Recently, different methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT \cite{Wallace23EDICT} and Null-text inversion \cite{Mokady23NullTestInv}. However, the above methods introduce considerable computational overhead. In this paper, we propose a new technique, named \emph{bi-directional integration approximation} (BDIA), to perform exact diffusion inversion with neglible computational overhead. Suppose we would like to estimate the next diffusion state $\boldsymbol{z}_{i-1}$ at timestep $t_i$ with the historical information $(i,\boldsymbol{z}_i)$ and $(i+1,\boldsymbol{z}_{i+1})$. We first obtain the estimated Gaussian noise $\hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i)$, and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot $[t_i, t_{i-1}]$ in the forward manner and the previous time-slot $[t_i, t_{t+1}]$ in the backward manner. The DDIM step for the previous time-slot is used to refine the integration approximation made earlier when computing $\boldsymbol{z}_i$. One nice property with BDIA-DDIM is that the update expression for $\boldsymbol{z}_{i-1}$ is a linear combination of $(\boldsymbol{z}_{i+1}, \boldsymbol{z}_i, \hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i))$. This allows for exact backward computation of $\boldsymbol{z}_{i+1}$ given $(\boldsymbol{z}_i, \boldsymbol{z}_{i-1})$, thus leading to exact diffusion inversion. Experiments on both image reconstruction and image editing were conducted, confirming our statement. BDIA can also be applied to improve the performance of other ODE solvers in addition to DDIM. In our work, it is found that applying BDIA to the EDM sampling procedure produces slightly better FID score over CIFAR10.

翻译：近期，为解决DDIM反演中的不一致性问题以支持图像编辑，研究者提出了多种方法，如EDICT \cite{Wallace23EDICT}和空文本反演\cite{Mokady23NullTestInv}。然而，这些方法引入了显著的计算开销。本文提出一种新技术——\emph{双向积分近似}（BDIA），能够以可忽略的计算开销实现精确的扩散反演。假设我们需利用历史信息$(i,\boldsymbol{z}_i)$和$(i+1,\boldsymbol{z}_{i+1})$估计时间步$t_i$处的下一扩散状态$\boldsymbol{z}_{i-1}$。首先获取估计的高斯噪声$\hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i)$，然后两次应用DDIM更新步骤：前向逼近下一时间段$[t_i, t_{i-1}]$上的ODE积分，后向逼近上一时间段$[t_i, t_{t+1}]$上的ODE积分。其中对上一时间段的DDIM步骤用于修正计算$\boldsymbol{z}_i$时先前的积分近似。BDIA-DDIM的一个优良特性在于，$\boldsymbol{z}_{i-1}$的更新表达式是$(\boldsymbol{z}_{i+1}, \boldsymbol{z}_i, \hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i))$的线性组合，这使得给定$(\boldsymbol{z}_i, \boldsymbol{z}_{i-1})$时能精确反向计算$\boldsymbol{z}_{i+1}$，从而实现精确扩散反演。我们在图像重建和图像编辑任务上进行了实验，结果证实了上述结论。除DDIM外，BDIA还可应用于改进其他ODE求解器的性能。本研究发现，将BDIA应用于EDM采样流程后，在CIFAR10数据集上获得了略优的FID分数。