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_cc_2024_Conference
Published: 2024-05-14 14:03:55

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.

Main paper: 
 introduction: 
 Diffusion models have gained significant attention in recent years. Early versions of Denoising Diffusion Probabilistic Models (DDPMs) [1] and score-based generative models [2] 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 a large number of sampling steps during image generation, which was a 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 is 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-Prompt (P2P) [6] editing. The core idea of P2P is to modify the cross-attention between text and images. For local edits, such as replacing a cat with a dog while keeping the background unchanged, only the cross-attention corresponding to the cat needs to be modified. For enhancing certain effects, the cross-attention corresponding to 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 errors 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) [8] 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 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) [11] and fixed-point inversion (FPI) [12] . However, AIDI and FPI only employ methods to solve fixed points of implicit functions without proving the existence of fixed points at 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 in DDIM inversion using the Banach fixed-point theorem [13] . 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-point 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 on 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 experimental 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 fixedpoint based editing.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: Listed in the introduction. 
preliminaries: ddpms and ddim inversion: 
 DDPMs model the probability distribution of a diffusion process by iteratively adding noise to a data sample until the data distribution becomes predominantly noise [1] . Subsequently, the data distribution is recovered from a random Gaussian noise through a reverse diffusion process. The diffusion process is typically regarded as a Markov chain starting from x 0 and obtaining x 1 , x 2 , ..., x T by adding noise. The specific process is outlined as follows:EQUATIONwhere schedule β 0 , β 1 , ..., β T ∈ (0, 1). x t can be expressed as a linear combination of x 0 and Gaussian noise ∼ N (0, I):EQUATIONIn 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 a 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 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:EQUATIONwhere θ represents a network that predicts added noise, z t denotes the compressed representation of x t in the latent space, t represents the time step, and p represents the encoding of a condition text. Now let's focus on DDIM inversion. DDIM inversion seeks to find a noise z T given the latent space representation z 0 corresponding to a given x 0 and the associated textual prompt p. This allows for the autoregressive reconstruction of z 0 . The process of DDIM inversion can be rewritten by Equation 3as follows:EQUATION≈ ᾱt ᾱt-1 z t-1 + √ 1 -ᾱt - (1 -ᾱt-1 )ᾱ t ᾱt-1 θ (z t-1 , t, p)Therefore, we can invert z 0 back to z T using Equation 4, allowing us to reconstruct z 0 using the reverse diffusion process. However, it can be observed that in Equation 4, z t is directly approximated by z t-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, 8] .Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: See section 7.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.• The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
theoretical support for fixed point in ddim inversion: 
 From the first equation of Equation 4, it is evident that finding z t can be regarded as solving a fixed-point problem for the implicit function with respect to θ (z t , t, p). Therefore, we can consider the right-hand side of the equation as a function with z t as a variable:EQUATIONThen, to find f (z t ) = z t , we can use z t-1 as the starting point for iteration. The iteration process can be expressed as follows:EQUATIONAfter multiple iterations, the fixed point of Equation 5 can be found. Now, we focus on the theoretical analysis of whether a fixed point exists in Equation 5. Let z i t and z j t represent any two points in the function f (z t ) at time step t, where i and j, with j greater than i, are within the range of 0 to N . We can obtain the following:EQUATION≤ √ 1 -ᾱt || θ (z i t , t, p) -θ (z j t , t, p)|| where || • || is L 2 norm.We know that in the reverse diffusion process, when predicting the z value at time step t -1 given the z value at time step t, an initial rough estimation of z 0 is made. This rough estimation of z 0 introduces some errors compared to the final obtained value of z 0 . However, the overall shape of the image is already quite similar, and as time t decreases, this rough estimation of z 0 gradually approaches the final obtained z 0 . This rough estimation of z 0 can be represented as follows:EQUATIONwhere z i 0 and z j 0 represent the predictions of time step zero. Although z i 0 and z j 0 obtained from z i t and z j t are very close, there might be some errors when t is not close to zero. Let's denote this error as ε 0 . Additionally, let's define the difference between z i t and z j t as ε t . With Equation 8, we can derive the following relationship:EQUATIONDue 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 |ε 0 | = k|ε t |, where 0 < k < 1. Moreover, from the perspective of ODE, the paths from z i t to z i 0 and from z j t to z j 0 can be viewed as two ODE trajectories with initial values of z i t and z j t , respectively, or as two integral curves of ODE given the initial values z i t and z j t . These integral curves non-tangent and non-intersecting [15] . Therefore, we can conclude that ε 0 and ε t have the same sign. We can also prove this by contradiction: if ε 0 and ε t 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 ε 0 = kε t , where 0 < k < 1. By substituting this into Equation 9, we obtain:EQUATION= k √ ᾱt z i t -z j t + √ 1 -ᾱt θ (z i t , t, p) -θ (z j t , t, p)By rearranging the Equation 10 and substituting it into Equation 7, we obtain:EQUATIONFrom Equation 2, it is known that ᾱt ∈ (0, 1). Therefore, it can be inferred that (1 -k √ ᾱt ) ∈ (0, 1). This proves that the implicit function f (z t ) satisfies the contraction mapping condition. By further applying the Banach fixed-point theorem [13] , we can conclude that the fixed point of f (z t ) exists and is unique.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes]Answer: [No] Justification: Our experiment is based on open-source code and data [9] . Guidelines:• The answer NA means that paper does not include experiments requiring code. Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
