DEALing with Image Reconstruction: Deep Attentive Least Squares
Authors: Mehrsa Pourya
Erich Kobler
Michael Unser
Sebastian Neumayer
Venue: ICML 2025 poster (2025)
Keywords: Inverse problems
data-driven priors
deep equilibrium
iterative refinement
spatial adaptivity.
Primary area: optimization->everything_else
TL;DR: DEAL is a novel image reconstruction framework that bridges traditional regularization and deep learning.

Abstract: State-of-the-art image reconstruction often relies on complex, abundantly parameterized deep architectures. We propose an alternative: a data-driven reconstruction method inspired by the classic Tikhonov regularization. Our approach iteratively refines intermediate reconstructions by solving a sequence of quadratic problems. These updates have two key components: (i) learned filters to extract salient image features; and (ii) an attention mechanism that locally adjusts the penalty of the filter responses. Our method matches leading plug-and-play and learned regularizer approaches in performance while offering interpretability, robustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a principled reconstruction approach.

DEALing with Image Reconstruction: Deep Attentive Least Squares
Mehrsa Pourya' Erich Kobler
Michael Unser'
Sebastian Neumayer
Abstract
Both terms are balanced by the hyperparameter À Rzo- Throughout this paper. x is the vectorized version of (color or grayscale) image of size (N, H x W).
State-of-the-art image reconstruction often relies complex, abundantly parameterized deep ar- chitectures. We propose alternative: data- driven reconstruction method inspired by the clas- sic Tikhonov regularization. Our approach itera- tively refines intermediate reconstructions by solv- ing a sequence of quadratic problems. These up- dates have two key components: (i) learned filters extract salient image features; and (ii) atten- tion mechanism that locally adjusts the penalty of the filter responses. Our method matches leading Boug-and-ple and learned regularizer approaches in performance while offering interpretability, ro- bustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a prineipled reconstruction approach. Code: https://iitub..oom/mmrsrpapp
From classic signal processing to the advent of deep learn- ing, substantial research efforts have focused on designing regularizers R. In the context data-driven methods, two primary approaches have emerged: (i) the explicit modeling of R: and (ii) the modeling of operators associated with R. such as its proximal operator, which is required in plug-and play (PnP) reconstruction (Venkatakrishnan et al. 2013). Following the explicit approach, the starting point for this work is the fields-of-experts (Roth & Black 2009). which
reads
No
Rm X 7
mes Ve (Wxx)).
(2)
c=1
Here, each W. E RHWxd convolves x with the filter template we E MM. where K denotes the spatial kernel size. Then, the nonnegative potentials va E C(R) are applied entry-wise to the W.x. Finally, the weights m. E [eM, 1]HW with éM >> 0 determine the (spatially varying) contribution of vi( W.)). Every component of (2) can be learned. Most implementations to date use m, 1. They differ in the parameterization of the and W. for the learning process (Chen et al.) 2014] Goujon et al.) 2024 Zach et al. 2024) or include nonlinear feature transforms (Li et al. 2020 Kobler al. 2022). While the learning of W. and d/. have been studied extensively, the local weights m. have received little attention far.
1. Introduction
Image reconstruction plays a fundamental role in compu- tational imaging and computer vision (McCann Unser 2019. Zeng 2001). The task to recover an unknown im- age of interest
Rd from noisy measurements y RM Their relation is modeled as y Hx, where the forward op- erator H E Ie M encodes the acquisition process. When H is ill-conditioned, one can resort to the regularized recon- struction
The use of spatially varying m, in is referred to as anisotropic regularization. For instance, m. M(y) can derived from the data y using heuristics (Chan et al. 2008 Grasmair & Lenzen 2010) or a neural network (Kofler ct 2023 Lefkimmiatis & Koshelev (023). When M extracts features from an estimated reconstruction, it is nat- ural to consider an iterative refinement of m Specifically, the reconstruction from ① can be fed back into M to ob- tain an improved m. for This leads to the attentive- reconstruction process
& E arg min (IIXX y112 +R(x)).
(1)
XERM
The data fidelity |Hx-yli controls the consistency of the reconstruction with the measurements. The regularizer R: Rd + R20 encodes prior information about the solu- tion and is also intended to make the problem well-posed.
'Biomedical Imaging group. EPFL Lausanne, Switzerland "Institute for Machine Learning, JKU Linz, Austria FFaculty of Mathematies, TU Chemnitz, Germany. Correspondence to: Mehrsa Pourya <echrsa.pourreeeeooeeeeea pourya@ppfl..h.
(I 1 XR)( (3)
X arg min XER
Proceedings of the nnnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
studied by Pourya et al. 2024 for t/e = I. We note that their scheme computationally demanding since each of

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DEAL for Image Reconstruction
their updates requires one to solve a problem that involves least absolute shrinkage and selection operator (LASSO).
denoisers lead a convergent scheme, but it is difficult to en- force these conditions in the learning (Reehorst & Schniter 2018 Hertrich et al., 2021). Recently, weaker conditions have been proposed by Pesquet et al. (2021): Hurault et al. 2022b).
Contribution Our contributions are as follows.
We simplify the updates 3 by choosing ve(+)) - (-) for the R from (2) Then, each update amounts to the solution of a linear equation, which we handle efficiently through the conjugate-gradien method.
Spatial Adaptivity An overview of spatially adaptive reg- ularizers (2) is given in Pragliola et al. (2023). Meanwhile, Hintermüller et al. (2017); Van Chung et al. 20177;; Koffer et al. (2023) focus on the total-variation (TV) regularizer Ruudin al. [992] as a special case, using either heuristics or deep learning to compute the m.- More general instances are considered by Lefkimmiatis & Koshelev (2023); ): Neu-
On the theoretical side, we establish the uniqueness of each update in (3) the existence of fixed point, a condition for the convergence of (3) and a stability result for the resulting reconstruction operator.
mayer et al. (2023); Neumayer & Altekrüger (2025). p. The first work deploys non-smooth potentials th and majoriza- tion minimization to solve the problem in .I. Similar to 33 this leads to series of quadratic problems. The latter works deploy differentiable the and solve (I) with acceler- ated gradient descent. All approaches have in common that they update the weights m, only once. In particular, they do not refine m, and the reconstruction iteratively as in
We learn the filters { [ nism M for M through denoising in such a way that the iterations (3) converge. Then, given an inverse problem with forward H. we deploy our method by tuning only two scalar hyperparameters: (i) the noise level; and (ii) the regularization strength A.
Nc. and the attention mecha-
Our experimental evaluations show results on par with state-of-the-art methods for various inverse problems.
Iterative Refinement Lenzen et al. (0144): Lenzen & Berger (2015) propose to iteratively refine the weights m. for TV and update them using some heuristic. For the gen- cral model (2) with U. = I. a refinement based on neural networks was considered by Pourya et al., (2024). Their M has a simple architecture comparable to ours. Iterative re- finements are also found outside of this setting, for example, in Saharia et al. (2023) for superresolution and in Darestani et al. 2024) for magnetic resonance imaging (MRI).
We empirically demonstrate the convergence and ro- bustness of our method. We also provide visual in- terpretations of the learned attention mechanism. Fig- is particularly striking: Each pixel of the denoised image is a weighted average of the noisy data, with the weights being adapted to the image structure. This qualitative aspect goes beyond numerical metrics and opens the door to well-performing and explainable image reconstructions.
Nonlocal Laplacians Quadratic potentials te lead to a symmetric positive semi-definite system as the optimality condition for (3) Another approach to such updates is (iterative) filtering with carefully designed graph Laplacians (Pang & Cheung, 2017). Recently, this was incorporated into deep architectures for image denoising (Zeng al.
2. Related Literature
Learned Regularization Classic regularizers for leverage sparsity in various domains, such as image gra- dients (Rudin et al. 1992) or wavelets (Mallat 1999). The parametric model introduced by RRohh & Black (2009)
2019 Valsesia et al. 2020 and scene flow (Teed & Deng
2021
has since spurred extensive research. Key areas of investiga-
Attention Mechanisms Originating in natural language processing (Vaswani et al., 2017), attention is now widely used in imaging (Dosovitskiy et al. 2020). Two main ap- proaches exist: (i) images are divided into patches and pro- cessed through scaled dot-product attention, as in SwinIR and Restormer (Liang et al.) [02T]] Zamir et al. (022): and (ii) attention is integrated into the architecture in the form of pointwise multiplications,
tion include learning paradigms (Chen et al. 20144] Efiland
let al. 2020). parameterization strategies (Zach et al. 2024).
and intrinsic properties like convexity (Goujon et al. 2023
2024). More complex architectures build upon strategies such as autoencoders (Li et al. 2020), algorithm unrolling (Kobler et al. 2022). adversarial training (Lunz al. 2018
Prost et al. 2021), and energy modeling (Zach et al. 2023
in attention ceaitiionnguided convo- lutional neural networks (CNN) (Tian et al. 2020). While patch-based methods effectively capture long-range depen- dencies, they introduce a significant computational cost due to the need to determine inner products across all patches. In contrast, pointwise attention more economical.
In parallel, implicit regularization methods were developed. Here, off-tth-sheelf denoisers have been incorporated into iterative reconstruction algorithms in place of the proxi-
mal operators /Venkatakrishnan et al. 2013 Romano et al.) 2017 Zhang et al. 2022). Non-expansive or homogeneous
2

