Recently, various methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT [36] and Null-text inversion [22]. 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 $\boldsymbol{z}_i$. A nice property of 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. It is demonstrated with experiments that (round-trip) BDIA-DDIM is particularly effective for image editing. Our experiments further show that BDIA-DDIM produces markedly better image sampling qualities than DDIM for text-to-image generation. 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 new SOTA performance over CIFAR10.

翻译：近期，研究者提出了多种方法以解决DDIM逆映射中不一致性问题，从而实现图像编辑，例如EDICT[36]和空文本逆映射[22]。然而，上述方法引入了显著的计算开销。本文提出一种名为"双向积分逼近"（BDIA）的新技术，该技术能够在计算开销可忽略的前提下实现精确扩散逆映射。假设我们需要利用历史信息$(i,\boldsymbol{z}_i)$和$(i+1,\boldsymbol{z}_{i+1})$估计时间步$t_i$处的下一个扩散状态$\boldsymbol{z}_{i-1}$。我们首先获得估计的高斯噪声$\hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i)$，随后两次应用DDIM更新过程：一次用于正向逼近下一时段$[t_i, t_{i-1}]$的ODE积分，另一次用于反向逼近前一时段$[t_i, t_{t+1}]$的积分。前一时段的DDIM步骤用于修正之前计算$\boldsymbol{z}_i$时作出的积分近似。BDIA-DDIM的优良性质在于，$\boldsymbol{z}_{i-1}$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$(\boldsymbol{z}_i, \boldsymbol{z}_{i-1})$时能够精确反向计算$\boldsymbol{z}_{i+1}$，从而实现精确扩散逆映射。实验证明，（往返）BDIA-DDIM对图像编辑尤为有效。进一步实验表明，BDIA-DDIM在文本到图像生成中产生的图像采样质量显著优于DDIM。除DDIM外，BDIA还可用于提升其他ODE求解器的性能。本研究发现，将BDIA应用于EDM采样过程可在CIFAR10数据集上取得新的最优性能。