Cross-Scale Self-Supervised Blind Image Deblurring via Implicit Neural Representation
Authors: Tianjing Zhang
Yuhui Quan
Hui Ji
Venue: NeurIPS 2024 poster (2024)
Keywords: Blind Image Deblurring
Self-Supervised Learning
Implicit Neural Representation
Primary area: machine_vision

Abstract: Blind image deblurring (BID) is an important yet challenging image recovery problem. Most existing deep learning methods require supervised training with ground truth (GT) images. This paper introduces a self-supervised method for BID that does not require GT images. The key challenge is to regularize the training to prevent over-fitting due to the absence of GT images. By leveraging an exact relationship among the blurred image, latent image, and blur kernel across consecutive scales, we propose an effective cross-scale consistency loss. This is implemented by representing the image and kernel with implicit neural representations (INRs), whose resolution-free property enables consistent yet efficient computation for network training across multiple scales. Combined with a progressively coarse-to-fine training scheme, the proposed method significantly outperforms existing self-supervised methods  in extensive experiments.

Cross-Scale Self-Supervised Blind Image Deblurring via Implicit Neural Representation
Tianjing Zhang', Yuhuí Quan', Hui Ji
Department of Mathematics, National University of Singapore 2 School of Computer Science and Engineering, South China University of Technology t tianjingzhangee wwweeuuuuu..... eduu,ddu, csyhquansscut.. .ew.ssctuuuuuuu.e..... en, matjhhnus edd.. sg
Abstract
Blind image deblurring (BID) is an important yet challenging image recovery problem. Most existing deep learning methods require supervised training with ground truth (GT) images. This paper introduces self-supervised method for BID that does not require GT images. The key challenge to regularize the training to prevent ever-fitting due to the absence of GT images. By leveraging an exact relationship among the blurred image, latent image, and blur kernel across consecutive scales, we propose an effective cross-scale consistency loss. This is implemented by representing the image and kernel with implicit neural representations (INRs), whose resolution-free property enables consistent yet efficient computation for network training across multiple scales. Combined with progressively coarse-to-fine training scheme, the proposed method significantly outperforms existing self-supervised methods in extensive experiments.
Introduction
Uniform blurring, a degradation commonly encountered in optics, leads to the loss of important details within captured image. Uniform blurring usually can described as the convolution:
y y= k
X
x + n,
(I)
where denotes the 2D discrete convolution operation, (y:x) is the pair of blurred and sharp (latent) images. k represents the blur kernel responsible for degradation, and n denotes measurement noise. For example, in digital photography, motion blur is caused by camera shake during exposure. When scene depth variation is small and camera movement is mainly translational within the image plane, motion blur can be approximated by the convolution model (I). with l representing camera motion. In fluorescence microscopy, specimens stained with fluorescent dyes and exposed to specific wavelength light often exhibit blurring due to light diffraction and optical path imperfections, such as optical component misalignment or lens aberrations. These factors obscure critical details, like micro-tubule arrangements in cells. Such blurring can also be modeled by ( In either case, both the latent image æ and blur kemnel k are unknown and must be estimated from the blurred image
BID aims at estimating (ks, .)), the pair of latent image and blur kernel, from the degraded image y- BID is a challenging non-linear inverse problem with many plausible solutions due to its inherent solution ambiguity. This ambiguity stems from the fact that the kernel can be decomposed into k == kj @ k2, suggesting that the pair (k1, 22 xx) are also viable solutions since = (k1 @ k2) 2 = k @ (k2 x). One such example the pair (8. y k, æ), where denotes the delta kernel, which gives trivial solution saying a blurred image is the convolution of a blurred image and a delta kernel. This ambiguity indicates the ill ill-posed-nes -ees- of the BID problem.
1.1 Discussion existing approaches for BID
The existing deep learning methods for BID can be roughly classified into the following categories:
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Supervised deep learning for BID with paired data: Supervised learning methods (e.g., 10
78 68 34 69 trains a neural network (NN) using paired data of blurred images (or blur
kernels) and GT images, to predict the latent image or/and blur kernel from an input blurred image. These methods mainly depend on large amount of paired data for training, limiting their application to specific scenarios where such paired data is challenging to collect.
Supervised deep learning for BID with un-paired data: Some works (e.g. 36 61] use
generative adversarial networks (GANs) to train on unpaired blurred and latent images. Although effective for domain-
images like faces and text, GANs perform poorly on general natural images due domain shift. Additionally, GAN-based methods often suffer from mode collapse and training instability, resulting in suboptimal outcomes.
BID with pre-trained generative model: These methods employ pre-trained generative models, such as diffusion models [1]. which are well-suited for processing images with structural consistency features like human faces. However, their reliance on pre-trained models to generate outputs that conform to the learned image distribution can lead to inauthentic results. This is particularly problematic in fields that require high fidelity, such as medical imaging and microscopy.
Self-supervised learning for BID: To circumvent the challenges of data collection and mitigate potential biases and inauthentic outcomes from generative models, growing body
9 18 (77) focuses on developing self-supervised deep learning approaches for BID that do not require training dataset or pre-trained model.
of research (e.g. 47 32
33
In this paper, we focus on sulf-supe BID, which is challenging due to the lack of GT images but offers many practical benefits. Most existing works self supfrvuuee BID are based the deep image prior (DIP) [57] of convolutional NNs (CNNs), which introduces the implicit prior from CNN for preferring structured patterns over random noise during training. The DIP-based self-supervised BID methods typically use two NN-based generators, Gh and Gz. to re- re-parameterize the blur kernel k and the latent sharp image æ. The generators then trained to maximize the likelihood of the blurred image y, minimizing the following self-supervised reconstruction loss:
Lar (Ok, O²) == [[9x(;; Ox)) @ Ga +x) - y112.
(2)
For instance, SelfDeblur 47 employed a CNN for Gu and Multi-Layer Perceptron (MLP) for Ga- MCEM 33 used U-Net for both Gn and Gu. and Zhuang et al. 77 used INR for both Gn and Gu-
Despite lacking access to GT images, these self-supervised BID methods perform competitively against supervised or pre-trained model based approaches, particularly with severely blurred images. However, their performance is less impressive on modestly blurred images and on real- world images. There is practical need for further studies to improve the performance of self-supervised BID across various blurring degrees and real-world images from different optics systems.
1.2 Main Idea and Contributions
Without GT images, self-supervised BID methods must address two vital questions for NN training:
1. Without accessing GT images, how to formulate a self supervisdd loss to teach the NN-based generators to accurately predict the latent image and kernel from only the blurred image?
2. The non-linear structure of BID makes training NN-based generators challenging. How can we efficiently train them to ensure accurate convergence to the latent images and kernels?
Our answer to Question a cross-scale loss function that leverages the cross-scale consistency of the estimates from consecutive scales for regularization. Our answer to Question 2 is a progressive cross-scale training scheme to training the NNs, enhancing training efficiency and ensuring the convergence to GT image/kernel.
Resolution-free INR for effective cross-scale interaction in BID: Our proposed method is built on re-parametrizattion of both latent images and kernels by INR (3 5T] INR represents signals as continuous functions rather than discrete valuel arrays, which our case is expressed as follows,
kii = (i. Qx) and =(i.j) 4. (i.j;
(3)
where (i,j) E 2R22 denotes the spatial co-ordinates of images/kernels, and O2. On denotes the weights of NNs Lg. Px for the kernel and image, respectively.

---

Using INR for representing images/kemels is due to its inherent resolution- property. By enabling the model to generate the prediction with higher/lower resolutions from the same learned model, INR facilitates seamless multi-scale processing and cross-scale interaction. In contrast, DIs-based NNs that directly maps noise to an image handle cross-scale interaction clumsily, requiring manual re-scaling or interpolation, which likely introduces artifacts in the prediction.
Self-supervised cross-scale loss for BID: Without GT images, the sole constraint have for training is = A @ æ. Additional priors for regularization are necessary to alleviate over-fitting. Historically, the down-sampled version of y. denoted as y1, for scale s, has often been used initiate the blur kernel estimate. However, it is important to note that the blurred image after down-sampling cannot be modeled by convolution between the down-sampled image and blur kernel:
(ar 2 22 & 2r/2 ? k42.
In this paper, we present a cross-scale constraint that accurately characterizes the connection between (y, æ k) at different scales. For instance, when s = 2, the cross-scale constraint is
3
(xle) @ (kL2) = 7 ((x 2 k) + Σ (x @ ga)l2) = (yle +
> a 2 9d)42),
where {ga)d=1 denote three quadrature mirror filters (QMFs) [59 of the kernel k. This constraint can be easily implemented with INR-based resolution-fres generators, and helps regularize the training of two generators to prevent likely over-fitting caused by the lack of GT images in loss function.
Progressive cross-scale learning for BID: A well-established practice traditional alternating iterative methods of BID to avoid convergence to trivial solutions involves solving the problem in coarse-to- manner 67 7 This means the kernel is initially estimated for the blurred image at a coarse scale, followed by propagation of the kernel estimate to finer scales. This approach has proven effective preventing iterations from converging to trivial solutions.
Despite its effectiveness, the coarse-to-fine strategy remains under-exploited in existing self- supervised BID methods, likely due to the resolution-fixed limitations of CNN MLP based re-parametrization. Utilizing INR's ss resolution-free properties, we introduce progressive learning strategy that begins training INR-based generators a coarser scale and then refines training finer scales. This scheme effectively addresses over- wver-fitting and ensures convergence to the truth.
Main contribution: Our main contributions summarized as follows:
Leveraging the resolution-ins ionlluioonnnddeeeddentt properties of INR for latent images/kernels, we propose supervupe cross-scale loss for training the NN without requiring GT images.
We introduce progressive multi-scale learning approach for BID, specifically designed to mitigate potential over-fitting due to the non-linear nature of the BID problem.
Extensive experiments conducted on variety of datasets reveals that our proposed method outper- forms existing self supe-vupe BID techniques as well as the supervised alternatives.
2 Related Works
Traditional non-learning methods for BID: Before deep learning became prevalent, regularization methods were the prominent approach for BID. These methods resolve solution ambiguities by imposing pre-defined priors images and blur kernels, such as image gradient sparsity [8] 4 24 5
677. image patch recurrence [54 [38] and Laplace priors in dark channels [42] [7]]. These methods
are based on some iteration scheme, and edge selection is effective technique for better robustness
and stability of the iteration [2
15| 71]. Since regularization methods can be recast as Maximum-A- A-Pottrrioi (MAP) estimators in Bayesian inference, another class of BID methods is derived from variational Bayesian estimators [35
65
41
63 2] as well as variational expectation maximization 22 [7]] The success of these methods requires rigorous tuning of paper-parmmeters related to priors. In contrast, deep learning based methods can automatically learn priors from data.
14 28
In recent years, there has been rapid progress in supervised deep learning methods for BID. By training over many pairs of blurred/truth images, these methods either explicitly estimate the blur kernel (e.g., [50] 40 34] or only estimate latent images (e.g..
Supervised deep learning for BID:

---

72)). The former is more efficient for handling uniform blurring. The latter is more general and can handle non- con-uniform blurring.
66
56
25
75 20
73
There are also methods that trained on unpaired dataset. Lu et al. (36] train the GAN for domain-specific deblurring. Wen et al. [61] propose a structure-aware deblurring method, and Chung et al. utilize generative priors from diffusion models to jointly estimate the blur kernel and latent images (13. Nevertheless, these methods still require GT images in unpaired data for training or depend on pre-trained models, which can be challenging and present difficulties in adaptation. In contrast, our method goes through a self-supervised manner, addressing this limitation.
Deep learning for BID with unpaired data or pre-trained model:
Self-supervised deep learning for BID: These methods address the issues of data collection and dataset bias in supervised methods. Built on the DIP prior for image/kernel, Ren el al. [47 intro- duced SelfDeblur, leveraging two NN-based generators trained through loss with optional TV regularization. The ensemble NN 9 aggregates deblurring outcomes from multiple NNs to improve performance. Li et al. [32] proposed using Monte-Carlo methods sample NN weights as an approximation of the MAP estimator of images and kernels. Li et al. (33] presented a self-supervised training scheme derived from the EM method. Dong et al. [8] combined the implicit prior from NN architecture and hand-crafted prior to regularize the NN training. Zhuang et al. [77] re-parametrized images and kernels by INR, and reply on the implicit prior induced by INR and early stopping for regularization. Built on resolution-free INR, we propose a cross-scale self self-supervised loss function and an efficient coarse-to- to-nine training scheme for supe-supe BID.
Most self-super BID methods are limited to handling uniform blurring, with Li et al. 33 being the only one capable of addressing both uniform and non-uniform blurring caused camera shake. Additionally, self-supervised super-resolution methods [55 also involve the estimation of blur kernel. While both estimate the blur kernel. However, these methods differ in their input: low-resolution images versus high-resolution images. Moreover, the blur degree in BID usually is much more severe than in super-resolution.
Coarse-to- estimation for BID: Coarse-to- schemes, which gradually refine the at different scales, have proven effective in traditional alternating iterative methods [65] 12 [52. The traditional multi-scale methods typically use estimates from coarser scales solely as initial estimates for finer scales. In contrast, our approach for self-supervised deep BID leverages the cross-scale interaction of estimates with exact relations, rather than approximations. Coarse-to-fiie scheme used recent supervised NN-based methods for BID (e.g., [39 56 74 (377 train the NN such that the estimates fit well the input image at different scales. Our multi-scale scheme is more for introducing new scale consistency loss function specifically tailored for self-supervised BID. This allows for more precise and consistent refinement of the estimations across scales.
INR for image recovery: In the domain of images, INR 10 37 43 encodes images NN weights, mapping coordinates to pixel values for a compact, continuous representation. There are also multi- scale extensions (27 49 of INR for more efficient image compression. It has been used for image restoration tasks such as image in-painting, denoising, and super-resolution [51] 21 64) Besides images, INR also has been used for encoding defocus blur kernels [66] and motion blur kernels (77). Our work is the first to exploit the resolution-free properties of INR for BID, leveraging these properties for multi-scale approach and cross-scale interaction.
3 Methodology
In this paper, we propose an INR-based progressive cross-scale self-supervised BID method. We first introduce the INR-based modeling for latent images and blur kernels, followed by the formulation of the self-supervised cross-scale loss function and the progressive learning strategy.
Double-INR model for BID: In our approach, the blur kernel and the latent image x are re- parameterized by two INR models, Tx and &a. respectively. Each model maps a spatial coordinate [ to a pixel value. Let ll,e In C 2² denote the sets of spatial coordinates within the feasible domain for the blur kernel k and the latent image æ, respectively. Then, they can be expressed as
{
k: [[[x]] = 4 (44: θk) x [||اa = 2 (1le Oz))
k[i,j]] = 4 ((.,j)) [i E Ipi
(4)
æ[i, ]] = P ([i [i E la:

---

where O₁ 2r denote the NN weights of k. a. The INR- re- ee-paamtterization 4 allows for the generation of kernel and image any coordinates, providing representation at arbitrary scales.
INR-based multi-scale representation: Let 4m denotes the standard down-sampling operator:
(i,m)) = kli m, (..j... mm) ml and (1,4))(i,,) = a:i m,j mm).
Then, form both the kernel and the image in dyadic pyramid, from the original scale and to Then define
coarser scales: Let (0) k, (0))
(x) 12, for S So- For any co-ordinate set II, we define its co-ordinate set at scale s as [[(*) {[ (2" E III. Then, using INR-based model 4 (e()) and (x()) scale can be expressed as
k (1() L2 and x²1)
(5)
(+()) k (2 th (2" 2k) and ac (+)
[
x ==2"T([)) T (2' O2)
(6)
Such multi-scale representation can facilitate the training of the INR-based generators for BID, allows for the generation of image details at arbitrary scales.
3.1 NN architecture of INR-based generators for kernel/image
Kernel generator: The kernel generator 4x in our approach is three-layer MLP with 128 feature nodes that takes coordinates normalized to [-1, 1] as input and adopts sinusoidal activation functions in every layer, following the Sinusoidal Representation Networks (SIRENs) [53]. The output layer employs a Softmax activation function to ensure the prediction satisfies the two physical constraints:
Non-negativity: A( 2 0 for all [i,5]:
Normalization:
k[i,j]=1.
Then, a Kernel Centering layer is applied address possible positional shifts in estimated kernels for better training stability. For a kernel E I "x satisfying 77. calculating its centroids (Cas Cy) by
Cy Σ (i,,), 1,3
Cx = Σ j --k(i..)
The kernel k is then shifted by [[H/2] Cy. [W/2] cz] to ensure it is centered geometrically.
Image generator: The image generator T adopts U-Net structure with five blocks, separated by down- or up-sampling layers and connected by skip connections. Each block integrates a sequence of Convolution, Batch Normalization, and ReLU. The NN concludes with 1 x 1 convolution layer followed by a Sigmoid layer to ensure the output image values remain within [o, 1]. To efficiently generate high-frequencies of images, following (51) the input spatial coordinates are first transformed into a higher dimensional space using high-frequency function 7(-) (sinusoidal):
7 (p) (p. sin (2 ... .......s........................ sin (2'-'mp) cos (2 (1-))p)),
(8)
where p : (i.j) represents the normalized coordinate values within [-1, 1), and T. is positive integer. Note this encoding operation is used only in 4g not in 2x-
3.2 Sele-Supervised scale consistency loss
Without GT images, the only readily available loss function to train the generators is the fitting loss:
(Ok. O.) = M, (y k @ x) = Ma th (Tx; O&) & Pa Oz),y)
(9)
where My(-) is some distance metric. Such fitting loss clearly is not sufficient to resolve solution ambiguities in BID. To address this, we introduce a scale consistency loss that regularize the training by enforcing cross-scale consistency of the estimation, which is based on the following proposition. Proposition 1. For a kernel (filter) k. let g1 92. gs denote its associated QMF filters (59) defined by
91 [m, ni] (-1)) ks (m, n), g2(m, n) (-1) mm, n), gs|m, n] = (-1)mm+ k (m, nl, (10) for any (m, ml] Ix. Then, we have the following relation between consecutive dyadic scales:
3
(x12) (kl2) 4 ((x 2 k) +
& ga) 42)
(11)
5

---

Proof. See Appendix A for the proof.
As seen from Proposition I the down-sampled blurred image yl2 does not equal the convolution of the down-sampled latent image æ/2 and the down-sampled kernel k/2, incurring additional term
Σ Ed= (xr @ 92) Therefore, down-sampled blurred images are fine for some initial estimation, not for regularizing the NN to obtain accurate estimation. To address this, based on Proposition [ introduce a scale consistency loss across two consecutive scales: for each scale 8,
L [ (kk az) MM (4(x)() (k( 12), (x)()
H2 +
Σ
9d H2)
(12)
Isd5a
_Mocd((((tti) @ (k((+11),, (x(*) @ K
(x
(+)
gd (5)
42). (13)
1S423
where 13
denotes the QMF filter bank of the kernel k(() defined by 10 and by 6
[ - (2 @k. and x (
[ Φ Oz)
Oz).
(14)
The scale-consistency loss 13 enforces cross-scale consistency of the estimations at different scales, providing additional regularization for training two INR-based generators.
3.3 Progressively coarse-to-fine training for BID
To address the non- non-linear nature of BID and avoid convergence to trivial solutions, we introduce a progressive learning strategy that trains the INR-based generators multiple scales. The training process consists of three stages at different scales. Define the fitting terms at different scale s by
1 = M, (yler, 2 (2 I((e; Ox) & 4 (2 -
(15)
The first stage serves as the initialization, operating the coarsest scale with the fitting loss c(()) The second stage progressive refines the training from the scale Sn to 0 with both the fitting loss and cross-scale consistency loss. Specifically, scale s, the loss is c/:) + Xx((1) where A a weight and The third stage is the final tuning stage scale with c(O) only. The training process is summarized Algorithm IIn our implementation,
((a) is the cross-scale consistency loss defined by 13 14
Algorithm 1: Self-supervised progressively coarse-to-fin training for BID
input: blurred image y
output: an estimated kernel k* and latent image x;
1: Initializing two generator NNs Q,, 4. with random weights O₂ and Q.i
(*).
2: Initial training: at the scale So. training the NNs with only the fitting loss fi
%% Progressively training the NNs
4: for +- So to () do
Training the NNs at the scale with the loss c(+) + X.
5: Final tuning: training the NNs at scale 0 with only the fitting loss (0)
Define K* (x Oj,) and x 2z O.).
Structural Similarity Index Measure (SSIM) 60 is used for (.) in 15
My = (n3 + 1 c1) (a of +2) -2 (2/ta14y + (20 C2).
For Mc(.) in 13 note that convolution process in 13) can be efficiently computed by by transform- ing the convolution operation into pointwise multiplication in its discrete Fourier transform (DFT), denoted by F(-). Thus, we define (-(-) in frequency domain with E -norm:
Mc(w,g) = [[F(x) - F1y)ll

