Exploring Fixed Point in Image Editing: Theoretical Support and Convergence Optimization
Authors: chen hang
Zhe Ma
Haoming Chen
Xuwei Fang
Weisheng Xie
Faming Fang
Guixu Zhang
Hongbin Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: fixed point; image editing
Primary area: machine_vision
TL;DR: We provide theoretical support for image editing methods that involve solving fixed points of implicit functions, and improve the fixed point loss in image editing.

Abstract: In image editing, Denoising Diffusion Implicit Models (DDIM) inversion has become a widely adopted method and is extensively used in various image editing approaches. The core concept of DDIM inversion stems from the deterministic sampling technique of DDIM, which allows the DDIM process to be viewed as an Ordinary Differential Equation (ODE) process that is reversible. This enables the prediction of corresponding noise from a reference image, ensuring that the restored image from this noise remains consistent with the reference image. Image editing exploits this property by modifying the cross-attention between text and images to edit specific objects while preserving the remaining regions. However, in the DDIM inversion, using the $t-1$ time step to approximate the noise prediction at time step $t$ introduces errors between the restored image and the reference image. Recent approaches have modeled each step of the DDIM inversion process as finding a fixed-point problem of an implicit function. This approach significantly mitigates the error in the restored image but lacks theoretical support regarding the existence of such fixed points. Therefore, this paper focuses on the study of fixed points in DDIM inversion and provides theoretical support. Based on the obtained theoretical insights, we further optimize the loss function for the convergence of fixed points in the original DDIM inversion, improving the visual quality of the edited image. Finally, we extend the fixed-point based image editing to the application of unsupervised image dehazing, introducing a novel text-based approach for unsupervised dehazing.

Exploring Fixed Point in Image Editing: Theoretical Support and Convergence Optimization
Chen Hang'; Ma²; Haoming Chen', Xuwei Fang', Weisheng Xie
Faming Fang'; Guixu Zhang'. Hongbin Wang"t 'Est China Normal University, Shanghai, 200062 China
2 Ant Group, Hangzhou, 310063 China
Bestpay Al Lab, Shanghai, 200080 China
Abstract
In image editing, Denoising Diffusion Implicit Models (DDIM) inversion has become widely adopted method and is extensively used in various image editing approaches. The core concept of DDIM inversion stems from the deterministic sampling technique of DDIM, which allows the DDIM process to be viewed as an Ordinary Differential Equation (ODE) process that is reversible. This enables the prediction of corresponding noise from reference image, ensuring that the restored image from this noise remains consistent with the reference image. Image editing exploits this property by modifying the cosssatttntionn between text and images to edit specific objects while preserving the remaining regions. However, the DDIM inversion, using the at time step / introduces errors between the restored image and the reference image. Recent approaches have modeled each step of the DDIM inversion process finding a fixed-point problem of an implicit function. This approach significantly mitigates the error in the restored image but lacks theoretical support regarding the existence of such fixed points. Therefore, this paper focuses on the study of fixed points in DDIM inversion and provides theoretical support. Based on the obtained theoretical insights, we further optimize the loss function for the convergence of fixed points in the original DDIM inversion, improving the visual quality of the edited image. Finally, we extend the fixed-point based image editing to the application of unsupervised image dehazing, introducing novel text-based approach for unsupervised dehazing.
time step to approximate the noise prediction
Introduction
Diffusion models have gained significant attention in recent years. Early versions of Denoising Diffusion Probabilistic Models (DDPMs) [T] and score-based generative models ( were capable of generating high-quality images and even surpassed the performance of Generative Adversarial Networks (GANs) [3] at that time. However, the initial diffusion models required large number of sampling steps during image generation, which was major bottleneck of diffusion models. Subsequent research focused on reducing the number of sampling steps without compromising the quality of generated images 4 During this period, DDIM [5] was proposed and widely used for its speed and flexibility in both deterministic and stochastic generation. When DDIM set to deterministic sampling, it can be viewed as an ODE process that is reversible. This property enables realistic image editing, leading to the introduction of Prompt-to-Prone idea of P2P is modify the cross-attention between text and images. For local edits, such replacing
(P2P) [6] editing. The core
'Equal contribution.
1 Corresponding Author, finfang @ cs.ccnu.edu.cn hongbin.whi whb caibb@antgrou.cm
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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cat with dog while keeping the background unchanged, only the cross cross- attention corresponding to the cat needs to be modified. For enhancing certain effects, the cross- atssetonon corresponding the modifying word is reweighted. However, P2P still suffers from unreliable results in realistic image editing, as it exhibits inconsistencies even when directly reconstructing the reference image without any edits. The main cause of this issue lies in the emror introduced during DDIM inversion, which will be discussed later in this paper. To address this problem, some methods introduce auxiliary variables to mitigate the issue, such as null-text inversion (NTI) 7 for learning null-text embeddings. Exact diffusion inversion (EDICT) [ uses two coupled vectors to invert each other in an alternating fashion, to achieve precise diffusion inversion. In recent methods, direct inversion (DI) [9] and inversion-free free editing (InfEdit) [10] decouple the source and target branches to avoid optimizing DDIM inversion and achieve satisfactory results. However, these methods do not address the error issue in DDIM inversion theoretically. This issue can be formulated as solving fixed points of implicit functions, which has been mentioned in accelerated iterative diffusion inversion (AIDI) [[]] and fixed-point inversion (FPI) [2]. However, AIDI and FPI only employ methods to solve fixed points of implicit functions without proving the existence of fixed points each sampling step of DDIM inversion. Only FPI indirectly demonstrates the convergence of fixed-point loss through numerical experiments on the used dataset. Such numerical experiments cannot rigorously prove the existence of fixed points. Therefore, in this paper, we rigorously prove the existence and uniqueness of fixed points DDIM inversion using the Banach fixed-poim theorem [3] Subsequently, based on this uniqueness, we identify flaws in the fixed-point loss used by AIDI and FPI and propose optimization. Finally, we explore the application of fixed-poite priet based image editing in unsupervised image dehazing. In summary, our contributions in this paper are as follows:
We theoretically prove that the Lipschitz constant in DDIM inversion is less than one. Based the Banach fixed-point theorem, we establish the existence and uniqueness of fixed points, providing theoretical support for image editing methods that involve solving fixed points of implicit functions.
Leveraging the uniqueness of fixed points, we demonstrate through theory and experimen- tal cases that the existing methods suffer from flaws in the fixed-point loss and propose optimization.
We extend the fixed-point based image editing approach to the task of unsupervised image dehazing and explore the feasibility of text-guided unsupervised dehazing through fixed- point based editing.
2 Preliminaries: DDPMs and DDIM Inversion
DDPMs model the probability distribution of a diffusion process by iteratively adding noise to data sample until the data distribution becomes predominantly noise [. Subsequently, the data distribution is recovered from random Gaussian noise through reverse diffusion process. The diffusion process is typically regarded as a Markov chain starting from Dy and obtaining I1, T2,.,.... by adding noise. The specific process is outlined as follows:
T q (IiT zo) = II q (x, 1)-1), q (22 2x-1) == N (1r) (I:VI- Bis,- - t=1
(1)
where schedule So G1 .,........ Br E (0, 1). 3e can be expressed as linear combination of zn and Gaussian noise c ~ 1),
3t Vanzo V1 oie, 0t a;:=1 1 -B,, 6 II Oss
In DDPMs, both the diffusion process and the reverse diffusion process are Markov chains, which require a large number of sampling steps during image generation. However, DDIM introduces method to accelerate sampling by decoupling the reverse diffusion process from the Markov chain, making it feasible to perform skip-step sampling and reducing the number of sampling steps [5). Simultaneously, the introduction of Stable Diffusion (SD) [14 alleviates the resource requirements for training diffusion models and improves inference speed. SD involves compressing the image