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DEAL for Image Reconstruction
3. Methodology
normalization. The data y and the forward H are problem- specific inputs that not learnable. We solve (8) by the conjugate-gradient (CG) algorithm with X as the initial guess. We use batched CG with at most Kin steps, where each sample terminates individually if its residue satisfies that |gxx++1
We specify the regularizer
using quadratic potentials as
No
Rm (x) = > m (Wx)))) = ( | (4)
bl2 S Ein for 6m O.
with the shorthands W- [ WJ and M Diag(my (m1; )Ne), where Diag returns a diagonal ma- trix whose diagonal entries are the argument vectors. This
3.1.2. MASK- MASE-GENNRA BLOCKK
To estimate M for (5) from local image structures, we use the shallow CNN with learnable nonlinearities
leads to a quadratic reconstruction problem with solutions
M(x) (Ø" W Wiix olp2 o W mix aipi o W (mass) (9)
(IIHX x112 ++II/M>x1))
& E arg min xERd
(5)
(see Figure | Right). Here, a denotes the model noise level and needs to be tuned outside the training regime. This choice of architecture inspired by anisotropic diffusion (Weickerl 1998, Bredies & Lorenz, 2018). where M is
The question then arises as how properly choose M and W. Regarding filter-based interpretation, W should be independent of y, while M modulates the response Wx at each location. Ideally, this modulation ought to depend on the structure of the (unknown) &. With abuse of notation, we introduce M: I + [eM, 1] NeJWW which leads to our attentive refinement scheme
typically computed from the gradients of a smoothed image using pixel-wise nonlinearities.
The first multi-conv layer W mask (see Section 3.1.3 in
extracts Ne spatial features using the same architecture W does in the reconstruction block. The two subsequent convo- lution layers W mix and W mix mix the Ne feature channels using kernels of size (3 3) without bias. The layers are connected via learnable pointwise nonlinearities you and y22, for which we follow Bohra et al. (2020). Specifically, each On is parameterized as a linear spline with N, equally dis- tributed knots on [0, r]] On (r, oc) we extend the splines linearly and enforce symmetry by setting ((x) (0(z) if æ < addition, we constrain them to be increasing for > 0. (The removal of either constraint has not led to significant performance improvement.) To guarantee the numerical stability of (8), the output of M must remain in (EM, 1]. Hence, we process each channel c E (1....,ccl individually as
Xk+1 E <((x,,)) y)
with Xg = 0 E Rdd
(6)
y(z,y) = arg min XERd
[Hx - -yl1--1 A1
The update 66) can be interpreted an infinite-depth neural network. Hence, we name it deep attentive least squares (DEAL) for image reconstruction. If x2 - 8, we get that & E y), namely, that X is a fixed-point of the operator We restrict < Kout and terminate the iterations (6)
77
when X6+11 - xs122 ||x I Cout with Cout > O.
3.1. Architecture
Next, we specify how the DEAL iterates (6) can be cast as a deep neural-network structure. [hoocesss the interplay of its essential building blocks (reconstruction and mask generation), which repeatedly exchange information.
(x) mmx(min((s((c(())))). 1).
(10)
where 53 is a symmetric linear spline as before. In contrast to the former splines, yP must be non-increasing on [0, oo). The underlying rationale is that M(x) should be close to 1 for small filter responses (constant image regions) and close to 0 for strong filter responses (salient edges). Following Goujon et al. (2024), we enable multilevel noise training using the positive scalings
3. RECONSTRUCTION BLOCK
At the heart of DEAL, the reconstruction block solves the spatially adapted optimization problem in (7) for given at- tentive weights M(x;) (see Section 3.1.2). The optimality
(o et((d) -(0010000) 10-9)
(11)
condition for the problem in is given by the linear system
with learnable linear splines N= By design of M, the first reconstruction block consists of a non-varying problem.
A = b,
(8)
with A = H M(xx)² and b HTy y. The multi-convolution block W (see Section 3.1.3) is learn- able. The hyperparameter À E R needs to be tuned for inverse problems outside the training regime. To avoid scaling ambiguities, we impose ||]|2 = by spectral
3.1.3 3. MULTI-CONV BLOCK
The Multi-Com block advocated by Goujon et al. (2023) consists of multiple convolution layers without nonlineari- ties in between. It enables the efficient construction of large
3

---

DEAL for Image Reconstruction
x
0
Mask-Generation Block
DEAL
X1+11
Mutti Conv Block W Mask
0
Mask-Generation Block
No x H
H: Forward Operator
M(xx)
y
Reconstruction Block
(3 Conv W
mi
(H H XW M(x.) H
Nr. x If
x
No
(3 3) Conv W
Stop Condition
#x
Yes
[+2 Tm
M(x)
Figure 1: Left: DEAL generates sequence of reconstructions X via 66) from the inputs y and H. initalization xo = 0, and hyperparameters a and A. When the stop condition is met, it returns & Right: Architecture of the mask-generation block.
receptive fields. We use three convolution layers, all with kernels of size (9 x 9). Thus, the effective field of view for this block is (25 x 25). For inputs with Ni channels, the number of output channels are 4.N.,. 8N.. and Ne af- ter each of these successive convolution layers. We set the group size and stride to one, and do not use bias. In all our experiments, we use No = 128. This block appears in two places: once in M(x) as W Mask and in the reconstruction block as W (i.e., the convolutions
with y = 10- This loss consists of three parts: (i) a squared error that enforces that the output matches the clean image; (ii) a squared penalty on the weight changes for the last two updates of (6); and (iii) accumulated second- order TV regularization of all learnable splines. The latter penalizes changes in the slopes of the splines and thereby promotes simpler splines (Bohra et al. 2020; Ducotterd et al. 2024). The second term in (2) vanishes if the gener- aled weights M(x²) converge. To promote convergence of 6 a fixed point, we sample Kout uniformly from [15, 60]
3.2. Training
(Anil al. 2022).
We learn the parameters I of DEAL for image denoising with additive white Gaussian noise (AWGN) of standard deviation on E [0, 50). The denoiser D (y) takes the data y and on as input and then returns the solution of (7) within most Kout iterations. We have chosen this simple training task for two reasons: (i) our learned model should also work for other inverse problems (often called univer- sality), as demonstrated by HHrrult et al. (0226) Goujon cet al. (2024); (ii) our choice leads to simplified computa- tions as H I. We provide details on initializations and hyperparameters with short ablation study in Appendix A
We minimize the loss 12 using Adam (Kingma & Ba 2015). At each step of the optimizer, we sample 16 patches of size (128 x 128) randomly from D. We have two training phases. First, we train the gray and color models for 70 000 and 40 000 steps, respectively, with an initial learning rate of 5 10-4 that is reduced 4 x 10 - by a cosine annealing scheduler. Then, we continue the training of the gray and color model for 10 100 and 5000 steps, respectively, with an initial learning rate of 2 x 10-1 that is reduced to 1 x 10-7 by annealing. We set cout Ein = 1 x 10-4 and limit the number of CG steps to Knn = 50. To select the best- rest-peroormig model, we evaluate its performance every 1000 steps and keep the checkpoint with the best validation performance. We use the set3 and set datasets validate the color and grayscale models, respectively.
Dataset and Loss As training set D {x
1m use
the images proposed in IZhang et al.
(2022). The images
x are corrupted by AWGN as x + F, and fed into Kout
DEAL, which leads to the sequence {xx 1==1 = Koat (y)
(((a))
of denoised images. To estimate the parameters 0 from the training data, we use the loss
Gradient Tracking We train DEAL through the deep equilibrium framework (Bai et al. 2019) in the Jacobian free mode. Specifically, perform most (Kout 1) it- erations without gradient tracking. Then, after convergence, we perform one additional update (6) with gradient track- ing. Here, an efficient backward path for the reconstruction block is crucial, specifically for 0gx as defined in (6). Since the backpropagation through the CG algorithm is pro-
((0)= = IE
X-D
||XKout - -x112
(12)
an vn-21(10,50 10,10
7 NG
| M(X 111] +
+
4

---

DEAL for Image Reconstruction
hibitively memory-expensive, we propose another approach.
Table 1: Denoising of BSD68 and CBSD68 images.
From
g
we have that
Gray
Color
X = y g LgX++11
(13)
On
5
15
25
5
15
25
BM3D
37.54
31.13
28.61
40.19
33.52
30.71
where L2 M(xx) W. The product rule leads
WCRR
37.65
31.20
28.68
0pxx+1 , Lg0gxx+1 A0g L1)xX-11
(14)
SARR
37.80
31.61
29.13
follows that
SAFI
37.90
31.56
29.05
A a d
(15)
DEAL (Ours)
37.85
31.61
29.16
40.33
33.95
31.31
ProxDRUNet
37.97
31.70
29.18
40.40
33,91
31.14
where the matrix Aş is the one from 8 and only the right- hand-side changes 1 X0 ( (L, L X We use nuto-differentiation to obtain the gradient estimate d&+1- We then find 0gx by solving (15) with CG.
DnCNN DRUNet
31.72
29.23
33.90
31.24
38.09
31.94
29.48 40.59
34.30
31.69
3.3. Inference
The Lipschitz estimate (16) is very conservative and T(..)) appears to often be even a local contraction. If (...) is contractive for every y E R.M, then we get the result of Theorem 4.4
Once the parameters are learned, deploy DEAL to gen- eral inverse problem by plugging the corresponding forward H and its adjoint H into the reconstruction block. This does not affect the mask-generation block. To adapt to the new task, we only tune two hyperparameters: the model noise level a and the regularization strength in (6). This requires a small validation with paired measurements and ground-truth images. Empirically, we observe that the tuning of À is more important than change in the noise level σ. A suggested value for a is 15. To ensure that we find a fixed point of (6), we set Kin = Kout = 1000, and choose the conservative stop criteria enn = 10 and
Theorem 4.4. Assume that ker(H) n ker(w)) {0} and M² Ex.Td y): Rd Ie is contractive in the sense that if (1 (x),,y)))))) S gl|x1 x2l12 with
< 1 implies that the iterations o converge to a unique fixed point & and that
| 8122 S G²- -|x| Xol12-
(17)
In particular, we have exponential convergence of (6 More- over, if x = T(x, v1) and 2 = T(2, y2), then it holds that
Cout = 10-5
1
H 2 yy y2112-
|| 2||
(18)
4. Theoretical Results
d
All the proofs are provided in Appendix B Proposition 4.1
5. Experimentss
guarantees the uniqueness of the updates
DEAL is trained with a denoising task. We include the results baseline benchmark. A key strength of universal methods like DEAL is their ability to adapt to different tasks with simple tuning of the hyperparameters. This is fast and requires only a small amount of task-specific data. To demonstrate the generalization capability, we present results on superresolution and MRI reconstructions, comparing DEAL selected (mostly universal) basclines. Additional anprriments-innludiin grayscale deblurring, hyperparame- ter sensitivity, and generalization given in Appendix C
Propusition 4.1. If ke) n ker (M(: W) = {0). then As positive-defnitte and (8) has a unique solution. More- over, if (xx) 2 emid, then A& 2 H H + EM W w and uniqueness holds if ker ker(H) ker(w)
(0).
The next results involve an estimate of the smallest eigen- value A = Amin (H + <M WT W) of Ag.
Lemma 4.2. Let X E Rdd If ker(H) n ker(W)== {0)
and M(x)² EdId, then T(x,.). R-M + Rd is Lipschitz-
continuous with constant [ 1/
Next, we establish the existence of fixed points for T(:y). Theorem 4.3. Assume that ker ke)(H) ) n ker(w)---re)e {0} and (x)) > ExId. Then, T(:y): : Rd + (0) maps into a ball around 0 with radius r
5.1. Grayscale and Color Denoising
We corrupt ground -oout images by adding white Gaussian noise with standard deviation o, E {5, 15, 25}. We provide in Table the average peak signal-to-noise ratios (PSNR) achieved by various methods over the images of the BSD68 set and the CBSD68 set, noticing that some approaches are implemented only for grayscale images. First, we in- clude (C)BM3D (Dabov et al. (007)) as a widely regarded
| /de If M (0) (c, NoNousHW is Lipschitz continuous with constant L, then T(,y) admits a fixed point and
L Hy
[ xx2,
X1
X2112-
(16)
X2
5

