To investigate the symmetry constraint in susceptibility tensor imaging.
Theory
The linear relationship between the MRI frequency shift and the magnetic susceptibility tensor is derived without constraining the tensor to be symmetric. In the asymmetric case, the system matrix is shown to be maximally rank 6. Nonetheless, relaxing the symmetry constraint may still improve tensor estimation because noise and image artifacts do not necessarily follow the constraint.
Methods
Gradient echo phase data are obtained from postmortem mouse brain and kidney samples. Both symmetric and asymmetric tensor reconstructions are applied to the data. The reconstructions are then used for susceptibility tensor imaging fiber tracking. Simulations with ground truth and at various noise levels are also performed. The reconstruction methods are compared qualitatively and quantitatively.
Results
Compared to regularized and unregularized symmetric reconstructions, the asymmetric reconstruction shows reduced noise and streaking artifacts, better contrast, and more complete fiber tracking. In simulation, the asymmetric reconstruction achieves better mean squared error and better angular difference in the presence of noise. Decomposing the asymmetric tensor into its symmetric and antisymmetric components confirms that the underlying susceptibility tensor is symmetric and that the main sources of asymmetry are noise and streaking artifacts.
Conclusion
Whereas the susceptibility tensor is symmetric, asymmetric reconstruction is more effective in suppressing noise and artifacts, resulting in more accurate estimation of the susceptibility tensor.
Thanks for putting together this manuscript and your continued work with STI. I had a couple of questions about the methodology and results.
Your implementation of aSTI included a minimum-norm term to address the A matrix rank. Have you investigated to what extent the minimum-norm term alone has on improving STI reconstruction (e.g. comparing aSTI verses a conventional STI reconstruction including the minimum norm regularisation)?
Following on from this, whether you would be able to provide any further insights into the origin of the asymmetric term in the susceptibility tensor. Discussing with colleagues, we initially considered whether it could arise from a component of the experimental image acquisition. However, if I am correct your simulations only included Gaussian noise, suggesting that the asymmetry originates from this contribution alone. Any further insight into how the asymmetric component arises would be appreciated.
Your question about the minimum norm is interesting. Just to clarify, our implementation for aSTI does not add any regularization terms. The reason we use minimum-norm is because the system matrix is low rank, so we choose the minimum-norm solution out of the possible solutions. We have not tried other options for choosing between the solutions but that would be an interesting thing to explore.
As you point out, for conventional STI (which is full rank), one could add an L2 regularization term for noise reduction. We have not tried this, but we suspect that it would play out similarly to L2 regularization in other domains, where it reduces noise at the cost of blurring.
This question about the source of asymmetry is a great one, and we do not have a full answer to it. As you point out, even just adding white Gaussian noise produces asymmetry. In the paper, we posit that any components of the measured signal beyond the contribution of the susceptibility has the potential to look asymmetric, causing it to be removed when we do aSTI. It would be interesting to further explore this question.