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into a latent space for diffusion instead of operating in the original pixel-level space. As a result, the reverse diffusion process can be expressed as follows:
Oi-- Ot
+ V1- (1 Et )0=1 eo (2t, L,
2t
where eg represents a network that predicts added noise, 2, denotes the compressed representation of in the latent space, represents the time step, and p represents the encoding of condition text.
Now let's focus DDIM inversion. DDIM inversion seeks to find noise zT given the latent space representation corresponding to given zo and the associated textual prompt This allows for the autoregressive reconstruction of 2p. The process of DDIM inversion can be rewritten by Equation as follows:
(1 Üz- Oktt
2t
1t-1 +
v1 Ct.
en (Zt, (,P)
(4)
04-1
0:-1
(1 Ö 1
21-1 +
vI at
69 (2(-1,t.p)
ar-1
ā=-1
Therefore, invert 2o back to 2r using Equation 4 allowing reverse diffusion process. However, it can be observed that in Equation 4 by 24-1. which inevitably introduces errors that accumulate over time. This inconsistency between the forward and reverse processes of the diffusion leads to poor quality in image reconstruction and editing 7 о
reconstruct 20 using the
2 is directly approximated
Theoretical Support for Fixed Point in DDIM Inversion
From the first equation of Equation 4 it is evident that finding fixed-point problem for the implicit function with respect to ea (2,, t)p Therefore, we can consider the right-hand side of the equation function with as variable:
be regarded solving
/ - -1-ā--1); ea (2t,
f(zx)
V
αt xi-1
2-
+
V1 11-arr
(5)
0:-1
Then, to find f(z) be expressed as follows:
we can use 2t-1 the starting point for iteration. The iteration process can
=1-1
3+1 f 0,
(6)
After multiple iterations, the fixed point of Equation can be found. Now, we focus on the theoretical analysis of whether a fixed point exists in Equation 5 Let of and 2f represent any two points in the function f(2,) at time step f, where i and j. with j greater than are within the range of N. We can obtain the following:
V1 ai V ā
(15(2)) - S(=2)
(ee((t;)))) - ca(2), 4PP)
$ <Vi
- co(2l,t,p)
where || is Le norm. We know that in the reverse diffusion process, when predicting the = value at time step given the value at time step initial rough estimation of 20 is made. This rough estimation of žu introduces some errors compared to the final obtained value of 2g. However, the overall shape of the image is already quite similar, and as time decreases, this rough estimation of gradually approaches the final obtained 2g. This rough estimation of can be represented as follows:
=0 = = v1 (.co(2), Voi. = v1 -(-caoo(eh.l)))) / Vol.
(8)
3

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where d and od represent the predictions of time step zero. Although z and 2g obtained from 2f and zf are very close, there might be some entors when is not close to zero. Let's denote this error as Eg. Additionally, let's define the difference between zf and zf et. With Equation following relationship:
we can derive the
- - = Vasso + - (<o(=), ( co(=),
Due to the fact that different initial values at time step can lead to similar 20. i.c., any position at time step & can converge to small range defined by the rough estimation of zo. we can obtain Jeol = klexl. where 0 < h < 1. Moreover, from the perspective of ODE, the paths from 2f to 2 and from = to =o can be viewed as two ODE trajectories with initial values of of and z/. respectively, or as two integral curves of ODE given the initial values zi and zy. These integral curves non-tangent and non-intersecting (5). Therefore, we can conclude that En and E, have the same sign. We can also prove this by contradiction: if Eo and have opposite signs, then the two ODE trajectories would have an intersection point. From that intersection point, there would exist two different paths in both forward and backward directions, which contradicts the deterministic sampling of DDIM. Consequently, we can deduce Eo = kEg, where obtain:
1. By substituting this into Equation 9
we
2) - =1 = kvaist + v1 ãr (eo(z), L,P) s0(=,,1,2)
(10)
= =koa 2 - 2 + +v- at (co(z)) (,p) - co co(
By rearranging the Equation 10 and substituting it into Equation 7 we obtain:
[lf(z)) - f(2f)1 & (1 (1-kvāi || =/111
(11)
From Equation 2 it is known that ã E (0, 1). Therefore, it can be inferred that (1 - kva E (0. 1).
This proves that the implicit function f(2)) satisfies the contraction mapping condition. By further applying the Banach fixed-point theorem 13 we can conclude that the fixed point of (zz) exists and is unique.
Convergence Optimization for Fixed Point in DDIM Inversion
In AIDI [ and FPI (12), determine the convergence of the fixed point obtained through iterative
processes in the implicit function f2t). the criterion used is f(x)==(+)))))))))))))) 2t. In FPI, numerical experiments conducted the given dataset, and it was concluded that convergence is achieved within 2 iterations. Furthermore, in Section 3. we have proved the uniqueness of the fixed point. Therefore, at
any position with time step i, convergence be reached towards the same point. We express this using Equation| 1T as follows:
[If(z)) - f(z))] < (1 kv a
|| 22
(If(2)) f(()) s (1 ,.)
|| &
(12)
[f((z)) - ff(ei) = kvā,)
|1 2
where 2/ = +,, and 2/ is the result obtained by iterating 2) in the implicit function f(z)). It also be considered that 2; is a new path distinct from 2f. Theoretically, the three cases of Equation 12 have the same convergence rate, which means they have the same Lipschitz constant. However, in practical computations, it is challenging for two different paths to converge exactly to the same fixed point, resulting in numerical errors. Consequently. this leads to the persistence of numerical discrepancies in the third inequality of Equation even at the end of the iteration.
l f(=\))l >E,
(13)
Incorporating the error into the third inequality of Equation 12
for for correction, have:
(]/(=)) f(2))l < (1 k kv |||| +l|l +
(14)
< (1 - k/a + +5) ||ا 2/11. 8>0 0

