Zero-shot Denoising via Neural Compression: Theoretical and algorithmic framework
Authors: Ali Zafari
Xi Chen
Shirin Jalali
Venue: NeurIPS 2025 spotlight ()
Keywords: compression-based denoising
zero-shot image denoising
neural compression
Primary area: general_machine_learning
TL;DR: We propose a theoretically-founded compression-based zero-shot image denoising method.

Abstract: Zero-shot denoising aims to denoise observations without access to training samples or clean reference images. This setting is particularly relevant in practical imaging scenarios involving specialized domains such as  medical imaging or biology. In this work, we propose the *Zero-Shot Neural Compression Denoiser* (ZS-NCD), a novel denoising framework based on neural compression. ZS-NCD treats a neural compression network as an untrained model, optimized directly on patches extracted from a single noisy image. The final reconstruction is then obtained by aggregating the outputs of the trained model over overlapping patches. Thanks to the built-in entropy constraints of compression architectures, our method naturally avoids overfitting and does not require manual regularization or early stopping. Through extensive experiments, we show that ZS-NCD achieves state-of-the-art performance among zero-shot denoisers for both Gaussian and Poisson noise, and generalizes well to both natural and non-natural images. Additionally, we provide new finite-sample theoretical results that characterize upper bounds on the achievable reconstruction error of general maximum-likelihood compression-based denoisers. These results further establish the theoretical foundations of compression-based denoising. Our code is available at: https://github.com/Computational-Imaging-RU/ZS-NCDenoiser.

Zero-shot Denoising via Neural Compression: Theoretical and algorithmic framework
Ali Zafari" ali. zafari@rutgere edu
Xi Chen"
Shirin Jalali shirin. wertiirrtttterec .dd
xi. chen150rutgers.cem
Rutgers University New Brunswick, NJ, USA
Abstract
Zero-shot denoising aims denoise observations without access training samples or clean reference images. This setting is particularly relevant in practical imaging scenarios involving specialized domains such medical imaging or biology. In this work, we propose the Zem-Shol Neural Compression Denoiser (ZS-NCD), novel denoising framework based on neural compression. ZS-NCD treats a neural compression network as an untrained model, optimized directly on patches extracted from single noisy image. The final reconstruction is then obtained by aggregating the outputs of the trained model over overlapping patches. Thanks to the built-in entropy constraints of compression architectures, our method naturally avoids overfitting and does not require manual regularization or early stopping. Through extensive experiments, we show that ZS-NCD achieves state-of-the-art performance among zero-shot denoisers for both Gaussian and Poisson noise, and generalizes well both natural and non-natural images. Additionally, we provide new finite-sample theoretical results that characterize upper bounds on the achievable reconstruction error of general -ikimumm-ikkelhood compression- based denoisers. These results further establish the theoretical foundations of compression-bait based denoising. Our code available at: https: //g thub. com/ Computations innalllltto--nlo Imaging- R0J/SS-NCDent
Introduction
Background and motivation Denoising is fundamental problem in classical signal processing and has recently gained renewed attention from the machine learning community. Let æ = (æ1,..., In)) e ET denote non-negative signal of length n, where signal not observable in many systems. Instead, we observe a noisy version = (yI...., Bn), where the observations are conditionally independent given æ, and each entry is distributed according a common conditional distribution:
II p(y: x.)).
We assume that the noise mechanism is memoryless (independent across coordinates) and homoge- neous (identical across entries). The goal of denoising algorithm is estimate from the noisy observations y. Given its prevalence in imaging and data acquisition systems, denoising has been central topic in signal processing for decades. Classical denoising methods rely explicit structural assumptions about the underlying signal æ often hand-crafted by domain experts [1, 2, 3, 4,5, 6, 8). In contrast, recent advances in machine learning have enabled a new class of data-driven denoising
Equal contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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overispping patches training: ((c,)(o)
set overiapping patches target training:
single moley image: y
Pe,y(ar
Zero-shot Learning Phase
$(1J)
Zero-shot Denoising Phase
Figure 1: Zero-Shot Neural Compression Denoiser (ZS-NCD). Learning phase: a neural compression model (architecture shown in Fig. 5 of the supplementary material) is trained overlapping patches extracted from single noisy image. Denoising phase: each pixel reconstructed by averaging predictions across neighboring patches processed by the trained model.
algorithms. These methods learn the optimal denoising function from data, leveraging statistical patterns directly from signal and noise distributions.
