# Generalized Consistency Trajectory Models for Image Manipulation

arXiv: https://arxiv.org/abs/2403.12510 · HTML: https://mlsanity.com/paper/2403.12510/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 26 of its 31 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [A Note on the Inception Score](https://mlsanity.com/paper/1801.01973/) (2018)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [EGSDE: Unpaired Image-to-Image Translation via Energy-Guided Stochastic Differential Equations](https://mlsanity.com/paper/2207.06635/) (2022)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)

## Built on by (most cited first, 11 of 11)

- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [OFTSR: One-Step Flow for Image Super-Resolution with Tunable Fidelity-Realism Trade-offs](https://mlsanity.com/paper/2412.09465/) (2024)
- [PT-Mark: Invisible Watermarking for Text-to-image Diffusion Models via Semantic-aware Pivotal Tuning](https://mlsanity.com/paper/2504.10853/) (2025)
- [LVTINO: LAtent Video consisTency INverse sOlver for High Definition Video Restoration](https://mlsanity.com/paper/2510.01339/) (2025)
- [Fast, faithful and photorealistic diffusion-based image super-resolution with enhanced Flow Map models](https://mlsanity.com/paper/2601.16660/) (2026)
- [Beyond and Free from Diffusion: Invertible Guided Consistency Training](https://mlsanity.com/paper/2502.05391/) (2025)
- [Align Your Tangent: Training Better Consistency Models via Manifold-Aligned Tangents](https://mlsanity.com/paper/2510.00658/) (2025)
- [Native Extrapolation Awareness in Flow-Based Conditional Generation](https://mlsanity.com/paper/2602.13061/) (2026)
- [Post-Hoc Guidance for Consistency Models by Joint Flow Distribution Learning](https://mlsanity.com/paper/2604.08828/) (2026)
- [γ-Bridge: A Look-Parametric Diffusion Bridge](https://mlsanity.com/paper/2607.22719/) (2026)

## Abstract (arXiv)

Diffusion models (DMs) excel in unconditional generation, as well as on applications such as image editing and restoration. The success of DMs lies in the iterative nature of diffusion: diffusion breaks down the complex process of mapping noise to data into a sequence of simple denoising tasks. Moreover, we are able to exert fine-grained control over the generation process by injecting guidance terms into each denoising step. However, the iterative process is also computationally intensive, often taking from tens up to thousands of function evaluations. Although consistency trajectory models (CTMs) enable traversal between any time points along the probability flow ODE (PFODE) and score inference with a single function evaluation, CTMs only allow translation from Gaussian noise to data. This work aims to unlock the full potential of CTMs by proposing generalized CTMs (GCTMs), which translate between arbitrary distributions via ODEs. We discuss the design space of GCTMs and demonstrate their efficacy in various image manipulation tasks such as image-to-image translation, restoration, and editing.