Phase-Conditioned Semantic Priors in MRI

• The paper introduces a generative model that integrates phase augmentation into MRI reconstruction to significantly improve image fidelity and robustness.
• The method employs both PixelCNN and diffusion-based priors, achieving higher PSNR and SSIM compared to magnitude-only or L1-wavelet approaches.
• Practical guidance is provided for scalable phase augmentation workflows that integrate seamlessly with linear (FISTA) and nonlinear (IRGNM) MRI reconstruction algorithms.

Phase-conditioned semantic priors are generative models specifically designed to regularize complex-valued inverse problems, notably in magnetic resonance imaging (MRI), by incorporating both magnitude and phase information within the prior learning process. Through a structured workflow—generation of phase-augmented complex training samples, training of a generative prior, and deployment as a regularizer in reconstruction—these priors enable semantic-level constraints conditioned on plausible phase, substantially improving image fidelity, robustness, and quantitative performance, especially under strong undersampling regimes (Luo et al., 2023).

1. Mathematical Foundation and Formulation

Phase-conditioned semantic priors model the empirical distribution of complex-valued MRI images. Let $m \in \mathbb{R}_+^{n \times n}$ denote the magnitude image and $\phi \in [-\pi, \pi]^{n \times n}$ the phase map. The complex-valued image is constructed as

$$x = m \cdot e^{i\phi}, \qquad x \in \mathbb{C}^{n \times n}.$$

Given undersampled multi-coil k-space data $y \in \mathbb{C}^{d \times N_{\rm coils}}$ and encoding operator $$E\colon \mathbb{C}^{n \times n} \to \mathbb{C}^{d \times N_{\rm coils}}$$, the Maximum-A-Posteriori (MAP) MRI reconstruction with a generative prior $p_\theta(x)$ is

$$\hat{x} = \arg\min_{x \in \mathbb{C}^{n \times n}} \frac12 \|E x - y\|_2^2 - \lambda \log p_\theta(x),$$

or, equivalently,

$$\hat{x} = \arg\min_x \|E x - y\|_2^2 + \lambda R(x),\quad R(x) = [-\log p_\theta(x)].$$

2. Phase Augmentation Strategy

To overcome the scarcity of phase annotations in clinical datasets, phase-conditioned priors leverage phase augmentation. The workflow begins with a magnitude-only dataset $\{m_i\}_{i=1}^M$ and augments each sample by synthesizing corresponding phase maps $\phi_i$. Two approaches are considered:

• Uniform phase sampling: $\phi_i(k, \ell) \sim \mathcal{U}[-\pi, \pi]$ independently.
• Diffusion-based phase sampling: $\phi_i$ sampled from a learned conditional prior $p_\psi(\phi \mid m)$, implemented via a conditional Langevin sampler with a pre-trained complex diffusion model $p_\psi(x)$.

For the latter, phase-augmented complex images are drawn by iteratively updating

$$x_n^{k+1} = x_n^k + \frac{\gamma}{2} \nabla_x \left[\log p_\psi(x_n^k) - \epsilon \| |x_n^k| - m \|_2^2 \right] + \sqrt{\gamma} z, \quad z \sim \mathcal{CN}(0, I),$$

until convergence, then extracting $\phi_i$ as the argument and $m_i$ as the modulus of $x_0$.

3. Generative Prior Architectures and Training

Two principal generative modeling frameworks are employed:

• PixelCNN-based prior:
  • Input: Two-channel images $$(\Re(x), \Im(x)) \in \mathbb{R}^{n \times n \times 2}$$.
  • Architecture: 24-layer gated causal convolutions with 64 features each, followed by a 10-component mixture of discretized logistics per pixel; $\approx 22$M parameters.
  • Loss: Negative log-likelihood,

  $$L_{\rm PNN}(\theta) = -\sum_{i=1}^{n^2} \log p_\theta(x^{(i)} \mid x^{<i}),$$

  where $x^{(i)}$ is the $i$-th raster-scan pixel.

• Diffusion-based prior (Score network):
  • Input: Noisy complex image $x_i$ at noise-level $i$.
  • Architecture: U-Net style score-based model (“Refine-Net”) with residual blocks and attention, channels [64, 128, 256]; $\approx 8$M parameters.
  • Loss: Denoising score matching (DSM),

  $$L_{\rm DSM}(\psi) = \sum_{i=1}^T \mathbb{E}_{x_0 \sim p_{\rm data}} \mathbb{E}_{x_i|x_0} \left[ \lambda_i \| s_\psi(x_i, i) - \nabla_{x_i} \log q(x_i \mid x_0) \|_2^2 \right],$$

  with $$q(x_i \mid x_0) = \mathcal{CN}(x_i \mid \alpha_i x_0, \sigma_i^2 I)$$, $T \approx 1000$.

