Researchers who are interested in phase-unwrapping and water-fat separation.Accurate phase-unwrapping is vital for MRI applications such as water-fat separation [1] and phase imaging [2]. Current phase-unwrapping algorithms are generally challenged by rapid phase variations, noise and disconnected regions. In this work, we propose a novel phase-unwrapping method based on the observation shown in Fig. 1. Let $$$\psi$$$ denote the phase map in Fig. 1b. The map of complex local difference (cLD), defined as $$$\sqrt{\sum_{m,n}\mid{angle}\left(\exp\left(1i*\psi\right)_{\left(m,n\right)}*\left(\exp\left(1i*\psi\right)_\left(i,j\right)\right)^*\right)\mid^2}$$$ where $$$(m, n)$$$ are pixel index in a k×k neighboring window centered at pixel $$$(i, j)$$$, is shown in Fig. 1c, in which high-intensity pixels correspond to locations with rapid variation of complex vectors, usually at tissue boundaries (as compared to Fig. 1a). Fig. 1d shows the map of phase local difference (pLD), defined as the root square of phase difference between pixel $$$(i, j)$$$ and it surrounding pixels $$$(m, n)$$$, where high-intensity pixels correspond to locations where phase wraps and/or varies rapidly. The red pixels in Fig. 1d are obtained by subtracting the cLD map from the pLD map, and correspond to locations where phase wraps. The proposed phase-unwrapping algorithm first clusters pixels into disconnected regions by thresholding the cLD map and then performs local polynominal surface fitting (LPSF) to unwrap phase with the knowledge of wrapping locations identified by thresholding the pLD map. For brevity, the proposed method is denoted as CLOSE (pixel Clustering and LOcal SurfacE fitting).

As shown in Figure 2, CLOSE first clusters the input phase map into two parts by cLD thresholding: disconnected blocks in which complex vectors change slowly, and residual pixels in which complex vectors change rapidly. After that, CLOSE sequentially performs intra-block, inter-block and residual-pixel phase unwrapping. For intra-block unwrapping, CLOSE segments each block into disconnected subblocks with smooth phase and residual pixels by pLD theresholding, and performs inter-subblock unwrapping and then residual-pixel unwrapping using LPSF. The unwrapping starts with the largest subblock, and proceeds with the closest subblocks and pixels. Inter-block unwrapping is implemented similar to inter-subblock unwrapping, and the residual-pixel unwrapping in the whole phase map is similar to residual-pixels in a block. In both block and subblock pixel clustering, an empirical threshold of π/4 was used.

To evaluate the performance of CLOSE under rapid phase changes, Gaussian phase surfaces with standard deviations (SD) of 50 pixels and height of 50, 100, and 200 rad were synthesized on a 256×256 grid (Fig. 3 first column), and Gaussian noise with SD of π/10 rad was added to these maps. To evaluate the performance under different SNRs, ring-shaped data with 12 sectors (magnitude increasing from 10 to 120) were synthesized on a 256×256 grid. The phase surface had a Gaussian shape with SD of 50 pixels and height of 20 rad. Gaussian noise with SD of 20 rad was added, resulting in SNRs in different vectors increasing from 0.5 to 6 with a step of 0.5.

To evaluate the performance of CLOSE in related MRI, two knee and one ankle 3-point spin-echo Dixon [3] datasets were obtained with a 0.35-T permanent magnet MR scanner (Xingaoyi Company, Ningbo, China). Thirteen volunteers were scanned with the following parameters: TE/TR = 28/580 ms, matrix size 256×256, FOV = 240×240 mm².

Figure 3 shows the simulation result under different phase variation levels. PRELUDE [2] produces correct phase maps at low variation levels (i.e., 50 and 100 rad), but fails at high variation level (i.e., 200 rad). In contrast, CLOSE generates correct unwrapping results under all three variation levels. Figure 4 shows that CLOSE successfully unwraps the simulated data even at a low SNR of 0.5, while PRELUDE-unwrapped results contain residual wraps when SNR is below 1.5.

Figure 5 shows the result of water-fat separation experiment. Residual wraps can be seen in PRELUDE-unwrapped phase maps, which leads to water/fat swaps, as pointed by arrows. CLOSE successfully unwrapped phase even in regions where phase varies rapidly and SNR is low, and produced water and fat images without significant swaps. Among all subjects studied, PRELUDE produced 15, 28 and 26 swaps, and CLOSE produced 2, 5 and 5 swaps out of the total 130 sagittal knee, 130 transverse knee and 65 sagittal ankle images, respectively.

CLOSE is a novel phase-unwrapping method based on the difference between local phase variation and local complex data variation. Both simulation and in vivo results demonstrate that CLOSE can correctly unwrap phase even in the presence of rapid phase variation, low SNR, and disconnected regions. CLOSE has great potential in phase-related MRI applications in practice.