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Category
Manmade
Natural
People
Saturated
Text
Average
Xu & Jia
65
19.23/.6554 23.03/0.554
25.32/0.852
14.79/0.563
18.56/0.7173
20. 18/0.708
Non- loooleaming
et al.
67
17.99/0.597 21.38/0.679 24.40/0.813
14.53/0.538
17.64/0.668
19.23/0.655
Zhong et al.
76
17.32/0.556 21.07/0.695
24.39/0.76]
14.86/0.602
15.86/0.532
18. 18.70/0.628
Michacli & Irami
38
17.43/0.418 20.70/0.511
23.35/0. 699
14, 14.14/0.491
16.23/0.468
18.37/0.518
Pan-DCP
42
18.59/0.594 22.60/0.698 24.03/0.772
16.52/0.632
17.42/0.619
19.89/0.666
Yan et al.
70
19.32/0.579 23.69/0.678 27.01/0.842
16.46/0.588
18.64/0.689
21 .0200.665
Yang & _
19.99/0.599 24.33/0.692
27.22/00866
17.04/0.605
20.35/0.762
21.79/0.704
DeblurGAN-v2 v2
25
15.93/0.321 18.95/0.429
21.53/0.668
13.79/0.480
14.82/0.519
17 17.04/0.490
Supervised
Kaufman & Fattal
20
18.94/0.517 22.05/0.586
27.05/0.833
15.18/0.599
17.85/0.717
20.22/0.650
MIMO-UNet
15.49/0.301 18.36/0.415
20.03/0.653
13.65/0.473
14.26/0.464
16.36/0.461
MPRNet
73
15.58/0.309 18.56/0.429
20.08/0.656
13.67/0.478
12.83/0.400
16.15/0.454
MPRNet
73
17.39/0.419 20.53/0.510
22.85/0.673
15.35/0.551
16.01/0.499
18.42/0.531
Restormer
72
15.63/0.324 18.55/0.433
20.29/0.665
13.70/0.499
13.40/0.451
16.31/0.474
Restormer
72
17.87/0.453 21.07/0.553
23.15/0.674
15.59/0.550
16.67/0.543
18.89/0.555
SeltDeblur
47
20.08/0.538 22.50/0.581
[27.4100
16.58/0.654
19.06/0.731
21.13/0.671
SelfDeblur 47
20.35/0.754 22.05/0. 709
2.5.94/0.88
16.35/0.636
20. 20.16/0.779
20.97/0.752
DEBID 0
19.62/0.692 24.12/0.807
(28.23/0.89)
17.12/0.692
19.44/0.711
21.71/0.751
Self Self- supervisdd
DEBID 9
22.14/0.803 26. 26.18/0.894.
118///89933..5515 31.25/00.23
18.43/0.714
23.00/0.822
24.20/0.831
MCEM 33
21.01/0.682 24.67/0.751
17/0.//.86.
16.63/0.651
70.51/0.760
22.20/0.741
MCEM
33
23.06/0.751 26.00/0.774 31.02/0.902
17.21/0.679
25.46/0.892
24.55/0.800
VDIP 18
20,97/0.66 647 24.51/0.770 27.53/0.
17.18/0.716
20.23/0.743
22.08/0.747
VDIP 18
22.86/0 868 26. 18/
30.76/0.927
18.55/0.727
27.24/0.927
25.12/0.869
Ours A
21.06/0.698 24.70/0.811
28.31/0.890
16.63/0.655
20.67/0.733
22.27/0.756
Ours
23.24/0.893 26.27/0.933
31.53/0.944
17.76/0.683
27.01/0.930
25.16/0.879
Table 1: Average PSNR/SSIM of the results for Lai al. dataset (26)- The methods marked with
A deblur the image by (23) using the estimated kernel, a standard protocol for evaluating kernel
estimation accuracy in BID. The methods marked with are retrained on the BSD-D dataset (48).
4 Experiments
Implementation details: The training consists of 5000 iterations across three stages. The first stage operates the coarsest scale So with 500 iterations. The second stage refines training from scale Sg to scale O, with 500 iterations per scale. The final stage tuning at scale for the remaining iterations. The NN is trained using the Adam optimizer with a batch size of 1. The initial learning rates for the image and kernel generators are set to x 10-3 and 5 x 10 respectively, decreasing to half their values every 2000 iterations. The weight À in the loss function is set 0.001 to keep the values of the two loss terms Leros and Lit in the same order. For comparison, we use results from the literature when available; otherwise, we use pre-trained models train from the provided code to achieve optimal performance. The code of the proposed method is available on Github
4.1 Evalution of motion deblurring on synthesized datasets
Two metries, PSNR (peak signal-to-noise ratio) and SSIM, are used for performance evaluation. Following SelfDeblur [47, we compute PSNR/SSIM after finding the best shift between GT and the result to handle shift ambiguity. In the tables. Bold in blue indicates the best among all supervised methods, Bold in black indicates the best, and underline the second-best among all GT-free methods.
Synthetic dataset with uniform blurring from Lai et al. [26]: This dataset consists of 100 images categorized into five groups: Manmade, Natural, People, Saturated, and Text, and covers 4 different kernels whose size ranges from 31 x 31 to 75 x 75. For this dataset, Sq is set to 2. As the focus of BID is on accurately estimating the blur kernel. Thus, two-stage evaluation protocol, as outlined by 47, is also used in test. Note that most supervised methods only generate clear images without kernels, and thus their evaluation is solely on their output images. The competing methods include 7 non-learning methods, 5 supervised methods, and 4 self-supervised methods. For all
https: : http:/://gttttaa ctm/t zhang nus ous/Deblur- INR
7

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Non-learning
Supervised
Metric
Cho
Lee
et al.
Whyte
Hirsch el al.
Vasu et al.
Yan
Yang
SRN
DeblurGAN
62
166
70
-v2
PSNR
28.98
27.34
28.07
27.77
29.89
29.61
29.22
27.06
26.97
MSSIM
0.933
0.796
0.848
0.852
0.927
N/A
N/A
0.840)
0.830
Supervised
Diffusion
Super-suee
Metric
Kaufman&
MPRNet PMPRNe|MIMO-US MIMO-UNel
Li al.
BlindDPS 13
SellDeblur
MCEM
VDIP
Ours
Fattal
20
73
11
31
47
33
[8
PSNR
30.17
26.32
25.34
26.89
24.02
25.85
30.26
29.55
30.69
MSSIM
0.915
0.827
0.791
0.837
0.702
0.792
0.940
0.922
0.942
Table 2: Average PSNR/MSSIM of the results from different methods the dataset Köhler's dataset
supervised methods, except for 20) both uniform and non-uniform blurring. For two recent methods, MPRNet [3] and Restormer 72
their models pre-trained the GoPro dataset
339 with
also include their models retrained on the BSD-D dataset 48 with only uniform blurring.
The comparison in Tab. [ shows that our method achieved the best performance. Note that for two supervised methods, MPRNet [73 and Restormer (72), their models trained on dataset with only uniform blur performs better on testing data with uniform blur, when compared to the ones trained on the dataset with non con-uniform blurring. However, this improvement is still not enough to match the performance of our method. The main reason that existing supervised methods target general blurring and overlook the physics prior of image formation, specifically the convolution model for uniform blurring. As a result, they underperform compared to our self-supervised method, which leverages this prior. This highlights the generalization issues inherent in supervised learning approaches, whose performance heavily depends the correlation degree between the training and testing data. Refer to Appendix F. for visual comparisons of different methods.
Synthetic dataset with modest non-uniform blurring from Köhler et al. [22]: The benchmark dataset Köhler [22] comprises 48 notion-blurred images which blurring is not exactly uniform. This dataset is for evaluating the robustness of BID method to handle modest non-ume unnuuofoomm blurring. Totally 17 methods are selected for comparison, and So is also set 2. Following the evaluation protocol in [33] we use the average PSNR and MSSIM (multi-scale SSIM) as the evaluation metrics. Tab. 2 shows that method outperforms all others in terms of both PSNR and MSSIM, indicating its robustness in handling modest non-uniform blur. Refer to Appendix F2 for visual comparisons.
4.2 Evaluation of motion deblurring on real-world datasets
Real-world dataset from Lai et al.'s [26]: : Lai el al.'s real-world dataset 266 consists of 100 real blurred images captured in diverse scenarios using various capturing settings. As no GT images for quantitative evaluation,
present only visual comparisons of many samples. Please refer to Appendix F3 for visual comparisons of different methods. Overall, our method generates the images with the best visual quality, consistent with its performance on the synthetic datasets.
Non-leamnn learnegrn
Supervised
Self-supervised
Xu & Jia Pan-DCP
Hu et al. DeepDeblur
DeblurGAN Restormer SelfDeblur
MCEM
Dataset
Metric
Ours
42
17
39
-v2 -v2
72
47
33
PSNR
27.14
27.22
26.41
27.87
28.70
28.96
27.92
28.17
28.31
RealBlur-J
SSIM
0.8303
0.7901
0.8028
0,8274
0.8662
0.8790
0.8420
0.8499
0.8511
PSNR
34.46
34.01
33.67
32.51
35.26
36.19
34.49
35.46
35.62
RealBlur-R
SSIM
0.9368
0.9162
0.9158
0,8406
0.9440
0,9570
0.9270
0.9436
0.9448
Table 3: Average PSNR/SSIM of the results on the RealBlur dataset 48
with small/modest blurring: Our proposed method is also evaluated in another real-world dataset published in [8): RealBlur-J and RealBlur-R, both containing
Real-world dataset RealBlur 48

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980 real-world blurry images. See Tab, B.b.BBre for the results. It can be seen that, unlike the experiments on synthetic datasets, there is a gap between the proposed method and supervised methods. This is understandable, our method cannot access GT images, whereas the supervised methods can. However, our method still outperforms all traditional methods and existing self-supervised methods. Refer to Appendix [F44 for visual comparisons of different methods.
4.3 Evaluation on microscopic deconvolution
In microscopic imaging, acquired images often suffer from blur due to optical limitations, out- of-focus elements, specimen motion, and the diffraction limit of light. Following (45), the test dataset consists of 120 images, covering 24 images from the test subsets "Confocal BPAE B" and "TwoPhoton MMCE". and includes 3 Gaussian point spread functions (PSFs) and 2 Poisson PSFs as the blur kernels. For the compared supervised methods, Restomer [22 and INIKNet [46), we retrained their models using the microscopic dataset 455 It can be seen from Tab. that our method outperforms all other supervsde methods and is comparable to the supervised methods trained on the microscopic dataset. Refer to Appendix F.5 for visual comparisons of different methods.
Supervised
Diffusion
Self-supervised
PSFs
Restormer
72 |NIKKNet
46
BlindDPS
13
SelfDeblur
47
MCEM 33
VDIP
18
Ours
Gaussian|
36.32/0.936
37.32/0.941
26.78/0.661
35.52/0.927
36. 12/.933 35.73/0.932
36.65/0.940
Poisson|
40.73/0.958
41.56/0.961
26. 26.94/0.667
39.88/0.950
39.26/0.945 38.47/0.94 942 40.71/0.960
Table 4: Average PSNR/SSIM of the results from different methods microscopic deconvolution.
Category
Manmade
Natural
People
Saturated
Text
Average
w/o Lemus
21.19/0.778 25.84/0.887
30.74/0.918
17.69 17.69/0.682
26.75/0.917
24.44/0.836
Single-scale
22.04/0.803 25.93/0.890
30.33/0.933
17.68 17.68/0.688
24.76/0.886
14.14/0.840
w/o Progressive
20.36/0.742 23.91/0.829
26.35/0.821
17.22/0.675
22.88/0.857
12.14/0
INR/CNN as + 4
16.62/0.370
26.21/0.865
28.83/0/877
17.03/0.668
23.34/0.776
23.01/0.767
MLP/INR as / =
19.88/0.661 19.47/0.479
26.77/0.718
16.07/0.667
15.66/0.537
19.17/0.521
MLP/CNN as 2 x
15.87/0.331 19.20/0.430
23.32/0.578
15.58/0.627
16.57/0.481
19/0..47
Multi-scale by direct |
18.64/0.616 22.61/0.722
25.95/0.781
16.64/0.601
20.65/0.861
20.89/0.716
Ours
23.24/0.893 26.27/0.933
/ 31.53/0.944
17.76/0.683
27.01/0.930
25.16/0.879
Table Ablation study of the proposed method in terms of of PSNR/SSIM.
4.4 Ablation study
The ablation study is conducted the Lai et al. dataset (26] The results are shown in Tab. 5
Effectiveness of self-supervised cross-scale consistency loss: To evaluate the gain from the proposed cross-scale consistency loss functions, we retrain the NN using only the fitting loss Cai for each scale, ie., without (w/o) Lesos Tab. shows an average gain of about 0.72 dB in PSNR with the cross-scale consistency loss, demonstrating its effectiveness.
Effective of progressive coarse-to-fine training:
We first examine the effectiveness of multi-scale training by performing the training only at the original scale. Tab. 5]shows about a I dB loss in PSNR, indicating multi-scale training's contribution. To evaluate the progressive training strategy, we train the NN with the sum of all loss functions at three scales, which resulting in a 3dB loss. This clearly indicates that progressive training is critical for effectively utilizing the multi-scale scheme. The reason is that the loss function at the coarser scale emphasizes lower frequencies since the image at coarser scale retains low but loses high frequencies. Thus, a NN trained on the sum of loss functions across 3 scales focuses more on low frequencies than one trained only at the finest scale, which fits both low and high frequencies. Therefore, progeesroees training scheme is more effective than joint multi-scale loss. In scale- progresreeee training, the estimation at the coarser scale provides

