Discussion and Conclusion

This report has presented a fast, fully automated, robust phase unwrapping algorithm that can be applied for phase maps of any dimension. The implementation of this algorithm for 2D and 3D MRI images has been demonstrated and the code is currently being used for EPI unwarping and rapid, automated shimming applications in fMRI.

There are two potential weaknesses in the current approach. The first is during the creation of the initial regions. If a single region includes two areas where the original phase differs by more than $2\pi$ then the algorithm can never successfully recover this original phase difference. Using the method of phase partitioning outlined in section 2.4 this is unlikely to happen in low resolution images (typical EPI images), although it has been noted to happen in high resolution images (voxel sizes of 1mm$^3$ or less) where small tracks of closely related phase voxels can be found in noisy areas, linking up larger, stable areas which should not be grouped in the same region. In practice we have overcome this by limiting the initial regions to be 2D (within plane only) for high resolution images. This dramatically reduces the likelihood of finding connecting tracks within noisy areas. In fact, we have not found any phase maps where two sizeable areas have been incorrectly connected using this method. Other methods, such as splitting regions into convex components (possibly by morphological operations) could also be pursued. However, the greater the number of initial regions, $N$, the longer the unwrapping takes to perform. In fact, applying a pre-processing stage which merged very small (single voxel or so) regions could result in a considerable speed-up for larger images.

The second potential weakness of the method is that the region merging algorithm attempts to solve the integer programming problem by ``greedy'' optimisation. That is, always choosing the best (least opportunity cost) candidate pair at each stage, and so can get stuck in local minima. Other optimisation methods could be used to solve the cost function minimisation instead, but although this method does not guarantee the optimal solution, it is very fast and we believe that it is a good compromise between speed and guaranteed accuracy. Moreover, unlike region-growing, the results of the region-merging algorithm do not depend on the location of any initial seed point.

Testing of the accuracy and robustness of the method was performed using a set of simulated images. These images contained various amounts of added Gaussian noise so that the performance of the algorithm at different SNRs could be determined. The results, quantified using the Mis-Classification Ratio, showed that the algorithm was extremely accurate and robust, for SNRs greater than 5. For most MR imaging methods the SNR is significantly better than this (our B0 mapping sequence had a SNR of 50), and therefore this method should be able to perform well with most MR phase data.

The software implementation of this phase unwrapping method is freely available (from www.fmrib.ox.ac.uk/fsl as a part (PRELUDE) of the EPI unwarping tool (FUGUE)) in both binary and source forms. It is hoped that this will enable more users in the MR community to develop applications where fast, automated phase unwrapping is necessary, and to participate in the evaluation and improvement of the current algorithm.