While learning-bassee based approaches achieve state-of-the-art performance and often outperform classical methods controlled settings, they face significant challenges in practice:
Supervision requirement Most soarning-basch methods require training set of paired samples {Vy æx, )=)) where æ; is clean signal and y; is its noisy counterpart. In practical scenarios such as medical imaging, {m are unavailable or prohibitively expensive expenssvve to obtain.
2. Data efficiency: These methods usually need lots of training data. Acquiring sufficient samples is difficult or costly, particularly in domains with strict data acquisition constraints.
To mitigate the reliance on paired clean and noisy samples, several self-supervised denoising methods have been developed that learn directly from noisy observations, without access to clean ground truth signals [9, 10, 11, 12, 13]. While these approaches alleviate the supervision requirement, they typically depend on access to large collections of noisy data and often yield suboptimal performance compared to methods trained with clean targets. Moreover, the absence of clean supervision necessi- tates the use of complex neural architectures and training schemes, which can make these methods computationally demanding and difficult to optimize in practice.
These challenges have sparked growing interest in zero-shot denoisers, which aim to recover clean signals from noisy observations without access to paired data or extensive noisy training data. Such methods are particularly appealing in domains where acquiring clean data is infeasible, and they offer the potential for deployable denoisers that adapt individual inputs with general purpose.
From neural compression zero-shot denoising Denoising algorichns-raaggnn from classical signal processing techniques deep learning methods methods-funddmeee fundamentally rely on the assumption that real-world signals are highly structured. Compression-basee denoising leverages this same principle, but rather than directly solving the inverse problem, instead performs lossy compression the

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Figure 2: Zero-shot denoising of Kodim05 with AWGN (o 25). Left: PSNR versus training iterations for zero-shot denoisers. Performance of BM3D [18] and Restormer [19] are included a classical baseline and supervised empirical upper bound, respectively. Right: Visual reconstructions with PSNR above each image. Compression-
denoising based JPEG- 2K [20] achieves inferior performance. Learning-basse zero-shot denoisers often struggle with either overfitting or high bias. DIP [21] and DD [22] require early stopping to avoid overfitting, ZS-N2S [12] and S2$ [23] struggle with high-resolution color images, and ZS-N2N [24] often produces noisy outputs with potential overfitting. BM3D tends to oversmooth the denoised image. In
contrast, ZS-NCD avoids these issues.
noisy observation y. under the hypothesis that the clean signal æ lies in lower-complexity subspace and is therefore more compressible.
In lossy compression, the goal is to represent signals from a target class using discrete encodings with minimal distortion. When applied noisy data, the intuition is that a lossy compressor-opennee at distortion level matched to the noise-will favor reconstructions close to the original clean signal. While this approach has a strong theoretical foundation [14, 15]. classical compression-bast based denoisers have shown limited empirical success, particularly for natural image denoising.
In this work, we revisit this idea in light of recent progress in neural compression, where learned encoders and decoders have demonstrated strong rate-distortion performance across a variety of image domains [16. 17). Building this foundation, we propose a zero-shot denoising method that we call the Zero-Shot Neural Compression Denoiser (ZS-NCD). Unlike traditional neural compression models that trained on large corpora of clean high-resolution images, ZS-NCD learns directly from a single noisy input image. Specifically, we extract overlapping patches from the noisy image, and train a neural compression network on those patches alone-withouu any clean supervision or prior dataset. Once trained, the denoiser is applied to all patches from the same image, and the final output is obtained by averaging the predictions overlapping regions. This approach illustrated in Figure 1.
Despite relying solely the noisy input and operating without supervision, ZS-NCD achieves state-of-the-art performance among zero-shot denoising methods across diverse noise models, and remains robust even inputs that lie outside the natural image distribution. We compare against the baselines in Figure 2, ZS-NCD shows superior performance in denoising and training stability.