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past lea
2003
eie path
500
400
193
103
Loaa frand 41
Lona Brand cl
Loas Irans 42
Lowu Irsedd a6.
pith koo
Ver oith 1oo
Inter set
022
g 230
230-
100
133
500
Figure 1: In the diffusion model with a step size of 50 steps, we selected the trend of fixed-point loss for the first 8 steps, while the trends for the remaining steps were consistent and are provided in the appendix A. In each step, we performed 10 iterations. The blue intra path loss represents f(2)) - 24 used in ATDI and FPI, while the orange inter path loss represents our optimized f(z)) f(2,). The trend of the intra path loss aligns with the findings reported in FPI, converging within 2 ~ 3 iterations. However, the inter path Ioss exhibits noticeable lag. Based on the proven uniqueness of fixed points in this paper, the convergence of the intra path loss does not indicate the convergence of fixed points. Instead, attention should be paid to the inter path loss to assess the convergence of fixed points.
It is not difficult to observe that if there are multiple paths and each path only focuses on the variation of the loss within the path, they will have the same convergence rate. However, when considering the loss between paths, there will be noticeable lag in the convergence rate, as shown in Figure !] The loss function f(=) - 2 used in AIDI and FPI precisely corresponds to the case of solely focusing on the variation within paths. As a result, the fixed points obtained when the f(z) ze loss converges may not be the optimal points. To optimize the original criterion for judging the convergence of fixed points, we modify it to f(z)= f(2) and define the convergence of the f(+) - f(2) loss as the convergence of fixed points. Through experimentation, we have also confirmed that the convergence of fixed points actually lags behind what is mentioned in FPI, as illustrated in Figure [I] In order to visually demonstrate the process of fixed point convergence, we randomly select two points from 2t as the axes. Subsequently, multiple paths are sampled by applying & 2 + c, ~ (O. 7). The results obtained from multiple iterations are plotted as trajectory lines, as shown in Figure
By optimizing the fixed point convergence criterion and the fixed-point loss, the computed fixed points will be more accurate. To further improve the precision of fixed points, is natural, as suggested by Figure 2 to consider selecting the final points from multiple paths, finding the cluster centers of these points, and using them as the results of the fixed points. However, this approach would increase the computational burden. Therefore, in subsequent experiments, we only consider the results obtained when the inter path loss converges.
5 Generalization of Fixed Point in DDIM Inversion to Image Dehazing
Image dehazing, as a subtask of low-level vision, primarily aims to remove the haze degradation from hazy images to recover clean images, thereby assisting high-level visual tasks [16 17 for improved performance. Hazy images are typically modeled as follows (18)
((x) = (x))(())))))))))))))))))())))[ + A(1 - ((x)),
(15)
where 1(x) is hazy image, (x) is the real scene to be recovered. t(x) e kk((x) is the medium transmission, where d(x)) is depth and ls is scattering coefficient of the atmosphere. A is the global atmospheric light.
Existing unsupervised image dehazing methods can broadly categorized into learning-based and non-learning-bart based approaches. I Learning-based methods primarily utilize unpaired images [19 20 21 to train networks for unsupervised dehazing, while non-learning-bast methods rely on image priors such dark channel prior [22) and rank-one prior (23). With the development of visual
5

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Irajectory Plat
Trajectory Pict
1.3
1244
83-
1242
Pone2
0.3
1240
13
1.38
-2.
-1.5
-0.
0.
0.5
1.0
14
14:00
1,77
1.470 Point 1
1.476
1.47R
POIn
5e 1,244
Irajectory
Trajectory Pict
22442
2.2188
Po1nt
2.248
12436
nocs
B10000
8.0001
0.0002
Pont
The axes were randomly chosen from two points in 2, The total number of iterations to compute the fixed points was 21, and the corresponding step numbers were labeled in each path in each subgraph. It can observed that the inter path loss starts to converge around the 7th step and beyond, indicating that the convergence of fixed points requires approximately 7 iterations.
Figure 2:
language models, some supervised multimodal dehazing approaches have emerged 24 However, there are few reports unsupervised multimodal dehazing. This paper mainly explores image- editing-based unsupervised dehazing, where the main principle is to achieve dehazing effects through cross-attention replacement of corresponding haze. Since the haze is replaced with null text, the null text optimization from NTI is necessary [7]. Otherwise, as the class-free guidance coefficient increases, the recovered image may collapse. The DDIM Inversion used in NTI does not involve fixed point correction. Therefore, in subsequent experiments, we will show that the use of fixed-poin optimized NTI can effectively alleviate the image collapse issue in dehazed images.
Experiments
This section will be divided into two parts. The first part focuses on image editing experiments, aiming to highlight that modifying the convergence criterion fixed points leads to better visual quality of the edited image compared to previous approaches. Additional techniques have also been introduced in AIDI and FPI to further enhance the visual effects of image editing. However, to avoid interference, only reproduced their fixed point code. For the dataset, selected PIE-Bench [9] in the context of image editing, which consists 700 images featuring 10 distinct editing types.
The second part comprises experiments on unsupervised image dehazing. Our experiments are based on the NTI approach, incorporating fixed points to alleviate the issue of image collapse in NTI based dehazed images. We directly utilized the real-world scenes provided in the RESIDE [25] dataset for our experiments. Finally, both parts of the experiment employed identical configurations, comprising a sampling step of 50 and a class-free guidance coefficient of 7.5. In the context of image dehazing, to preclude any potential disruption of haze-free regions, no additional text descriptions were introduced. The original prompt solely 'haze', while the target prompt designated as '0'

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Input
P2P
f(zt)
2t
f(z))
f(2t)
a slanted rusty mountain bicycle the building
an orange van with surfboards on hop
youny old woman is holding dog
on pointing of kitchen
digital art of brown hair with flying butterfly
serious angry mon
2, denotes the intra path loss, while f(x) f(&) signifies the optimized inter path loss. In the prompt, red signifies additions, whereas strikethrough indicates deletions. More results can be found in appendix A.2
Figure 3: f(+)
6.1 Image Editing
In the experiments on image editing, we primarily focused on the P2P and its variations: P2P with fixed points and P2P with improved convergence of fixed points. P2P serves as baseline to demonstrate the effectiveness of fixed points. The proposed optimization is compared to the original fixed-point method to showcase the improved visual quality of the edited images. We evaluate the effectiveness of our proposed optimization from two perspectives. Firstly, directly compare the imaging results of image editing, as shown in Figure and Table [ In
fixed point convergence
7

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Input
f(z))
f(x)) f(2,)
Figure 4: The autoregressive reconstruction result derived from noise subjected DDIM inversion.
Table 1: Performance results for image editing and reconstruction, with evaluation metrics drawn from PIE-Bench ( presented across seven indicators spanning three dimensions. Bold indicate the best results.
Structure
Background Preservation
CLIP Similariy
Method
Distancey 10
PSNR
LPIPS, 10
MSF 10
SSIM, 102
Whole
Edited
Image Editing Image Reconstruction
P2P
69.953 70.753
17.87 17.73
208. 89/ 210.78
219.88 225.78
71.14/ 70.91
25.01 23.669
22..4/21.29
[(x)
69.521 / 70.495
17.83 / 17.73
208, 98 210.76
221.48/ 225.60
71.4/ 70.91
25.19 / 23.66
22.53 / 21.33
[ia)) f(a))
69.190/70.000 70.004
208.88 210.71
71.15/0.99
23.80
22.58 / 21.37
Figure 3 we selected some results that exhibit noticeable differences before and after optimization. It can be observed that optimizing the convergence of fixed points leads improved performance and robustness. Secondly, we also evaluate the three methods in terms of autoregressive reconstruction of the images, as shown in Figure 4 and Table [ It can be observed that using the optimized fixed points yields reconstructed images that have higher similarity to the input images.
6.2 Image Dehazing
In the unsupervised image dehazing experiments, we pri- marily compared the NTI approach with NTI incorporating fixed points. From the experimental results depicted in Figure 5 and Table 2 it can be observed that the inclu- sion of fixed points in NTI effectively mitigates the image collapse issue in certain dehazed images. As mentioned carlier, in unsupervised dehazing, where the attention for haze is replaced with null text attention, not using NTI would result in image collapse when the class-free guid- coefficient increases. Therefore, also included examples of image collapse in Figure 5
Table 2: Results of dehazing in real- world scenes, evaluation conduct- ed utilizing Non-Reference Image Quali- ty Assessment (NR-IQA). Bold indicate the best results.
Method
NR-IQA
NTI
27.5879
4,1499
NTI w fixed point fised point NT1
24.1297
3.4777
37.1827
7.7027
Limitations, Prospects and Impact
Although the fixed-point loss achieves better visual quality after optimization, there still room for improvement. Strategies proposed in AIDI and FPI can be employed in image editing to further enhance the consistency between edited images and reference images. However, in this study, we did not reproduce the mentioned strategies in order to avoid interference with the optimization of the fixed-point. We now primarily discuss the limitations of unsupervised image dehazing and other image restoration tasks. In unsupervised dehazing, although some haze can be effectively removed