---

DEAL for Image Reconstruction
Original
Measurement
WCRR 229.51, 0.85)
SAFI (29, ,9,9 0.86)
DEAL (Ours) (30.07, 0.86)
ProxDRUNet (30.13, 0.85)
DRUNet (30.49, 0.88)
Figure 2: Denoising of the castle image for 25. For each reconstruction (PSNR, SSIM) is provided.
Xg 6.12
X1 20.26
X2 22.98
X10 29.87
X40 30.07
X6s 30.07
10-4
M
Xx.
10-3
10 10-5
30
25
PSNR
20
20
40
60
M(xo)
M(x:)
M(x²)
M(X10)
M(
M(Xos)
Figure 3: Left: Solution path and whinnel-wise averages M the weights for DEAL iterations, exemplified with the castle image and on = 25. Right: Plot of the residual values and PSNR over the number of outer iterations.
classic baseline. We also evaluate against WCRR (Goujon et al. 2024), unadaptive field-of-expert model of the form (2) that employs weakly convex potentials the We also in- clude its data-adaptive counterpart SARR (Neumayer et al.
better than the DRUNet-based approaches in how it retains struetures such the rip of the tower. In Figure we pro- vide the solution path associated to ( the averages M of the masks M(x;), and two convergence plots. Specifically, the weights M extract the image structure, leading to lower regularization cost at edge features.
2023 Neumayer & Altekrüger] 2025). Regarding the refine-
ment perspective (3). we include SAFI (Pourya et al. 22024). which utilizes vic & instead of fz. Lastly, we include the deep-leaming- based approaches DnCNN (Zhang al. 2017a DRUNet (Zhang al. 2022). and ProxDRUNet (Hrrault et al. 2022a). DRUNet is among the state-of-the- art denoisers, and ProxDRUNet is constrained version of it that is more amenable to PnP reconstructions. DEAL outperforms existing spatially adaptive methods and closes the gap to DRUNet-based approaches while having 30 times fewer parameters. We provide qualitative results in Figure where also provide the structural similarity-inde -index metric (SSIM). In the magnified part, we can see that DEAL does
5.2. Color Superresolution
Here, the forward H involves two steps: a blurring of the image through convolution with a known kernel, followed by downsampling with s E (2,3) that reduces the number of 2D measurements by factor of s2, As further degra- dation, an AGWN of standard deviation o, is added the data. Specifically, we investigate two setups: (i) bicubie kernel; and (ii) four Gaussian kernels with standard devia- tions (0.7, 1.2, 1.6, and 2.0) as in (Zhang et al. 2022). We
6

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DEAL for Image Reconstruction
Original
Measurement
DEAL (Ours)
report in Table the average reconstruction PSNR for the center-cropped CBSD68 images and the number of parame- ters for all models. This evaluation is used by Hurault et al, (2022a), and differs from the one of Zhang et al. (2022) ). In Figure 4 we provide visual comparison. Overall, DEAL has a nice complexity-periiee performance tradeoff.
(28.2) 0.83)
To deploy DEAL, we set 15 in the architecture trained for denoising and only tune the scalar hyperparameter in 7 to the noise level on, keeping it fixed across the su- perresolution tasks. This requires only few validation images. Specifically, we found X {0.1, 2.5, 5.5) for on E {0, 2.55, 7.65, 12.75). respectively. An explicit formula for is investigated Table of Appendix C
We tune the other universal methods based on the recommen- dations of their papers. DPIR (Zhang al. 2022)) represents
the state-of- the-art among PnP methods, but is tied to fixed number of iterations. ProxDRUnet (Hurault et al. (022aa addresses this by constraining DRUnet, which empirically ensures convergence. The PnP method IRCNN (Zhang et al. 2017b) uses a more lightweight CNN instead. While diffu- sion models are strong image generation, DiffPIR (Zhu et al. 2023 cannot match DPIR this superresolution task.
ProxDRUNet (27.95 0.82)
IRCNN (27.27 0.79)
DPIR (27.99 0.83)
PSNR of
The two (bicubic for and 3) end-to-end trained SwinIR transformer models (Liang et al. (2021) perform best in their noiseless training regime, but degrade significantly when the setting changes (see also Table 7 iin Appendix (C.T)
There are no interpretable hyperparameters for adaptation, and retraining requires significantly more data compared the minimal tuning of universal approaches. This lack of generalization is major concern in practice.
25
5.3. MRI Reconstruction
Now, we deploy DEAL for MRI tasks. Specifically, tackle the single- and 15-coil MRI setups detailed by by|Goujon| (203)). There, the ground truth consists of knee images from the fastMRI dataset (Knoll et al.) 2020). both with fat suppression (PDFS) and without fat suppression (PD). The forward H involves k-fold subsampling in the Fourier domain and corruption by AGWN with on 0.002. For each of the four evaluation tasks, tune the hyperparameters of all methods. In Table 3 report the PSNR values centered (320 320) patches of the remaining fiftyy test images. We compare against the popular TV regularization, the convex CRR regularizer Goujon al. 2023). its weakly convex extension WCRR, and the ProxDRUNet. be deployed without task-specific training. In Table 4 we report the computation times for several methods V V100-SXM2-32GB GPU. We are significantly faster than the iterative refinement approach SAFI and get close to the (non-adaptive) WCRR baseline, Qualitative results are given in Figure of Appendix
41(0 1)
~ I
= =
20
30
(Xa)
ten images
Figure+ 4: Top: Superresolution task with s = 2 and on 2.55. Bottom: Dependence on initialization.
Trodk Eng WWW
Conerd
methods are universal and can
Tesla
Figure 5: Visualizations of W W. Only one eigenvalue is zero.

---

DEAL for Image Reconstruction
Table 2: Superresolution results on (256 x 256) center-cropped CBSD68 dataset. The number #0 of parameters is given in millions. SwinIR trained with a bicubic kernel and then also deployed for the others.
Category
#
PSNR (s = 2)
PSNR (s=3)
bicubic
4 different kernels
bicubic
4 different kernels
Noise on
2.55
7.65
12.75
2.55
7.65
12.75
DEAL
Explicit Reg
0.85
29.91
27.99
26.58
25.75
26.83
26.20
25.27
24.59
Prox-PnP
Conv. PnP
32.64
27.93
26.61
25.79
26.13
25.29
24.67
IRCNN
PnP 0.19
29.84
26.97
25.86
25.45
26.74
25.60
25.72
24.38
DPIR
PnP
32.64
29.63
27.79
26.58
25.83
26.70
26.05
25.27
24.66
DiffPIR
Diff. Model
93.56
29.73
27.84
26.48
25.63
SwinIR
End to End
11.75
30.88
24.56
22.84
20.73
27.76
22.41
21.24
19.53
5.4. Computational Scalability of DEAL
Table 3: PSNR values for the MRI experiment.
Here, we investigate the scalability of DEAL. To do so, we focus on single-coil MRI experiment with 8-fold Carte- sian masks of the size of the image, where only around 12 percent of the entries are nonzero. We use four images of sizes 2562, 5122, 1024?, and 2048?. The largest image is extracted from a high-resolution MRI brain image (Martinez ct al. 2023). We perform our experiments on Tesla V100- SXM2-32 GB GPU. Table 5 we report the time and memory usage for each of the images. DEAL is consistently faster with smaller memory footprint than Prox-DRUNet, which is the most similar method in terms of universal- ity, performance, and convergence properties. Noteworthy, Prox-DRUNet fails for the image of size 2048?.
f-lo single coil
8-fold multi-coil
PD
PDFS
PD
PDFS
Zero-fill (HT y)
27.40
29.68
23.80
27.19
TV
32.44
32.67
32.77
33.38
WCRR
35.78
34.63
35.57
35.16
SARR
36.25
34.77
35.98
35.26
SAFI
36.43
34.92
36.06
35,36
DEAL (Ours)
36.59
34.92
36.21
35.32
ProxDRUNet
36.20
35.05
35.82
35.12
PaP-DrCNN
35.24
34.63
35.11
35.14
Table 4: Time (seconds) for the MRI experiment.
4-fold single coil
8-fold multi-coil
6. Interpretability and Robustness
PD
PDFS
PD
PDFS
We conclude empirically from Figures 4 and I0 that DEAL is not tied to a specific number Kout of iterations (6). In particular, doing more updates does not degrade the per- formance, unlike many PnP methods such as DPIR. The convergence to a fixed point (see Theorem 4.3) occurs for all experiments. In particular, both the relative error and the
WCRR
12
20
9
8
SAFT
436
470
388
326
DEAL (Ours)
14
17
22
18
ProxDRUNet
113
38
170
105
PSNR converge. To demonstrate the robustness regarding initialization, we instantiate the superresolution task. As see in Figure 4 DEAL converges (in about 10 steps) to the same reconstruction, independently of the initialization, which is in accordance with Theorem 4.4 rors for this task are given in Figure 11 of Appendix [C.4 where the empirical convergence is emphasized once more.
as in (2.. This is practical certification for Proposition 4.1
Our attention mechanism is multiplicative, see ( rather than based on a traditional key-query parameterization. We present two striking interpretations of the mechanism. For simplicity. we focus on the castle-denoising example from Figure 2 with the final solution X In Figure 6 we illus- trate two learned filters within W. and the corresponding responses W XR. The associated weights (masks) m. mx(x)) are well-adapted to the structural features captured by these filters. In effect, the mask suppresses the image structures in the final squared responses (m,( @ (w, * x+))² which leads to a reduced regularization cost in (4) Hence, DEAL preserves salient structures. This is desirable as the image structure should not contribute to the cost.
The relative er-
We present visualizations for all parts of our architecture in Appendix D Remarkably, find mostly finite differences and their higher-order counterparts at various scales within W (see Figure 1 in the appendix). These filters extract the salient features of the input. The impulse response of WT W and its Fourier transform are given in Figure 5 Empirically, we observe that ker ker(W) = span(1a) with
8