This paper introduces a zero-shot image denoising framework based neural compression. Our main contributions are:
Paper contributions
zero-shot denoising algorithm using neural compression. We propose a fully unsupervised method that trains neural compression network on image patches of the noisy input. It does not rely on clean images, paired datasets, or prior training on the target distribution. It is architecture- agnostic and leverages only the structure present in the observed noisy image.
Theoretical results connecting denoising and compression performance, We establish finite- sample upper bounds on the reconstruction error of the proposed compression-ban maximum likelihood denoisers, for both Gaussian and Poisson noise models.
3

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Extensive empirical validation. We demonstrate that our method achieves state-of-ther of-thh--rttrt art perfor- mance among zero-shot denoising techniques across a range of noise models and datasets.
Related work
Self-supervised and zero-shot denoising Supervised learning- based denoisers such as DrCNN [25] and Restormer [19] achieve statc-of-the-arl performance across various noise models, but require large datasets of paired clean and noisy images- often impractical in real-world settings. To avoid clean images, self-supervised methods have been proposed, including Noise2Noise [10], Noise2Self [12], Noise2 [11], Noise2Same{26i and Noise2Score [13], which only use noisy images for training. However, their reliance on large noisy datasets remains a limitation. Zero-shot denoisers address this by training on a single noisy image. These include (i) untrained networks like DIP [27] and Deep Decoder [22], and (ii) single-image adaptations of self-supervised methods, e.g,, N2F [28], ZS-N2N [24), ZS-N2S [12] and its augmented variant with ensembling. S2S [23]. More re- cently, DS-N2N [29] improves ZS-N2N by further upsampling the downsampled paired noisy images. Pixel2Pixel [30] boosts the performance by using non-local similarity approach. DIP-based models avoid masking and leverage full-imaae context, but require early stopping under- patierr-rrattiiizationn to avoid overfitting. Self-supervised variants suffer from masking- masking-induced induced information loss. Hybrid approaches, such as masked pretraining-base based method [31], uses external datasets for training and perform zero-shot inference, thus falling outside the zero-shot setting studied here.
Neural compression Learning-basse baasid lossy compression, often referred as neural compression. uses an autoencoder architecture combined with an entropy model to estimate and constrain the bitrate the bottleneck [32, 33]. These methods have significantly outperformed traditional codecs, particularly in image [32, 34, 35, 36, 37] and video [38, 39, 40] compression. In addition to these standard settings, several works have also explored applying neural compression to noisy data, either by adapting neural compression models for more efficient encoding of noisy images [41, 42, 43, 44].
Compression- based denoising Compression based denoising leverages the insight that structured signals are inherently more compressible than their noisy counterparts. This connection was for- malized by Donoho [14], who introduced the minimum Kolmogorov complexity estimator, and further refined by Weissman et al. [15], showing that, under certain conditions both the signal and the noise, optimal lossy compression of a noisy follal-followed by suitable post- post-processing asymptotically achieve optimal denoising performance. Prior these theoretical developments, early empirical methods such as wavelet-basee schemes [45, 46, 47] and MDL-inspired heuristics [48] had explored this principle. Nevertheless, traditional compression- based denoisers have generally underperformed in high-dimensional settings such as image denoising.
Learning- Leaseing-based joint compression and denoising using neural compression has been explored in recent works [49, 50], where the goal is to achieve lower rate in compression. The empirical application of training neural compression for AWGN denoising was also proposed in [51]. Training the neural compression models in these works requires a dataset of images. In contrast, our proposed ZS-NCD is two-step denoiser based neural compression, trained on single noisy image. It achieves state-of-the-arr performance across both AWGN and Poisson noise models. Moreover, we contribute new theoretical results that advance the foundations of compression-base denoising.
3 Compression-bssed denoising: Theoretical foundations
Lossy compression Let @ Fn denote the signal class of interest, such vectorized natural images of fixed size. A lossy compression code for Q is defined by an encoder-decoder pair (f.g) g). f: Q -> (1,... ...... 2R), and g (1, (1 - R The performance of a lossy code is characterized by: i) Rate R, indicating the number of distinct codewords; ii) Distortion 6, defined as the worst-case per-symbol mean squared error (MSE) over the signal class:
suP=eQ !
p(f())))
The set of reconstructions produced by the decoder forms the codebook:
C {g(i) : 1,....2f1 28g R".

