Improving Data Fidelity via Diffusion Model-based Correction and Super-Resolution

TL;DR

Numerical experiments demonstrate that the proposed method effectively enhances low-fidelity, low-resolution data by correcting numerical errors and noise while simultaneously improving resolution to recover fine-scale structures.

Abstract

We propose a unified diffusion model-based correction and super-resolution method to enhance the fidelity and resolution of diverse low-quality data through a two-step pipeline. First, the correction step employs a novel enhanced stochastic differential editing technique based on an imbalanced perturbation and denoising process, ensuring robust and effective bias correction at the low-resolution level. The robustness and effectiveness of this approach are validated theoretically and experimentally. Next, the super-resolution step leverages cascaded conditional diffusion models to iteratively refine the corrected data to high-resolution. Numerical experiments on three PDE problems and a climate dataset demonstrate that the proposed method effectively enhances low-fidelity, low-resolution data by correcting numerical errors and noise while simultaneously improving resolution to recover fine-scale structures.

Figures (11)

• Figure 1: Diagram of the two-step pipeline. The Correction step (blue arrow) removes various biases from LFLR data $\mathbf{u}^l_i, i=1, \cdots, n$ to recover HFLR data $\tilde{\mathbf{u}}^h$. The restriction operator $\mathcal{R}$ (black arrow) maps the HFHR data $\mathbf{u}^h$ to its HFLR version $\tilde{\mathbf{u}}^h$ and the Super-Resolution step (red arrow) performs a pseudo-inverse mapping to enhance the resolution of HFLR data $\tilde{\mathbf{u}}^h$ and reconstruct the HFHR data $\mathbf{u}^h$.
• Figure 2: Diagram of the imbalanced perturbing and denoising (IPD) process. The process begins with perturbing the LFLR data $\mathbf{u}^l$ using forward time SDE \ref{['eqn:diff']} to produce perturbed data $\mathbf{u}^l(t_1)$, whose distribution $p(\mathbf{u}^l(t_1))$ aligns with the perturbed HFLR distribution $p(\tilde{\mathbf{u}}^h(t_2))$. As a result, the perturbed data $\mathbf{u}^l(t_1)$ can be treated as a sample from distribution $p(\tilde{\mathbf{u}}^h(t_2))$. Subsequently, the Probability Flow (PF) ODE maps take this perturbed data to the HFLR data $\tilde{\mathbf{u}}^h$. The first row illustrates how an LFLR data point is corrected to produce the HFLR data, while the bottom two rows depict the evolution of distributions throughout the process.
• Figure 3: Box plot of comparison the TVD between correction obtained by Algotirhm \ref{['alg:dc']} under different settings and reference. The settings include IPD and BPD, search ending times ($T_e$) and four metrics $\mathcal{M}$: MMD, MELRu, MELRw, and $\mathcal{W}_2$.
• Figure 4: Comparison of correction obtained by Algorithm \ref{['alg:dc']} using BPD (select $t^*$ by Algorithm \ref{['alg:t']}) and IPD (select $t_1^*, t_2^*$ by Algorithm \ref{['alg:t1t2']}) with MELRw metric and $T_e=0.2$ for LF data polluted with White and Pink noise. The red line represents the LF data, the yellow line corresponds to the correction using BPD, the blue line represents the correction using IPD, and the dashed green line shows the reference HF data.
• Figure 5: Corrections using IPD with the MELRw metric and $T_e=0.2$. The top three rows display corrections of LF solutions produced by three numerical solvers: the Fourier spectral method, Godunov scheme, and Lax-Wendroff scheme. The bottom three rows show corrections of LF data polluted by White, Pink, and Brown noise, respectively. The red lines represent the LF data, the blue lines show the correction using IPD, and the dashed cyan lines indicate the reference HF data.

Theorems & Definitions (10)

• Theorem 1
• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof
• Theorem 1
• proof