Feihong Liu^{1,2}, Geng Chen^{2}, Jun Feng^{1}, Pew-Thian Yap^{2}, and Dinggang Shen^{2}

We present a method of effective phase-correction of diffusion-weighted images with the goal of obtaining real-valued signals with zero-mean Gaussian distributed noise. Our method estimates the noise level locally and is hence well-suited for spatially-varying noise.

**Purpose**

Magnitude-based diffusion MRI is superimposed by a noise floor, which causes bias in microstructural estimation. Through phase correction, the noise floor can be removed and the signal can be Gaussianized to have zero-mean Gaussian noise distribution. Existing phase correction methods typically employ smoothing filters to estimate the background phase^{1,2}. However, these filters typically fail to consider the spatially nonstationary nature of the noise and hence result in biased outcomes^{3}. In this abstract, we propose an anisotropic diffusion filtering approach, called multi-kernel filtering (MKF), to extract the smooth background phase by adaptively considering the spatial variability of noise. MKF generates multiple filtering kernels to fit local regions more closely and yields data with noise that is more Gaussian after phase correction.

The first step in phase correction is to determine the background phase $$$\varphi_{\text{bg}}({x})$$$ from the real and imaginary parts of the diffusion weighted (DW) image (i.e., $$$I_{r}$$$ and $$$I_{i}$$$):

$$\varphi_{\text{BG}}({x}) = \arctan{\frac{f(I_{\text{i}}({x}))}{f(I_{\text{r}}({x}))}},\label{eq2}$$

where $$$x$$$ is a voxel location, and $$$f(\cdot)$$$ denotes a noise-removal filtering operator. The phase-corrected image is $$$I^{\text{c}}(\vec{x})={I}_{\text{r}}^{\text{c}}({x})+j{I}_{\text{i}}^{\text{c}}({x})$$$ with

$$\begin{cases}\begin{aligned}I_{\text{r}}^{\text{c}}({x}) &= M({x})\cos({\varphi({x})-\varphi_{\text{BG}}({x})}),\\ I_{\text{i}}^{\text{c}}({x}) &= M({x})\sin({\varphi({x})-\varphi_{\text{BG}}({x})}),\end{aligned}\end{cases}\label{eq:e3}$$

where $$$\varphi_{\text{BG}}({x}) = \arctan{\frac{{I}_{\text{i}}({x})}{{I}_{\text{r}}({x})}}$$$ and $$$M(x)= \sqrt{\text{I}_{\text{r}}^{2}(x) + {I}_{\text{i}}^{2}(x)}$$$.

To better account for spatially-varying noise in phase correction, we employ Multi-Kernel Filtering (MKF). Unlike bilateral filtering (BF), MKF generates spatially-varying filtering kernels that adapt to local image characteristics employing Expectation Maximization (EM) clustering algorithm. EM clustering aggregates the voxels $$$\{I({x}_1), \ldots, I({x}_l), \ldots, I({x}_n)\}$$$ into locally coherent clusters, each of which is modeled using a Gaussian distribution. Thus, the whole image can hence be modeled using a Gaussian mixture model (GMM), which MKF determines hierarchically using EM clustering algorithm. The clusters across scales represented using a cluster tree. Clusters in parent layers provide contextual information that can help leaf nodes in generating filtering kernels that can adapt to local image characteristics^{5}. Smoothing is performed within these leaf clusters based on the generated kernels.

We generated synthetic DW images data using Phantom$$$\alpha$$$s^{4} with gradient directions identical to the real data described below. The background phase was synthesized according to the method described in literature^{1}. Spatially, the non-stationary Gaussian noise was added to both real and complex parts of data. DW images were acquired using a SIEMENS 3T Magnetom Prisma MR scanner with acquisition parameters, TR =$$$2500 \text{ms}$$$, TE=$$$89 \text{ms}$$$, FoV=$$$210\times 210~\text{mm}^2$$$, matrix size=$$$140\times 140$$$, and $$$b=750,1500,3000~\text{s/mm}^2$$$ with a total of $$$64$$$ diffusion directions.

Figure 1 shows that MKF yields improved filtering performance as measured by Mean Absolute Error (MAE) and Structural SIMilarity (SSIM), compared with TV^{2}, wTV^{3}, and MPPCA^{4}, indicating that incorrect filtering can lead to errors in phase rotation and eventually inaccurate Gaussianization of the signal. This is confirmed by Figure 2, where phase-corrected imaginary images, which should only contain Gaussian noise, are shown. The Gaussianity of the noise was evaluated using Kolmogorov–Smirnov test (KS-test) with $$$p\leq 0.05$$$ indicating that the noise is non-Gaussian. Figure 3 shows the number of non-Gaussian voxels and the $$$p$$$-value distribution, again confirming that MKF gives the best performance in Gaussianization of the signal. Figure 4 shows the phase-corrected real-valued images (averaged of six repetitive scans). MKF results in fewer artifacts than TV, wTV, and MPPCA.

1. Pizzolato, Marco, et al. "Noise floor removal via phase correction of complex diffusion-weighted images: Influence on DTI and Q-space metrics." International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2016.

2.Eichner, Cornelius, et al. "Real diffusion-weighted MRI enabling true signal averaging and increased diffusion contrast." NeuroImage 122 (2015): 373-384.

3. Pizzolato, Marco, and Rachid Deriche. "Automatic and Spatially Varying Phase Correction for Diffusion Weighted Images." 26th annual meeting of the International Society for Magnetic Resonance in Medicine (ISMRM). No. EPFL-CONF-234471. 2018.

4.Veraart, Jelle, et al. "Denoising of diffusion MRI using random matrix theory." NeuroImage 142 (2016): 394-406.5.Bar, Moshe. "Visual objects in context." Nature Reviews Neuroscience 5.8 (2004): 617.

5.Bar, Moshe. "Visual objects in context." Nature Reviews Neuroscience 5.8 (2004): 617.

Fig.1 Filtering of synthetic data with $$$b=750,1500,3000~\text{s/mm}^2$$$ and a total of $$$64$$$ gradient directions plus $$$5$$$ b0 volumes. MKF outperforms other methods for both MAE and SSIM across different $$$b$$$-values.

Fig.2 (a) Phase-corrected imaginary images for $$$b=0,750,1500,3000~\text{s/mm}^2$$$ from top to bottom row. The images given by MKF and MPPCA have fewer structures than TV and wTV, indicating better filtering performance. (b) The negative-log $$$p$$$-values are given by KS-tests performed based on $$$5\times5$$$ blocks. Non-Gaussian voxels are marked as white.

Fig.3 (a) The number of voxels that are non-Gaussian indicated by KS-test. (b) The $$$p$$$-value distribution.

Fig.4 Phase-corrected real-valued images for $$$b=0,750,1500,3000~\text{s/mm}^2$$$ from top to bottom. MAG denotes the magnitude images.