---

strong initialization for the finer scales. The final result is obtained by applying the fitting term exclusively at the finest scale.
INR (coordinate NN) vs. MLP//CN (image-to-image NN) This ablation study is to evaluate the necessity of using INR for re-parameerizzttion of image and/or kernel, instead of using the NN that maps an image to an image. In the study, we separately replace our -NR-based kerne/iimaee generator with MLP/CNN based representation adopted in [47]. whose down-sampled versions of images/kernels at different scales are generated by standard down-sampling process.
Direct down-sampling vs. using multi-scale grid in INR: To verify the benefit of resolution-fiee free property of INR, we also consider generating multi-scale versions of image/kernel by standard down-sampling using bilinear interpolation, CNN-based representation. This is referred to as "Multi-scale by direct 4" Tab. S shows that the performance of multi-scale by standard down-sampling process very poor. Clearly multi-scale representation by standard downsampling process is not as effective as resolution-free INR.
their high resolution version from INPs, same to
4.5 Comparison of computational effciency
The computational effciency of the propoposed method is compared to three most related self- supervised BID methods. The results are reported in terms of running time, number of parameters, and memory usage, when processing 256 x 256 image with 31 x 31 blur kernel NVIDIA 3090 RTX GPU. See Table 6 for the details. It can be seen that the proposed method achieves a balance between computational cost and deblurring performance.
Methods
SelfDeblur 47
MCEM
33
VDIP
Ours
Running time (s)
219.71
226.31
245.04
213.02
Number of parameters (k)
3427.2
2409.0
3523.2
2342.4
Memory usage (GB)
6.31
1.27
9.19
1.82
Table 6: Model complexity comparison.
4.6 More details, studies, experiments and visual comparisons the appendix
The appendix includes (1) ablation study on setting scale Sg, (2) details on parammee settings, (3) details on NN architecture, (4) more comparison on running time; (5) visualization of some sample results, (6) visualization of intermediate results, and (7) experiments on Levin et al." 's dataset [28
5 Discussion and Conclusion
This paper introduces a self-supervised method for BID, which does not require GT images. Lever- aging the resolution- cooluttioo-feeeeeee representation of INR for image/kernel, propose a supervupe cross-scale consistency loss function and a progressive coarse-to- carse-to-finee training strategy. The proposed method significantly outperforms existing methods in extensive experiments.
There are two limitations of the proposed self-supervised method. The first is the computational cost for processing large number of images, the method requires training the model for each individual sample. A potential solution is to explore the usage the proposed techniques in meta-learning or testing-time adaptation. This would allow the proposed technique to rapidly adapt pre-trained model instead of train the NN from scratch. The second limitation is that the proposed method is only applicable handle uniform blurring, it relies on the convolution model. Extending this approach to handle non- non-uniform blur will be another important direction for future research.
10

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Acknowledgments
Yuhui Quan would like to acknowledge the support from National Natural Science Foundation of China under Grant 62372186, Natural Science Foundation of Guangdong Province under Grants 2022A1515011733 and 2023A 15501284,,, and Fundamental Research Funds for the Central Uni- versities under Grant x2jsD2230220. Hui Ji would like to acknowledge the support from Singapore MOE Academic Research Fund (AcRF) Tier Research Grant 1-8000000000000
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6 Appendix
Below we provide additional details and results which are not presented in the main manuscript.
A Proof of Proposition
Proof. The proof is based on the convolution theorem and the properties of the DFT. Without loss of generality, we assume that image a and the kernel & are of size 2MM x 2N. Let æ. and kl22 denote the down-sampled images of æ and k. respectively. Let F denote the discrete Fourier transform (DFT), Recall that for signal R2Mx2N its DFT is defined as:
2M 2-
F())[m, =
Σ
2|e.jle 12 23r -2x 2
t=
==
Then, for a down-sampled version of the signal ø12, whose DFT is
M-IN-1
F(z(2)(m,n)
Σ
Σ
=|20,
2j|e
12m W W 12m RJ.
E= 1=0
The DFTs of the down-sampled signals 2.42 and k42 can be expressed as follows: For each frequency [m, ro], we have
1 (F(zr) 22m, 2m] F(x) [2m + M, 2m]
F [m. =
4
(16)
F(x) [2m, 2n NJ + F(x)) [2m M, 2n N)),
F Fkk2)) [m. - (F(k) [2m, 2m + F(k) [2m + M.n]
(17)
F(k) 12m,2n NJ (k) 12m M, 2n N)),
and
F Fyl2) [m.n] = (F(y) [2m, 2n] + F(y) 12m + M,n
4
(18)
+ F(y) 12m, 2n + N] + Fly) 12m + M, 2n + NJ),
By convolution theorem, for y = x & we have that
F(y)(m,n] = F()) (mk))m,
Thus,
F (U)2) (m, m|
Σ ==
F(w) 12m 2M,2n 2n + +jN]- F(k) [2m EM, 2n j.
(19)
Applying convolution theorem on 2.12 @ k42 again, we have
F(x12 & [m,) n| = F(xl2)mm, n| F(kl2)(m, n)..
(20)
Plugging in
16 and 17
into the above equation, have
F(z12 @ k22) m, n)
F(x) 12m + LM, 2n + jMJ
ΣΣ F(k)
[2m + eM, 2n + jNl
16
0= 0=0
1= ==
(21)
Define 3 filters g1. 02.92.90 93 by
91 m, n
(-1 k n
92 m,
n*k m.
(22)
93 (,, n
(- 1) mm ke [mm
16

---

Then, theirs DFTs are given by
F(g ) m, ni = F(k) m + M, nl F(g2 m, n F(k)
m, n + N]
(23)
F(ga) m, n = =F(k) mm M, N]
Then, the expression of the right-hand side of Equation 21
can be rewritten as:
F(æ @ k4)) n| = F ((2z @ k)42) [m, n. +
F ((< @ ga)t2) (m.
(24)
d=1
By convolution theorem, we have then
(# @ k42) 4
1
k))2) ((x ((x))))2)
Σ
((x @ (allal
d=1
The proof is complete.
B Ablation study on performance impact by different maximum scale So
In this ablation study, we investigate how different choices scale So impact the performance on motion deblurring tasks. The experiments conducted using Lai al.'s 's dataset. We compare results using the same framework with different Sg values: single-scale, two-scale, three-scale, and four-scale (i.e., So E (0, 1, 3)). Each setting undergoes a total of 5000 training epochs, with each stage except the last one consisting of 500 training epochs.
As shown in Tab. 7 multi-scale frameworks consistently outperform single-scale setups. Among the three-scale model demonstrates the most significant improvement. However, the four-scale model shows only a slight improvement over the single-scale setup. This minimal gain can be attributed the initial stages the four-scale framework, where the target image, resized to 1/8 of the original size, is far from the true image. This highly erroneous initialization causes more harm than benefit, hindering the NN's convergence to the true image.
multi-scale configurations,
Category
Manmade
Natural
People
Saturated
Text
Average
Single-scale (Sp 0) 22.04/0.803 25.93/0.890
30.33/09933
17.68/0.688)
74/../0.886
24, 14/0.000
Two-scale (Sp 22.87/0.86 866
25.89/0.888
30.54/0.921
17.89/0.693
26.72/0.919
24.78/0.857
Three-scale (Sg 2) 23.24/0.893
26.27/0.933
31.53/0.944
17.76/0.683
27.01/0.930
25.16/0.879
Four-scale (Su 22.89/0.866 25.04/0.8666 86644/0.6666.. 30.21/0.899 1w.01//99997777....
17.71/0.699
25.21/0.903
24.21/0.847
Table 7: Scale variation study the proposed architecture in terms of PSNR/SSIM on the dataset Lai et al. [26]. Bold for best performers and underline for second-best performers.
C Implementation Details
C.1 Hyper-parameter settings for different Datasets
The following Tab. g summarizes the hyper-parameters, specifically the range kernel sizes and the coarsest scale S, used in model across various datasets. These parameters selected based the specific characteristics of each dataset, including image size.
C.2 Details of image generators
Please see Tab. 9 for the details of the architectures used for image generative network 4 In Tab. the "CBR" denotes the subblock for three successive layers: Comn2d4-BachNNom24kRR... The "CBR/Down" means modifying the Conv2d layer in "CBR" with stride (2, 2), which performs downsampling operations. The "Up" means the upsampling layer with bilinear interpolation. The "Cat(m,n)" denotes the concatenation of the outputs from layers No.m and No.n.
17

