Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping.

We demonstrate in both simulated and real cases the effect that undersampling of a three-dimensional (3D) wrapped phase distribution has on the geometry of phase singularity loops and their branch cut surfaces. The more intuitive two-dimensional (2D) problem of setting branch cuts between dipole pairs is taken as a starting point, and then branch cut surfaces in flat and ambiguous 3D loops are discussed. It is shown that the correct 2D branch cuts and 3D branch cut surfaces should be placed where the gradient of the original phase distribution exceeded pi rad voxel(-1). This information, however, is lost owing to undersampling and cannot be recovered from the sampled wrapped phase distribution alone. As a consequence, empirical rules such as finding the surface of minimal area or methods based on the wrapped phase gradient will fail to find the correct branch cut surfaces. We conclude that additional information about the problem under study is therefore needed to produce correct branch cut surfaces that lead to an unwrapped phase distribution with minimum local errors. An example with real data is provided in which downsampled phase contrast magnetic resonance imaging data are successfully unwrapped when the position of the vessel walls and the physical properties of the flowing blood are taken into account.