A novel phase‐unwrapping method based on pixel clustering and local surface fitting with application to Dixon water–fat MRI

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The performance of phase‐related MRI techniques, such as chemical shift mapping 1, magnetic susceptibility mapping 4, blood flow imaging 6, and brain phase imaging 8, usually depends on the accurate acquisition of phase information. The phase calculated from the complex MR data is generally wrapped into the range of (−π, π], and phase unwrapping is usually required to recover the underlying true phase. A large number of phase‐unwrapping algorithms have been proposed 2. Phase‐unwrapping methods are generally based on the assumption that the underlying true phase is smooth and the phase difference between adjacent pixels is less than π. If this assumption is satisfied, then the true phase map can be easily obtained 10. In the presence of strong noise and rapid phase changes, the true phase difference between adjacent pixels may be above π 2. In this scenario, accurately recovering the underlying true phase is a nontrivial task, especially for two and higher dimensional data. A further potential challenge of phase unwrapping is the presence of disconnected areas in the region of interest (ROI). Current phase‐unwrapping algorithms tackle these challenges using various approaches, which can be divided primarily into two categories: path‐following approaches 15 and minimum‐norm methods 20.
Path‐following approaches exploit the phase gradient of adjacent pixels to realize line integration in the entire map, according to Itoh's theory 24. If poles exist, then the unwrapped result will depend on the integration path. Most path‐following methods optimize the integration path to resolve the inconsistency caused by poles 2. For example, in Goldstein's branch cut algorithm 15, poles in the wrapped image are identified and connected by branch cuts, and the unwrapped phase is calculated by the line integral along a path that does not cross any branch cuts. Phase unwrapping based on the region‐growing approach can be considered a kind of path‐following method. In region‐growing methods, a pixel in a region with relatively uniform phase is selected as the starting seed, and phase information from already unwrapped regions is then used to determine the location of the growing pixel and to predict the correct phase value of this pixel. The performance of region‐growing methods depends on the selection of seeds and growing pixel (ie, growing path) 25. Xu and Cumming 25 proposed to guide the unwrapping path along the most reliable directions by checking the consistency of phase predictions. A quality map represents the quality of phase in each pixel in the given image, and can be used to guide the region‐growing path. Existing quality maps include the local first‐order phase derivative variances 27, the local second‐order partial derivative variances 26 of the phase, and the magnitude information of each pixel 28. In the path‐dependent approaches, errors occurring at any point along an integral or region‐growing path will propagate to the unwrapping of the following pixels, and accumulate to produce severe errors in results.
Minimum‐norm methods achieve phase unwrapping by minimizing the difference between the local derivative of the true phase and that of the wrapped phase 10. The simplest minimum‐norm phase‐unwrapping approach is perhaps the least‐squares method, which minimizes the sum of the squared difference between the derivatives of the wrapped and estimated phase 2. The minimum‐norm approach can be improved by introducing weights to the cost function or injecting a mask to mask out inconsistent pixels 2. The true phase in the entire map can be modeled as an empirical mathematical function, such as a polynomial models 29, truncated Taylor series 21, and Markov model 31. With these models, phase unwrapping is translated into a parameter‐estimation problem.

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