---

DEAL for Image Reconstruction
Method
(256 x 256)
(512 x 512)
(1024 x 1024)
(2048 x 2048)
Time (8)
Mem. (GB)
Time (s)
Mem. (GB)
Time (s)
Mem. (GB)
Time (s)
Mem. (GB)
DEAL (Ours)
6.4
0.38
36.0
1.51
173.0
6.03
1800.0
24.11
Prox-DRUNet
18.5
0.87
62.1
3.38
240.0
13.35
NA
NA
Table Runtime and memory footprint in terms of different image sizes for MRI reconstruction.
(W. W.xi )² (w. x
mx()))
(m,
(wo xx))
X2 A
A e
Filter channel e 37
.05
.15
.05
Filler channel c 93
0
10
10-3
10
Figure 6: Learned filters. Left to right: Two filters Wa along with (Wxx)); corresponding masks m. (xx); and adapted squared response.
Figure 7: adaptive-averaging interpretation. Left: Measure-
ments y. Middle: Reconstruction X with focus on its wihh entry (in red). Right: Weight associated to A at posi- tion with [x| - Top to bottom: Three regions of the castle image.
Next, we describe our second interpretation. Since we are in the denoising setting (H I), we have that x = A with - I+ AW M(x=11) W. Therefore, the map- ping from the measurements to the solution X₃ is linear. More precisely, the nth component of the vectorized recon- struction X is a weighted average of the measurements y with the weight given by the nth row of A To extract to the nth unit vector c.. In Fig- ure we see that A A Cn aligns well with the structure of the neighborhood around the nthh pixel. This indicates that spatial information is encoded into A during the refinements. For instance, directional averaging is observed in the first and second rows Figure 7 while the image of its third row exhibits little structure and DEAL averages over a larger region, with an emphasis on the center pixel. Thus, at equilibrium. DEAL acts as an adaptive averaging mechanism, intelligently averaging the noisy measurements with weights that emerge from our iterative refinements.
processing ideas, which blended with recent advances in deep learning, particularly, infinite-dppth networks. It con- sists of two parts: (i) an iterative refinement of intermediate reconstructions based on a least-square-type problem; and (ii) recurrent attention mechanism that adapts the prob- lem spatially. We achieved competitive performance on different tasks while being able to provide interpretability, universality, and theoretical guarantees.
this row, we apply A -1 -
So far, we have only trained DEAL on denoising task. If sufficient data are available, it appears possible to fine-tune all components of DEAL to further improve its performance. Moreover, DEAL is designed for the E2 data fidelity, and devising extensions for other data-fider data-fidelity terms is an inter- esting direction of future work.
Impact Statement
7. Conclusion
We aim to advance the field image reconstruction by com- bining deep-learning tools with classic proceesin techniques. We provide an architecture for the solution of
We have presented deep attentive least squares (DEAL) for image reconstruction. DEAL builds upon classic signal-

---

DEAL for Image Reconstruction
inverse problems that is interpretable and leads to compet- itive reconstruction quality. To the best of our knowledge, our work does not raise any ethical or societal concerns.
Dosovitskiy, A., Beyer, L.,, Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, TT., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., et al. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2020.
Acknowledgements
M.P. and M.U. acknowledge support from the European Research Council (ERC Project FunLearn) under Grant 101020573 and by the Swiss National Science Foundation, S.N. and E.K. acknowledge support from the DFG within the SPP2298 under project number 543939932. E.K acknowledges support from the Austrian Science Fund (FWF) project number 10.55776/COE12.
Ducotterd, S., Goujon, A., Bohra, P., Perdios, D., Neumayer, S., and Unser, M. Improving sipschitz-constrand neural networks by learning activation functions. Journal of Machine Learning Research, 25:1-30, 2024.
Grant
Effland, A., Kobler, E., Kunisch, K., and Pock, T. Varia- tional networks: An optimal control approach to early stopping variational methods for image restoration. Jour- nal of Mathematical Imaging and Vision, 62(3):396-416,
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DEAL for Image Reconstruction
A. Ablation Studies
For the linear splines 91, 422, and (P3 that appear M, we fix r 3, N2 31. Moreover, we initialize (P and (22 as the absolute value |- and (p as e- motivated by classic anisotropic diffusion (Weickent 1998). Fach 8, in (11) has 14 knots in the range [-], They are initialized as the constant function 3. We choose o = on in M, where on the standard deviation of the noise associated with a sample. We set À = x(a.) with a learnable spline x. defined using 52 knots in [-1,51 51] initialized as the identity. The denoisers are strongly tied, particularly, W and M must cope with various settings.
We used Nc 128 number of filters. We observed that reduction of Nc to 64 or 32 degrades the performance for denoising on 25 by around 0.1 and 0.15, respectively. We also observed that going beyond the filter size (9 x 9) in the Multi-Conv block does not improve the performance. The proposed constraints and initializations for the learning of the nonlinearities stem from the learning of such parts without any constraints and with zero initialization.
Since we used a shallow CNN for mask generation, we could not achieve good performance with conventional nonlinearities. Therefore, we use the deep-spline framework of (Bohra et al. 2020). We initially learned the splines with zero initialization and no constraints. There, observed that (0i and (22 tended toward symmetric potentials, while the last de resembled cutoff functions. These observations lead splines is fewer than 2000. Adding more spline knots (parameters) did not improve performance.
proposed configuration. The total number of parameters for all the learnable
The attention mechanism is crucial for DEAL. Without it (M = Identity), the filter bank collapses to a single high-order Laplacian and leads to a huge drop in performance, as much as about 4dB in denoising for dn = 25 with the grayscale images.
B. Proofs
B.1. Proof of Proposition 4.1
Proof. Assume that there exists x E Rdd {0} E ker(A&), so that x Agx 0. By definition of As. this implies that x H Hx = O and x W (3 Wx = 0. Hence, we get that x E ker(H) and x E ker(M(xx W), which is a contradiction. For (² Ž easld, we estimate A, > H + W WTW W and the uniqueness as in the first part.
B.2. Proof of Lemma
Proof. Let & = y1) and é = (x),2), so that Ax² = H yı and Ax2 = Hyy. y2. This implies that Ax - 2)
H (y1 - y2) and we estimate
H d
|| 2112 5
bly y2ll2.
(19)
B.3. Proof of Theorem 4.3
Proof. First, investigate the range of T(:y). By definition of T(sy), it holds for any Rd that
[Hyla
[xxx ||A H y/12
(20)
X
For the second part, we want to apply the Brouwer fixed- fixed-point theorem. To this end, we must prove that T(.y) is continuous. Let X1, X2 E Rdd. x = T)x,,y). and X3 Txxx, y). Then, holds that
A A2 =
A A122 A2x² A1
x1 X2 = A ' A1)
(21)
Incorporating (20 and the normalization
W/|2 = 1, we further infer that
||*|
*2l12 < |A AA- ||
M (2x
<L Hy Xx2 - x 112.
(22)
Lipschitz-continouus Hence, Lippiiiiiiiiiiiiiiiooooooooooo and fixed point exists.
13

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DEAL for Image Reconstruction
B.4. Proof of Theorem 4.4
Proof. Due the Banach fixed-point theorem, the exponential convergence rate holds. To estimate the difference of x (xx, and y2. the contractivity of y:) and Lemma 4.2 get that
x y (1T( T(2,y2)ll2 < v1) y1.. (1T(),))) T(2,
H 2 yy y2l12.
S 21|2 +
(23)
d
From this, we readily infer
18
C. Additional Experiments
C.1. Sensitivity to Hyperparameter Tuning and Generalization
Based observations in various setups, always choose the model noise level 15. In contrast, À must be adapted to the data noise level. This is typical for variational regularization, where À scales with the square of the data noise level o We investigate this closer for multi-coil MRI reconstruction at data noise level o, 0.002. We give in Table 6 the reconstruction PSNR for different model noise levels and regularization strengths X. Indeed, the performance depends primarily on Interestingly, even a tenfold change in maintains a reasonable reconstruction quality. In general, a higher leads to blurred images, while lower À does not remove the artifacts.
Table 6: Hyperparameter Effect.
Table 7: PNSR across kernels.
Method
Bicubic
A
B
C
D
0.01
0.1
10
DEAL
29.91
29.59
29.76
28.57
27.21
5
27.0
33.77
32.25
29.28
SwinIR
30.88
25.72
25.85
24.50
23.66
15
32.5
33.88
31.8
28.31
25
33.05
33.61
31.42
27.75
50
32.96
33.05
30.96
26.75
Regarding the adaptation of À to the noise level we can also use theoretically motivated closed-form formula for X. As we see in Table
g the performance drop is marginal compared to the fine-tuned case.
Table 8: Average reconstruction PSNR over 4 kernels for superresolution with 2 centered-cropped CBSD68 data.
Method
On 2.55
o, 7.65
on 12.75
DEAL (fine-tuned) DEAL (A = 0.1 + 0.035a
27.99
26.58
25.75
27.97
26.57
25.75
To provide further evidence for the good generalization of DEAL, we perform superresolution with a downsampling rate of s 2 without noise, where the SwinIR trained on the bicubic kernel and À is fine-tuned on the bicubic task. Then, we apply SwinIR and DEAL to new kernels with no further change for both models. Table the kernels A-D are Gaussian kernels with different standard deviations (0.7, 1.2, 1.6, and 2.0). Then, the reconstruction PSNR the center-cropped CBSD68 images is given. We conclude that SwinIR needs retraining for new kernels, while DEAL still performs well in the presence of kernel mismatch with the hyperparameters being tuned on the bicubic kernel and deployed on the Gaussian ones.
14

