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Exploring Fixed Point in Image Editing: Theoretical Support and Convergence Optimization
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Proves existence and uniqueness of fixed points in DDIM inversion via a contraction argument, proposes an inter-path fixed-point convergence loss (f(zt)-f(ẑt)), and applies fixed-point-based NTI to unsupervised image dehazing.
Problem
DDIM inversion approximates z_t by z_{t-1}, introducing accumulating errors that hurt image reconstruction and editing; prior fixed-point methods (AIDI, FPI) lack proof that fixed points exist and use a convergence criterion that may not identify true fixed points.
Approach
Shows the DDIM inversion step map is a contraction (Lipschitz constant 1 - k·sqrt(alpha_bar_t) < 1), so Banach's theorem gives unique fixed points; because of uniqueness, replaces the intra-path convergence criterion f(zt)-zt with an inter-path criterion f(zt)-f(ẑt) using a perturbed path ẑt = zt + ε.
Key results
- On PIE-Bench, the inter-path fixed-point criterion f(zt)-f(ẑt) improves P2P editing structure distance from 69.953 to 69.190 and PSNR (whole) from 25.01 to 25.31 versus P2P baseline.
- On real-world RESIDE dehazing, NTI with fixed point lowers BRISQUE from 27.5879 to 24.1297 and NIQE from 4.4499 to 3.4777.
- Fixed-point inter-path loss is reported to converge at about 7 iterations, lagging the intra-path loss which converges in 2-3 iterations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | Structure Distance×103 ↓: 69.521 / 70.495 -> 69.190 / 70.046 | Helped |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | PSNR ↑: 17.83 / 17.73 -> 17.87 / 17.74 | Mixed |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | LPIPS×103 ↓: 208.98 / 210.76 -> 208.88 / 210.71 | Helped |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | MSE×104 ↓: 221.48 / 225.60 -> 219.85 / 225.50 | Helped |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | SSIM×102 ↑: 71.14 / 70.91 -> 71.15 / 70.91 | Mixed |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | CLIP similarity Whole ↑: 25.19 / 23.66 -> 25.31 / 23.80 | Helped |
| Use inter-path loss instead of intra-path loss for fixed-point convergence (ablation) | P2P fixed-point image editing · PIE-Bench | CLIP similarity Edited ↑: 22.53 / 21.33 -> 22.58 / 21.37 | Helped |
| Add fixed-point inversion to NTI | NTI dehazing · RESIDE real-world scenes | BRISQUE ↓: 27.5879 -> 24.1297 | Helped |
| Add fixed-point inversion to NTI | NTI dehazing · RESIDE real-world scenes | NIQE ↓: 4.4499 -> 3.4777 | Helped |
Surprising findings
- The intra-path loss f(zt)-zt converges within 2-3 iterations while the inter-path loss lags, suggesting prior convergence criteria stop before fixed points are accurate.
Limitations
- Dehazing results show artifacts due to imprecise haze attention maps localized in deeper UNet layers.
- Authors note the approach needs more computation for fixed-point iterations and they did not report error bars or statistical significance.
- Priors such as dark channel and depth weighting can cause image collapse in some cases.
Experimental setup
Datasets: PIE-Bench RESIDE
Benchmarks: PIE-Bench RESIDE
Metrics: Structure distance PSNR LPIPS MSE SSIM CLIP similarity BRISQUE NIQE
Baselines: P2P f(zt)-zt fixed-point criterion (AIDI/FPI) NTI NTI w fixed point
Builds on: DDIM Prompt-to-Prompt null-text inversion AIDI FPI
Closest papers by meaning
- Tuning-Free Inversion-Enhanced Control for Consistent Image Editing (AAAI 2024 · similarity 0.89)
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- Inversion-Free Image Editing with Language-Guided Diffusion Models (CVPR 2024 · similarity 0.88)
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