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Compression- based denoising We propose compression based denoising as structured maxi- mum likelihood (ML) estimation. Given a noisy observation y I p(y: æ;) and a lossy compression code ( for Q, the compression-baser MI. denoiser solves
C arg min [(c; y), where - -ig c.).
:=
This formulation leverages the fact that clean signals, by virtue their structure, are more compress- ible than their noisy counterparts. Therefore, the most likely codeword under the noise model, when selected from a codebook designed to represent clean signals, serves as a natural denoising estimate. This ML-based view unifies denoising across noise models and provides a principled way to select reconstructions from a discrete, structure-aware prior.
In the of AWGN: y + z, where NO0 I. the described denoiser simplifies to:
i = arg min y
cll2-
That is, denoising corresponds to projecting the noisy observation onto the nearest codeword.
Poisson noise commonly arises in low-light and photon-limited imaging scenarios. In this setting, each is modeled as Poisson random variable with mean az;; ~ Poisson(azz.). Under this model, the compression-based MI. denoiser simplifies
a arg mineee E (ac, y log c.).
While the loss function in (2) is statistically well-motivated, it is more sensitive to optimization issues than its Gaussian counterpart due to the curvature and nonlinearity of the log term. To improve robustness and simplify optimization, we also consider alternative loss based a normalized squared error between с and the rescaled observations:
arg mineee ||c- - 1
Theoretical analysis We begin by analyzing the performance of compression- caapression-based ML denoising under AWGN, The following result provides a non-asymptotic upper bound on the reconstruction error in terms of the compression rate and distortion. All proof's can be found in Appendix A.
Theorem 1. Assume that x: E Q and let (f g) denote a lossy compression for Q that operates at rate R and distortion 8. Consider y x + z, where ~ (O, of I,). Let å denote the output of the compression-based denoiser defined by (f.g) in (1). Then,
1 Vn
(2 (1 (1+2/T),
o 2012 S V + 20,
(4)
with a probability larger than 2 22++2
This bound decomposes the denoising error into two terms: distortion term Vo, which reflects the approximation quality of the compression code, and rate-dependent term that scales with the square root of the code rate R. The latter captures the likelihood concentration around the clean signal in high-probability regions of the noise distribution. Notably, the result holds aon-asymptotically and does not assume the code is optimal, only that provides a distortion-ò covering of Q. This highlights that even non-ideal compression codes can enable effective denoising, provided the rate-distortion tradeoff is call-cali.rated.
To better understand the implications of Theorem 1, in the following corollary, we focus on the special case of k-sparse signals.
Corollary 1 (AWGN, sparse signals). Let Qn denote the of k-sparse vectors R" satisfying ||0||00 < 1. Fix a parameter n E (0, 1), and suppose = x + where z ~ NO0, o² Then, there exists family of compression codes such that, when used with the denoiser defined in (1), with a probability larger than 1 moloogg//k the estimate & satisfies
|| Øll2 S ,,,C h logg In
(5)
vn
where C = 2(1 2 2 lm and nn logg Inga &+t c+1)
5

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Corollary provides probbprobybylityy bound on the normalized error sides and having (va + vo)² < 2(a + b), it follows that, with high probability
vn ||
2/12. Squaring both
1 | x 20 k logz 7m n n
2
Thus, up to universal constants, the dominant term in the upper bound scales as o log( xx) This matches the known minimax rate for estimating k-sparse signals in Gaussian noise when k/n + 0; see [52]. Determining whether the residual term is an artifact of our proof optimality is an interesting problem. Finally, while the comparison above is high-probability rather than in expectation, one can integrate the tail bound from the proof to obtain an expected-risk bound.
real barrier to
We next extend our analysis to sienal-dependent noise model. Poisson noise is particularly relevant in imaging applications such microscopy and astronomy, where photon counts vary with signal intensity. Unlike Gaussian noise, Poisson observations induce a non non-linear likelihood surface, making analysis more delicate. Theorem 2 and 3 establish performance guarantees for compression based Poisson denoising, using both exact ML formulation and a practical squared-error surrogate.