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DEAL for Image Reconstruction
C.2. Grayscale Deblurring
Here, evaluate the performance of the DEAL approach on a grayscale-debluringg task. We use the same setup as DPIR for this experiment (Zhang al., 2022). This includes two blur kernels of sizes (17 17) and (27 27) from (Levin et al. 2009) and additive Gaussian noise with on 2.55 and on 7.65. In Table report the PSNR of the reconstructions for the Set3 images, namely Cameraman, House, and Monarch. We set the model noise level a 15 and À (0.5,2.5) for the two given AWGN noise levels. We also compare with model-based EPLL (Zoran & Weiss 2011 and the learning approach FDN that is specific to deblurring (Kruse et al. 2017). We observe that we are consistently the second-bes method on this task after DPIR. We provide visual comparison with DPIR in Figure
Table 9: PSNR for grayscale deblurring.
2.55
7.65
17x17
27x27
17x17
27x27
C.man House Monarch C.man House Monarch C.man House Monarch C.man House Monarch
EPLL
29.18
32.33
27.32
27.85
28.13
22.92
24.82
28.50
23.03
24.31
26.02
20.86
DEAL (Ours)
31.72
35.20
32.77
31.64
35.03
32.48
27.89
32.24
28.26
27.79
32.11
28.15
FDN
29.09 29.75
29.13
28.78
29.29
28.60
26.18
28.01
25.86
26.13
27.41
25.39
IRCNN
31.69
35.04
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Original
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DEAL (Ours) (32.48, (184)
DPIR (32.99, 0.95)
Original
Figure 8: Both kernel and deblurring results for noise level 0, 2.55.
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DEAL for Image Reconstruction
C.3. Visualizations for MRI Reconstruction
We provide in Figure 9 visual reconstruction examples obtained with the different methods from Table the solution path and the convergence plots for DEAL in Figure 10
We also provide
Original
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DEAL (Ours) (34.08,0.86))
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Figure 9: Eight-fold multi-coil MRI reconstructions for PD image of the knee.
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Figure 10: Solution and mask path for the 8-fold multi-coil MRI reconstruction for a PD image of the knee.
C.4. Convergence Plot for Superresolution
In Figure II
we represent the convergence plot for the superresoluion task on the setup of Figure
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Figure 11: Convergence plot for the superresolution task with 2 and noise level on 2.55 for different initializations.
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DEAL for Image Reconstruction
D. Visualization of Model Components
Now, we inspect the components our learned grayscale model. In Figure 13 we depict the impulse response of W, namely, the equivalent convolution kernels of the block(a.k.a filters). The convolutions of W mask have similar structure, as we see in Figure 14 In addition, we depict the learned splines in Figure 12 For &e we visualize three noise levels {5, 15, 25) and channels (44, 93, 99). The three visualized channels correspond to vertical edge filters of various scales, see Figure 13 They resemble threshold functions that set high responses zero. This results in less regularization in the regions that have high responses to the filters that are often activated by the image structures. This is a desirable behavior as the structure of the image should not contribute to the regularization cost. Additionally, the widths of the last spline og are increasing for all channels with respect to the noise level ø. Thus, more regularization performed at higher noise levels. Moreover, we show the channel-wise average of the masks for the noisy and the denoised castle image in Figure [5]
The masks remove the undesirable contribution of the image the filter responses Wx of the regularizer. This results in lower penalization of the edges and yields sharper solutions.
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Figure 12: Learned splines in the mask-generation network M (Figure
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DEAL for Image Reconstruction
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Figure 13: Effective convolution kernels for the Multi-Conv block W the grayscale model. All plots use the same range, where neutral gray corresponds to zero.
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DEAL for Image Reconstruction
Figure 14: Effective convolution kernels for the Multi-Conv block W mask in the grayscale model. All plots use the same range, where neutral gray corresponds to zero
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DEAL for Image Reconstruction
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Figure 15: Filter responses and masks. From left to right: channel-wise average of (i) the squared response the noisy image (top) and the solution of the castle denoising problem (bottom); (ii) the masks computed on the noisy image (top) and the solution (bottom): (iii) corresponding adapted responses.
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Reviews:
[{"review_id": "D64Et9xIxG", "reviewer": "Reviewer_dGc6", "summary": "Summary\n\nThis paper proposes a novel Maximum a posteriori (MAP) method for solving linear inverse problems with Gaussian noise. The proposed method is based on Fields-of-Experts (FoE) regularization [1]. The authors suggest using quadratic potentials and learning the remaining parameters of the FoE regularization. The proposed algorithm is implemented as a neural network, and the regularization parameters are learned through end-to-end training. \n\nMain findings:\n\n- Using quadratic potential significantly accelerates the algorithm, enabling the effective learning of the regularization parameters.\n- The regularization parameters learned from a denoising task can be directly applied to other linear inverse problems, such as super-resolution and MRI reconstruction, with minimal hyperparameter tuning. \n- The proposed approach achieves reconstruction performance comparable to plug-and-play (PnP) methods.\n\n[1] Roth, Stefan, and Michael J. Black. \"Fields of experts: A framework for learning image priors.\" 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05). Vol. 2. IEEE, 2005.\n\n## Update After Rebuttal\n\nI want to thank the authors for their detailed response and for the effort they have put into addressing the concerns raised in the initial review. In light of the additional results and explanations provided by the authors, I have updated my score to **“4: Accept.”**", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and valuable comments. To summarize our response:\n- We addressed the discrepancy of our numbers with the DPIR paper (due to cropping), we also added denoising results for the DPIR setup to underline the validity of our results.\n- We compare with a state-of-the-art neural network for super-resolution to offer a balanced view of DEAL's performance. \n- We adapt the writing of our paper to highlight the key points of the comments.\n\nIf our responses have addressed your concerns, we would greatly appreciate it if you could reconsider your rating.\n\n> 1. The key novelty of the proposed method...\n\nIndeed, denoising is not the main performance benchmark, and we will adjust the writing to emphasize this more. In particular, our results for super-resolution and MRI are far more competitive and on par with SOTA *universal* (trained on denoising and applicable to other tasks) methods. Regarding novelty, beyond its easy extension to various tasks, DEAL bridges classic regularization with deep learning, offering interpretability, theoretical guarantees, and fewer parameters — all key advantages over other ML models. For this, our novel attention mechanism embedded into the FoE model is crucial; learning only the FoE components substantially deteriorates the performance (see response point 2 to reviewer Q6VC).\n\n> 2. Since the proposed method requires...\n\nInitially, we compared our method with SOTA PnP approaches, ensuring fair hyperparameter tuning across all evaluation tasks (see response point [Experimental Designs Or Analyses] of reviewer on6j). To showcase the benefits of universality, we adapt the super-resolution experiment to include an end-to-end trained transformer SwinIR [3], designed for a bicubic blur kernel. As expected, it is by far the best method within its training regime. However, its performance decreases substantially for other blur kernels and under the addition of noise. Likewise, for MRI, the literature shows that such models perform poorly outside of their training regime (e.g., DuDoRNet in Table 1 of [5]). Unfortunately, we could not find a well-performing end-to-end model for our specific setup (multi-coil, downsampling factor, noise). Generally, this lack of generalization for end-to-end methods is a major concern, and we highlight this with the new results in our paper.\n\nIn the following table, #θ denotes the number of parameters in million, the triplets after PSNR are (s, $\\sigma_n$, used kernels) where avg4 is the average of four different kernels. The two best numbers are bolded for each column. \n\nMethod|Category|#θ|PSNR (2,0,bicubic)|PSNR (2,2.55,avg4)|PSNR (2,7.65,avg4)|PSNR (2,12.75,avg4)|PSNR (3,0,bicubic)|PSNR (3,2.55,avg4)|PSNR (3,7.65,avg4)|PSNR (3,12.75,avg4)|\n|-|-|-|-|-|-|-|-|-|-|-|\n|DEAL|Explicit Reg|**0.85**|**29.91**|**27.99**|**26.58**|25.75|**26.83**|**26.20**|**25.27**|24.59|\n|Prox-PnP|Conv. PnP|32.64|-|**27.93**|**26.61**|**25.79**|-|**26.13**|**25.29**|**24.67**|\n|IRCNN|PnP|**0.19**|29.84|26.97|25.86|25.45|26.74|25.60|25.72|24.38|\n|DPIR|PnP|32.64|29.63|27.79|**26.58**|**25.83**|26.70|26.05|**25.27**|**24.66**|\n|DiffPIR[4]|Diff. Model|93.56|29.73|27.84|26.48|25.63|-|-|-|-|\n|SwinIR[3]|End to End|11.94|**30.88**|24.56|22.84|20.73|**27.76**|22.41|21.24|19.53|\n\n> 3. There appears to be a discrepancy...\n\nLet us clarify the confusion. There are two different evaluation setups: we use 256x256 center-cropped images, as performed in ProxDrunet [2], while DPIR [1] uses full-size images. Our results match Table 1 of ProxDRUNet [2]. To cross-check the performance, we reran the evaluations within the setup of DPIR. Here, our results match exactly Table 2 of the DPIR paper [1]. Our chosen evaluation is detailed in Section 5.1, and we will revise the paper to avoid future confusion caused by the different setups.\n\nThe two best numbers are bolded for each column. \n\n|Method|σₙ=5|σₙ=15|σₙ=25|\n|-|-|-|-|\n|BM3D|40.19|33.52|30.71|\n|DEAL(Ours)|40.33|**33.95**|**31.31**|\n|Prox-DRUNet|**40.40**|33.91|31.14|\n|DNCNN|-|33.90|31.24|\n|DRUNet|**40.59**|**34.30**|**31.69**|\n\n> 4. The reported results for DPIR...\n\nWe use the setup of [2] and our results are following their Table 4. There are three differences with Table 8 of DPIR: (i) we report the metrics on 256x256 center-cropped CBSD68 images; (ii) the noise levels are different; (iii) we report the average over the 4 kernels (a-d). The configuration is described in Section 5.2, and we will adapt the writing to prevent future confusion caused by the different setups.\n\nRefs.\n\n[1] Zhang et al. \"PnP image restoration with deep denoiser prior.\", 2021.\n\n[2] Hurault et al. \"Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization.\", 2022.\n\n[3] Liang et al. \"Swinir: Image restoration using swin transformer.\", 2021.\n\n[4] Zhu et al. \"Denoising diffusion models for PnP image restoration.\", 2023.\n\n[5] Song et al. \"Solving inverse problems in medical imaging with score-based generative models, 2021.", "claims_and_evidence": "I think yes, the claims made in the submission supported by evidence.", "methods_and_evaluation_criteria": "I have some concerns regarding the evaluation.\n1. The key novelty of the proposed method seems to be its ability to (almost) directly apply a model trained on denoising to other inverse problems. If this is the case, evaluating the model on denoising tasks may not fully capture its main contribution. It's useful to see how the proposed denoiser compares to state-of-the-art alternatives. However, these results could be presented as baseline comparisons rather than performance evaluations.