We present a unified probabilistic formulation for diffusion-based image
editing, where a latent variable is edited in a task-specific manner and
generally deviates from the corresponding marginal distribution induced by the
original stochastic or ordinary differential equation (SDE or ODE). Instead, it
defines a corresponding SDE or ODE for editing. In the formulation, we prove
that the Kullback-Leibler divergence between the marginal distributions of the
two SDEs gradually decreases while that for the ODEs remains as the time
approaches zero, which shows the promise of SDE in image editing. Inspired by
it, we provide the SDE counterparts for widely used ODE baselines in various
tasks including inpainting and image-to-image translation, where SDE shows a
consistent and substantial improvement. Moreover, we propose SDE-Drag -- a
simple yet effective method built upon the SDE formulation for point-based
content dragging. We build a challenging benchmark (termed DragBench) with
open-set natural, art, and AI-generated images for evaluation. A user study on
DragBench indicates that SDE-Drag significantly outperforms our ODE baseline,
existing diffusion-based methods, and the renowned DragGAN. Our results
demonstrate the superiority and versatility of SDE in image editing and push
the boundary of diffusion-based editing methods.

我们提出了一种统一的概率形式用于扩散式图像编辑，其中潜变量以任务特定的方式进行编辑，并且通常偏离原始随机微分方程或常微分方程（SDE 或 ODE）引起的相应边际分布。代之以定义了一个相应的 SDE 或 ODE 进行编辑。我们在公式中证明了两个 SDE 之间的边际分布的 Kullback-Leibler 散度逐渐减小，而 ODE 的散度在时间趋近于零时保持不变，这表明了 SDE 在图像编辑中的优势。受此启发，我们针对各种任务（包括修复和图像到图像转换）提供了常用 ODE 基准的 SDE 对应项，其中 SDE 显示出一致而显著的改进。此外，我们还提出了基于 SDE 公式的简单而有效的点内容拖曳方法 SDE-Drag。我们创建了一个具有自然图像、艺术图像和 AI 生成图像的挑战性基准（称为 DragBench）进行评估。对 DragBench 的用户研究表明，SDE-Drag 在很大程度上胜过了我们的 ODE 基准、现有的基于扩散的方法以及著名的 DragGAN。我们的结果证明了 SDE 在图像编辑中的优势和多功能性，并推动了基于扩散的编辑方法的发展边界。