SNR) – 20 induced by a point source. (v, vi) Susceptibility in image space obtained by truncated k-space division with the threshold – 0 and 0.1. As a consequence of the dipole kernel zero behavior in the cone surface neighborhood, there is substantial noise propagation from the field measurements into the susceptibility estimate (40), as illustrated in an example of reconstruction by direct division (v and vi). A little noise added in the phase map (peak SNR – 20) leads to a totally corrupted susceptibility image that bears no physical resemblance to the true susceptibility source.

Inline graphic — The ill-posedness of the dipole inverse problem. The unit dipole field in sagittal section (i) and its surface rendered contour (ii). (iii) The zero cone surfaces of the dipole kernel in k-space. (iv) Field map derived at signal-to-noise ratio (SNR) – 20 induced by a point source. (v, vi) Susceptibility in image space obtained by truncated k-space division with the threshold – 0 and 0.1. As a consequence of the dipole kernel zero behavior in the cone surface neighborhood, there is substantial noise propagation from the field measurements into the susceptibility estimate (40), as illustrated in an example of reconstruction by direct division (v and vi). A little noise added in the phase map (peak SNR – 20) leads to a totally corrupted susceptibility image that bears no physical resemblance to the true susceptibility source.