convergence optimization for fixed point in ddim inversion: 
 In AIDI [11] and FPI [12] , to determine the convergence of the fixed point obtained through iterative processes in the implicit function f (z t ), the criterion used is f (z t )-z t . In FPI, numerical experiments were conducted on the given dataset, and it was concluded that convergence is achieved within 2 ∼ 3 iterations. Furthermore, in Section 3, we have proved the uniqueness of the fixed point. Therefore, at any position with a time step t, convergence can be reached towards the same point. We express this using Equation 11 as follows:EQUATION||f (z j t ) -f (ẑ j t )|| ≤ 1 -k √ ᾱt ||z j t -ẑj t ||,where ẑi t = z i t + , ∼ N (0, I), and ẑj t is the result obtained by iterating ẑi t in the implicit function f (z t ). It can also be considered that ẑi t is a new path distinct from z i t . 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 12 even at the end of the iteration.EQUATIONIncorporating the error into the third inequality of Equation 12for correction, we have:||f (z j t ) -f (ẑ j t )|| ≤ 1 -k √ ᾱt ||z j t -ẑj t || + ε (14) ≤ 1 -k √ ᾱt + δ ||z j t -ẑj t ||, δ > 0 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.1. In each step, we performed 10 iterations. The blue intra path loss represents f (z t ) -z t used in AIDI and FPI, while the orange inter path loss represents our optimized f (z t ) -f (ẑ t ). The trend of the intra path loss aligns with the findings reported in FPI, converging within 2 ∼ 3 iterations. However, the inter path loss 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 a noticeable lag in the convergence rate, as shown in Figure 1 . The loss function f (z t ) -z t 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 t ) -z t 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 t ) -f (ẑ t ) and define the convergence of the f (z t ) -f (ẑ t ) 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 1 . In order to visually demonstrate the process of fixed point convergence, we randomly select two points from z t as the axes. Subsequently, multiple paths are sampled by applying ẑi t = z i t + , ∼ N (0, I). The results obtained from multiple iterations are plotted as trajectory lines, as shown in Figure 2 .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, it 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. 
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] :EQUATIONwhere I(x) is hazy image, J(x) is the real scene to be recovered. t(x) = e -kd(x) is the medium transmission, where d(x) is depth and k is scattering coefficient of the atmosphere. A is the global atmospheric light.Existing unsupervised image dehazing methods can be broadly categorized into learning-based and non-learning-based approaches. Learning-based methods primarily utilize unpaired images [19, 20, 21] to train networks for unsupervised dehazing, while non-learning-based methods rely on image priors such as dark channel prior [22] and rank-one prior [23] . With the development of visual language models, some supervised multimodal dehazing approaches have emerged [24] . However, there are few reports on unsupervised multimodal dehazing. This paper mainly explores imageediting-based unsupervised dehazing, where the main principle is to achieve dehazing effects through cross-attention replacement of corresponding to 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-point 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 of 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, we only reproduced their fixed point code. For the dataset, we selected PIE-Bench [9] in the context of image editing, which consists of 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 was solely 'haze', while the target prompt was designated as '∅'. Figure 3 : f (z t ) -z t denotes the intra path loss, while f (z t ) -f (ẑ t ) 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.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 a baseline to demonstrate the effectiveness of fixed points. The proposed optimization for fixed point convergence 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, we directly compare the imaging results of image editing, as shown in Figure 3 and Table 1 . In 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 to 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 1 . It can be observed that using the optimized fixed points yields reconstructed images that have higher similarity to the input images. In the unsupervised image dehazing experiments, we primarily 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 inclusion of fixed points in NTI effectively mitigates the image collapse issue in certain dehazed images. As mentioned earlier, 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 guidance coefficient increases. Therefore, we also included examples of image collapse in Figure 5 . 
limitations, prospects and impact: 
 Although the fixed-point loss achieves better visual quality after optimization, there is 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 from images, there are still issues with artifacts. For example, in the middle column of Figure 5 , some artifacts can 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 5 . It can be seen that the attention map corresponding to the haze semantics only provides a 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 [27] , 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 to be weighted into each layer of crossattention. However, the cross-attention at different layers may have little correlation with the final haze attention map in terms of morphology. We attempted to 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.Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [No] Justification: Our experiment was conducted according to the benchmark of PIE-Bench [9] . Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
conclusion and future work: 
