Confident ODE Editing (CODE)

• The paper introduces CODE, a training-free approach that combines deterministic ODE inversion with Langevin correction to robustly restore and edit corrupted images.
• It decouples inversion depth and likelihood correction, enabling precise control over the trade-off between input fidelity and generated realism.
• Empirical results across datasets like CelebA-HQ and LSUN demonstrate that CODE achieves lower FID and higher PSNR compared to traditional SDE-based methods.

Confident Ordinary Differential Editing (CODE) is a method for image synthesis and restoration that enables robust editing guided by noisy or Out-of-Distribution (OoD) images within the diffusion model framework. CODE introduces a pipeline based on probability-flow ordinary differential equations (ODEs), which provides deterministic mappings between input images and latent representations, paired with controlled likelihood ascent using Langevin dynamics. This approach requires no task-specific training or handcrafted modules and is compatible with any pre-trained diffusion model. Positioned at the intersection of conditional image generation and blind image restoration, CODE is designed to maximize the likelihood of the input under the model prior while maintaining fidelity to the possibly corrupted input, introducing a principled alternative to traditional blind restoration techniques (Delft et al., 2024).

1. Problem Setting and Theoretical Goals

CODE addresses the problem of editing or restoring images given a single "guidance" image $x_0$ that may be corrupted, noisy, or otherwise OoD relative to the diffusion model's training distribution. The goal is to output an image $\tilde{x}_0$ that:

• Remains faithful to $x_0$: measured via perceptual or pixelwise similarity metrics such as $L_2$, PSNR, or SSIM.
• Is realistic: the output lies close to the model's learned data manifold, assessed via Fréchet Inception Distance (FID) or perceptual distances like LPIPS.

A principal challenge is managing the fidelity–realism trade-off: Adding noise and denoising (as in SDEdit) may enhance realism but sacrifices input fidelity, while insufficient noise injection may trap the model in a non-natural image mode. CODE explicitly decouples "noise level" (controlled by inversion depth $L$) from "correction strength" (handled by the Langevin step size $\epsilon$), providing more granular and independent control over this trade-off.

2. Mathematical Principles: ODE-Based Editing and Latent Correction

The generative foundation of CODE is the score-based diffusion model, typically instantiated as a Variance-Preserving SDE:

$$dx = -\frac{1}{2} \beta(t)x\,dt + \sqrt{\beta(t)}\,dW_t$$

with a drift term given by the score estimate $s_\theta(x, t) \approx \nabla_x \log p_t(x)$.

The associated probability-flow ODE is:

$$dx = [f(x, t) - \frac{1}{2}g^2(t)\,s_\theta(x, t)]\,dt$$

where $f(x, t) = -\frac{1}{2}\beta(t)x$ and $g(t) = \sqrt{\beta(t)}$. CODE utilizes a discretized ODE inversion, equivalent to DDIM with $\sigma_t = 0$, ensuring deterministic bijective mapping between $x_0$ and latent $x_L$ for any $L \in [0, T]$.

After ODE inversion to a chosen latent depth, CODE applies Langevin dynamics in the latent space:

$$x_{L, k+1} = x_{L, k} + \epsilon \cdot s_\theta(x_{L, k}, L) + \sqrt{2\epsilon} \cdot \eta, \quad \eta \sim \mathcal{N}(0, I)$$

The step size $\epsilon$ modulates the correction strength toward high-density regions without over-injecting noise, directly influencing realism/fidelity independently of $L$.

3. Confidence Interval–Based Clipping (CBC)

ODE inversion from corrupted or OoD images can produce intermediate latent values $x_t$ containing highly improbable pixel intensities under the forward diffusion process. To suppress the impact of such outliers, CODE introduces confidence interval–based clipping (CBC):

Given the forward process $$x_t \sim \mathcal{N}(\sqrt{\alpha_t}\,x_0,\, (1-\alpha_t)I)$$ with $x_0 \in [-1, 1]$, CODE applies per-coordinate clipping:

$$x_t^{\text{clipped}} = \text{Clip}\left(x_t,\; -\sqrt{\alpha_t} - \eta \sqrt{1-\alpha_t},\; +\sqrt{\alpha_t} + \eta \sqrt{1-\alpha_t}\right)$$

where $\eta$ is chosen (e.g., $1.7$–$2.0$) so that the interval covers at least 95% of mass for $\eta \geq 2$ ($$P[\text{in interval}] \geq \Phi(\eta) - \Phi(-\eta)$$). This clipping aggressively removes unlikely pixels before latent-space correction, mitigating the effect of strong corruptions and masks.

4. Comparison to SDE-Based Editing and Empirical Analysis

In contrast to SDEdit, which injects noise to a latent $x_L \sim q(x_L|x_0)$ and reverses stochastically via DDPM, CODE offers:

• Deterministic inversion: ODE-based mapping from $x_0 \rightarrow x_L$ incurs no additional fidelity loss.
• Langevin correction: Targeted log-likelihood ascent in latent space, divorced from forward noise injection.
• Decoupled trade-off control: Inversion depth $L$ governs how far the image is mapped into latent space; correction strength $\epsilon$ controls the degree of movement toward the model prior.

Empirically, using 47 types of corruption on CelebA-HQ, CODE achieves substantial performance improvements:

Method	FID↓	PSNR-Input↑	LPIPS-Source↓
Inputs	143.5	—	0.48
SDEdit	47.8	18.74	0.32
CODE	30.7	19.61	0.30

CODE yields uniformly lower FID at equal or higher PSNR; the trade-off curve for FID vs. $L_2$-distance uniformly dominates that of SDEdit. Qualitative results show CODE reconstructs fine structure and plausible semantics under severe corruptions (e.g., fog, masking), preserving both identity and image realism. The method generalizes across datasets (CelebA-HQ, LSUN Bedroom, LSUN Church). CBC is shown via ablation to be critical for handling extreme outliers; ODE inversion alone or CBC alone is inadequate for optimal performance (Delft et al., 2024).

5. Pipeline Structure and Hyperparameterization

The full CODE algorithm consists of:

1. ODE Inversion: Deterministically projects $x_0$ to $x_L$ via probability-flow ODE, with CBC applied to suppress out-of-range pixels.
2. Langevin Correction: In latent $x_L$, performs $N$ Langevin steps, with multi-latent annealing possible via steps at varying latent levels $\{L_j, ..., L_1\}$, step-size $\epsilon_0$, annealing factor $\alpha$, and $K$ annealing rounds.
3. ODE Decoding: Projects corrected latent back to image space via the reverse ODE.
4. Confidence Parameter Tuning: Hyperparameter $\eta$ for the CBC interval size; latent depths and correction schedule are user-controlled for fine realism–fidelity balance.

This pipeline is fully training-free and blind with respect to the corruption, relying solely on a pre-trained score network and no ground truth target assumptions.

6. Significance and Context

CODE provides a robust, flexible solution to conditional image editing with corrupted or OoD input guidance, resolving key limitations in existing SDE-based approaches. The decoupling of inversion depth and likelihood correction allows independent steering of output fidelity and realism. Deterministic ODE mapping eliminates unnecessary stochasticity, reducing output variance, and improving reproducibility and metric scores (PSNR, SSIM). The confidence interval–based clipping step significantly strengthens robustness to extreme corruptions, offering a blind restoration mechanism that does not require task-specific design.

This suggests CODE is a principled alternative in the landscape of blind image restoration and conditional generation, applicable across diverse domains and degradations. Its training-free, model-agnostic character and empirical robustness across metrics underscore its utility for practitioners requiring flexible fidelity–realism control in image synthesis workflows (Delft et al., 2024).