Training is conducted on either a small complex dataset ($\approx 1$k images) or a large phase-augmented magnitude dataset ($\approx 80$k slices), producing corresponding priors $P_{SC}, D_{SC}$ (small) and $P_{LC}, D_{PC}$ (large).

4. Integration into MRI Reconstruction Algorithms

The learned priors are incorporated as regularizers in both linear and nonlinear MRI reconstruction schemes:

• Linear PICS (Proximal-Gradient/FISTA):
  • Objective: $$\min_x \frac12 \|E x-y\|_2^2 - \lambda \log p_\theta(x)$$.
  • FISTA iteration:

  $$\begin{aligned} z^k &= x^k + \frac{k-1}{k+2}(x^k - x^{k-1}), \ x^{k+1} &= \operatorname{prox}_{\tau R}\left(z^k - \tau E^*(E z^k - y)\right), \end{aligned}$$

  with the proximal operator for $R(x) = -\log p_\theta(x)$ approximated by a single gradient step.

• Nonlinear IRGNM/NLINV (Joint Image and Coil Sensitivity Estimation):
  • Joint forward model $F(x, c)=y$ solved by Gauss–Newton iterations,

  $$\min_{\delta x, \delta c} \frac12 \| F'(x^k, c^k)[\delta x; \delta c] + F(x^k, c^k) - y \|_2^2 + \alpha_k R(x^k + \delta x) + \beta_k \mathcal{W}(c^k + \delta c),$$

  with a two-stage scheme: conjugate gradient (CG) with Tikhonov prior for early iterations, then FISTA with the learned prior for refinement.

5. Quantitative Evaluation

Experimental assessments include various undersampling and reconstruction settings:

Prior/model	PSNR (5×)	SSIM (5×)	Comments
Magnitude-only $P_{SM}$	39 dB	0.95	Phase artifacts
Small complex $P_{SC}$	43 dB	0.98	Clean magnitude and phase
Large complex $P_{LC}$	44 dB	0.99	Fewer outliers, robust
$L_1$-wavelet	38 dB	0.94	Inferior at higher undersampling

Further results:

• Nonlinear NLINV reconstructions reflected similar trends; magnitude-only priors introduced ghosting, especially in linear PICS.
• The large training set reduced PSNR/SSIM variance across slices, particularly at 4× and 6× undersampling.
• 3D TurboFLASH (8.2×) blind reader scores (1–5 scale): coil-combination reference 5.0, $L_1$-wavelet PICS 3.2±0.6, diffusion prior PICS 4.1±0.5, NLINV 3.8±0.6.

6. Practical Guidance and Insights

• Priors trained exclusively on magnitude data are fundamentally limited in recovering phase, leading to smoothing artifacts in both magnitude and phase reconstructions.
• Phase augmentation using a small set of complex-labeled examples (employing a diffusion prior) enables the exploitation of large existing institutional databases for learning rich, complex-valued priors.
• PixelCNNs provide more accurate likelihood modeling but are computationally intensive; diffusion priors offer an effective trade-off between sample fidelity, speed of gradient evaluation (approximately 100 evaluations per reconstruction), and robustness.
• Effective protocol: phase-augment with a few thousand complex-labeled samples, generate multiple (5–10) phase realizations per magnitude slice, then train the generative prior on the expanded set.
• The learned prior can be used as a drop-in regularizer in FISTA or IRGNM frameworks through proximal-gradient integration of the learned score or log-likelihood gradient.
• Nonlinear IRGNM/NLINV is preferable at high undersampling rates (>5×) to mitigate aliasing where linear models are insufficient.
• Phase augmentation generalizes prior learning, obviating the requirement for paired magnitude-phase ground truth.

7. Significance and Implications

Phase-conditioned semantic priors expand the utility of MRI deep learning workflows by leveraging abundant magnitude-only image archives for high-fidelity, complex-valued prior construction. This facilitates improved MR image reconstruction, particularly under aggressive undersampling, without dependence on extensive paired complex datasets. A plausible implication is that this methodology could be generalized to other imaging settings where phase information is important yet underrepresented in available datasets (Luo et al., 2023).