MR phase data provides useful information both alone and in combination with the magnitude component.¹ In terms of post processing, current state-of-the-art phase processing methods fall into two paradigms: phase-unwrapping-based methods first convert the phase information from a wrapped cyclic field to an unwrapped linear field on which traditional image processing operations are applied; homodyne filtering techniques,² on the other hand, use linear filtering operations on the complex signal first, later extracting the phase. The commonality between these two paradigms is that filtering is always performed on a linear or Euclidean topology, either on the unwrapped phase or the complex signal, which is not optimal as phase is inherently cyclic. This work presents a novel non-linear phase filtering approach based on continuous max-flow theory,³ which processes phase using a cyclic topology, rather than as linearly unwrapped or complex number. We apply this novel phase processing technique to channel phase data prior to channel combination and compare the results with optimized homodyne high-pass filtered images (HHPF), and a robust technique called phase unwrapping using recursive orthogonal referring (PUROR).⁴

Imaging: Healthy volunteers were imaged at 7T using a 16-channel head coil. Three-dimensional whole-brain multi-echo gradient echo images were acquired (6 echoes, TR/TE/Echo spacing: 40/3.77/4.1 ms, matrix: 380x340x102, FOV: 190x170x127.5 mm). The channel data were saved for later processing.

Optimization: The phase processing framework proposed is called cyclic continuous max-flow (CCMF) in that the phase gradient is determined over a cylindrical manifold, which removes the necessity of unwrapping while representing the phase explicitly. This maintains the inherent topology of the phase, avoiding artifacts that result from linearizing said topology. Given the raw image phase, $$$θ_0(x)$$$, the optimization problem used to reconstruct a smoothly varying version of the phase, $$$θ(x)$$$, is:
$$\underset{θ(x)}\min∫_{\Omega}(D_{θ(x)}(x)+S_{θ(x)}(x)|∇θ(x)|)dx$$
$$$S_{θ(x)}(x)$$$ is the smoothness term, and $$$D_{θ(x)}(x)$$$ is the data term. The CCMF functional is addressed using primal-dual optimization and augmented Lagrangian multipliers³ and is given in Fig 1. A more detailed explanation of this algorithm is provided in an associated technical report.⁵ In the proposed framework, $$$S_{\theta(x)}(x)$$$ is a constant. $$$D_{θ(x)}(x)$$$ is defined using a truncated Gaussian likelihood model:

$$D_θ(x)=\begin{cases}-ln(P(θ_0(x)=θ)),&\text{if}\,|θ_0(x)-θ|<0.95\\\infty,&\text{else}\end{cases}$$
where $$$P(θ_0(x)=θ)$$$ is the probability of the complex image having a phase of $$$θ$$$ before the addition of circularly symmetric independent complex Gaussian noise. The truncation ensures that does not have any singularities that are not already present in the original phase. The original, low-pass ($$$θ(x)$$$) and high-pass ($$$θ_0(x)-θ(x)$$$) images are given in Figure 2. $$$\theta(x)$$$ is interpolated from 40 indicator functions, $$$u_\theta(x)$$$, to ensure sufficient intensity resolution. The time required for a single slice was under 5 s for CCMF compared to 150 ms for HHPF, both of which are faster than phase unwrapping. The benefit of this framework over ones based on Euclidean topologies is that it is robust to unwrapping errors while allowing for more complex data terms, such as those involving truncation, which limit the effect of singularities and wrapping artifacts.

Channel Combination: The individual channel phase data were processed with both CCMF and HHPF (filter size equal to 30% of FOV). An unwrapped⁴, high-pass filtered⁷ dataset was generated on a channel-by-channel basis as well. All processed phase images were combined using the inter-echo variance channel combination technique.⁶

This work presents the application of a novel non-linear phase filtering approach to generating accurate phase images from multi-channel MR phase data. The results from the CCMF algorithm were comparable to HHPF results, with superior performance in challenging regions of the brain (near air/tissue interface). Figure 3 illustrates a representative slice from one set of volunteer data. The CCMF results show more robust performance in terms of vessel conspicuity (arrows in Fig. 3). The raw channel-combined phase data (complex sum) are presented in Figure 4 to illustrate the extent of phase wrapping resolved by the CCMF and HHPF algorithms. It is evident that CCMF resulting phase images address the phase wraps more effectively at short (Fig. 4 row 1) and especially at long echo times (Fig. 4 rows 2 and 3). The arrows in Fig. 4 highlight areas where veins are obscured in the HHPF, but are clearly seen in the CCMF-processed images.By processing phase information using its inherent topology, higher quality high-pass phase maps can be reconstructed in a relatively short amount of time. This improves the visualization of structures such as veins in channel-combined phase images at the periphery and around the sinuses where additional phase artifacts are often present as a result of linearizing the phase topology.