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Input
NTI
NTI Fixed
Point
Fixed Point w/o
NTI
Haze
Attention Map
Figure 5: Imaging results of image dehazing under real-world scenes in RESIDE dataset.
from images, there are still issues with artifacts. For example, in the middle column of Figure some artifacts be observed. We believe that one reason is the imprecise attention map corresponding to the haze. We present the matched attention map in last line from Figure It can seen that the attention map corresponding to the haze semantics only provides rough match, far from pixel-level alignment. After analyzing the process of attention map generation, we found that the attention map mainly resides in the deeper layers of the Unet. This implies that pixel-level fine-grained alignment cannot be achieved, which also hinders the direct transfer of image editing to some fine-grained image restoration tasks such as image deraining.
We have summarized some experiences in attempting to improve the performance of unsupervised image dehazing. A more straightforward approach is to fine-tune the dehazing task using existing mature techniques such as Lora (26) and Adapter 277, enabling the model to match more accurate haze attention maps. Another strategy that does not require training is to introduce image priors to weight the haze attention map. Since the attention map is primarily obtained by summing and averaging multiple layers of cross-attention, the priors need be weighted into each layer of cross- attention. However, the cross-attention at different layers may have little correlation with the final haze attention map in terms of morphology. We attempted
directly weight the dark channel prior [22] and depth estimation prior (28) into each layer of cross-attention and found that this direct weighting method can improve the restoration results of some images but may also cause some images to collapse (see appendix A.3) Therefore, designing a weighted algorithm specifically for priors is
necessary. In addition, similar to other generative models (29 30 31,, the ability of image editing can be exploited to generate deceptive and harmful content, and the fixed-point based image editing may potentially exacerbate the negative impact of deep generative models for malicious purposes.
Conclusion and Future Work
This paper fills the theoretical gap in the fixed- sreddppoint theory of DDIM inversion for image editing, providing theoretical support for image editing methods that employ fixed points. Additionally, based on the unique conclusion regarding the existence of fixed points obtained in this paper, optimize the convergence criterion for fixed points and propose the inter path loss as a measure for assessing fixed-poni convergence, supported by both theoretical analysis and experimental results. Through

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experiments conducted on PIE- PIE-Bench, we demonstrate improved visual quality after optimization. Finally, we extend the application of fixed-point based image editing to unsupervised image dehazing and analyze the limitations and prospects of image restoration tasks. In our future works, we will delve further into axed-point computations and unsupervised semantic based image restoration. Regarding fixed points, our exploration will focus on reducing computational overhead associated with their calculation and enhancing the precision. Concerning unsupervised image restoration, we will primarily examine ways to improve the corrective impact of priors imprecise semantic attention maps, thereby achieving superior image restoration outcomes.
Acknowledgements
This work was supported by the National Key R&D Program of China (2022ZD0161800) and the National Natural Science Foundation of China under Grant 62271203. Besides, this work were supported by Ant Group and Bestpay Al Lab.
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[20] Bosheng Ding, Ruiheng Zhang, Lixin Xu, Guanyu Liu, Shuo Yang. Yumeng Liu, and Qi Zhang. U 2 net: Unsupervised unified image dehazing and denoising network for single hazy image enhancement. IEEE Transactions on Multimedia, 2023.
[21] Yang Yang, Chaoyue Wang, Risheng Liu. Lin Zhang, Xiaojie Guo, and Dacheng Tao. Self-augmented unpaired image dehazing via density and depth decomposition. In In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 2037-2046, 2022.
[22] Kaiming He, Jian Sun, and Xiaoou Tang. Single image haze removal using dark channel prior. IEEE transactions on pattern analysis and machine intelligence, 33(1))2334 2253, 2010.
[23] Jun Liu, Ryan Wen Liu, Jianing Sun, and Tieyong Zeng, Rank-one prior: Real-time scene recovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.
[24] Ziwei Luo, Fredrik K Gustafison, Zheng Zhao, Jens Sjölund, and Thomas B Schön. Photo-realistic image restoration in the wild with controlled vision-language models. arXiv preprint wrXiv:240.009732, 2024.
[25] Boyi Li, Wenqi Ren, Dengpan Fu, Dacheng Tao, Dan Feng. Wenjun Zeng, and Zhangyang Wang. Bench- marking single-image dehazing and beyond. IEEE Transactions Image Processing, 28(1):492 505. 2018.
[26] Edward Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation large language models. arXiv preprint arXiv:2106.00 09685, 2021.
[27] Neil Houlsby, Andrei Ciurgiu, Stanislaw Jastrzebski. Bruna Morrone, Quentin Laroussilhe, Andrea Ges- mundo, Mona Attariyan, and Sylvain Gelly. Parameter-efficceet transfer learning for nlp. In International conference on machine learning, pages 2790-2799, PMLR, 2019.
[28] Lihe Yang. Bingyi Kang, Zilong Huang. Xiaogang Xu, Jiashi Feng, and Hengshuang Zhao. Depth anything: Unleashing the power of large-scale unlabeled data. arXiv preprint arXiv:2401. 10891, 2024.
[29] Yilun Xu, Ziming Liu. Max Tegmark, and Tommi Jaakkola. Poisson flow generative models. Advances in Neural Information Processing Systems, 35:1 117766116779, 2022.
[30] Cheng Lu, Yuhao Zhou Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775-55 5787, 2022.
[31] Alexandros Graikos, Nikolay Malkin, Nebojsa Jojic, and Dimitris Samaras. Diffusion models as plug-and- and- play priors. Advances in Neural Information Processing Systems, 35:14715 -4728755-44728 2022.
11

---

Figure 6: The trend of fixed-point loss for 50 steps.
A Appendix
A.1 Loss Trend for 50 steps
We provide the complete loss trend in Figure 6 It can be observed that the intra path loss exhibits noticeable hysteresis. Since the primary purpose is to demonstrate the hysteresis rather than illustrate the final convergence position, did not use logarithmic scaling on the coordinate axes.
A.2 More Experiments
12

---

Input
P2P
f(21) 2t
f - f(2t)
progintl__pompt [round] cake with orange fresting wooden plate'
"editing prompt": "a [square] cake with orange fresting on wooden plate'
prigptt "a
ppompt":
[cat] sitting on
wooden choin"
"editing prompt"
[dog] sitting on
wooden choir"
"original prompt":
"blue light, block
and white (cot) is
playing with
flower".
promptt
"blue light. block and [dog] is playing with a flower'
ornginol_prompp""
"an [orange] cat
sitting top of fence",
saiting_prompee
[black]
sitting on top of
fence"
*oniginal_peen prompt": "a [dog] lyying down on a white background",
"editing_ppeeee prammtt__rroppt" " ta [lion] is loying down white background*
"original_ prompr":
"meat [balls] on
white plate".
"meat [sushi] on plate"
original prompt":
the statue of
liberty holding
[torch]'.
ediiing prompt":
"the stotue
liberty holding a
[flower]'
"original_promme prompt"" [white] horse running in the sunset" "editing prrting_roompt [golden] horse running in the sunset"
13