 This paper fills the theoretical gap in the fixed-point 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, we optimize the convergence criterion for fixed points and propose the inter path loss as a measure for assessing fixed-point convergence, supported by both theoretical analysis and experimental results. Through experiments conducted on 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 fixed-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 on imprecise semantic attention maps, thereby achieving superior image restoration outcomes.  
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification:Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [No] Justification: The dataset and reference code are both open-source and free to use for research purposes.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 
new assets: 
 Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: Guidelines:• The answer NA means that the paper does not release new assets.• Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. • The paper should discuss whether and how consent was obtained from people whose asset is used. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: 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, we did not use logarithmic scaling on the coordinate axes."original_prompt": "a [round] cake with orange frosting on a wooden plate", "editing_prompt": "a[square] cake with orange frosting on a wooden plate" "original_prompt": "a [cat] sitting on a wooden chair", "editing_prompt": "a[dog] sitting on a wooden chair" "original_prompt": "blue light, a black and white [cat] is playing with a flower", "editing_prompt": "blue light, a black and white [dog] is playing with a flower" "original_prompt": "an [orange] cat sitting on top of a fence", "editing_prompt":"an [black] cat sitting on top of a fence" "original_prompt": "a [dog] is laying down on a white background", "editing_prompt": "a [lion] is laying down on a white background""original_prompt": "meat [balls] on white plate", "editing_prompt": "meat [sushi] on white plate""original_prompt": "the statue of liberty holding a[torch]", "editing_prompt":"the statue of liberty holding a[flower]" "original_prompt": "a [white] horse running in the sunset", "editing_prompt": "a[golden] horse running in the sunset""original_prompt": "[black] chair in a conference room", "editing_prompt": "[blue] chair in a conference room" "original_prompt": "a fluffy dog with a blue leash sitting in the [grass]", "editing_prompt": "a fluffy dog with a blue leash sitting in the [ground]""original_prompt": "a [meerkat] puppy wrapped in a blue towel", "editing_prompt": "a [lion] puppy wrapped in a blue towel" "original_prompt": "[purple] tulips in vase", "editing_prompt": "[yellow] tulips in vase""original_prompt": "a woman in a [jacket] standing in the rain", "editing_prompt": "a woman in a [blouse] standing in the rain" "original_prompt": "the city of dresden, germany, europe", "editing_prompt": "[a sunny day of] the city of dresden, germany, europe" "original_prompt": "a cute dog holding a [red] heart", "editing_prompt": "a cute dog holding a [pink] heart" "original_prompt": "a water droplet hangs from a string of lights", "editing_prompt": "a water droplet hangs from a string of [red] lights" "original_prompt": "a german shepherd dog stands on the grass with mouth [closed]", "editing_prompt": "a german shepherd dog stands on the grass with mouth [opened]""original_prompt": "a golden retriever [holding a flower] sitting on the ground in front of fence", "editing_prompt": "a golden retriever sitting on the ground in front of fence""original_prompt": "three white [dumplings] on brown bowl", "editing_prompt": "three white [sushi] on brown bowl" "original_prompt": "white [tiger] on brown ground", "editing_prompt":"white [cat] on brown ground" "original_prompt": "two birds sitting on a branch", "editing_prompt": "two [origami] birds sitting on a branch" "original_prompt": "a[gray] horse in the field", "editing_prompt": "a [white] horse in the field""original_prompt": "a [stream] in a lush green forest with rocks", "editing_prompt": "a [road] in a lush green forest with rocks" "original_prompt": "a group of [pink] flowers hanging from a tree", "editing_prompt": "a group of [red] flowers hanging from a tree" "original_prompt": " [smoke]", "editing_prompt": "[fire]""original_prompt": "mountain landscape with [flowers and] withered grass", "editing_prompt": "mountain landscape with withered grass" "original_prompt": "a little girl with [green] eyes and flowers in a blue dress", "editing_prompt": "a little girl with [blue] eyes and flowers in a blue dress" "original_prompt": "sunset over a field with clouds and a [bright] sky", "editing_prompt": "sunset over a field with clouds and a [dark] sky" "original_prompt": "a bird standing on [clods]", "editing_prompt": "a bird standing on [eggs]""original_prompt": "a camera and a notebook on a [bed]", "editing_prompt": "a camera and a notebook on a [table]" "original_prompt": "a [yellow] apple sitting on top of a wooden table", "editing_prompt": "a [red] apple sitting on top of a wooden table " "original_prompt": "ocean with [clouds] in the sky", "editing_prompt": "ocean with [stars] in the sky" "original_prompt": "the [almond] tree is blooming in the spring", "editing_prompt": "the [violet] tree is blooming in the spring""original_prompt": "two [red] berries on a tree branch with a blue sky in the background", "editing_prompt": "two [purple] berries on a tree branch with a blue sky in the background""original_prompt": "a vase of colorful flowers on a [wooden] table", "editing_prompt": "a vase of colorful flowers on a [plastic] table " "original_prompt": "a painting of a woman holding a [pink flower]", "editing_prompt": "a painting of a woman holding a [teddy bear]" "original_prompt": "a rose on the [rail]", "editing_prompt": "a rose on the [wood]""original_prompt": "a woman with [flowers] in her hair", "editing_prompt": "a woman with [snakes] in her hair" "original_prompt": "a photo of a man and woman sitting on the ground", "editing_prompt": "a [colorful] photo of a man and woman sitting on the ground""original_prompt": "woman with [brown] hair", "editing_prompt": "woman with [blue] hair" We present success cases and failure cases of incorporating depth prior and dark channel prior, as shown in Figure 8 . 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 1 in this case.• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 