\n2. Since the proposed method requires fine-tuning of hyperparameters $\\lambda$ and $\\sigma$ for each task, comparing solely to plug-and-play methods may not provide a comprehensive perspective. I suggest including comparisons with recent state-of-the-art neural architectures specifically trained for super-resolution and MRI reconstruction. This would offer a more balanced view of the method's performance. \n3. There appears to be a discrepancy in the reported PSNR values for DRUNet on the CBSD68 dataset. The reported PSNR values  (33.85 and 31.21 for CBSD68 with sigma 15 and 25 respectively) are approximately 0.45dB lower than those reported in the original DRUNet paper [1], which lists values of 34.30 and 31.69 under the same conditions. It would be helpful to clarify this inconsistency.\n4. The reported results for DRIP in the color super-resolution experiment appear significantly lower than those presented in the original DRIP paper [1] (as shown in Table 8). Additional clarifications on the experimental setup and any differences in implementation would help understand this gap.\n\n[1] Zhang, Kai, et al. \"Plug-and-play image restoration with deep denoiser prior.\" IEEE Transactions on Pattern Analysis and Machine Intelligence 44.10 (2021): 6360-6376.", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "Please refer to the Methods And Evaluation Criteria section for further details.", "relation_to_broader_scientific_literature": "The key contributions of the manuscript are related to the literature about PnP methods.", "essential_references_not_discussed": "I did not come across related works that are essential for understanding the key contributions of the paper but are not currently cited.", "other_strengths_and_weaknesses": "Please refer to the Methods And Evaluation Criteria section for further details.", "other_comments_or_suggestions": "I have no further comments.", "questions_for_authors": "I have no further questions for the 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": "WXiTyyci2V", "reviewer": "Reviewer_yde4", "summary": "This paper presents Deep Attentive Least Squares (DEAL), a novel image reconstruction method that bridges traditional signal processing and modern deep learning. DEAL formulates reconstruction as an iterative least squares problem with spatially adaptive regularization, where an attention mechanism dynamically modulates the regularization weight based on the local image structure. The approach efficiently solves a sequence of quadratic problems using a learned multi-convolution filter and a conjugate gradient solver, ensuring both interpretability and computational efficiency. The authors provide rigorous theoretical analysis, proving the uniqueness of solutions, convergence guarantees, and robustness to initialization. Experimentally, DEAL is evaluated on denoising, super-resolution, and MRI reconstruction, achieving competitive results while using significantly fewer parameters than state-of-the-art deep learning models. The study highlights DEAL’s interpretability, universality, and theoretical guarantees, positioning it as a promising alternative to heavily parameterized deep models for inverse imaging problems.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s feedback. We added more experiments and comparisons to highlight DEAL’s generalization and scalability (see response point 2 to reviewer dGc6 and response point [Questions For Authors] to reviewer Q6VC) and provided additional details on the attention mechanism (see responses point 2 and 4 to reviewer Q6VC). Our code repository will be released with the final paper.", "claims_and_evidence": "1. Claim: DEAL provides a principled approach to image reconstruction, combining classic signal processing techniques with deep learning insights.\nEvidence: DEAL is formulated as an iterative least-squares problem with spatially adaptive regularization. The authors draw connections to traditional Tikhonov regularization and modern plug-and-play (PnP) methods, demonstrating how DEAL balances interpretability and performance.\n\n2. Claim: DEAL achieves convergence to a unique fixed point, ensuring stability and robustness.\nEvidence: The authors provide rigorous theoretical guarantees (Propositions 4.1–4.4), proving uniqueness under mild conditions and showing that the iterative updates lead to a contraction mapping under certain assumptions. Empirically, DEAL consistently converges in experiments, regardless of initialization.\n\n3. Claim: DEAL outperforms traditional spatially adaptive regularization methods and achieves performance close to deep learning-based approaches while using significantly fewer parameters.\nEvidence: Extensive experiments on denoising, super-resolution, and MRI reconstruction show that DEAL surpasses non-adaptive regularization methods like WCRR and SARR. Furthermore, Table 1 and Table 2 demonstrate that DEAL’s PSNR scores approach those of DRUNet-based models, despite having 30× fewer parameters.\n\n4. Claim: The learned attention mechanism effectively suppresses regularization in structured image regions, preserving details.\nEvidence: The authors provide visualizations (Figures 7 and 8), showing how learned masks adapt to image structures. The masks reduce regularization in smooth areas while enforcing it in noisy regions, preventing excessive smoothing of important features.\n\n5. Claim: DEAL is robust to hyperparameters and initialization, making it a reliable reconstruction method.\nEvidence: Experiments in Figure 5 and Appendix C.3 show that DEAL consistently converges to the same solution regardless of initialization. The model is also shown to be less sensitive to hyperparameter tuning compared to deep learning-based methods.\n\n6. Claim: DEAL generalizes well across different inverse problems without task-specific retraining.\nEvidence: Unlike deep neural networks that require retraining for each task, DEAL is directly applied to denoising, super-resolution, and MRI reconstruction by simply adjusting two hyperparameters (σandλ). The results in Table 3 and Table 4 confirm its versatility.\n\nThe claims in this paper are well-supported by both mathematical theory and empirical validation. The combination of theoretical convergence guarantees, extensive benchmarking, and qualitative visualizations strongly substantiates DEAL’s effectiveness and robustness for inverse imaging problems", "methods_and_evaluation_criteria": "Methods：\nThe authors of this paper propose the Deep Attentive Least Squares (DEAL) method to address the image reconstruction problem, integrating classical regularization techniques with modern deep learning approaches:\n\n1. Formulation as an Iterative Least Squares Optimization with Adaptive Regularization \n\nDEAL formulates the image reconstruction problem as an iterative least squares optimization incorporating a spatially adaptive regularization term. In each iteration, a conjugate gradient (CG) algorithm is employed to solve the quadratic optimization problem efficiently. The regularization term is dynamically adjusted via learned attention weights, allowing the model to adapt to different image regions and enhance reconstruction quality.\n\n2. Spatially Adaptive Regularization via an Attention Mechanism\n\nA shallow convolutional neural network (CNN) is used to compute attention weights, which modulate the local impact of the regularization term. The CNN extracts multi-scale features and applies pointwise nonlinear transformations to learn an adaptive weight distribution. This learned attention mechanism allows DEAL to reduce regularization in structured regions while enhancing regularization in noisy areas, thereby preserving critical image details.\n\n3. Multi-Convolution Module for Efficient Feature Extraction\n\nDEAL employs a multi-layer convolutional structure to extract spatial features efficiently. These convolutional layers do not incorporate nonlinear activation functions, ensuring a large receptive field while maintaining interpretability. This architecture is used in both the reconstruction process and the mask generation module, allowing for shared feature representation.\n\n4. Training on a Denoising Task to Improve Generalization\n\nTo enhance generalization, DEAL is trained on an additive white Gaussian noise (AWGN) denoising task, where the noise level is set as \nσ_n∈[0,50]. The training data includes both grayscale and color images, utilizing the BSD68 and CBSD68 datasets. The loss function consists of three components:\n①Mean Squared Error (MSE): Ensures consistency between the reconstructed output and the ground truth image.\n②Stability Penalty on Attention Weights: Suppresses unstable variations in the regularization adjustment.\n③Total Variation Regularization on Learned Activation Functions: Prevents excessive complexity in the nonlinear transformations.\n\n5. Applicability to Various Inverse Problems\n\nOnce trained, DEAL can be directly applied to a range of inverse problems, including denoising, super-resolution, and MRI reconstruction, without requiring additional training. The model only requires tuning of two hyperparameters—regularization strengthλand noise levelσ to adapt to different tasks.\n\nEvaluation Criteria\n\nThe authors evaluate DEAL using both quantitative and qualitative metrics, comparing its performance across multiple inverse imaging tasks.\n\n1. PSNR\n\nPSNR is the primary metric used to assess image quality and is widely applied in inverse imaging tasks.A higher PSNR value indicates a reconstruction closer to the ground truth image.Tables 1, 2, and 3 report the PSNR results of DEAL for denoising, super-resolution, and MRI reconstruction.\n\n2. SSIM\n\nSSIM measures the structural similarity between the reconstructed and ground truth images, evaluating perceptual quality.Figures 3 and 5 demonstrate that DEAL preserves fine details better than competing methods.\n\n3. Convergence and Stability\n\nTheoretical analysis (Theorems 4.3 and 4.4) proves that DEAL has a unique solution and guarantees convergence.Experimental results in Figure 4 show that DEAL consistently converges under different initialization conditions.Unlike Plug-and-Play (PnP) approaches (e.g., DPIR), where performance may degrade with excessive iterations, DEAL steadily converges to a fixed point, ensuring stable performance.\n\n4. Comparison with State-of-the-Art Methods\n\nDEAL is compared against the following methods:\n①Classic methods: BM3D, Total Variation (TV) regularization.②Learned regularization approaches: WCRR, SARR, SAFI.③Deep learning models: DnCNN, DRUNet, Prox-DRUNet.\nTables 1, 2, and 3 show that DEAL achieves performance close to DRUNet-based methods while using only 1/30 of the parameters, highlighting its advantage in computational efficiency.\n\n5. Computational Efficiency\n\nTable 4 reports the computational time for MRI reconstruction, showing that DEAL is significantly faster than the iterative refinement method SAFI and comparable to the non-adaptive WCRR baseline.Figure 5 further illustrates DEAL’s efficiency in super-resolution tasks, demonstrating faster and more stable convergence compared to PnP-based methods.\n\nThe authors provide rigorous theoretical guarantees for Deep Attentive Least Squares (DEAL), demonstrating its uniqueness, convergence properties, and stability under certain conditions. The theoretical foundation is based on iterative least squares optimization with spatially adaptive regularization.", "theoretical_claims": "The authors provide rigorous theoretical guarantees for Deep Attentive Least Squares (DEAL), demonstrating its uniqueness, convergence properties, and stability under certain conditions. The theoretical foundation is based on iterative least squares optimization with spatially adaptive regularization.\n\n1、Uniqueness of the Solution\nClaim: DEAL ensures a unique solution at each iteration, provided that the problem satisfies specific conditions.\nSupporting Theory: Proposition 4.1 states that if the intersection of the null spaces of H and M(x_k)W contains only the zero vector, i.e., ker(H)∩ker(M(x_k)W) = {0}, then the system matrix A_k is positive definite, ensuring the uniqueness of the solution in equation (8).\n\n2、Lipschitz Continuity and Stability of the Iterative Process\nClaim: The iterative update operator T(x, y) is Lipschitz continuous, ensuring stability across iterations.\nSupporting Theory: Lemma 4.2 proves that if ker(H) ∩ ker(W) = {0} and M(x)^2 is greater than or equal to epsilon_M times the identity matrix, then T(x, y) is Lipschitz continuous with a bounded Lipschitz constant.\n\n3、Existence of a Fixed Point\nClaim: DEAL converges to a fixed point, ensuring stability and reliability in practical applications.\nSupporting Theory: Theorem 4.3 establishes that T(x, y) maps into a bounded region and has at least one fixed point, provided that M(x)^2 is Lipschitz continuous with a constant L.