Fast Unwrapping using Discrete Gradient Evaluation (FUDGE): an analytical correction to the Laplacian-based phase unwrapping technique for discrete data.

Amanda Ching Lih Ng^{1}, Meei Pyng Ng^{2}, Sonal Josan^{3}, Shawna Farquharson^{4}, Claire Mulcahy^{4}, and Roger J Ordidge^{1}

The phase of the complex-valued MRI gradient echo data contains information on local magnetic field changes. The phase is implicitly wrapped into the range $$$\small{\left[-\pi,\pi\right)}$$$. In order to recover the field information, the phase needs to be unwrapped. Laplacian-based phase unwrapping [1-3] is a commonly used technique, particularly as a pre-processing step in Quantitative Susceptibility Mapping [4]. The method unwraps phase by calculating the Laplacian of the unwrapped phase, $$$\small{\phi_u}$$$, using the wrapped phase, $$$\small{\phi_w}$$$:

$$\phi_u = \nabla^{-2} \left( \cos \phi_w \nabla^2 \left( \sin \phi_w \right) - \sin \phi_w \nabla^2 \left( \cos \phi_w \right) \right), \tag{1}$$

where $$$\small{\nabla^2}$$$ and $$$\small{\nabla^{-2}}$$$ are the Laplacian and inverse Laplacian operations, computed using the time derivative property of the Fourier transform. Whilst this is analytically correct, it was derived with the assumption of a continuous signal and employing the continuous Fourier transform. These assumptions do not translate well to discrete signals and the use of the discrete Fast Fourier Transform (FFT), which are present in the context of MRI phase data and computation. In fact, the FFT of the discrete Laplacian operator is commonly used to calculate the Laplacian operations. Yet, even when employing this modification, the resulting unwrapped phase (for 1D) becomes

$$\phi\left[x\right] = \nabla^{-2} \left[ \sin\left(\phi_{w}\left[x+1\right]-\phi_{w}\left[x\right]\right) - \sin\left(\phi_{w}\left[x\right]-\phi_{w}\left[x-1\right]\right) \right] ,\tag{2}$$

where $$$\small{x}$$$ is the discrete image space coordinate. From this, it can be seen that a change of 0 is indistinguishable from a change of $$$\small{\pi}$$$,resulting in incorrect unwrapped phase values. We present an analytically correct alternative to this method, by correctly assuming the discrete nature of the data and computations.

We derived a discrete version of Laplacian-based unwrapping, where the second order partial derivative is

$$\frac{\delta^2\phi_{u}}{\delta t^2} = \tan^{-1}\frac{\sin\left(\phi_{w}\left[t+1\right]-\phi_{w}\left[t\right]\right)}{\cos\left(\phi_{w}\left[t+1\right]-\phi_{w}\left[t\right]\right)}-\tan^{-1}\frac{\sin\left(\phi_{w}\left[t\right]-\phi_{w}\left[t-1\right]\right)}{\cos\left(\phi_{w}\left[t\right]-\phi_{w}\left[t-1\right]\right)},\tag{3}$$

the 3D Laplacian is defined as$$ \nabla^2 \phi_{u} = \frac{\delta^2\phi_{u}}{\delta x^2} + \frac{\delta^2\phi_{u}}{\delta y^2} + \frac{\delta^2\phi_{u}}{\delta z^2}\tag{4}$$and the inverse Laplacian is calculated using the Discrete Cosine Transform:

$$\phi_{u}\left[x,y,z\right] = \text{DCT}^{-1}\left(\left[2\cos\left(\frac{\pi}{N}\left(k-1\right)\right)-2\right]^{-1}\text{DCT}\left(\nabla^2 \phi_{u}\right)\right).\tag{5}$$

We shall refer to the original algorithm as Laplacian-based Uwrapping using Continuous K-space (LUCK) and the new algorithm as Fast Unwrapping using Discrete Gradient Evaluation (FUDGE). We applied both methods to two numerical simulation datasets and an experimental in vivo MRI gradient echo data. The numerical simulations comprised: a phase image with spheres of varying phase values; a phase image containing a sphere (diameter=5mm, susceptibility=0.3ppm, B0=3T, TE=20ms). In vivo MRI brain data from a healthy volunteer (35yo, female) was acquired on a 7T research MRI scanner (Siemens, Erlangan, Germany) in accordance with local ethics (fully flow-compensated GRE, TE=7.65ms,15.3ms, TR=18ms, FA=13$$$^{\circ}$$$, iPAT=3, matrix=366$$$\scriptsize{\times}$$$316$$$\scriptsize{\times}$$$224, voxel=0.6mm isotropic, TA=7:42min). iLSQR [5] was used to calculate QSM for the second numerical simulation.

[1] Schofield, Marvin A., and Yimei Zhu. 2003. “Fast Phase Unwrapping Algorithm for Interferometric Applications.” Optics Letters 28 (14): 1194–96. doi:10.1364/OL.28.001194.

[2] Li, Wei, Alexandru V. Avram, Bing Wu, Xue Xiao, and Chunlei Liu. 2014. “Integrated Laplacian-Based Phase Unwrapping and Background Phase Removal for Quantitative Susceptibility Mapping.” NMR in Biomedicine 27 (2): 219–27. doi:10.1002/nbm.3056.

[3] Li, Wei, Bing Wu, and Chunlei Liu. 2011. “Quantitative Susceptibility Mapping of Human Brain Reflects Spatial Variation in Tissue Composition.” NeuroImage 55 (4): 1645–56. doi:10.1016/j.neuroimage.2010.11.088.

[4] Wang, Yi, and Tian Liu. 2015. “Quantitative Susceptibility Mapping (QSM): Decoding MRI Data for a Tissue Magnetic Biomarker.” Magnetic Resonance in Medicine 73 (1): 82–101. doi:10.1002/mrm.25358.

[5] Li, Wei, Nian Wang, Fang Yu, Hui Han, Wei Cao, Rebecca Romero, Bundhit Tantiwongkosi, Timothy Q. Duong, and Chunlei Liu. 2015. “A Method for Estimating and Removing Streaking Artifacts in Quantitative Susceptibility Mapping.” NeuroImage 108 (March): 111–22. doi:10.1016/j.neuroimage.2014.12.043.

Figure 1. Numerical simulation images and line profiles demonstrate substantial errors in the LUCK-processed phase and exact results from the FUDGE processing

Figure 2. Errors produced in unwrapped phase by LUCK translate to inaccurate susceptibility after QSM processing.

Figure 3. In vivo brain data: FUDGE produced less error in the unwrapped phase compared to LUCK.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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