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Dataset
Kernel Size Range
Coarsest Scale Sp
Lai et al.'s datasct 26
31 31 73 x 75
2
Köhler et al. 's dataset
22
27 x 27 131 x 131
2
Lai el al. 's real dataset
26
27 x 27 - 99 x 99
2
Realblur datascts
48
27 x 27 - 79 x 79
2
Microscopic datasets
45
5 x - 13 x 13
Levin ef al. datasct 28
11 x 11 - 27 x 27
Table Kernel size ranges and coarsest scale settings for various datasets.
Block
Channels
No.
Block
Channels
No.
Block
Channels
Block
Channels
Encoder
Decoder
Encoding by Eq
CBR/Down
Up
CBR
128
CBR
128
10
CBR
128
18
Cat(14,17)
144
26
Up
CBR/Down
128
11
CBR
16
19
CBR
128
27
Cat(S_26)
144
CBR
12
CBR/Down
Up
CBR
128
CBR
16
13
CBR
128
21
Cat(11.20)
144
29
Up
CBR/Down
128
14
CBR
16
22
CBR
128
Cut(2,29)
128
CBR
15
CBR/Down
Up
CBRx2
128
CBR
16
16
CBR
128
24
Cat(8.23)
144
32
Sigmoid
C
Table 9: Architecture details of image generative NN 2 in the proposed method
D
Additional comparison of computational cost
Tab. I0 reports the rumning times for processing 256 256 image with a 31 x 31 blur kernel, comparing severe existing BID methods, including those beyond self-supervised approaches. The inference time is measured using an NVIDIA 3090 RTX GPU. Unlike supervised methods, which require considerable time for model training but is very fast processing images using pre-trained model, our approach involves training model for each specific image that needs processing. This makes our method similar to traditional iterative methods, focusing on case-by-case basis rather than relying on a pre-trained model. The running time for processing an image is comparable to many iterative optimization methods and existing self-supervised learning solutions. One of future works would be on how to reducing this per-image training time, possibly through the implementation of meta learning testing- time model adaption.
Non-leamnnn
Supervised
Dillusion based
Selí-S Sell-Supervised
Sun et af. Jin al.
Yan et al.
Yang Ji
MPRNer Restormer
BlindDPS
SelfDeblur
MCEM
VDIP
Ours
70
70
72
+7
Time(s)
113.98
35.84
1242.97
354.03
0.21
0.25
286.55
219.71
226.33
245.04
213.02
Table 10: Time comparison with existing BID methods when processing a 256
256 image
E Broader impacts
The proposed self-supervised learning method for deblurring images has the potential to impact wide range of applications, including surveillance security, scientific research, and digital media restoration. By improving the accuracy and clarity of images, our research can facilitate deeper insights and more effective interventions in these fields.
In surveillance, higher clarity images can enhance public safety by providing more detailed visual information. However, this also raises concerns about privacy and the potential for mass surveillance. In scientific research, improved image quality can lead to better data and more significant discoveries, though there is risk that overly processed images could misrepresent the original data. In digital media restoration, while the technique helps preserve cultural heritage, it also poses the risk of altering historical records.
18

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Despite these possible concerns, our goal is to contribute to enhancing image clarity in critical areas such as public safety, scientific research, and cultural preservation. We emphasize the responsible application and continuous improvement of this technology to mitigate potential risks and maximize its positive impact.
Visual comparisons of the results in the experiments
F.1 Visual comparisons on Lai et al. 's dataset
In this section, we visualize the results of different methods on some examples from Lai et al. dataset (26), which is known for its severe blurring effects. Figs. [D-ddmmnstttteee that method consistently produces results with sharper details and fewer artifacts compared to existing methods, showcasing its effectiveness in addressing significant blur challenges. In contrast, supervised learning methods trained on external datasets yield poor quality results, highlighting the limited generalization performance of supervised approaches when dealing with complex real-world blurring.
Blurry image
Cho et af.
TP
Xu & Jia
Xu et al.
67
Michaeli Trani
38
Perrone & Favaro 44
Pan-DoP Pan-DCP 42
Kaufman & Fattal 20
MPRNet 2R
MIMO-UNe 1
Restormer ?P!
SelfDeblur 47
MCEM 33
VDIP IB
Ours
GT
Figure 1: Visual results the dataset of Lai al. 26
19

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Blurry image
Chu et al.
12
Xu & Jia 63
Xu al.
67
Michaeli & Irani
Perrone & Favaro
Pan-DCP AB
Kaufman & Fattal
2T1
MPRNet 7k
MIMO-UNel II|
Restormer 12
SelfDeblur 4Y
MCEM 33
VDIP 18
Ours
GT
Figure 2: Visual results on the dataset of Lai et al. (66
20

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wow Parties devekosed
Parbles
Parties Emiiigation derecoeed climabe
climate
supppe
Blurry image
Cho et ol. 1P
Xu & Jia
65
Xu et al. 67
Te Parbies
Be Parves
JEVER COURIONA
IT dase oring Countrie Parbies
Parties
Michaeli Irani
38
Perrone & Favaro
Pan- DCP
42
Kaufman Fattal 20
BEIEG Parties
AOGORACCUIERR Parties
developing countrie
Robdon
Parties
Sccione
Parties
Smniiigation developed
suppor
climabe
MPRNet
MIMO-UNet
Restormer 72
SelfDeblur 47
developing countrie
developing
anason countrie
developing
countrie
OITY TOTOO
Parties
Parties
Parties
emibiiaaioo climate
Emibigablon
developed
mibiiaation
develope sped
instuties aose
climate
suppre
iclimabe
MCEM 3
VDIP
GT
Figure Visual results on the dataset of Lai et al.
26
Blurry image
Pun-DCP [42
MIMO-UNet M
SeliDeblur 47
MCEM
VDIP
Ours
image
Figure 4: Visual comparison on the dataset of Lai et al. 26
21

---

Blurry image
Cho 12
Xu & Jia
Xu 67
Michaeli & Irani
38
Perrone & Favaro (44
Pan- DCP 42
Kaufimnn & Fattal 20
MPRNet 73
MIMO-UNet
Restormer
SelfDeblur +7
2012
MCEM 33
VIIP I8
Ours
GT
Figure 5: Visual results on the dataset of Lai et al. 26
Blurry image
Pun-DCP 42
MIMO-UNet |||
SelfDeblur 47
mlll
MCEM 33
VDIP L8
Ours
GT image
Figure 6: Visual comparison on the dataset of Lai et al. 26
22

---

F.2 Visual comparisons on Köhler et al. 's dataset
this section, we display the visual results on Köhler el al. 's dataset [22]. which is characterized by its non-uniform blurring effects. From Fig. F [ our method demonstrats satisfying performance on this challenging dataset, effectively handling the intricate and variable blurring effects. Further- more, we compare our results with those obtained using the Diffusion-bass method, BlindDPS [33. revealing its difficulty in adapting to blurring effects and image types that deviate from its pre-trained model's s training data.
Blurry image
Cho &Loc 12
Kaufman & Fattal
20
BlindDPS LB
Selfebllu 47
MCEM 3
Ours
GT image
Figure 7: Visual comparison on the dataset of Köhler er al.
1
1
III
Blurry image
&Lee
Kaufinnn &
20
BlindDPS
11
18
SelfDeblur
MCEM
Ours
image
Figure 8: Visual comparison on the dataset of Köhler et al.
22
F.3 Visual comparisons on Lai et al. real dataset
Fig. 9 iillustrates a comparison between our method and other competitive self-supervised methods on Lai et al. 's real-world data, where our method consistently generates higher quality images.
23

---

J0316
Be an
L Leggnn
Legend
IT Legend AEIT
IT Legend
ENIT
Blurry image
SellDeblur
47
MCEM KK]
Ours
Figure 9: Visual comparison the real dataset of Lai al.
26
F.4
Visual comparisons on RealBlur dataset
In this section, we present visual comparison of the results from state-of-the-ard supervised learning methods 77 72) and our approach on the RealBlur dataset. Although our method achieves slightly lower average PSNR and SSIM values across the entire dataset compared to these representative supervised learning methods, the visual evidence presented in Fig. Iemoontttas that our results are often sharper. In other words, the gap between our method and the supervised methods is smaller in terms of visual quality than indicated by the quantitative metries.
Blurry image
MIMO-Unet LDI
MPRNet T8
Restormer
Ours
GT image
Figure 10: Visual comparison with supervised methods on challenging cases from RealBlur-J 48
24

---

F,5 Visual comparisons on Microscopic dataset
This section provides comparison between our method and other methods in the task of microscopic deconvolution. Our approach consistently restores images with more texture and details.
Blurry
INIKNet
SelfDeblur
BlindDPS
MСEM
VDIP
Ours
GT
Figure 11: Visual comparison on the microscopic deconvolution. All images are originally grayscale hut different colormap is used to better highlight the differences among the various reconstructions
G Visual comparison on the ablation study
This section provides comprehensive visual comparison for the ablation study in Fig. 12] illustrating the impact various configurations on our model's performance. These visuals serve supplement to the detailed quantitative results discussed in the main text, found in Sec. 4.4
Blurry image
w/o Leros
Single-Scale
w/o Progressive
INR/CNN as agg
MLP/INR 9g/0n. MLP/CNN 0//02
Multi-scale J.
Ours
GT image
Figure 12: Visual comparison on the ablation study.
25

---

H Visualization of Intermediate Results
Beration=500
Ieraatinn= 1000
Ieration= 1500
Iteration 5000
Blurry image
1
(0) 1
(0) [
GT image
Figure 13: Intermediate results of estimated blur kernel and latent image at end of each stage.
Apart from validating the efficacy of our approach through the quality of the final results, is also interestting to see the intermediate status of the proposed method. To this end, we use "manmade 01 from Lai et al. 's dataset [26] as case study, visualizing intermediate results better understand the progress through different stages to the final output. As shown in Figure [3] in the first stage, we obtain an initial representation the smallest scale during initialization. In the second and third stages, we achieve results at larger scales with finer details. In the final stage, the neural network builds on the groundwork laid by the preceding stages to refine the output, ultimately generating the desired clear image.
I Experiments on Levin et al. 's dataset 28 with small kernel size.
Levin et al.' 's dataset contains 32 images generated by convolving 4 clear images using 8 motion-blle kernels and adding Gaussian white noise with s.t.d. 1%. The size of these kernels is small, ranging from 11 11 to 27 x 27. Following 47), besides PSNR and SSIM, Error Ratio (30 is also used as quantitative metric. From Tab. [1 approach performs well across all three metrics. For visual
comparisons, please see Figs. [4 15 where our results more closely resemble the GT image, with fewer artifacts and more details.
Non-lcaming
Metric
known k
Krishnan el al.
Cho&Lee [2]
Levin el al.
Xu & Jia
Sun el al. [54]
Zuo el al.
Pan-DCP
24
29
65
78
42
PSNR
34.53
29.88
30.57
30.80
31.67
32.99
32.66
32.69
SSIM
0,949
0.866
0.896
0.909
0.916
0.933
0.933
0.928
Error Ratio
1.000
2.452
1.711
1.772
1.489
1.284
1,250
1.255
Supervised
Diffusion-
Self Self-Supervise
Metric
SRN*
MPRNet 773
Restormer 72
BlindDPS 13
SelfDeblur [47
MCEM
VDIP
Ours
[56
18
33
PSNR
23.43
26.21
27.78
14.91
33.07
32.81
33.12
33.74
SSIM
0.712
0.795
0.838
0.367
0.931
0.927
0.929
0.938
Error Ratio
6.086
3.050
2.633
21.96
1.196
1.273
1.188
1.185
Table 11: Average PSNR/SSIM the results from different methods on the dataset Levin et al. dataset (2). The method marked with # is retrained with synthesized datasets using eight blur kernels
the dataset Levin et al. 28
26