---

"oniginal__pner prompt":
[black] chair in conference room
"editing prompt [blue] choir conf erence room
"original_ prompt": "a
fluffy dog with
blue leash sitting in
the [gnass]"
editing prompt":
fluffy dag with
blue leash sitting
the [ground]"
"original_prome""
[meerkat] puppy wrepped blue towel',
"editing prompt": "a [lion] puppy wrapped blue towe!"
"original_ppee prompt":
[purple] tulips vase
tediting__pni p [yellow] tulips vase
Begginal_poopptte "a womon in a [jocket]
standing in the rain'.
"editing_pronme promppt t
woman in a [blouse] standing the rain'
originollprempp"" "the city of dresden, germany, "editinng prompt": "(a sunny day of) the city dresden. germany, europe"
"original_promppt" p 'a
cute dog holding
[red] heart",
"editing_ prommptt 'a
cute dog holding
[pink] heart"
"original_ :roppptt_ppoppt 'a water droplet hongs
from a string
lights'.
"editing prompt":
water deoplet hangs
from string [ned] lights"
14

---

"original prompt": "a germon shepherd dog stands on the with mouth [clesed]".
"editing promppt: 'a
german shepherd dog stands on the grass with mouth [opened]"
"original_ progptt 'a
golden retriever (holding a flower]
sitting the ground front of fence"
"editing_ prompt" 'a golden retniever
sitting on ground in front of fence"
proomp" "three white
[dumplings] on brown bowl". "editing_prommee
three white [sushi]
brown bowl
p "white [Tiger] on brown ground". eediting_promptt "white [cat] on brown ground'
"original_ t
birds sitting on branch"
"editing__pree p:
"wwo [origami] birds
sitting on branch"
"originol promppth "o [gray] horse the field".
*dditing prompt": to [white] horse the field'
'original_pomee " [stream] in lush
green forest with
rocks".
"editingg_prmppt""
[road] in lush green
forest with rocks'
Drigiial_promptee "c group of [pink]
flowers hanging from tree'.
"editing__pron prompt":
group of (red)
flowers hanging from
15

---

Bniginal__romppe [smoke]] ediiting "editing_prompt"" [fire]
"originol_prome" prompt"" mmuunaain landscape
with (flowers and]
withered gross",
"mountain landscape with withered gross"
"original_ promppt': "a
little girl with
[green] eyes and
flowers in blue
dress".
editing_prempp" 'a little ginl with [blue]
eyes flowers
blue dress"
"original_ prompt""
"sunset field
with clouds and [bright] sky". "editing_prompt";
"sunset over with clouds and a [dork]
"originol_ =prompt:: "o bird standing [clods]]
lediting lediting_prompe ppompptt 'o bird standing [egg=]'
"original_prompt" "a camena and a
notebook [bed]'
edittng_prompp"" 'a
cameea and
notebook [table]'
Banhgina__propptt
[yellow] apple sitting
on top of a wooden
table'
"editing prompt":
[ned] apple sitting on
of wooden table"
"original_ prompt":
"oceon with [clouds]
the sky".
"editing prompt":
occeon with [stors] in the sky
16

---

"original prompt":
"the [almond] tree blooming in the spring".
pediting_proppt'. "hee [violet] tree is blooming the spring"
"original prompt": "two [red] bernies on tree
blue sky in the
background",
"editing prompt":
"two [purple] bernies
tree branch
blue sky
background"
"original promptt: "o
of colorful
flowers
[woeden] table"
prrmmiggppmmpt..... 'a
vase of colorful
flowers on [plastic]
table"
paroginon_ppoopt" "a pointing of woman holding [pink flower]:. editiing promptttttptt...m 'a pointing holding a (teddy bear
"originol_pre _prompt "o the [rail]". "editing__pee prrritt___p_p_______ 'a
rose the [wood]"
"original_ prompt": "a woman with [flowers) her hair" "editing_pee p woman with [snakes] her hair"
"originol_ppene "pompptt 'o photo of man and woman sitting ground".
bediting_prompt"" "a [colorful] photo
man and womon sitting the ground"
"original_prone prompt "woman with [brown] hair" "editiing_poompt" : Womon with [blue]
hair"
Figure The image editing results PIE-Bench.
17

---

input
fixed point
depth prior
dark channel prior
Success Cases
Fallure Cases
Figure 8: The class-fres guidance coefficient is set to The red boxes highlight noticeable changes, which are beneficial in the success cases but detrimental in the failure cases.
A.3 Dark Channel and Depth Prior
We present success cases and failure cases of incorporating depth prior and dark channel prior, as shown in Figure & In the experiments, we directly utilized the fixed point without NTI. To prevent the collapse caused by an increased class-free guidance coefficient, we set the class-free guidance coefficient to this case.
18