Theorem 2. Consider the same setup of lossy compression in Theorem Assume that for any æ E Q, T4 E (Tmin Fmax), where 0 < Fmin < Tmax < 1. Assume that V1..... Un are independent with y1 Poisson(aar) Let é denote the solution of (2). Let C1 //axx/ (F and
C2 min
" 3 In (VT+ vn). Then, with probability larger than 1 2-
1 > - 2 C15 C2
R
(6)
na
Theorem Consider the same setup as in Theorem 2. Let & denote the solution of (3). Let C = 4//m2(/1+ v Then, with a probability larger than 2-
R
Remark Theorems 2 and show that, in the case of Poisson noise, minimizing either the ML loss function the computationally efficient MSE loss function can recover the signal. This result is also consistent with our simulations reported later in Section
Zero-shot compression-based denoiser
We refer to general class of learning-based denoisers that operate by compressing noisy images using neural compression as the Neural Compression Denoiser (NCD). In this framework, denoising is achieved by identifying low-complexity reconstruction from the output of neural compression model. In the previous section, we characterized the performance of such denoisers in a setting where the compression code is fixed in advance, either learned from external data designed using classical methods, and applied independently of the noisy input. This setup is not zero-shot, as it relies on prior training code design. Inspired by this idea, we now propose a fully unsupervised variant: the Zero-Shot Neural Compression Denoiser (ZS-NCD). In ZS-NCD, a neural compression network trained directly patches extracted from single noisy image, without access to clean targets external data. This section describes the ZS-NCD architecture and optimization procedure in detail.
Proposed zero-shot denoiser: ZS-NCD Let P(G)) Rhx= + & denote the patch extraction operator, which returns h x le patch whose top-left corner is at pixel (i,j). Let fe, and ge, denote the encoder and decoder networks, parameterized by weights 0, and 02, respectively, Define I as the set of all coordinates (i,j) E (I,..., h k + 1} fl....,w k + 1} from which a valid & x k patch can be extracted.
Given single noisy image y, the ZS-NCD is trained to minimize the following patchwise objective:
(0, O2) arg min (0).02) (i.j)ZZ
Σ (Lx (Jo, (P())) (v)) Alog P(Jo, (P())) (8)
where (Joo (P(,,))()) denotes the likelihood (or entropy model) of the latent code produced by the encoder, and À >> O is hyperparameter controlling the trade-off between fidelity and compressibility.

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Algorithm Finding Lagrangian coefficient
I: Initialize: A >> 0, tol > 0. Ç & (0, 1)
2: for k 0, 1.... A.....Kaax. - do
3: Estimate a(k) from (9).
Compute residual p(A) [|a[(] y/
5: Compute scale factor = (r(() no ?-
[8] & tol then
75 14
break else
32-
A 1100/(1++1 8>
(100(1 + c18)) otherwise
220
1000
1250
1300
10: end
Figure 3: Effect of A in denoising Mouse
11:
end for
Return: X² = ((8)
Nuclei image.
In (8), -= k2, and the function Lx RKK x R R. is a distortion loss determined by the noise model, defined in Section 3. For example, in the AWGN case, Lx corresponds to the squared & norm of the distance between a noisy patch and its neural compression reconstruction. Note that fo. maps the input into a discrete latent space, which is non-differentiable and thus incompatible with standard gradient-basee based optimization. To address this, we follow the neural compression framework of [32], using continuous relaxation during training (e.g., uniform noise injection) and applying actual discretization only at test time. The entropy term is modeled using a factorized, non-parametric density [34].
After training, the denoised image obtained by applying the encoder and decoder to each patch and averaging the overlapping outputs. For each pixel (i,j), let ((i,)) C I denote the set of patch locations such that P(r.))) includes the pixel (i.j). The final estimate at location (i.j) is given by
((,)))
(fo, (P(r)) ((-
where [Irg])] denotes the number of patches covering pixel (i,j), and -((,)) denotes the (a, b)-th pixel of the patch output. For interior pixels away from the boundary, [Z K. As shown later in Section 5, this aggregating of reconstructed patches significantly enhances denoising performance.
Setting the hyperparameter X The ZS-NCD objective in (8) includes a hyperparameter, X, which balances reconstruction fidelity and compressibility. Interpreted through the lens of lossy compression, varying À allows the model to explore different rate-distortion trade-offs. However, in the context of denoising, our goal is not compression but accurate signal recovery from the noisy observation y. This raises the central question: how should X be selected to optimize denoising performance? In the following, we explain our approach for setting À under both AWGN model and Poisson noise model.