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gurv
Sury
Gut
CA
CAN
CAn
Blurry image
Krishnan
2
Cho al. 1P
Levin 20
Surv
Surv
Surv
Surv
CAn
lilll CAnt
builli CAN
biilli CAn
Xu & Jia 6ui
Sun et al.
S4
Zuo et al.
78
Pan-DCP 42
yot
Yurv
Surv
CAn
CAn
MPRNet
73
Restormer
BlinDDPS L5
SelfDeblur 47
Surv
Sury
Surv
Surv
CAn
CAN
Sailli CAn
Bailli CAn
B
MCEM 38
VDIP 18
Ours
GT
Figure 14: Visual comparison on the dataset Levin et al.
28
27

---

Blurry image
Krishnan
2
Cho al. 1P
Levin
Xu Jia 651
Sun et al.
54
Zuo et al.
78
Pan-DCP 50
MPRNct 73
Restormer
BlinDDPS L5
SelfDeblur 47
MCEM 38
VDIP 18
Ours
GT
Figure 15: Visual comparison on the dataset Levin er al.
28
28

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Reviews:
[{"review_id": "RAzD0t99H3", "reviewer": "Reviewer_QZrB", "summary": "This paper presents an approach to solve blind image deconvolution. The authors use multiscale Implicit Neural Representations (INRs) to depict both the latent image and the blur kernel. In addition, they propose a method that incorporates a cross-scale consistency loss and a progressive scale optimization process. Experimental results demonstrate superior performance on small-scale simulated datasets and competitive results compared to state-of-the-art methods on large-scale and real-world datasets.", "questions": "\\- Minor points:\n\n    \\- In Algorithm 1, please indicate whether the loop end condition in step 4 includes ‘0’.\n\n    \\- On line 259, the sentence “the test dataset consists of 120 images“ appears to be repeated.", "limitations": "Blind deconvolution, when limited to the uniform case, may not be particularly useful for real-world problems. While the authors have acknowledged this, it would be beneficial to include experiments demonstrating the effectiveness of the proposed method’s components on widely-cited non-uniform datasets, such as GoPro [40]. This would provide valuable guidance for future research in the field.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The authors propose a cross-scale loss function that compensates for the inaccuracies introduced by simply downscaling the latent image and the blur kernel, and then calculating their convolution at the lower scale. This is achieved through the use of Quadrature Mirror Filters (QMFs).\n    \n2. The authors conduct ablation studies, providing evidence of the effectiveness of each component within the proposed method.\n    \n3. The proposed method outperforms other considered unsupervised methods in synthetic uniform blind deconvolution.", "weaknesses": "1. There exist prior works on Multiscale Implicit Neural Representations. The authors should provide a more detailed discussion on the similarities and differences between their proposed method and these previous works. For instance:\n    \n\n- “PINs: Progressive Implicit Networks for Multi-Scale Neural Representations” by Landgraf, Zoe, Alexander Sorkine Hornung, and Ricardo S. Cabral, presented at ICML 2022.\n    \n- “Miner: Multiscale Implicit Neural Representation” by Saragadam, Vishwanath, et al., presented at ECCV 2022.\n    \n\n2. The method appears to require at least one hyperparameter (the number of scales) to be set based on the target dataset. This suggests that the method is not entirely blind.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for appreciating our work and your valuable comments. See below for the responses to the concerns and questions.\n\n\n**[W1]** *There exist prior works on Multiscale Implicit Neural Representations. The authors should provide a more detailed discussion on the similarities and differences between their proposed method and these previous works. For instance, \"PINs\" (Landgraf et al., ICML 2022) and \"Miner\" (Saragadam et al., ECCV 2022).*\n\nThanks for the references. The concepts of \"progressive\" and \"multi-scale\" have different definitions in PINs/Miner and our work, as they have different aims and applications. PINs and Miner focus on designing **NN architectures** for INRs to achieve higher image compression ratios. In contrast, our goal is to develop new **training scheme and loss functions**, utilizing the resolution-free property of an off-the-shelf INR, for solving the blind image deconvolution problem. Specifically,\n\n+   \"Progressive\" in PINs refers to the progressive encoding of frequencies  in NN design. In our approach, \"progressive\" refers to training the NN from a coarse scale to a finer scale in the image domain, which pertains to the training process rather than NN  design.\n\n+   \"Multi-scale\" in PINs refers to employing a multi-scale implementation w.r.t. different frequency spectra for positional encoding and a hierarchical MLP to encode these spectra. In Miner, the image is represented in a multi-scale pyramid with correspondingly varying sizes of MLPs. Again, both approaches concern NN design. In contrast, our approach involves utilizing the cross-scale property of convolution and progressive training from coarse scale to fine scale, focusing solely on  training.\n\nWe will include such a discussion in the revision.\n\n--- \n\n**[W2]** *The method appears to require at least one hyperparameter (the number of scales) to be set based on the target dataset. This suggests that the method is not entirely blind.*\n\nFollowing common definitions, in the context of image deblurring, \"blind image deconvolution\" refers to estimating both the kernel and the image from a blurred image, as opposed to \"non-blind image deconvolution,\" which refers to estimating only the image from a blurred image and a known kernel. Thus, while \"blind\" in general image restoration may refer to the automated setup of all parameters, in our context, it specifically refers to the unknown nature of the blur kernel. We will clarify it  in the revision. In addition, the number of scales is dependent on the image size of the dataset, which is easy to set in practice.\n\n--- \n\n**[Q1]** *In Algorithm 1, please indicate whether the loop end condition in step 4 includes '0'.* \n\nWe appreciate the reviewer's attention to detail. The loop end condition in step 4 of Algorithm 1 does  include '0'. We will make it clear in the revision. \n\n--- \n\n**[Q2]** *On line 259, the sentence \"the test dataset consists of 120 images\" appears to be repeated.* \n\nThank you for pointing out the typo. We will correct it in the revision.    \n\n--- \n\n**[Limitation]** *It would be beneficial to include experiments demonstrating the effectiveness of the proposed method’s components on widely-cited non-uniform datasets, such as GoPro [40]. This would provide valuable guidance for future research in the field.* \n\nThanks for the comments. Blind deblurring has three types of configurations: uniform blurring, non-uniform blurring of static scenes, and dynamic scenes with moving objects. To the best of our knowledge, there is no self-supervised learning method for deblurring dynamic scenes yet. While our method is optimized for uniform blurring, we also evaluate its performance on both synthetic datasets with non-uniform blurring (Köhler dataset in Section 4.1) and real-world non-uniform blurring datasets of static scenes (RealBlur in Section 4.2). The results show the robustness of our approach to non-uniform blurring of static scene. GoPro, a dataset for dynamic scene deblurring, is not included for evaluation, as the image formation model used in our approach cannot model the blurring caused by moving objects. We will discuss this limitation and provide a few examples on the GoPro dataset in revision.", "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": "MMdjNkLogQ", "reviewer": "Reviewer_ihYN", "summary": "This paper proposed a self-supervised method for blind image deconvolution (BID). The main idea is to introduce the implicit neural representation (INR) technique for representing both the blur kernel and the image, such that they can be parameterized at different scales using a single model. With such a INR representation, supervision can be enforced in a cross-scale way to enhance the efficacy of the model. Extensive experiments have been conducted to demonstrate the effectiveness of the proposed method.", "questions": "1. To my knowledge, the performance of fully supervised methods highly depends on the training data. So I would like to know how the supervised methods were trained on the Lai, Kohler, and Levin datasets, since only test images are contained in them.\n2. The method is constructed based on the uniform blurring assumption, and thus it should be discussed why it can perform well on the Kohler dataset which focuses on non-uniform blurring.\n3. Typos: In Line 31, $\\delta,y=k\\otimes x$ should be $\\delta\\otimes y=k\\otimes x$.", "limitations": "The authors discussed the limitations of the method, which are reasonable.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Introducing INR to the studied BID problem is new and brings benefits in cross-scale representation.\n2. Various datasets have been considered in experiments, which makes the results convincing.", "weaknesses": "1. The presentation of this manuscript should be improved. For example, a large part of Section 1.2 can be put into Section 3.\n2. Experiments are insufficient in the following sense:\n- A recent self-supervised BID method [1] is not compared.\n- The cross-scale supervision training strategy can also be applied other self-supervised methods by with simple downscale operation, which is not tested.\n- The use of the loss functions is not well justified. For example, it is not clear what the performance could be if other more popular ones are used, such as L2 and L1 loss.\n- Model complexity comparison in addition to the running time in Appendix D should be provided.\n\nReferences:\n\n[1] D. Huo et al. Blind Image Deconvolution Using Variational Deep Image Prior. IEEE Transactions on Pattern Analysis and\nMachine Intelligence, 45(10): 11472-11483, 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable comments. Please see below for our responses.\n\n**[W1]** *The presentation of this manuscript should be improved.* \n\nThanks for the feedback, we will improve the organization by moving some parts of Sec. 1.2 into Sec. 3.\n\n--- \n\n**[W2.1]** *Comparison with Recent Self-Supervised BID Method VDIP [Dong et al., PAMI, 2023]:* \n\nThanks for the reference. Please see Table 1 for the comparison in PSNR(dB)/SSIM over 4 datasets. The results show that ours generally outperforms VDIP on all datasets, especially on real-world datasets. We also present visaul comparison in our attanched PDF file. It can easily observe that the proposed method can better restore the blurry images.  Besides, we compare the model complexity between two in Table 2, which shows that our method has significant smaller memory usage and model size. \n\n*Table 1: Average PSNR(dB)/SSIM of the results from VDIP and Ours* \n| Method        | Lai et al.’s Dataset | Köhler’s Dataset | Levin et al.’s Dataset | Microscopic Dataset|\n|:-------------:|:------------------:|:----------------:|:----------------------:|:-------------------:|\n| VDIP   | 25.12 / 0.869      | 29.58 / 0.922    | 33.12 / 0.929          |    37.10 / 0.937 |\n| Ours    | **25.16 / 0.879**      | **30.69 / 0.942**    | **33.74 / 0.938**          | **38.25 / 0.948** |\n\n*Table 2: Model complexity of VDIP and ours*\n| Method| Params (k) |Memory Usage (GB)|Running Time (s)|\n|:-------------:|:------------------:|:----------------:|:----------------------:|\n| VDIP   |3523.2|9.19|245.04|\n| Ours   |2342.4|1.82|213.02|\n\nWe will include this comparison in the revision.\n   \n--- \n\n**[W2.2]** *The cross-scale supervision training strategy can also be applied to other self-supervised methods by with simple downscale operation:*\n\nThanks for comments. The study you suggested is already done in the ablation study from lines 276-280, \"INR (coordinate NN) vs. MLP/CNN (image-to-image NN),\" with results reported in the \"MLP/CNN\" row of Table 5 of the paper. In this study, we replaced INR with a CNN (image-to-image) and used bi-linear down-sampling for the cross-scale loss, which actually is what you suggested for the existing work \"SelfDeblur[50]\". The results indicate its much worse performance than the  original single-scale version of \"SelfDeblur[50]\" with PSNR: 18.19 (cross-scale) vs. 20.97 (single-scale).  It shows that, the resolution-free property of INRs is crucial for effectively exploiting the potential of cross-scale consistency loss. We will clarify this in the revision.\n\n--- \n\n**[W2.3]** *The use of the loss functions is not well justified. For example, it is not clear what the performance could be if other more popular ones are used, such as L2 and L1 loss:* \n\nThe metric used in our work is not the focus. Thus, our method employs SSIM Loss, aligning with the most relevant previous works, such as MCEM[32], SelfDeblur[50], and VDIP. The latter two initially trained the NN with L2 loss and switched to SSIM loss after 500/2000 iterations. See Table 3 below for experiments replacing SSIM Loss with L2 Loss, which shows around a 1dB decrease in PSNR across different methods.\n\n*Table 3: Average PSNR(dB)/SSIM of the results from different methods on the Lai et al.'s dataset.*\n   | Loss Function | SelfDeblur  | VDIP  | MCEM  | Ours |\n   |:-------------:|:----------:|:--------:|:----:|:----:|\n   | L2 Loss       | 20.97/0.752| 23.97/0.818 | 23.21/0.791 | 24.28/0.827 |\n   | SSIM Loss     | **22.39/0.793** | **25.16/0.869** | **24.55/0.800** | **25.16/0.879** |\n\n\n--- \n\n**[W2.4]** *Model Complexity Comparison:* \n\nThanks for the suggestion. Please refer to the global response for the response. \n\n\n--- \n\n**[Q1]** *To my knowledge, the performance of fully supervised methods highly depends on the training data. So I would like to know how the supervised methods were trained on the Lai, Kohler, and Levin datasets, since only test images are contained in them:*\n\nPlease refer to the global response for the details.\n\n--- \n\n\n**[Q2]** *The method is constructed based on the uniform blurring assumption, and thus it should be discussed why it can perform well on the Kohler dataset which focuses on non-uniform blurring:*\n\nThe Köhler dataset is generated by taking pictures of an image (the ground truth) posted on a board. The degree of non-uniform blurring is not severe compared to the overall blurring degree. Uniform blurring can provide a reasonably effective approach to handling its blurring effect. The experiment on the Köhler dataset is commonly used to test the robustness of uniform deblurring methods on images with modest non-uniform blurring. We will clarify this in the re vision to prevent any potential confusion.", "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": "wynwVl3qUk", "reviewer": "Reviewer_PNcB", "summary": "This paper introduces a self-supervised method for BID that does not require GT images. By leveraging an exact relationship among the blurred image, latent image, and blur kernel across consecutive scales, this paper propose an **effective cross-scale consistency loss** implemented by representing the image and kernel with **implicit neural representations** (INRs), whose resolution-free property **enables consistent yet efficient computation for network training at multiple scales**. The experimental part  verifies that proposed method outperforms some existing self supervised methods on several datasets.", "questions": "- There are some confusing points in Tab.1: (1) The results of the supervised method are of little reference value for they are trained on non-convolutional modeling datasets(GoPro). (2) According to the ablation experiment, even without the main contribution (1) mentioned in line 73, the PSNR(24.44) is almost the same as the second best method(MCEM) (24.55). The uniform blur setting is inconsistent with some existing methods (such as the non-uniform modeling used by MCEM). Are those self-supervised method retrained for comparison? \n- The proposed uniform blurring method is better than MCEM in the non-uniform deblurring experiment (Tab.2), which is a bit contradictory. Is this an advantage brought by individual sample training rather than the method itself?\n- In the multi-scale training process, what is the effect of x and k estimated in the previous scale on the next scale? From the image, it can be understood that coarse-to-fine is getting clearer. How does k change in this process? In addition, in Tab.5, the average PSNR of training only at the original scale is two points higher than that of training with at three scales, indicating that the optimization direction of BID at different scales is inconsistent. In this case, why does training with progressive scales bring better performance?