---

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24


Reviews:
[{"review_id": "EUetLKDrz7", "reviewer": "Reviewer_cPeB", "summary": "Recent methods treat each step of DDIM inversion as a fixed-point problem to reduce errors, but they lack theoretical support. This paper addresses this gap by making the following contributions: This paper theoretically proves that the Lipschitz constant in DDIM inversion is less than one. By applying the Banach fixed-point theorem, it establishes the existence and uniqueness of fixed points, thus providing the necessary theoretical foundation for image editing methods involving implicit functions. It leverages the uniqueness of fixed points to highlight flaws in existing fixed-point loss methods through theoretical analysis and experimental cases, subsequently proposing optimizations. Furthermore, this paper extends the fixed-point based image editing approach to the task of unsupervised image dehazing, and explores the feasibility of text-guided unsupervised dehazing through fixed-point based editing.", "questions": "My question is included in the weaknesses section. I would be willing to raise the rating if my concern is addressed.", "limitations": "Please check weakness", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "1. This paper proves the existence and uniqueness of fixed points in DDIM inversion using the Banach fixed-point theorem.\n2. It identifies flaws in the loss functions AIDI and FPI used during inversions.\n3. The authors modify the loss function to $f(z_t) - f(\\hat{z}_t)$", "weaknesses": "1. In Section 3, the statement \"Due to the fact that different initial values at time step $t$ can lead to similar $z_0$, i.e., any position at time step $t$ can converge to a small range defined by the rough estimation of $z_0$, we can obtain $|\\epsilon_0| = k|\\epsilon_t|$, where $0 < k < 1$,\" might be assertive. This assertion is especially unconvincing when $t$ is large. A rigorous proof for this part would be beneficial.\n\n2. The experiment only compares NTI and P2P, omitting the results of AIDI and FPI. A more extensive and fair comparison should be conducted. Additionally, the image editing results are unsurprising due to the nature of the inversion-based method, and the human editing results appear to be tweaked.\n\n3. The optimization for the image editing task occurs only during inversion. However, this method inherently offers less controllability over the output compared to the input. How does this method address this limitation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer cPeB for devoting time to this review and providing valuable comments.\n\n**Weaknesses:**\n> **W1:** This assertion is especially unconvincing when $t$ is large.\n\n**A:** The premise of this assertion is the DDIM deterministic sampling without perturbation. Under this premise, it is theoretically the case that any position at any time step t  will converge to the same position, which implies that the mean of $\\varepsilon_0$ is 0 and the variance is very small. This property also allows the diffusion model to be generalized to other tasks such as image restoration[1],[2]. Therefore, the first half of the assertion is valid.  Regarding the statement \"$|\\varepsilon_0| = k|\\varepsilon_t|$, where $0< k < 1$\", we can provide a proof by contradiction. Suppose $k \\geq 1$, and when $t$ is large, we let $t = T$. In this case, ${\\varepsilon_t} \\sim \\mathcal{N}(0, 2I), {\\varepsilon_0} \\sim \\mathcal{N}(0, 2k^{2}I)$, which implies that the variance of the distribution of $\\varepsilon_0$ is greater than $2I$, which clearly contradicts the premise. Furthermore, ${\\varepsilon_0} \\sim \\mathcal{N}(0, 2k^{2}I)$ means that the pre-trained diffusion model is not a convergent model. Hence, the assertion is valid.   \n*References:*  \n[1].Wang J, Yue Z, Zhou S, et al. Exploiting diffusion prior for real-world image super-resolution[J]. International Journal of Computer Vision, 2024: 1-21.  \n[2].Sun H, Li W, Liu J, et al. Coser: Bridging image and language for cognitive super-resolution[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 25868-25878.\n\n> **W2:** The experiment only compares NTI and P2P, omitting the results of AIDI and FPI. The image editing results are unsurprising.\n\n**A:** Our response to this question is as follows:\n1. The purpose of this paper is to provide theoretical support for AIDI and FPI, and to improve the fixed-point loss they use. The relationship between this work and theirs is complementary, not competitive. To achieve better performance in image editing, AIDI uses Blended Guidance and Stochastic Editing, while FPI uses Prompt-aware Adjustment. However, neither AIDI nor FPI have open-sourced their code, and reproducing these methods is not the focus of this paper. \n\n2. \"the image editing results are unsurprising\" is due to the fact that we used the same number of iterations for all images at different time steps, which may have led to some images not converging fully at certain time steps. However, this is still sufficient to demonstrate that the fixed-point loss used by AIDI and FPI is not perfect, and that there is room for improving the performance of AIDI and FPI. This paper also provides suggestions on how to further improve the performance, as mentioned in lines 155-159, where it states that if we can find the cluster centers, the performance will be further improved. Finally, the key point of this work is meant to provide theoretical support for AIDI and FPI and improvements to the fixed-point loss used by AIDI and FPI, rather than to compete with their methods. We hope this clarifies the relationship and the purpose of this work. \n\n> **W3:** This method inherently offers less controllability over the output compared to the input.\n\n**A:** The main application of the fixed-point loss is in DDIM inversion, and this is also the focus of this work. If one wishes to have more control over the output, the Blended Guidance method used in AIDI could be a suitable approach. The Blended Guidance method adjusts the guidance scale during sampling. It increases the guidance scale for regions corresponding to the positive attention map, and decreases the guidance scale for regions corresponding to the negative attention map. This allows for more modifications to the areas that need to be edited, while making fewer changes to the areas that do not require editing.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EyusQVuh5t", "reviewer": "Reviewer_5f4S", "summary": "The paper proves the existence and uniqueness of fixed points in DDIM inversion using the Banach fixed-point theorem. It identifies flaws in existing fixed-point loss functions and proposes optimizations to improve convergence and visual quality of edited images. It also introduces a novel text-based approach for unsupervised image dehazing using fixed-point based editing.", "questions": "how does the proposed fixed-point based dehazing approach compare with state-of-the-art dehazing techniques in terms of performance and computational efficiency?", "limitations": "The authors have addressed the limitations of their work, along with potential negative societal impacts.", "rating": 5, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper addresses the inconsistencies in image editing results caused by errors introduced during the DDIM inversion process. It provides a solution to enhance the reliability and quality of edited images with the proposed fixed-point optimization. The paper also gives a thorough analysis of DDIM inversion, identifying key issues and proposing optimization strategies for it. This detailed analysis helps to undetstand the challenges in the context of image editing. The paper also outlines clear directions for future work.", "weaknesses": "- the performance gains the paper proposes are minor. It might not be worth the complexity of the approach.\n- the paper doesn't seem to have thoroughly tested its method on various challenging situations, especially for image dehazing.\n- the paper acknowledges that proposed method may introduce additional computational overhead when optimizing for fixed point convergence. However, specific quantitative measures of this overhead are not provided, so further analysis is needed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer 5f4S for devoting time to this review and providing valuable comments.\n\n**Weaknesses:**\n> **W1:** The performance gains the paper proposes are minor.\n\n**A:** The reason why the performance gains are minor in image editing is that across the entire dataset, we used a fixed number of iterations to reach the fixed-point. This may have resulted in some images not converging fully at certain time steps. It's important to note that the focus of this paper is on providing theoretical support for the fixed-point and optimizing the fixed-point loss. Therefore, using a fixed set of variables for the experiments is more equitable. However, in the case of unsupervised image dehazing, the performance gains are significant, as shown in the last two columns of Figure 5 in the paper. The use of the fixed-point effectively avoids the collapse of the recovered images.\n\n> **W2:** The paper doesn't seem to have thoroughly tested its method on various challenging situations.