ChordEdit: One-Step Low-Energy Transport for Image Editing

• One-step T2I models provide unprecedented synthesis speed and motivate real-time text-guided image editing.Source: §1 · p. 2
• Training-free simple-drift editing fails in one-step models through severe object distortion and loss of non-edited-region consistency.Source: §1 · p. 2
• ChordEdit recasts editing as dynamic optimal transport and replaces erratic drift differences with a time-weighted Chord Control Field.Source: §1 · p. 2
• The resulting low-energy field supports a single large integration step and optional proximal refinement for stronger target semantics.Source: §1 · p. 2
• ChordEdit demonstrates state-of-the-art efficiency while maintaining high background preservation and semantic fidelity on PIE-bench.Source: §1 · p. 2

• Prior diffusion and flow-based editors often require iterative multi-step inversion or few-step acceleration, making real-time interaction infeasible.Source: §2 · p. 3
• Training-free differential editors are stable when averaged over multiple steps but collapse in the single-step limit because of high energy and variance.Source: §2 · p. 3
• SwiftEdit achieves one-step performance with a dedicated inversion network but sacrifices model-agnostic flexibility.Source: §2 · p. 3

• A pre-trained T2I model induces a conditional probability flow with drift v(x_t,t,c), where x_t is the image state and c is the text condition.Source: §3.1 · p. 3
• Editing transports a source image sampled under c_src to a target image sampled under c_tar by modifying the source flow with an instantaneous residual.Source: §3.1 · p. 3
• The ideal editing field aligns the source and target conditional dynamics in continuous time.Source: §3.1 · p. 3
• Because arbitrary-time states are unavailable, ChordEdit queries a synthetic noisy proxy and forms a target-minus-source observable residual through a time-only linear map.Source: §3.2 · p. 3
• The observable formulation applies to drift, noise-prediction, velocity, score-to-drift, x0, and consistency parameterizations through suitable time-only linear maps.Source: §3.2 · p. 3, §3.3 · p. 3

4. ChordEdit

• ChordEdit combines dynamic optimal transport with observable model fields to estimate a low-energy control field for stable one-step editing.Source: §4 · p. 3, §4.1 · p. 3

4.1. OT View: Editing as an Estimation Problem

• The ideal transport field is inaccessible, while directly using the noisy observable field is unstable for single-step integration.Source: §4.1 · p. 3

4.2. Chord Control: A Low-Energy Local Estimator

• ChordEdit estimates a locally smoothed field by minimizing a strictly convex quadratic surrogate over a short temporal window.Source: §4.2 · p. 4
• The practical Chord Control Field uses causal temporal averaging of observable fields at t and t−δ, producing a one-sided kernel smoothing of the naive field.Source: §4.2 · p. 4
• Temporal averaging contracts field, time-derivative, and spatial-gradient norms, suppressing high-energy spikes and tightening explicit-Euler stability.Source: §4.2 · p. 4
• The chord field satisfies C_cho ≤ C_nai, reducing local truncation error and improving the stability margin for a one-step integration.Source: §4.2 · p. 4

4.3. Proximal Refinement

• An optional single forward-pass proximal refinement amplifies target semantics without reinversion and remains separate from transport energy measurements.Source: §4.3 · p. 4

4.4. Algorithm

• The default single-noise implementation computes all intermediate variables in parallel, rendering transport 1-NFE.Source: §4.4 · p. 5

Experimental Setup

• PIE-bench contains 700 512×512 editing examples across 10 categories, evaluated for background fidelity and semantic alignment.Background fidelity uses PSNR and MSE on non-edited regions.Source: §5.1 · p. 5

Quantitative Results

• ChordEdit achieves state-of-the-art efficiency while preserving competitive editing quality, including less than half SwiftEdit’s VRAM use.It is also reported as 19× faster than FlowEdit and over 208× faster than Direct Inversion.Source: §5.2 · p. 5

One-Step Editing

• At NFE=1, transport-only ChordEdit achieves high PSNR, while optional proximal refinement further improves semantic alignment.The full method balances transport stability with semantic enhancement without model-specific training, inversion, or protective masking.Source: §5.2 · pp. 5–6

Qualitative Results

• Qualitative comparisons show prompt-faithful edits with strong background preservation and fewer artifacts or identity failures than competing methods.A user study reports preferences of 42.5% for editing semantics and 48.3% for background preservation.Source: §5.2 · p. 6
• Energy visualizations associate the naive field’s high energy with artifacts and background corruption, whereas ChordEdit produces a stable low-energy field.The comparisons use SwiftBrush-v2 and SD-Turbo results.Source: Figure 8 · p. 7, Figure 9 · p. 7