\n- According to the existing conclusions in the BID field in lines 94-98 of the paper, the second contribution point of the paper summarized in lines 107-108 is not important. In addition, the use of multi-scale coarse-to-fine can alleviate the ill-posed problem of multiple solutions. There is no analysis or literature references provided though some work has been done before indeed.\n- In Lines 191-192, it claims the role of the cross-scale estimation consistency is to limit the ambiguous solution of BID. Though a rational analysis of the approach and ablation experiments are provided, readers may wonder what is the motivation for designing this regularization term and its actual impact in BID process. If there is some intuitive analysis or explanation, it will be easier for them to accept.\n- For the standard downsampling implemented by the coordinate index in line 165, is interpolation  need  when the image size is not even? Is the downsampling used in the ablation experiment to compare image-to-image NNs done in the same way or with more common interpolation methods such as bicubic or bilinear?", "limitations": "The authors have  adequately addressed the limitations and potential negative societal impact of their work as follows:\n- Computational cost for processing a large number of images, as the method requires training the model for each individual sample.\n- The proposed method is only applicable to handle uniform blurring, as it relies on the convolution model. \n\nConsidering the practical application value and the single-image training and reasoning method proposed, the paper should compare the computational cost of the single-image reasoning results and the comparison with existing methods.\n\nAs mentioned above, it is necessary to add discussions about different setting and it is confusing that proposed method performs better than MCEM in non-uniform setting.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper proposes a new method on the existing self-supervised niform BID task, which is complete and sound. The main contribution is the cross-scale estimation consistency constraint, and provides a complete theoretical proof of the rationality of the regularization term. The cross-scale consistency is used to train progressive scale IRNs from convergencing to trivial solutions.  which has a certain originality and provides an inspiring new analytical way for BID.\n- The paper conducts in-depth theoretical and experimental analysis on uniform-based deconvolution, verifies the superiority of the proposed method through experimental comparison on two types of motion blurred image synthetic data and real data, and combines sufficient ablation experiments to verify the impact of each module of the framework on performance.\n- Aside from a few details, the submission is clearly written and generally well organized, and the supplementary materials are accompanied by a large amount of quantitative and visualization materials.\n- The authors are careful and honest about evaluating both the strengths and weaknesses。", "weaknesses": "- The INRs framework and progressive learning mechanism on which the paper is based are common in image restoration, the degree of innovation is average except for cross-scale estimation consistency constraint.\n- With the development of BID or the larger blind image restoration area, the degradation settings of different methods are somewhat different. When describing the paper’s setting (Eq.1), in addition to comparison methods, introducing the existing blurring degradation settings (uniform kernel/non-uniform kernel/implicit modeling) and their differences and application value is necessary. The discussion of this part is missing, which causes readers/researchers confused by different method settings and applicability.\n- The related work section of the paper does not fully cite and introduce recent learning-based self-supervised or frequency estimation degradation and some important works, such as KernelGAN(NeurIPS2019), FCA(AAAI2021), S2K(NeurIPS2021), etc. These solutions also involve deconvolution and propose estimation methods.\n- Some descriptions of the method principles or experimental settings are vague or problematic, see Questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the comments. Please see below for the responses.\n\n**[W1]** *The INRs framework and progressive learning mechanism on which the paper is based are common. The degree of innovation is average except for cross-scale estimation consistency constraint:* \n\nWe agree that the cross-scale consistency constraint is our main contribution. Our innovation with INRs lies in being the first that exploits their resolution-free property for image restoration. It is critical for utilizing cross-scale consistency, as shown in our ablation study. Direct down-sampling is ineffective. Our second contribution is progressive training, which, though established in other areas, is first effectively implemented in self-supervised deep learning for BID using INRs.\n\n---\n**[W2]**  *The discussion of degradation settings of different methods is missing* \n\nThanks for the suggestion. There are typically 3 settings: uniform blurring, non-uniform blurring of static scenes, and non-uniform blurring of dynamic scenes with moving objects. Due to space limitations, we focus on uniform blurring. In the revision, we will add more details on the methods regarding their degradation settings and practical applications.\n\n---\n**[W3]** *The missing reference, such as KernelGAN(NeurIPS2019), FCA(AAAI2021), S2K(NeurIPS2021), etc. They also propose estimation methods:* \n\n\nThanks for the references. These aim for super-resolution (SR), not blind deblurring. While both estimate the blur kernel, they have different inputs: low-resolution images (SR) versus high-resolution images (BID). They also deal with different blurring kernels: approximately isotropic (SR) versus strongly anisotropic (BID). While they might benefit each other in basic ideas, the techniques in one cannot be easily applied to the other. We will add these references in the revision.\n\n---\n**[Q1]** *The results of the supervised method are of little reference value for they are trained on non-convolutional modeling datasets:* \n\nPlease refer to the global response for the response. \n\n---\n**[Q2]**  *Even without the contribution (1) mentioned in line 73, the PSNR is almost the same as the second best method (MCEM):*\n\nNote that we have 2 contributions: (1) cross-scale consistency constraint and (2) progressive multi-scale training. Without progressive training, the performance is 2dB lower than MCEM. With only progressive training, ours does not surpass MCEM either. The cross-scale consistency constraint provides a further 0.7 dB gain, letting ours to noticeably outperform MCEM. This justifies the importance of combining both (1) and (2).\n\n---\n**[Q3]** *Are those self-supervised method retrained for comparison?* \n\n Most referenced self-supervised methods do not use pre-training; they train a NN directly on the test sample to get results. MCEM uses a piece-wise convolution model with codes for both non-uniform and uniform blurring. The results are either cited from their papers (SelfDeblur[50] and DEBID[11]) or generated using their codes (MCEM[32]).\n\n---\n**[Q4]** *Why the proposed one outperforms MCEM, and is this an advantage  by individual sample training rather than the method itself?* \n\nBoth MCEM and ours train individual NNs on each sample. The Köhler dataset's non-uniformity is modest and can be approximated with a convolution model. For general non-uniform datasets like Lai et al.'s, our method does not outperform MCEM. We will clarify this in the revision. \n\n---\n**[Q5]** *Regarding the effect of x and k:* \n\nThe estimation, $x$ and $k$, from the coarser scale, provides the initialization of the finer scale. Like  the image,  the kernel gets closer to the truth as the process progresses. Please refer to Appendix J for details.\n\n---\n**[Q6]** *Regarding why training with progressive scales bring better performance:*\n\nThe loss function at the coarser scale emphasizes lower frequencies since the image at coarser scale retains low but loses high frequencies. Thus, a NN trained on the sum of loss functions across 3 scales focuses more on low frequencies than one trained only at the finest scale, which fits both low and high frequencies. This motivates our scale-progressive training scheme over the joint multi-scale loss. In progressive training, the estimate from the coarser scale serves as a good initialization for the estimation at the final scale. The final result is achieved by using only the fitting term at the finest scale. \n\n---\n**[Q7]** *Regarding the importance of the second contribution, and the references:*\n\nWhile the coarse-to-fine approach has been effective in traditional iterative methods (e.g.[1,2]), we are the first to apply it in deep self-supervised learning for BID. Direct down-sampling in deep learning is ineffective, as shown in Table 5. Our contribution is using resolution-free INR to implement this coarse-to-fine scheme successfully.\n    \n[1] Li et al. \"Unnatural l0 sparse representation for natural image deblurring\". CVPR, 2013\n\n[2] Yang and Ji. \"A variational EM framework with adaptive edge selection for blind motion deblurring\". CVPR, 2019.\n\n---\n**[Q8]** *Regarding the role of the cross-scale estimation consistency:*\n\nFirst, the cross-scale consistency provides additional constraints for regularization. Secondly, it ensures that the estimate from the coarse scale is more accurate than using the existing one,  ${x} \\otimes {k}) \\downarrow_2 \\neq {x} \\downarrow_2 \\otimes {k} \\downarrow_2$, which provides a better initial estimate for the final scale. \n\n---\n**[Q9]** *Regarding the interpolation and down-sampling*\n\nIn our INR-based approach, interpolation isn't needed for uneven image sizes, as INR uses a coordinate NN providing a resolution-free kernel/image. For comparison experiments on image-to-image NNs, we used bilinear downsampling to compute consistency loss and generate low-scale images. We will clarify this in the revision.\n\n---\n**[Limitation]** *Computational cost of reasoning* \n\nPlease refer to the global response for the reply.", "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": "ndcCdgpRT5", "reviewer": "Reviewer_UEmh", "summary": "This paper proposes a blind image blurring method which reparametrizes the latent images the blurring kernel by the implicit neural representations (INRs). In addition, the authors also propose a cross-scale consistency loss. The authors validate the effectiveness of their method on several datasets.", "questions": "I only see very limited novelty of this paper. The proposed idea is similar to the one in Ref1. The only contribution I can see is the cross-scale consistency loss. I list the details in the [Weaknesses].\n\nRe1: Zhuang, Zhong, et al. \"Blind image deblurring with unknown kernel size and substantial noise.\" International Journal of Computer Vision 132.2 (2024): 319-348.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The cross-scale consistency loss sounds an interesting idea.\n2. The presentation of this paper is good.", "weaknesses": "1) The idea of using a deep neural network for latent images and blurring kernel reparametrizatrion is not new. The current proposed method is very similar to the one in ref1. But I do not see the authors have discussed the differences between the proposed method and the method in Ref1. Thus, I doubt the novelty of this proposed method.\n\n2) The current design, the authors use different network architectures to generate images and kernel. But how to control or balance the complexity between latent images generator and the kernel generator?  The authors need to make it more clear why use these specific network design for images and kernel.\n\n3) If the batch size is 1, will it be better to use other norms instead of batch norm?\n\n4) The authors should also compare their method with the one in Ref1 as the Ref1 seems to be the more recent one compared to other self-supervised learning method.\n\n5) The authors should also show how the generated kernel looks like. Is the generated kernel feasible? Is it close to the ground-truth kernel?\n\nRe1: Zhuang, Zhong, et al. \"Blind image deblurring with unknown kernel size and substantial noise.\" International Journal of Computer Vision 132.2 (2024): 319-348.\n\n\n=========\nscore has been revised after reading rebuttals.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks the reviewer for the comments. See below for our responses to the concerns and questions.\n\n**[W1]** *The idea of using a deep neural network for latent images and blurring kernel reparametrizatrion is not new. The current proposed method is very similar to the one in ref1. But I do not see the authors have discussed the differences between the proposed method and the method in Ref1. Thus, I doubt the novelty of this proposed method.*\n\nThanks for the reference. We never claimed that using NN-based representation for image/kernel is our  contribution. NN-based image/kernel representation started with DIP, and there has been an extended list of such self-supervised methods for image restoration, including [1]. Our main contribution is to further improve the performance of such an approach for BID by introducing a self-supervised cross-scale consistency loss, along with a progressive training scheme, both built on the resolution-free property of INRs.\n\nOur work is very different from [1]. While both utilize INRs, the main differences include:\n(a) We propose a cross-scale consistency constraint for regularization of BID, while [1] uses L1/L2 norm regularization, early stopping, and other tricks for regularization.\n(b) [1] does not consider a multi-scale-based training scheme, whereas we propose a multi-scale progressive training scheme.\n(c) The INP is used the same way as CNN in [1], while we utilize the resolution-free property of INPs for implementing the proposed constraint and progressive training.\n\n[1] Zhong et al. \"Blind image deblurring with unknown kernel size and substantial noise\". IJCV, 2024.\n\n--- \n\n**[W2]** *The current design, the authors use different network architectures to generate images and kernel. But how to control or balance the complexity between latent images generator and the kernel generator? The authors need to make it more clear why use these specific network design for images and kernel:* \n  \nSimilar to most existing works on self-supervised BID,  our approach employ different architecture and model sizes for the kernel and image. Given that image size is generally much larger than kernel size, the model size for representing an image needs to be considerably larger than that for a kernel. Therefore, within our coordinate NN, we use a U-Net with parameter no. 2292(k) for the image, and a three-layer MLP with parameters no. 50(k) for the kernel. This design choice is consistent with previous methods [2, 3].\n\n[2] Ren et al.  \"Neural blind deconvolution using deep priors.\"  CVPR, 2020.\n    \n[3] Chen et al. \"Self-supervised blind image deconvolution via deep generative ensemble learning.\"  TCSVT, 2023\n\n--- \n\n**[W3]** *If the batch size is 1, will it be better to use other norms instead of batch norm?*\n\nThank for pointing it out. We agree that the usefulness of batch BN becomes limited when the batch size is 1, which also happens in existing works [2,4,5], We initial experiments on adopting LayerNorm shows no impact in the results.  We will elaborate it more in revision.\n\n[4] Li et al. \"Self-supervised blind motion deblurring with deep expectation maximization.\" CVPR, 2023. \n\n[5] Ulyanov et al. \"Deep image prior.\" CVPR. 2018.\n\n--- \n\n**[W4]** *The authors should also compare their method with the one in Ref1 as the Ref1 seems to be the more recent one compared to other self-supervised learning method.* \n\nThere is no code available in the public domain for [1], and we also didn't receive the code for evaluation after  contacting the authors of [1]. Additionally, the experiments conducted in [1] do not include any dataset with the same configuration as evaluated in this paper. Considering the complex regularization used in [1], it is challenging to make a fair comparison.\n\n--- \n\n**[W5]** *The authors should also show how the generated kernel looks like. Is the generated kernel feasible? Is it close to the ground-truth kernel?* \n\nWe would like to point out that, we have included visual results of the generated kernels in the Appendix, displayed in the top-left corner of the corresponding images. The estimated kernels are very close to the ground-truth kernels. Please refer to our Appendix, specifically Fig. 1--Fig. 12, for these visual comparisons.", "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:
Dear AC and reviewers,