\n\n**A:** In terms of fixed-point based image restoration, we have only conducted experiments on image dehazing, and the specific reasons for this are explained in lines 226-231 of the paper. The reason is that haze, as a pervasive degradation, is relatively easier to capture the corresponding attention map at the semantic level, as shown in the last row of Figure 5 in the paper. Capturing the semantic-level attention map is heavily dependent on the pre-trained diffusion model used, and in this work, we have relied on the Stable Diffusion v1.4 model from the diffusers library. We have also experimented with Stable Diffusion v1.4 for image desnowing and deraing, but found that it was unable to accurately capture the attention maps corresponding to the rain and snow regions. Therefore, we believe that to extend the fixed-point based image restoration to other degradations, a diffusion model specifically trained to capture the attention maps for the corresponding degradations would be required. Finally, the primary focus of this work is to provide the theoretical support for the existence and uniqueness of the fixed-point, as well as the optimization of the fixed-point loss. We have also aimed to demonstrate the potential of the fixed-point in image restoration tasks, as exemplified by the image dehazing results.\n\n> **W3:** Specific quantitative measures of this overhead are not provided, so further analysis is needed.\n\n**A:** The Reviewer 5f4S make a valid point regarding the additional computational overhead. We have therefore included the runtime information for reference. The additional computational overhead is primarily incurred during the DDIM inversion process. Specifically:   \n  (1) The original fixed-point loss takes approximately 0.26s per individual time step.  \n  (2) The improved fixed-point loss takes approximately 0.61s per individual time step.  \n  (3) The total time to edit a single image using the original fixed-point loss is approximately 41s.   \n  (4) The total time to edit a single image using the improved fixed-point loss is approximately 60s. \n\nIn both cases, the total number of time steps is 50. We hope this additional information helps to address your concern regarding the computational overhead.\n\n**Questions:**\n> **Q1:** How does the proposed fixed-point based dehazing approach compare with state-of-the-art dehazing techniques.\n\n**A:** Since the focus of this paper is on the theoretical support and convergence optimization, the performance is lagging behind the state-of-the-art dehazing techniques. The main reason for this performance gap is the inherent limitation of the pre-trained diffusion models in accurately capturing the attention maps corresponding to the degradations. As shown in the last row of Figure 5, the haze attention map is not entirely precise. To address this issue, we would need to utilize lower-level attention maps and train an additional diffusion model that can accurately capture the degradation-specific attention maps. In this work, our aim has been to demonstrate the potential of the fixed-point based image restoration approach in terms of its semantic-level interpretability, which we believe is a key advantage over other methods. Moreover, another key advantage is that with the ability to precisely capture the corresponding degraded attention maps, our method can be applied to arbitrary datasets without the need for additional training.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fp8sVEEfKp", "reviewer": "Reviewer_Cq8S", "summary": "The paper concerns with the fixed points of the DDIM inversion scheme, which is central to many image editing methods. The authors first establish that the DDIM inversion process at any given time step $t$ exhibits a unique fixed point by demonstrating that the corresponding functional has a Lipschitz constant strictly smaller than 1.\n\nFurthermore, the authors reveal that the commonly employed stopping criterion $f(z_t)-z_t$ in the fixed-point inversion process might not guarantee convergence to the optimal solution. To address this, they introduce a new optimization criterion, $f(z_t)-f(\\hat z_t)$, where $\\hat z_t$ represents a perturbed version of $z_t$,  creating a new optimization path. Since the fixed point is unique, both the original and perturbed paths should converge to the same point. Thus, the new criterion ensures convergence to solutions that better approximate the desired fixed point, yet at the expense of a few additional iterations per time step.\nThe authors of demonstrate the proposed method through experiments on image editing and unsupervised dehazing tasks.", "questions": "* Please address the weakness above.\n* Can the theoretical analysis presented lead to more efficient algorithms than the straightforward Picard iteration method?\n* Can any theoretical statements be made regarding the change in convergence rate under the proposed optimization loss?", "limitations": "* While I appreciate the extensive discussion on the application of dehazing, I would appreciate further discussion on the proposed underlying method (or optimization loss), which, in my view, is the central focus of the paper.", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The paper is well-written. The authors provide a clear motivation and necessary background, and their derivations are easy to follow.\n* The contributions of the paper are solid. The presented fixed-point analysis is simple yet insightful, and the new optimization criterion is both creative and easy to implement.\n* The experiments, while not extensive, are sufficient to demonstrate the proposed method.", "weaknesses": "* While I believe the  the theoretical conclusions derived  in Section 3 are correct, there are several inaccuracies that must be addressed to ensure the soundness of the derivations. Specifically, $z_t$ and $f(z_t)$ are vectors, hence, the inequality in (7) is not clear. How it is defined? It is pointwise ineuqliaty? Furthermore, it seems that the authors assume the difference between the scores on the right-hand side of (7) is non-negative, which needs to be justified.\nContinuing in this regard, the differences on both sides of (11) are vector differences, making the inequality unclear. I believe Inequality (11) should be  $||f(z_t^i)-f(z_t^j)||\\leq (1-k\\sqrt{\\bar{\\alpha}}) ||z_t^i-z_t^j||$.\n\n* The motivation and intuition behind equation (12) should be explained prior to its presentation. Currently, the purpose of this equation is unclear upon initial reading.\n\n* The implications of inequality (14) should be clarified. Why is this inequality interesting? How does it affect fixed-point convergence, and what role does $\\delta$ play in this process? Are there any conditions on $\\delta$ that need to be assumed to ensure proper convergence?\n\n* In contrast to previous sections, I find Section 5 to be lacking necessary mathematical derivations, explanations, and background on NTI. In its current form, the section provides only very high-level details, obscuring the paper's contribution in this context.\n\n* In the image editing experiments, the proposed method appears to offer only marginal improvement, both visually and quantitatively. In the dehazing experiments, the method seems to provide a marginal visual improvement, but it does demonstrate a notably more stable recovery process.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer Cq8S for devoting time to this review and providing valuable comments.\n\n**Weaknesses:**\n> **W1:** Is Inequality (7) pointwise ineuqliaty? Inequality (7) and (11) are missing $||\\cdot||$.\n\n**A:** Inequality (7) is a pointwise inequality. Additionally, Inequalities (7) and (11) should have the $|\\cdot|$ notation, similar to Inequality (12). We apologize for the oversight that caused any inconvenience. We have also adopted your suggestion to replace the $|\\cdot|$ notation in Inequality (12) with $||\\cdot||$, and have added the $||\\cdot||$ notation to Inequalities (7) and (11) as well. We are grateful for your careful review and helpful feedback.\n\n> **W2:** The motivation and intuition behind equation (12) should be explained prior to its presentation.\n\n**A:** The intuition behind Inequality (12) is that in the absence of numerical errors, the path $\\hat{z}_t$ and the path $z_t$ will converge to the same point, and the difference between the paths $\\hat{z}_t$ and $z_t$ will also converge to $0$. The convergence rate of the three is the same in the ideal case.\n\n> **W3:** The implications of inequality (14) should be clarified. What role does $\\delta$ play in this process? \n\n**A:** Inequality (14) demonstrates that in the non-ideal case, the convergence of the difference between paths will be slower than the convergence of the difference within a single path. In other words, when using the intra-path loss, while it may appear to have converged, we cannot be certain of the distance from the ideal fixed point.  However, The inter-path loss provides a reference for this. The role of $\\delta$ is to reflect the fact that in the presence of numerical errors, the inter-path loss will converge more slowly than the intra-path loss. As a result, the inter-path loss can provide a perspective that is closer to the ideal fixed point. The experiments shown in Figure 1 also validate this point. In general, the $\\delta$ value is a very small number. Even in the presence of numerical errors, the function would still converge to a small interval. If the $\\delta$ value were to disrupt the contraction mapping property, it would lead to the function failing to converge to a small interval, which would be contradictory to the conclusions proved in Section 3.\n\n> **W4:** I find Section 5 to be lacking necessary mathematical derivations, explanations, and background on NTI.