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 consistently better performance over four pre-trained models.

翻译：近期，针对DDIM逆过程中存在的不一致性问题，研究者提出了多种图像编辑方法，例如EDICT[36]和空文本逆过程[22]。但上述方法引入了显著的计算开销。本文提出一种新技术——双向积分逼近（BDIA），能够以可忽略的计算开销实现精确扩散逆过程。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$t_i$处的下一扩散状态$\boldsymbol{z}_{i-1}$。首先获得估计高斯噪声$\hat{\boldsymbol{\epsilon}}(\boldsymbol{z}_i,i)$，然后两次应用DDIM更新过程：分别在前向方向上逼近下一时间区间$[t_i, t_{i-1}]$的ODE积分，以及在反向方向上逼近上一时间区间$[t_i, t_{t+1}]$的积分。其中对上一时间区间的DDIM步骤用于修正计算$\boldsymbol{z}_i$时先前作出的积分逼近。BDIA-DDIM的一个优良性质是：$\boldsymbol{z}_{i-1}$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$\boldsymbol{z}_{i+1}$，从而实现了精确扩散逆过程。实验证明，（往返）BDIA-DDIM在图像编辑中尤为有效。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。此外，BDIA不仅适用于DDIM，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采样过程时，在四个预训练模型上均获得了一致的性能提升。