Decision: 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.

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: 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.

2. 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.

3. 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?
questions: questions: My question is included in the weaknesses section. I would be willing to raise the rating if my concern is addressed.
limitations: limitations: Please check weakness
paper_summary: paper_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.
strength_weakness: strength_weakness: - the performance gains the paper proposes are minor. It might not be worth the complexity of the approach.
- the paper doesn't seem to have thoroughly tested its method on various challenging situations, especially for image dehazing.
- 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.
questions: 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: limitations: The authors have addressed the limitations of their work, along with potential negative societal impacts.
paper_summary: paper_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.

Furthermore, 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.
The authors of demonstrate the proposed method through experiments on image editing and unsupervised dehazing tasks.
strength_weakness: strength_weakness: * 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.
Continuing in this regard, the differences on both sides of (11) are vector differences, making the inequality unclear. I believe Inequality (11) should be  $||f(z_t^i)-f(z_t^j)||\leq (1-k\sqrt{\bar{\alpha}}) ||z_t^i-z_t^j||$.

* 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.

* 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?

* 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.

* 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.
questions: questions: * Please address the weakness above.
* Can the theoretical analysis presented lead to more efficient algorithms than the straightforward Picard iteration method?
* Can any theoretical statements be made regarding the change in convergence rate under the proposed optimization loss?
limitations: 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.
paper_summary: paper_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.
strength_weakness: strength_weakness: 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.
2. The paper does not report the error margins or statistical significance tests for the experimental results, which limits the interpretability of the results.
3. 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.
4. I am very curious whether the method proposed by the author is as robust to some complex nonlinear degradation as to simple linear degradation.
questions: 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: limitations: None