\n\n**A:** We apologize for the lack of sufficient explanation and background regarding the NTI. NTI involves optimizing the null-text embedding as a learnable parameter. Its fine optimization compensates for the inevitable reconstruction error caused by the classifier-free guidance component, and this is crucial for image restoration. From the perspective of the mathematical formulation, the classifier-free guidance prediction is defined as: \n$$\n\\tilde{\\varepsilon_{\\theta}}(z_t,t,C, \\varnothing)=w \\cdot {\\varepsilon_{\\theta}}(z_{t}, t, C)+(1-w) \\cdot \\varepsilon_{\\theta}(z_{t}, t, \\varnothing)\n$$\nWhere $C$ represents the text embedding, $\\varnothing$ is the null-text embedding, $z_{t}$ denotes the input at timestep $t$, and $w$ is the guidance scale parameter. The primary optimization target of NTI is the $\\varnothing$ term. We hope this provides the necessary background and explanation to clarify the NTI.\n\n> **W5:** In the image editing, the proposed method appears to offer only marginal improvement.\n\n**A:** The marginal improvement in image editing is primarily due to the fact that we chose to use a fixed number of iterations. This may have resulted in some cases where the solution did not fully converge to the ideal fixed point. Nevertheless, there is still room for performance improvement. As mentioned in lines 155-159 of the paper, one potential strategy is to compute multiple inter-path losses and then select the cluster center of the converged points as the final ideal fixed point.\n\n**Questions:**\n> **Q1:** Can the theoretical analysis presented lead to more efficient algorithms than the straightforward Picard iteration method?\n\n**A:** The fixed point computation in this work is based on the Picard iteration method. If more efficient algorithms are desired, we would recommend exploring Aitken's acceleration method or Steffensen's method. Aitken's acceleration can provide faster convergence than the basic Picard iteration by extrapolating the sequence of iterates. Steffensen's method is another technique that achieves higher order convergence compared to the Picard iteration.\n\n> **Q2:** Can any theoretical statements be made regarding the change in convergence rate under the proposed optimization loss?\n\n**A:** Regarding the change in convergence rate, we only provided a rough illustration of the slight increase in the Lipschitz constant using the inequality (14) to demonstrate the change in convergence rate. Given that the observed change in convergence rate in the experiments is subject to the influence of numerical errors, which may vary across different scenarios, we did not provide further theoretical statements. However, we can offer an intuitive statement. If we use the original fixed-point loss, which takes a single-path perspective, the convergence rate would be consistent with the ideal case, as there are no other references. When the intra-path loss changes minimally, it would consider the fixed point to be reached. In contrast, the multi-path perspective introduces the inter-path loss. Even when the intra-path loss changes very little, the inter-path loss may still exhibit a decreasing trend. Furthermore, even if the intra-path loss is zero, due to numerical errors, the inter-path loss may not be exactly zero. This implies that the inter-path loss provides additional information indicating that we still need to travel a small distance to reach the ideal fixed point. As a result, the inter-path loss may converge slightly slower than the intra-path loss in the presence of numerical errors.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6LveTeLGXA", "reviewer": "Reviewer_eqay", "summary": "This paper provides important theoretical support and practical optimization in the field of image editing, especially the fixed point problem in the DDIM inversion process. Through the application in image editing and dehazing tasks, the effectiveness and generalization potential of the fixed point theory are demonstrated. The paper has performed well in theoretical contribution and experimental verification.", "questions": "I don't particularly understand the specific theoretical derivation in the article, but I think the experimental part still has room for improvement. I will decide my final rating based on the author's rebuttal and the opinions of other reviewers. Overall, I think this article is interesting.", "limitations": "None", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Based on theoretical findings, the authors optimized the loss function in the original DDIM inversion and improved the visual quality of image editing.\n2. The fixed point theory is applied to the image dehazing task, and unsupervised dehazing methods based on text guidance are explored, showing the generalization potential of the fixed point theory.\n3. Experiments on the PIE-Bench dataset verify the effectiveness of the optimized fixed-point convergence criterion and demonstrate the improved image editing quality.", "weaknesses": "1. Although the paper discusses fixed-point computational optimization, no detailed computational resource requirements (i.e. GFLOPs, param.) are provided, which may affect the reproducibility of the experiments.\n2. The paper does not report the error margins or statistical significance tests for the experimental results, which limits the interpretability of the results.\n3. The experiments mainly focus on image editing and dehazing tasks, and it may be necessary to verify the generalization ability of the fixed point theory in a wider range of image restoration tasks.\n4. I am very curious whether the method proposed by the author is as robust to some complex nonlinear degradation as to simple linear degradation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer eqay for devoting time to this review and providing valuable comments.\n\n**Weaknesses:**\n> **W1:** No detailed computational resource requirements (i.e. GFLOPs, param.) are provided.\n\n**A:** Our approach relies on the Stable Diffusion v1.4 model, which is available through the Diffusers library. As such, the GFLOPs and parameter counts can refer to the official documentation of Stable Diffusion v1.4. Of course, we have also calculated the GFLOPs and parameters of each module in Stable Diffusion v1.4 ourselves, and the following data is for reference only:  \n| | vae| unet| text_encoder| \n| :---: | :---: | :---: |:---:| \n|parameter| 83.65M| 859.52M | 123.06M  |\n| GFLOPs| 1773.5  | 222.3    |0.12    | \n\n> **W2:** The paper does not report the error margins or statistical significance tests for the experimental results.\n\n**A:** The Reviewer eqay are correct that we should have included these important measures to provide a more rigorous evaluation. Upon your recommendation, we have conducted additional runs and calculated the mean values along with the error margins. The updated results are as follows:\n| | Distance   | PSNR   | LPIPS   | MSE   | SSIM   | Whole   | Edited   |\n| :---: | :---: | :---: |:---:| :---: | :---: | :---: | :---: |\n|Image Editing | 69.181$\\pm$0.001 | 17.88$\\pm$0.01  | 208.87$\\pm$0.02  | 219.86$\\pm$0.05    |  71.24$\\pm$0.18   | 25.33$\\pm$0.016   |  22.59$\\pm$0.001  |\n| Image Reconstruction |  70.247$\\pm$0.20  | 17.80$\\pm$0.07    | 210.70$\\pm$0.03     | 225.47$\\pm$0.01    |  70.90$\\pm$0.05 | 23.81$\\pm$0.004    |  21.39$\\pm$0.02  |\n\n> **W3:** It may be necessary to verify the generalization ability of the fixed point theory in a wider range of image restoration tasks.\n\n**A:** We have addressed this issue in lines 226-231 of the paper. The primary reason for the relatively poor performance on tasks like rain and snow removal is that the pre-trained diffusion model we utilized is not accurate in capturing these specific types of degradation. The pre-trained Stable Diffusion v1.4 is not sufficiently adept at generating accurate attention maps for these weather-related artifacts. To address this limitation, the solution would be to train a diffusion model that is specifically designed to capture the attention maps corresponding to rain and snow. And it requires the use of shallow, pixel-level attention maps that can better identify the characteristics of these weather-related degradations.\n\n> **W4:** I am very curious whether the method proposed by the author is as robust to some complex nonlinear degradation as to simple linear degradation.\n\n**A:** The key to the robustness against different types of degradations lies in the accuracy of the attention map matching. Based on our experimental observations, Stable Diffusion v1.4 was able to capture the attention maps for degradations such as rain, snow, and fog. However, the issue lies in the fact that the default Stable Diffusion v1.4 model uses deep-level attention maps, which are not precise enough to accurately capture the attention maps for rain and snow. As a result, our method's performance on tasks like snow and rain removal was not as effective as we would have liked. Regarding other types of degradations, such as blur, shadows, occlusions, and noise, we have not yet explored them by using Stable Diffusion v1.4. Therefore, we are uncertain to capture the attention maps for these degradations. To achieve more robust results, we believe the solution would be to retrain a diffusion model that is specifically designed to capture the attention maps corresponding to the various types of degradations.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper received a Weak Accept, 2 Borderline Accept and a Reject. The reviewers found the theoretical contributions of the paper interesting. They also raised a number of concerns: 1) Some missing details in the analysis and 2) limited experiments and comparisons.
The authors engaged in discussions with the reviewers and most reviewers found the answers satisfactory. One reviewer is still concerned with a potential limitation in the theoretical analysis. The AC has read all the reviews and the discussion. The AC finds that the authors have addressed most concerns and recommends the paper for acceptance.
Decision: Accept (poster)