Recently, different 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 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.

提出了一种新技术，名为双向积分逼近（BDIA），以实现可忽略的计算开销的精确扩散反演。该技术在图像编辑方面具有重要意义，同时还可以改善其他ODE求解器的性能。

双向积分逼近下的精确扩散反演