Cheng Junying^{1,2}, Mei Yingjie^{2,3}, Chen Maodong^{2}, Wang Changqing^{1,2}, Liu Xiaoyun^{1}, Chen Wufan^{1,2}, and Feng Yanqiu^{2}

Current phase-unwrapping algorithms are generally challenged by severe noise, rapid-varying phase or disconnected regions. We present a novel phase-unwrapping method by using phase jumps detection and local polynomial surface fitting. The proposed method first segments the whole phase map into blocks by exploiting the phase jumps that are automatically identified. Intra-block wrapping may still exist if the true phase difference between adjacent pixels exceeds π inside a block. To address potential intra-block wraps, we further segment each block into subblocks using the phase partition method, and perform inter-subblock unwrapping using the block-growing method. Simulation and in vivo Dixon water-fat separation experiments were implemented to evaluate the performance of the proposed method, with comparisons to PRELUDE and CLOSE. This method has great potential in phase-related MRI applications in practice.

Robust phase-unwrapping is of vital importance with application for Dixon
water-fat separation^{1} and quantitative
susceptibility mapping^{2}.
However, phase-unwrapping remains challenging due to severe noise, rapid-varying
phase or disconnected regions. In this work, we present a novel phase-unwrapping
method for Dixon water-fat separation by using phase jumps detection and local
polynomial surface fitting (LPSF).

**Methods**

* The proposed algorithm:* The local difference of
phasor reflects the local variation of true phase (LDTP), and is theoretically less
than or equal to the local difference of wrapped phase (LDWP)

The
extension of the proposed method from 2D to 3D is straightforward. The LDWP, LDTP
and quality map are calculated using a 26-neighborhood window, the voxels are
grouped into 3D blocks, and the underlying true phase was approximated by a 3D LPSF
model. Subblocks are limited
within plane to avoid that a single subblock includes two areas where true
phase differs by more than 2π. The proposed method was performed on simulation and in
vivo Dixon water-fat separation data, with comparison
to phase region expanding
labeler for unwrapping discrete estimates
(PRELUDE)^{6} and (phase-unwrapping
algorithm based on pixel clustering and local surface fitting ) CLOSE^{3, 7}.

** Simulations:** In simulation
experiment, the magnitude of simulation data was created with 13 sectors. The
magnitude of sectors first decreased from 70 to 10, and then increased to 70
with an interval of 10. The phase is obtained by
the following formulas

$$phase= 500\times \left ( \frac{x}{256}-0.5 \right )\times exp(-5\times ((\frac{x}{256}-0.5)^{2}+(\frac{y}{256}-0.5)^{2}))$$

Zero-mean Gaussian noise with a SD of 20 was added to the real and imaginary parts of the synthesized complex data.

** In vivo study: **The 3-point Dixon water-fat
separation datasets

Figure 2 shows the unwrapped results of the simulation using the three methods. The results generated by the proposed method have no obvious residual wraps, and the mean and SD of misclassification ratio is 0.26 ± 0.07%.

The representative phase-unwrapping and water-fat separation results of a sagittal ankle slice in the 0.35 T in vivo Dixon water-fat datasets are shown in Figure 3. Figure 4 shows the unwrapped images and water-fat separation results of a representative coronal slice in the 3.0 T 3D knees data. No obvious residual wraps and water-fat swaps are found in the results produced by the proposed method. Among all subjects studied, the number of slices with swaps produced by PRELUDE and CLOSE are separately 194 and 39, while that produced by the proposed method is 9 out of the total knee and ankle images.

**Discussion **

This work presents a phase-unwrapping method by using phase jumps detection and LPSF. The proposed method realizes phase-unwrapping by exploiting the region-growing LPSF approach, which is insensitive to noise and can acquire accurate unwrapped results even when adjacent true phase difference exceeds π. Additionally, the variance of the second-order partial derivatives as a quality criterion is combined with stack technique to improve the unwrapping path of residual pixels, and this will reduce error propagation and accumulation.

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