We sincerely appreciate the reviewers for their constructive comments, as well as their time and effort in evaluating our manuscript. 
Please find below our clarifications on some common concerns and questions.

--- 

 **Main contributions**

Our work presents two main contributions:

+   A novel cross-scale consistency constraint for regularizing self-supervised BID.
+   A progressive coarse-to-fine training scheme to effectively alleviate overfitting of NN caused by solution ambiguities.

The main differences between the proposed one and existing works are:

+ The usage of Implicit Neural Representation (INR) for representing kernels/images is not our main contribution, our INR-related contribution is the first to exploit the resolution-free property of coordinate-NN for image restoration. This resolution-free property is crucial for the effectiveness of our proposed constraint and training scheme. Our ablation study demonstrates that direct down-sampling on image-to-image NN does not work for the proposed constraints and training scheme.

+   While there are existing works on multi-scale INR for image processing, their aims and focuses are different. For general image representation, the multi-scale approach focuses on NN design for compact representation, rather than on training or loss functions. In super-resolution (SR), the task differs significantly from BID, although both involve kernel estimation.

+    Progressive training has also been employed in traditional iterative methods, but our work represents the first implementation in a deep learning-based approach. A straightforward downsampling-based generalization to image-to-image NNs does not work; the resolution-free property of INR is crucial to its success.


Although our method is specifically designed for BID, the concept of utilizing the resolution-free property of INR has potential applications in developing  multi-scale approaches to other  image restoration tasks.

--- 

**Comparison to supervised methods**

We sincerely thank the reviewers for pointing this out. As this paper's focus  is on developing a powerful self-supervised method, our experiments focus on the performance gains  of our method achieves over existing self-supervised methods. Consequently, we have directly cited the results of supervised methods from related works [21, 32] for consistency. These results were obtained from the models trained on the GoPro dataset [40], which covers non-uniform blurring of dynamic scenes.

We agree that the performance of a supervised method will be impacted  by training data. In response, we have re-trained two representative supervised methods, Restormer [73] and MPRNet [74], on a dataset with only uniform blurring, the BSD-D dataset provided by "Real-blur" [1]. This dataset contains 20,000 image pairs generated by 40 synthetic uniform motion blur kernels using the method from [Schmidt et al. CVPR'16]. Please refer to Table 1 below for the results.


*Table 1: Average PSNR/SSIM of the results from compared supervised methods and Ours*
| Method     | Training data| Lai et al.’s Dataset | Köhler’s Dataset | Levin et al.’s Dataset | 
|---------------|--------------------|--------------------|------------------|------------------------|
| Restormer[73]    |Non-uniform |16.31 / 0.474      | 27.61 / 0.828    | 27.38 / 0.838          |    
| Restormer*[73]    |Uniform| 18.89 / 0.555      | 28.25 / 0.852    | 30.30 / 0.896          |    
| MPRNet[74]    |Non-uniform |    16.15 / 0.454   |  26.32 / 0.827  | 26.21 / 0.795         |     
| MPRNet*[74]    |Uniform|     18.42 / 0.531   |  27.91 / 0.848  |     28.35 / 0.850      | 
| Ours    |- |**25.16 / 0.879**      | **30.69 / 0.942**    | **33.74 / 0.938**          

It can be seen that a supervised method trained on a dataset with only uniform blur performs better on testing data with uniform blur, when compared to one trained on the dataset with non-uniform blurring. However, this improvement is still not enough to match the performance of our method. The main reason is that existing supervised methods aim to handle general blurring and thus they do not utilize the physics prior of image formation, specifically the convolution model for uniform blurring. Consequently, they do not perform as well as our self-supervised method, which leverages this physics prior. 

--- 

**Comparison of Model complexity & computational cost**

We would like to first point out that all existing self-supervised BID methods train an individual NN for each sample for inference. There is no model pre-training time. Please see Appendix D for the comparison of inference time among different methods. See also Table 2 below for the comparison  of different methods when processing a  image of size $256\times 256$ with a  blur kernel size $31\times 31$. The results show that while all methods have similar running times and model sizes, MCEM and ours have noticeably smaller memory usage.

*Table 2: Model complexity comparison*
| Methods | SelfDeblur [50]  | MCEM [32] | Ours  |
|-|-|-|-|
| Running time (s) | 219.71 | 226.33| 213.02|
| #Params (k)| 3427.2| 2409.0| 2342.4|
| Memory usage (GB)| 3.61 | 1.27  | 1.82  |

We will add these results in the revision.

[1] Rim et al. "Real-World Blur Dataset for Learning and Benchmarking Deblurring Algorithms", ECCV, 2020.

Meta-review: This paper receives two borderline accept and two weak accept. This paper proposes a self-supervised method for blind image deconvolution. In the rebuttal, the authors have adequately addressed most of the reviewers' major concerns.  A more detailed motivation for why INR and the progressive cross-scale can benefit self-supervised learning is needed in the revised version.
Decision: Accept (poster)