An MRI signal is composed of real and imaginary components. Mathematically, a complex image can be expressed as *s*(*m*,*n*) = |*s*(*m*,*n*)|exp(iϕ(*m*,*n*)), where |*s*(*m*,*n*)| denotes the magnitude image and ϕ(*m*,*n*) denotes the phase image. Magnitude images are widely used, while the phase components are often discarded. However, phase images can provide useful information about field inhomogeneity (1, 2), velocity of blood flow (3), or the chemical shift between water and fat (4). Unfortunately, extracting the values of physical parameters from phase images is not trivial because phase ϕ(*m*,*n*) is defined in the range (−π,π]. If the values of physical parameters vary across a wider range, the corresponding phase values will be outside this interval and are wrapped back into the range, resulting in so-called phase wrapping. Thus, the main difficulty of utilizing phase information is that the relationship between the physical parameters and the wrapped phase values is unknown. To obtain this information, a procedure called *phase unwrapping* is needed. A variety of different phase unwrapping algorithms have been developed. Most of them make the assumption that the true phase difference between two adjacent pixels is smaller than π. If this condition is satisfied, the unwrapped phase can be easily determined. However, in MRI this condition may be violated by noise, rapidly varying phase values, or even phase discontinuity. Phase discontinuity may be encountered in velocity-encoded imaging and field mapping (5). In this case, phase unwrapping becomes a difficult problem.

Existing phase unwrapping methods (some of which are 3D) can be classified into four different categories: path-following (5–10), cost function optimization (11–14), Bayesian (15), and parametric modeling methods (16). Since the presented method belongs to the first category, a brief introduction of path-following methods follows.

Path-following methods apply line integration over a phase gradient map. If there are no poles (17), the integration path is arbitrary. Otherwise, the phase unwrapping result is dependent on the integration path. Most path-following methods attempt to handle this inconsistency by optimizing the integration path: Goldstein's branch cut algorithm (6), for example, identifies the poles and connects them with branch cuts. The phase can then be unwrapped along any path that does not cross the branch cuts according to the residue theorem. Buckland (10) used a minimum-cost-matching method to find the set of cuts that minimizes the total cut length. Chavez et al. (5) developed a method that detects “cutlines” and distinguishes between noise-induced poles and signal undersampling poles based on the length of the “fringelines.” Quality-guided algorithms rely on quality maps to guide the integration path; they unwrap the high-quality pixels first and avoid the low-quality pixels until the end of the integration procedure. In MRI phase unwrapping, phase derivative variance is a useful quality map in practice, but in some cases classifies as “low quality” the areas that have rapidly changing phase, but high signal-to-noise ratio (SNR). Several hybrid quality maps are also available but they suffer from similar problems (7).

Xu and Cumming (8) proposed a region-growing algorithm for interferometric synthetic aperture radar phase unwrapping. This method uses the phase information from unwrapped neighbor pixels to predict the correct phase of the new pixel to be unwrapped. A reliability check is also applied to make sure that the phase unwrapping follows a robust path. Based on this, Zhu et al. (9) presented a quality-guided fitting plane algorithm, which uses phase derivative variance as a quality map. Plane fitting in a 3 × 3 window also reduces computation time and increases the reliability.

This work presents a phase unwrapping algorithm called UNwrapping using Region grOwing and Local Linear estimation (UNROLL). This algorithm combines a region-growing technique using pixel stacks, used for image domain phase correction (18, 19), with the local plane fitting scheme mentioned above. The variance of the second-order partial derivatives of the phase, rather than the phase derivative variance, is applied as a quality map. As follows from the results presented and discussed below, this quality map is able to correctly classify the areas with a rapidly changing phase, but a high SNR, as “high quality.” When this is combined with the region-growing strategy using pixel stacks, unwrapping can be done pixel-by-pixel and the fitting plane can be extended to an *n* × *n* window, which is important to improve reliability in the presence of strong noise.