Chord Control Field

• As integration steps approach one, the naive field’s energy spikes and PSNR collapses, while ChordEdit remains stable and Pareto-dominates the naive baseline.The comparison uses δ=0 for the naive baseline and δ=0.15 for ChordEdit.Source: §6.1 · p. 7

Noise Samples

• Increasing noise samples produces negligible marginal returns for ChordEdit, whose n=1 Pareto front remains nearly unchanged through n=4.Figure 10 qualitatively confirms the same pattern.Source: §6.2 · p. 8, Figure 10 · p. 8, Figure 11 · p. 8
• ChordEdit’s single-noise configuration remains stable and Pareto-dominates the naive baseline even when the naive method uses four samples.The results attribute this behavior to geometric control and intrinsic variance reduction.Source: §6.2 · p. 8

Transport and Refinement

• The Chord field prioritizes consistency with 23.89 PSNR, while proximal refinement raises CLIP-Edited from 21.87 to 22.96.This ablation separates high-fidelity transport from semantic amplification.Source: §6.3 · p. 8, Table 2 · p. 8

Model Agnosticism

• On SD-Turbo, ChordEdit improves PSNR from 21.38 to 22.20 and CLIP-Edited from 21.96 to 22.96 over the naive baseline.Table 3 reports consistent improvement across tested models.Source: §6.4 · p. 8, Table 3 · p. 8

Conclusion

• ChordEdit replaces the naive high-energy drift difference with a temporally smoothed Chord Control Field that enables a single large integration step.The field preserves non-edited regions while maintaining semantic alignment.Source: §7 · p. 8
• ChordEdit achieves 0.38s runtime and low VRAM use while maintaining high fidelity and strong semantic alignment.The conclusion reports these results alongside comparisons with naive and other one-step methods.Source: §7 · p. 8
• The method remains robust across models and seeds, even with a single noise sample, supporting real-time high-fidelity editing.The stated scope is generative image editing for creative and assistive applications.Source: §7 · p. 8

Related Work

• Prior editing methods trade off inversion, training, or multiple inference steps, while ChordEdit targets training-free, inversion-free, single-step editing.Fast one-step T2I backbones provide the foundation but have nonlinear dynamics that complicate control.Source: A. Full Related Work · pp. 11–12

Our Method

• ChordEdit constructs a control field in the observable residual domain to average and stabilize the high-energy signal.The field is paired with proximal refinement.Source: A. Full Related Work · p. 12, §B.1 · p. 12

Optimal-Transport Formulation

• The formulation treats ut as the complete vector field transporting the source density ρ1 to the target density ρ0.No reference flow or additive residual field is introduced.Source: §B.1 · p. 12, §B.2 · p. 13

Estimator Derivation

• The causal estimator combines noisy local measurements, short-window homogeneity, and a recursive quadratic energy prior.Under these assumptions, the closed-form estimator is exact before first-order online approximations.Source: §B.1 · p. 12, §B.3 · p. 13, §B.4 · p. 13
• The online approximation has O(δ) error under standard smoothness, while measurement noise enters through the observable surrogate.Density-weighted effective durations can replace scalar time weights without changing the closed form.Source: §B.5 · p. 13

Model-Agnostic Mapping

• A linear time-dependent map Bt converts model outputs into a unified velocity domain, with shared-noise residuals enforcing a fixed noisy state.The supplementary derivations cover noise-, velocity-, x0-, and v-prediction parameterizations.Source: C. Unified Comparison-Domain Map and… · p. 13, §C.1 · p. 13, §C.2 · p. 13

Implementation

• Continuous-time derivatives ˙α(t) and ˙σ(t) are approximated using first-order finite differences.For example, ˙α(t) ≈ (α(t) − α(t − δ))/δ.Source: §C.5 · p. 15

Numerical Stability

• Maps containing α(t) in the denominator become numerically unstable as α(t) approaches zero near t ≈ 1.This instability affects maps such as A(ϵ)_t and threatens numerical conditioning near the endpoint.Source: §C.5 · p. 15
• ChordEdit queries the field at t and t − δ values bounded away from t = 1, keeping α(t) nonnegligible and B_t well-conditioned.An example uses t = 0.90 and δ = 0.15.Source: §C.5 · p. 15

Theoretical energy contraction

• Temporal smoothing by convolution makes the Chord Control Field an L2-energy contraction of the observable proxy field.The contraction applies to the total Benamou–Brenier kinetic energy and can be strict for non-degenerate smoothing kernels.Source: D. Energy Contraction Property of the… · pp. 15–16
• The chord field generates a dynamic flow with strictly lower or equal kinetic energy than the naive proxy field.This follows from convex averaging and Jensen’s inequality applied to the estimator.Source: D. Energy Contraction Property of the… · p. 16

