Exact Diffusion Inversion via Bidirectional Integration Approximation

Recently, various methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT [39] and Null-text inversion [23]. 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 negligible computational overhead. Suppose we would like to estimate the next diffusion state z{i-1} at timestep ti with the historical information (i, zi) and (i+1, z{i+1}). We first obtain the estimated Gaussian noise epsilon(zi, i), and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot [ti, t{i-1}] in the forward manner and the previous time-slot [ti, 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 zi. A nice property of BDIA-DDIM is that the update expression for z{i-1} is a linear combination of (z{i+1}, zi, epsilon(zi, i)). This allows for exact backward computation of z{i+1} given (zi, z_{i-1}), thus leading to exact diffusion inversion. We perform a convergence analysis for BDIA-DDIM that includes the analysis for DDIM as a special case. It is demonstrated with experiments that BDIA-DDIM is effective for (round-trip) image editing. Our experiments further show that BDIA-DDIM produces markedly better image sampling quality than DDIM and EDICT for text-to-image generation and conventional image sampling.