Posterior Continuation with Noise-Conditioned Frequency Exposure for Diffusion Inverse Problems

Technical Summary: Posterior Continuation with Noise-Conditioned Frequency Exposure for Diffusion Inverse Problems

Problem Statement
Diffusion posterior sampling addresses inverse problems by combining pretrained diffusion priors with measurement-consistency guidance. However, existing training-free methods often fail under severe degradations (e.g., strong motion blur, heavy downsampling) or ill-conditioned operators. The core limitation identified is that current samplers inject measurement information uniformly across all frequencies and noise levels, ignoring the noise-dependent reliability of signal components. At high noise levels, clean estimates derived from the diffusion model contain score-induced errors. When full-band likelihood gradients are computed from these inaccurate estimates, they can be misaligned with the local posterior geometry, causing structural drift and artifacts (e.g., ringing, texture wash-out). Conversely, low-frequency structures remain relatively stable even at high noise levels.

Methodology
The authors propose a Posterior Continuation Framework based on Noise-Conditioned Frequency Exposure. The central principle is that measurement frequencies should be exposed according to the instantaneous diffusion noise level, rather than following a fixed schedule based on solver step indices.

The framework operates through three main stages at each noise level $\sigma_i$ in the reverse-time trajectory:

1. Diffusion Prediction: A pretrained diffusion solver first estimates a clean image $\hat{x}_{0,i}$ from the noisy sample $x_{\tau_i}$.
2. Noise-Conditioned Frequency-Limited Likelihood Refinement: Instead of enforcing full-band consistency immediately, the method constructs an intermediate posterior that emphasizes frequency bands reliable at the current noise level.
   • A frequency mask $M_{\omega_i}$ is applied in the Fourier domain, where the cutoff radius $\omega_i$ expands as the noise level $\sigma_i$ decreases.
   • A mixed measurement objective $L_i$ interpolates between band-limited consistency (emphasizing reliable low frequencies) and full-band consistency.
   • Langevin dynamics are used to refine the estimate $\hat{x}_{0,i}$ under this frequency-limited likelihood, producing a refined estimate $x^{ref}_{0,i}$. This prevents high-frequency score errors from corrupting the update at high noise levels.
3. Post-Refinement Haar-Domain Commitment: To address the issue that band-limited likelihoods leave high-frequency detail coefficients weakly constrained (potentially leading to operator-dependent artifacts), a commitment rule is applied after refinement.
   • The estimated image and the refined image are transformed into the Haar wavelet domain.
   • Coarse corrections (low-pass Haar coefficients) are committed immediately ($w_{i,c}=1$).
   • Refined details (high-pass Haar coefficients) are gated by a schedule $w_{i,d}$ that synchronizes with the frequency exposure. Details are only adopted as the corresponding measurement frequencies become "exposed" (i.e., as $\sigma_i$ decreases and $\omega_i$ increases).
   • The fused estimate is then re-noised to proceed to the next step in the trajectory.

Key Contributions

1. Posterior Continuation View: The paper introduces a framework where measurement bandwidth is coupled to the instantaneous diffusion noise level, creating a solver-independent exposure schedule. This contrasts with prior frequency-aware methods that rely on discrete step indices.
2. Theoretical Characterization: The authors provide a theoretical analysis showing how score-induced errors in clean estimates lead to likelihood-gradient mismatches that scale with noise levels. They demonstrate that frequency truncation effectively removes the contribution of unexposed high-frequency bands to this mismatch.
3. Practical Training-Free Sampler: A concrete implementation combining diffusion prediction, frequency-limited likelihood refinement, and a Haar-domain commitment rule. This design ensures that coarse structures are stabilized early while fine details are deferred until they are reliably identifiable.
4. Empirical Performance: The method achieves competitive-to-state-of-the-art performance across super-resolution, inpainting, and deblurring tasks on FFHQ and ImageNet. Notably, it achieves up to a 5 dB PSNR improvement on motion deblurring compared to strong baselines.

Results
Experiments were conducted on four linear inverse problems: 4× super-resolution, motion deblurring, Gaussian deblurring, and random inpainting.

• Quantitative: The proposed method ("Ours") consistently outperforms baselines (including DPS, DDNM, DDRM, FGPS, and DCDP) in PSNR, SSIM, and LPIPS. The gains are most significant on ill-conditioned operators like motion deblurring (e.g., 36.69 PSNR vs. 29.66 for DAPS on FFHQ).
• Qualitative: Visual comparisons show better preservation of global structure and object identity, with sharper local textures and suppressed ringing artifacts compared to methods that apply full-band guidance too early.
• Ablation Studies: Removing frequency continuation leads to drift and artifacts at high noise. Removing the Haar commitment step degrades detail quality, confirming that both components are necessary for stable posterior sampling.

Significance and Claims
The paper claims that the instability of existing diffusion posterior samplers is not merely algorithmic but structural, arising from a mismatch between noise levels and the reliability of frequency components. By decoupling the exposure of measurement information from solver-specific step counts and tying it to the physical noise scale, the method provides a principled way to navigate the posterior geometry. The significance lies in demonstrating that noise-conditioned frequency exposure and schedule-synchronized detail commitment are essential for recovering fine details without introducing artifacts in challenging inverse problems. The work offers a training-free solution that improves robustness across diverse degradation types without requiring task-specific retraining.