\n\n4、Convergence Guarantee and Rate of Convergence\nClaim: The iterative process exhibits exponential convergence under contraction mapping conditions.\nSupporting Theory: Theorem 4.4 states that if T(x, y) is a contraction mapping, meaning that for two inputs x1 and x2, the update operator satisfies the inequality:\n||T(x1, y) - T(x2, y)|| ≤ q * ||x1 - x2|| with q < 1,\nthen DEAL converges exponentially to a unique fixed point x-hat, with a decay rate proportional to q raised to the power of (k-1).\n\n5、Stability with Respect to Input Perturbations\nClaim: DEAL is stable under variations in the input measurements y, meaning that small perturbations in the input lead to bounded changes in the reconstruction.\nSupporting Theory: Equation (18) states that for two different inputs y1 and y2, the difference between their corresponding reconstructions is bounded as follows:\n||x-hat - z-hat|| ≤ (1 / (1 - q)) * (||H|| / lambda_epsilon) * ||y1 - y2||.\nThis result implies that DEAL produces consistent and stable reconstructions even when faced with measurement noise or variations in input data.", "experimental_designs_or_analyses": "The experimental design in this paper is well-structured and aligned with the research objectives, ensuring a comprehensive evaluation of DEAL across multiple inverse imaging tasks. Below is an assessment of the key experimental components:\n\n1、The study evaluates DEAL on denoising, super-resolution, and MRI reconstruction, aligning well with its iterative least squares formulation and spatially adaptive regularization framework.\n\n2、The authors use BSD68, CBSD68, and fastMRI datasets and compare DEAL against BM3D, TV, WCRR, SARR, SAFI, DnCNN, DRUNet, and Prox-DRUNet, ensuring a fair and comprehensive benchmark.\n\n3、DEAL is trained on an AWGN denoising task with σ_n∈[0, 50], using Adam optimization and cosine annealing, but lacks detailed sensitivity analysis on hyperparameters.\n\n4、The results are quantitatively evaluated using PSNR (Tables 1-3), SSIM (Figures 3 & 5), and convergence analysis (Figure 4), but statistical significance tests (e.g., confidence intervals) are not reported.\n\n5、The study provides clear training details and evaluation protocols, demonstrating robustness across different conditions, but code availability is not explicitly mentioned.", "relation_to_broader_scientific_literature": "The contributions of this paper are closely connected to existing research in the field of image reconstruction. The authors propose the DEAL method, which combines the fields-of-experts model (Roth & Black, 2009) and PnP methods (Venkatakrishnan et al., 2013), integrating traditional signal processing and modern deep learning techniques to introduce a novel image reconstruction method. Compared to previous learning-based regularization methods, the proposed DEAL method offers significant advantages in terms of interpretability, computational efficiency, and convergence.\nThe paper correctly cites relevant literature, covering important studies in image reconstruction and regularization, such as the learning paradigms discussed by Chen et al., 2014 and Effland et al., 2020, as well as the parameterization strategies addressed by Goujon et al., 2023 and Zach et al., 2024. The authors also reference research on autoencoders (Li et al., 2020) and algorithm unrolling (Kobler et al., 2022), accurately highlighting the relationship and distinctions between DEAL and these methods.\nIn terms of innovation and impact, the paper contributes by combining classical signal processing techniques with deep learning methods, proposing a new approach that integrates iterative refinement and attention mechanisms. Unlike deep learning methods like DRUNet, DEAL provides convergence and robustness guarantees, with clear advantages in computational efficiency and interpretability. The innovative combination in DEAL positions it with broad potential applications and influence in the field of image reconstruction.\nOverall, the paper correctly cites relevant literature and proposes an innovative and promising solution in the field of image reconstruction.", "essential_references_not_discussed": "The paper does not overlook any critical related research, and all core references relevant to the DEAL method are appropriately cited. The authors correctly reference key works in the field of image reconstruction and regularization, such as the fields-of-experts model and PnP methods, and do not neglect any prior research crucial to this work. So the paper properly attributes existing methods and cites relevant literature without significant omissions.", "other_strengths_and_weaknesses": "Strengths:\nThe paper excels in originality by proposing DEAL, a novel approach that effectively combines traditional signal processing with modern deep learning techniques, providing a clear solution to challenges in image reconstruction. The method’s theoretical guarantees for convergence and robustness add significant value, making it stand out among other methods in the field. The contributions are highly significant, offering improvements in interpretability, computational efficiency, and performance. The paper’s clarity is strong, with well-defined methodology, clear explanations, and comprehensive experimental results that demonstrate the efficacy of the approach.\n\nWeaknesses:\nWhile the contributions are important, there is a slight limitation in the generalization of the method to other tasks beyond the presented inverse problems, such as super-resolution or more complex imaging scenarios. Additionally, although the paper is well-written, some aspects could benefit from further elaboration, such as the details of the learned attention mechanism and its practical applications in different domains. Despite this, the overall structure and writing are clear and accessible, making the paper easy to follow.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "CtbMxcsWEE", "reviewer": "Reviewer_on6j", "summary": "This paper presents a least-square-type image reconstruction method. It is formulated as a conjugate gradient method. It consists of of an iterative refinement process with two main components: one that estimates the reconstructed image and a another that generates a mask, modulating the response of the prior filter in a spatially adaptive manner for its following iterations. Similar to plug-and-play methods, the proposed model is trained on the denoising task and can be universally applied to other image reconstruction tasks given the forward operator and a validation set on the target task to set its hyperparameters, including the standard deviation of noise and the weight assigned to the prior term. The strength of the proposed method is two fold. It achieves near SOTA performance comparable to similarly universally pretrained methods with a relatively low computational complexity, while also ensuring robustness and convergence to a unique solution.\n\n## update after rebuttal\nI thank the authors for responding to my main concerns. I will maintain my recommendation to accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's time and feedback and are glad that DEAL is recognized as a significant improvement in the performance-efficiency trade-off within image reconstruction techniques. In summary, \n- We clarified our choice of comparison methods.\n- We added diffusion models and transformers as additional comparisons for super-resolution.\n- We provided insights into hyperparameter sensitivity.\n\n>  [Experimental Designs Or Analyses] Regarding the super-resolution task, would it be fare to expect the compared methods to account for the considered addition of noise?\n\nFor super-resolution, we use the exact same setup as the ProxDRUNet and use their given optimal hyperparameters that are different for each noise level. For IRCNN and DPIR, formulas for optimal hyperparameters are given within their codes based on the noise level for the super-resolution task. For MRI, we follow the WCRR paper's hyperparameter tuning setup with ten validation images to adapt all methods. \n\n> 1. Why those specific methods and why only those and not others?\n\nThe competing methods are selected based on code availability and similar experimental setups. Due to their flexibility, and since our method is also universal, we mainly focused on PnP approaches. \n\nSuper-resolution: Here, DPIR is the SOTA among PnP methods. DPIR is tied with a fix number of steps and its performance will degrade if iterated more. By constraining the underlying DRUnet, ProxDRUnet addresses this problem and obtains convergence guarantees. Instead of a DRUnet, IRCNN deploys a lightweight CNN. Now, we added the end-to-end trained transformer SwinIR [1] and the diffusion model DiffPIR [2].\n\nMRI: For medical grayscale images, we also compare with classical TV and the SOTA explicit learned regularization methods WCRR, SARR, and SAFI.\n\n> 2. How does the performance compare to SOTA deep learning-based methods without theoretical guarantees...\n\n We added comparisons to the transformer-based SwinIR and diffusion models for superresolution. End-to-end models like SwinIR are highly sensitive to their training tasks and degrade significantly when conditions change. Diffusion models excel in image generation, but the current diffusion-based reconstruction methods often perform worse than DPIR for image super-resolution in terms of PSNR metric (see Table 2 of the DiffPIR paper [3]). Our superresolution results confirm these findings (see response point 2 of reviewer dGc6 for the updated table).\n\n\n> 3. Is it reasonable to also test the performance without searching for the considered hyperparameters? In other words, as an additional baseline, how bad does the performance get without access to a validation set?\n\n Regarding the two hyperparameters, we can always set the model noise level $ \\sigma = 15 $ based on our observations in different setups. In contrast, $\\lambda$ must be adapted to the data noise level. This is typical for variational regularization, where $\\lambda$ inversely scales with the data noise level $\\sigma_n^2$. We investigate this closer for multi-coil MRI reconstruction at data noise level $\\sigma_n = 0.002$. Indeed, the performance depends primarily on $\\lambda$. Interestingly, even a 10-fold change in $\\lambda$ maintains reasonable reconstruction quality. In general, higher $\\lambda$ leads to blurred images, while lower $\\lambda$ does not remove the artifacts.\n \n In the following table, for the mentioned MRI setup, the reconstruction PSNR for the different choices of the model noise level $\\sigma$ and regularization strength $\\lambda$ is given. \n| σ \\ λ|0.01|0.1|1|10|\n|-|-|-|-|-|\n| 5|27.00|33.77|32.25|29.28|\n|15|32.50|33.88|31.80|28.31|\n|25|33.05|33.61|31.42|27.75|\n|50|32.96|33.05|30.96|26.75|\n\n\nRefs.\n\n[1] Liang et al. \"Swinir: Image restoration using swin transformer.\", 2021.\n\n[2] Zhu et al. \"Denoising diffusion models for plug-and-play image restoration.\", 2023.", "claims_and_evidence": "- Proof for theoretical claims are provided in the supplementary material.\n    \n- Convergence: The authors refer to Theorem 4.3 and illustrate examples on how convergence to a fixed point occurs for both the relative error and the resulting PSNR of the reconstruction.\n    \n- Robustness: An example of robustness to various forms of initialization for the 2x super-resolution task is provided.\n    \n- Interpretability: The authors present a couple of examples of learned filters for the denoising task and offer reasonable interpretations of how the associated masks are adapted to the structural features captured by those filters.\n    \n- Performance: Near SOTA results are demonstrated on multiple tasks including, denoising, super-resolution and MRI.", "methods_and_evaluation_criteria": "Yes, they are commonly considered evaluation criteria in the literature.", "theoretical_claims": "I did not go through all the proofs provided in the supplementary material.", "experimental_designs_or_analyses": "Regarding the super-resolution task, would it be fare to expect the compared methods to account for the considered addition of noise?", "relation_to_broader_scientific_literature": "The proposed least-square-type image reconstruction method builds on classical regularizers that focus on sparsity in image gradients and parametric models. The method utilizes implicit regularization techniques using plug-and-play denoisers and spatial adaptivity, which are allow it to achieved improved performance. Iterative refinement methods and nonlocal Laplacians contribute to its robustness and convergence. Presenting near state-of-the-art performance with low computational complexity makes it well-positioned among efficient neural network architectures for image reconstruction.", "essential_references_not_discussed": "N.A.", "other_strengths_and_weaknesses": "- Originality: The particular approach to formulate the problem and solve it with confidence in convergence to a unique solution seems original.