Comments: title: Post-Rebuttal
comment: We appreciate the reviewer eqay's recognition of our work. Regarding the reviewer's questions, we respond as follows: 
- Its ability to capture details appears to be insufficient. Have the authors attempted any experiments in low-light image enhancement area?  
Since the pre-trained Stable Diffusion V1.4's the semantic attention map is at deeper layers, its ability to capture fine details is somewhat limited.  We have mainly experimented with removing rain, snow, and haze, and have not tried low-light image enhancement. However, if low-light image enhancement also relies on semantic awareness and the attention map also can be captured, it should be effective.
- Why, then, is rain more challenging to address?  
The main reason is that the attention map is based on deeper layers, which results in the attention map having a much smaller size compared to the original image. This makes it difficult to accurately capture raindrops or small snowflakes.
- Have the authors considered the non-uniformity of real-world hazy images?  
We believe the reviewer is asking about the scenario where the $t(x) = e^{-kd(x)}$ in the equation $I(x) = J(x)t(x) + A(1-t(x))$ has $k$ as a variable. We have tested this scenario and found it to be effective for removing haze. This haze is generated by equipment in an indoor factory setting, and the purpose is to remove the obstruction caused by these haze conditions. However, since this real-world scenario involves the commercial confidentiality of our collaborators, we did not include it in the paper.
Thank you to the authors for the rebuttal, which has addressed my concerns to some extent.

According to the authors' description, the proposed method seems to focus more on semantic-level processing, while its ability to capture details appears to be insufficient. However, many tasks seem to have a strong semantic demand, such as low-light image enhancement. Have the authors attempted any experiments in this area? Merely providing some analysis without concrete examples seems like a limitation to me. On the other hand, it seems that dehazing tasks are more complex in terms of physical imaging compared to deraining tasks. Why, then, is rain more challenging to address? Have the authors considered the non-uniformity of real-world hazy images?
We appreciate the reviewer Cq8S's recognition of our work. Regarding the reviewer's questions, we respond as follows:  
- How was this specific number of iterations determined?  
We referred to the strategies used in AIDI and FPI, which observed the loss convergence behavior through some data and then determined a fixed number of iterations. Their observations showed that the intra-path loss can converge in about 3 iterations, so the experiments using intra-path loss used 3 iterations. The inter-path loss was observed to converge in about 6 iterations, so we used 6 iterations. This is a relatively fair configuration.
- Was there an ablation study conducted to investigate its impact?  
We experimented with gradually increasing the number of iterations from 3 to 6, as well as decreasing from 6 to 3, in 50 time steps for image editing. The performance was between the 3-iteration and 6-iteration results, with the former performing slightly better than the latter. We had only saved the results for the former in our previous experiments.  
|  Distance   | PSNR  | LPIPS  | MSE  | SSIM  | Whole  | Edited  |
|  ----  | ----  | ----  | ----  | ----  | ----  | ----  |
| 69.46  | 17.84 | 208.91 | 220.44 | 71.14 | 25.25 | 22.55 |
- Why not allow the process to run until approximate convergence is achieved, based on some predefined stopping criteria?  
We did try setting a threshold to stop the process. However, the threshold that works for different images and time steps can vary, and the manually set thresholds took a long time to reach in some cases. For those that could not reach the threshold, we stopped at 50 iterations. But we found the experimental results on some metrics were not as good as the fixed iteration approach, and it took longer. The main reason was that the manually set thresholds might be higher than the ideal values in some cases. Therefore, we did not mention this strategy in the paper, and instead recommend using the clustering center of multiple path iterations. Considering a fairer comparison with AIDI and FPI strategies, time efficiency, and verifying the defects of intra-path loss, we decided to use the fixed iteration approach.
- It is crucial for theoretical contributions to demonstrate some practical merits as well.  
Since the gains of fixed points are limited in image editing, we added extra unsupervised dehazing experiments in the paper to further demonstrate the practical value and potential of fixed points in other tasks.
I thank the authors for their rebuttal, which addressed most of my concerns satisfactorily.  However, an issue that remains, and was also raised by other reviewers, is the marginal improvement in image editing quality. The authors attributed this to the use of a fixed number of iterations.

My follow-up question is how this specific number of iterations was determined. Was there an ablation study conducted to investigate its impact? Additionally, why not allow the process to run until approximate convergence is achieved, based on some predefined stopping criteria?

While I appreciate that the focus of the paper is primarily theoretical, I believe it is crucial for theoretical contributions to demonstrate some practical merits as well. This could include faster convergence, improved stability, or enhanced image quality.