# Exact Diffusion Inversion via Bi-directional Integration Approximation

arXiv: https://arxiv.org/abs/2307.10829 · HTML: https://mlsanity.com/paper/2307.10829/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 34 of its 39 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [StyleGAN-XL: Scaling StyleGAN to Large Diverse Datasets](https://mlsanity.com/paper/2202.00273/) (2022)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [DragDiffusion: Harnessing Diffusion Models for Interactive Point-based Image Editing](https://mlsanity.com/paper/2306.14435/) (2023)
- [EDICT: Exact Diffusion Inversion via Coupled Transformations](https://mlsanity.com/paper/2211.12446/) (2022)
- [An Edit Friendly DDPM Noise Space: Inversion and Manipulations](https://mlsanity.com/paper/2304.06140/) (2023)

## Built on by (most cited first, 30 of 44)

- [Taming Rectified Flow for Inversion and Editing](https://mlsanity.com/paper/2411.04746/) (2024)
- [Arbitrary-steps Image Super-resolution via Diffusion Inversion](https://mlsanity.com/paper/2412.09013/) (2024)
- [On Exact Inversion of DPM-Solvers](https://mlsanity.com/paper/2311.18387/) (2023)
- [ROBIN: Robust and Invisible Watermarks for Diffusion Models with Adversarial Optimization](https://mlsanity.com/paper/2411.03862/) (2024)
- [UniEdit-Flow: Unleashing Inversion and Editing in the Era of Flow Models](https://mlsanity.com/paper/2504.13109/) (2025)
- [Lightning-Fast Image Inversion and Editing for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2312.12540/) (2023)
- [How I Warped Your Noise: a Temporally-Correlated Noise Prior for Diffusion Models](https://mlsanity.com/paper/2504.03072/) (2025)
- [BELM: Bidirectional Explicit Linear Multi-step Sampler for Exact Inversion in Diffusion Models](https://mlsanity.com/paper/2410.07273/) (2024)
- [Videoshop: Localized Semantic Video Editing with Noise-Extrapolated Diffusion Inversion](https://mlsanity.com/paper/2403.14617/) (2024)
- [Source Prompt Disentangled Inversion for Boosting Image Editability with Diffusion Models](https://mlsanity.com/paper/2403.11105/) (2024)
- [IMPUS: Image Morphing with Perceptually-Uniform Sampling Using Diffusion Models](https://mlsanity.com/paper/2311.06792/) (2023)
- [GenPO: Generative Diffusion Models Meet On-Policy Reinforcement Learning](https://mlsanity.com/paper/2505.18763/) (2025)
- [Analyzing and Mitigating Model Collapse in Rectified Flow Models](https://mlsanity.com/paper/2412.08175/) (2024)
- [Visual Prompting for One-shot Controllable Video Editing without Inversion](https://mlsanity.com/paper/2504.14335/) (2025)
- [Gradient-free Decoder Inversion in Latent Diffusion Models](https://mlsanity.com/paper/2409.18442/) (2024)
- [Efficiently Access Diffusion Fisher: Within the Outer Product Span Space](https://mlsanity.com/paper/2505.23264/) (2025)
- [ERDDCI: Exact Reversible Diffusion via Dual-Chain Inversion for High-Quality Image Editing](https://mlsanity.com/paper/2410.14247/) (2024)
- [Oscillation Inversion: Understand the structure of Large Flow Model through the Lens of Inversion Method](https://mlsanity.com/paper/2411.11135/) (2024)
- [TimeWak: Temporal Chained-Hashing Watermark for Time Series Data](https://mlsanity.com/paper/2506.06407/) (2025)
- [Frequency-Calibrated Membership Inference Attacks on Medical Image Diffusion Models](https://mlsanity.com/paper/2506.14919/) (2025)
- [GenPO++: Generative Policy Optimization with Jacobian-free Likelihood Ratios](https://mlsanity.com/paper/2606.06967/) (2026)
- [Task-Oriented Diffusion Inversion for High-Fidelity Text-based Editing](https://mlsanity.com/paper/2408.13395/) (2024)
- [DIVE: Inverting Conditional Diffusion Models for Discriminative Tasks](https://mlsanity.com/paper/2504.17253/) (2025)
- [Don't Forget your Inverse DDIM for Image Editing](https://mlsanity.com/paper/2505.09571/) (2025)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction](https://mlsanity.com/paper/2506.18290/) (2025)
- [Contrastive Diffusion Alignment: Learning Structured Latents for Controllable Generation](https://mlsanity.com/paper/2510.14190/) (2025)
- [On Exact Bit-level Reversible Transformers Without Changing Architectures](https://mlsanity.com/paper/2407.09093/) (2024)
- [Improving Diffusion-Based Image Editing Faithfulness via Guidance and Scheduling](https://mlsanity.com/paper/2506.21045/) (2025)
- [Editable Noise Map Inversion: Encoding Target-image into Noise For High-Fidelity Image Manipulation](https://mlsanity.com/paper/2509.25776/) (2025)

## Abstract (arXiv)

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 bi-directional integration approximation (BDIA), to perform exact diffusion inversion with neglible computational overhead. Suppose we would like to estimate the next diffusion state boldsymbolzi-1 at timestep ti with the historical information (i,boldsymbolzi) and (i+1,boldsymbolzi+1). We first obtain the estimated Gaussian noise hatboldsymbolε(boldsymbolzi,i), and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot [ti, ti-1] in the forward manner and the previous time-slot [ti, tt+1] in the backward manner. The DDIM step for the previous time-slot is used to refine the integration approximation made earlier when computing boldsymbolzi. A nice property of BDIA-DDIM is that the update expression for boldsymbolzi-1 is a linear combination of (boldsymbolzi+1, boldsymbolzi, hatboldsymbolε(boldsymbolzi,i)). This allows for exact backward computation of boldsymbolzi+1 given (boldsymbolzi, boldsymbolzi-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.