\n    \n- Clarity: The provided illustrations for the experiments clarify the claims, at least for a the limited cases considered.\n    \n- Significant: Based on the identified related works, the proposed methods is a significant improvement in terms of the trade-off on performance and efficiency, while maintaining theoretical convergence properties.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "1. The authors have chosen only a handful of baselines for the comparisons. Why those specific methods and why only those and not others?\n    \n2. How does the performance compare to SOTA deep learning based methods without theoretical guarantees, such as transformers and diffusion-based methods that claim to solve multiple tasks including those considered in the experiments?\n    \n3. Is it reasonable to also test the performance without searching for the considered hyperparameters? In other words, as an additional baseline, how bad does the performance get without access to a validation set?", "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": "6R3rOPyNBi", "reviewer": "Reviewer_Q6VC", "summary": "This paper introduces Deep Attentive Least Squares (DEAL), a novel data-driven image reconstruction method that bridges traditional regularization techniques with modern deep learning. The authors propose an alternative to complex, highly parameterized deep architectures by leveraging the principles of classic Tikhonov regularization. The key idea is to iteratively refine intermediate reconstructions by solving a sequence of quadratic optimization problems. The method achieves state-of-the-art performance comparable to leading plug-and-play and learned regularizer approaches, while offering additional benefits such as interpretability, robustness, and provable convergence behavior.\n\n## Update After Rebuttal\n\nI would like to thank the authors for their detailed response and their efforts to address the issues raised in the initial review. I will maintain my recommendation to weak accept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s time and feedback and are glad that DEAL is recognized as creative and original. To summarize our responses:\n- We added a paragraph to the related work focusing on attention mechanisms. \n- We performed a new experiment to survey the scalability of DEAL for large medical images. \n- We provided more ablation on the impact of the attention mechanism. \n\n> 1. The font size in Figures 1 and 2 is somewhat small...\n\nWe will enlarge the font-sizes and improve the resolution of the Figures. \n\n> 2. While the paper includes some ablation studies ...\n\nTo highlight the impact of the attention mechanism and the spline parameterization, we added the following ablation:\n\n- The attention mechanism is crucial for DEAL. Without it (M= Identity), the filter bank collapses to a single high-order Laplacian, leading to a huge performance drop e.g. around 4dB in denoising for $\\sigma_n =25$ on our grayscale testset.\n\n- Since we use a shallow CNN for mask generation, we cannot achieve good performance with conventional nonlinearities. Therefore, we use the deep spline framework of [1]. We initially learned the splines with zero initialization and no constraints. There, we observed that $\\varphi_1$ and $\\varphi_2$ tended toward symmetric potentials, while the last $\\{\\phi_c\\}$ resembled cut-off functions. These observations lead to our proposed setup for these splines as described in the paper. The total number of the parameters for all the learnable splines is less than 2000 and adding more spline knots (parameters) does not improve performance.\n\n> 3. Although DEAL demonstrates competitive performance ...\n\nWe augmented our tables and added more metrics including the number of parameters to further clarify our comparisons with SOTA methods (see the updated table in point 2 reviewer dGc6). We also provide a new large-scale experiment for the MRI setup to showcase the scalability of DEAL (see response point [Questions For Authors]). \n\n> 4. Given the close connection between DEAL and attention mechanisms...\n\nWe will add this paragraph to the related work section: \n**Attention Mechanism**: Originally popular in natural language processing, attention mechanisms are now widely used in image processing. Two main approaches exist: (i) patch-based attention, where images are divided into patches and processed using scaled dot-product attention, as in SwinIR [2] and Restormer [3]; and (ii) point-wise multiplicative attention, which integrates attention directly into the architecture, as in attention-guided CNNs [4]. While patch-based methods effectively capture long-range dependencies, they introduce significant computational overhead due to the need to compute inner products across all patches in a latent space. In contrast, point-wise attention avoids this, making it a more efficient alternative. Our approach to implementing attention is more aligned with methods that rely on point-wise multiplication, rather than the key-query parameterization of transformers. In particular, the way attention is incorporated in DEAL is highly interpretable (see Section 6).\n\nWe will add more references to the paper's version and adapt the writing of Section 6 to emphasize more on the interpretability of our attention mechanism. \n \n> [Questions For Authors]: While DEAL is shown to be efficient for moderate-sized problems...\n\nWe designed a new MRI experiment to address the questions. We use four images of sizes 265x256, 512x512, 1024x1024 and 2048x2048. The largest one is extracted from a high-resolution MRI brain image [5]. The smaller ones are obtained by bicubic downsampling. For the MRI setup, we use 8-fold Cartesian masks of the size of the image, where only around 12 percent of the mask entries are non-zero. We perform our experiments on a TeslaV100-SXM2-32GB GPU. There, DEAL can successfully handle the reconstruction of an image of size 2048x2048. In contrast, Prox-DRUNet fails at this size. To provide more insights, we add the following table in the paper, where we report the time and memory usage for each of the images in the aforementioned setup. DEAL is consistently faster than Prox-DRUNet with less memory usage. \n\n| Method| 256x256 (Time s)| 256x256 (Mem. GB)| 512x512 (Time s)|512x512 (Mem. GB)|1024x1024 (Time s)| 1024x1024 (Mem. GB)|2048x2048 (Time s)|2048x2048 (Mem. GB)|\n|-|-|-|-|-|-|-|-|-|\n|DEAL (Ours)|6.4|0.38|36.0|1.51|173.0|6.03|1800.0|24.11|\n|Prox-DRUNet|18.5|0.87|62.1|3.38|240.0|13.35|NA|NA|\n\nRefs. \n\n[1] Bohra et al. \"Learning activation functions in deep (spline) neural networks.\", 2020.\n\n[2] Liang et al. \"Swinir: Image restoration using swin transformer.\", 2021.\n\n[3] Zamir et al. \"Restormer: Efficient transformer for high-resolution image restoration.\", 2022.\n\n[4] Tian et al. \"Attention-guided CNN for image denoising.\" 2020.\n\n[5] Martinez et al. \"BigBrain-MR: a new digital phantom with anatomically-realistic magnetic resonance properties at 100-µm resolution for magnetic resonance methods development.\" 2023.", "claims_and_evidence": "The claims made in the submission are well-supported by clear and convincing evidence, including theoretical proofs, extensive experiments, and comparisons with state-of-the-art methods.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate and well-aligned with the problem of image reconstruction. The use of standard datasets, metrics, and comparisons with state-of-the-art methods ensures that the evaluation is rigorous and meaningful.", "theoretical_claims": "The theoretical claims are well-supported by correct proofs, the authors provide a rigorous theoretical foundation for their method. The theoretical claims like Proposition 4.1 (uniqueness of updates), Lemma 4.2 (Lipschitz continuity), Theorem 4.3 (existence of fixed points), and Theorem 4.4 (exponential convergence) are supported by standard mathematical tools such as positive definiteness, Lipschitz continuity, and the Banach fixed-point theorem. These claims are thoroughly explained in both the main text and the supplementary material.  Some assumptions, like ker(H)∩ker(W)=0 in the manuscript are reasonable and empirically supported too.", "experimental_designs_or_analyses": "The experimental design and analyses are generally sound, but additional ablation studies and scalability tests could further strengthen the paper:\n1. A more detailed analysis of the impact of the attention mechanism and spline parameterization\n2. More results to show scalability of DEAL", "relation_to_broader_scientific_literature": "DEAL's contributions are well-grounded in the broader literature, building on and extending prior work in iterative refinement, learned regularization, and theoretical analysis. The novel idea of iterative refinement with learned filters and an attention mechanism is particularly noteworthy and has the potential to inspire future research in this area.", "essential_references_not_discussed": "While the paper provides a solid foundation, it could benefit from a more detailed discussion of Attention Mechanisms in Image Restoration in the related works section. For instance, citing seminal works such as \"Attention-Guided CNN for Image Denoising\" would help contextualize the proposed method within the broader landscape of attention-based approaches in image restoration.", "other_strengths_and_weaknesses": "Strengths: DEAL combines traditional regularization with deep learning in a novel way, this is a creative and original approach.  The paper is well-written and clearly presents the method, theoretical analysis, and experimental results. The visualizations and supplementary material enhance the clarity of the presentation.\n\nWeakness: Some details could be further refined and supplemented.", "other_comments_or_suggestions": "1. The font size in Figures 1 and 2 is somewhat small, which may hinder readability. Enlarging the font or providing higher-resolution versions of these figures would improve the reader's experience.\n2. While the paper includes some ablation studies (e.g., filter size, number of filters), a more comprehensive analysis of the impact of different components (e.g., attention mechanism, spline parameterization) could further strengthen the claims.\n3. Although DEAL demonstrates competitive performance, there is still a noticeable gap compared to methods like DRUNet (2022). To better highlight DEAL's advantages, the authors could include a table comparing the number of parameters, computational efficiency, and performance metrics (e.g., PSNR, SSIM) between DEAL and state-of-the-art methods. This would provide a clearer picture of DEAL's strengths, particularly in terms of efficiency and scalability.\n4. Given the close connection between DEAL and attention mechanisms, it would be beneficial to add a dedicated section on attention mechanisms in image restoration in the related works. This would help situate DEAL within the broader context of attention-based approaches and highlight its unique contributions.", "questions_for_authors": "While DEAL is shown to be efficient for moderate-sized problems, I’m curious about its performance on very large-scale datasets (e.g., high-resolution medical images).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper presents a well-supported approach to image reconstruction, combining classic regularization with deep learning. After reviewing both the critiques and the rebuttal, I believe the concerns raised by the critical reviewer have been adequately addressed. The method shows strong theoretical grounding, solid performance, and practical benefits like robustness and interpretability. I’m recommending a weak accept, but ask the authors to revise the final version to clarify comparisons with task-specific baselines, address any inconsistencies in reported results, and provide more detail on computational cost and scalability.
Decision: Accept (poster)