Numerical error and stability

• Ccho ≤ Cnai, so the chord-controlled path has a global error bound no larger than the naive path for the same step size.The result follows from contraction of field magnitude and consistency terms while preserving the same explicit-Euler stability condition.Source: D. Energy Contraction Property of the… · p. 18, E. Gap to the Optimal Control Field · pp. 21–22

Risk reduction

• For an appropriately chosen non-degenerate kernel, the chord estimator achieves strictly lower risk than the naive estimator.The smoothing trades bias against variance, with the risk bound combining an O(δ2) bias contribution and an O(δ−1)σ2 variance contribution.Source: E. Gap to the Optimal Control Field · pp. 18–19

Empirical validation

• ChordEdit’s smoothed field remains stable as the step count approaches one and strictly Pareto-dominates the naive baseline across reported trade-off spaces.The reported comparisons cover semantic alignment alongside structural or background preservation, while increasing Monte Carlo samples provides negligible marginal returns for ChordEdit.Source: §F.1 · p. 22, §F.2 · p. 22

Analysis of Chord Time t

• Increasing chord time t generally strengthens CLIP-Edited alignment but slightly reduces background fidelity.The reported comparison spans t values such as 0.80 to 1.00.Source: §F.3 · p. 23
• The t = 0.90 setting provides a robust balance between semantic strength and preservation.Source: §F.3 · p. 23

Analysis of Refinement Time tc

• Increasing proximal refinement time tc monotonically improves CLIP alignment while increasing LPIPS distortion.The reported sweep increases tc from 0.1 to 0.5.Source: §F.3 · p. 23
• The tc = 0.30 default strengthens the target semantic while limiting the refinement’s fidelity cost.The analysis describes higher tc as more prone to overediting non-target regions.Source: §F.3 · p. 24

Joint Analysis of δ and λ

• Across preservation metrics, the naive δ = 0 baseline is worst, while increasing δ monotonically improves LPIPS, MSE, SSIM, and PSNR.Source: §F.4 · p. 24
• Increasing λ directly strengthens semantic alignment at the expected cost of decreased background fidelity, producing a predictable control trade-off.Source: §F.4 · p. 25

Quantitative validation

• ChordEdit consistently outperforms the naive field in structural and perceptual fidelity across three fast generative models.The comparison covers InstaFlow, SwiftBrush-v2, and SD-Turbo using structural and perceptual metrics.Source: G. More Quantitative Results · pp. 25, 27
• The Chord Control Field preserves background consistency across pixel-space PSNR, structural SSIM, and deep-feature LPIPS measures.Source: G. More Quantitative Results · p. 26

Transport and refinement

• NFE=1 transport prioritizes fidelity, whereas NFE=2 refinement provides stronger overall semantic alignment.The framework separates consistency-preserving transport from semantic-boosting proximal refinement.Source: G. More Quantitative Results · pp. 26–27

User control

• The step scale λ controls edit strength, with larger values producing progressively stronger target semantics.Source: Figure 20 · p. 27

Societal impacts

• High-fidelity real-time editing can make deceptive content and misinformation more convincing.The authors identify misuse as a risk of realistic, structurally consistent edits.Source: §I · p. 28

Study design

• The study recruited 150 participants for randomized four-way blind comparisons of ChordEdit, InfEdit, FlowEdit, and SwiftEdit.Participants viewed the original image, prompt, and four edited results.Source: J. User Study · p. 28
• Participants judged semantic alignment and preservation quality, including naturalness, background preservation, and artifacts.Source: J. User Study · p. 29

Results

• The study collected 4,500 votes for each criterion and found a clear preference for ChordEdit.The total equals 150 participants × 30 prompts for each criterion.Source: J. User Study · p. 29
• ChordEdit received 42.5% of semantic-alignment preferences, ahead of FlowEdit at 25.3%, InfEdit at 19.6%, and SwiftEdit at 12.6%.Source: J. User Study · p. 29
• ChordEdit received 48.3% of preservation-quality preferences, exceeding InfEdit’s 35.4%, FlowEdit’s 7.1%, and SwiftEdit’s 9.2%.The study reports artifacts and distortion as reasons other methods were penalized.Source: J. User Study · p. 29

Conclusion

• ChordEdit was the only method to rank first in both semantic alignment and preservation quality.Source: J. User Study · p. 29