Case I: Gaussian noise. Let & denote the output of the ZS-NCD denoiser, and consider the AWGN with (0, o?. Then,
model y
(10) the first term is the noise variance, and the second is the true denoising error. While z and à not fully independent, they are intuitively weakly correlated in successful denoising regimes, where the estimate ã depends only indirectly on the noise. Thus, the cross term is expected to be small: _ 2 (x æ)] R3 This approximation suggests that, ideally, under low-noise regimes, lly - expected to be close to o Based on this insight, we propose simple and effective heuristic for choosing A: select À such that 1
E - - E - x
2 = + =11 = 2)).
n ly x 212 is closest to the known noise variance o². This procedure can be implemented efficiently via tree-based search strategy, described in Algorithm 1. To apply Algorithm 1, one needs an estimate the noise power o3. This a well-studied problem and there exist robust algorithms for estimating the variance of noise [2, 53]. For example, in [2], it is shown that the noise power can be estimated from the median of the absolute differences wavelet coefficients.
Finally, we observe that the performance of 7S-NCD relatively robust to the choice of X. For instance, on the Nuclei dataset (Figure 3), 7S-NCD outperforms the -tate-of-thhe-ar shro learning-
7

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based denoiser, ZS- NS-Nosse2Noise, across wide range of À values, for both o; = 10 and 02 = 20. A similar approach can also be applied the case of Poisson noise, well.
Case H: Poisson noise In the case of Poisson noise, in addition to estimating X, we need to estimate a, which is used to normalize the measurements, in both the MSE-based and the MI.E-based methods. Note that in the case of Poisson noise, [[[y]]] = aT;; and therefore, with high probability, 2 1/1 a En Ji. Using this observation, and assuming that n -i=1 24 0.5, we estimate a as = Σ'= Dt-- We then use the estimated noise level á to normalize both MSE and MLE based optimization for denoising Poisson noise. See Table I for the result of this noise parameter estimation on a sample image. Having an estimate of noise parameter a, we then follow a similar procedure used in the of AWGN to set the parameter A. Specifically, we write the MSE between normalized y and the denoised image & as
E 11
212
E
Ily/a
x
+
x
212
Σ I4
x
a 212 + E az)² - 2]].
(1I)
Again, assuming that we are in low-noise regime, i... the second and the third terms in (11) are close to zero, and using 1 -i=1 2; s 0.5 approximation, it follows that _E |ly +|2 ~ 2n This implies that, in the case Poisson noise, we can still use Algorithm I' find X, after updating [|ā((k) - y no² [[=(k) - yy1 za We empirically observe that MSE fo,ā) in training our 1 networks is close to 2n
as reported Table which indicates that 2a is good approximation the MSE that can be used as a threshold for selecting X. When & is not known, we obtain the estimate á,
and use 2&
as a valid threshold set A.
Table I: Analyzing the estimation of Poisson noise parameter for Barbara image in Setl1 (MSE values reported in terms of PSNR).
estimated à empirical MSE(g//o..) [dB]
1/(2a) [dB]
1/(2á) [dB]
25
23.02
17.12
16.98
16.63
50
46.05
19.66
20.00
19.64
5 Experiments
In this section, we evaluate the denoising performance of ZS-NCD on both synthetic and real-world noise, across natural and microscopy images. We compare against representative zero-shot denoisers, including both traditional and learningbbased methods. All baselines are dataset-fre free, i.c., they operate solely on the noisy image to be denoised. For non-learning methods, we include JPEG-2K and BM3D. Although rarely used as denoising baseline, JPEG-2K provides a useful point of denoising, as it represents a fixed, pre-defined
comparison from the perspective of compression-basee compression code. For learning-based methods, we evaluate Deep Image Prior (DIP) [21], Deep Decoder (DD) [22], Zero Noise2Self (ZS-N2S) [12], Self2Self (S2S) [23], and Zero-Shot Noise2Noise (ZS-N2N) [24].
Due to instability in training for several baselines, we report their best achieved performance (with early stopping or model selection), whereas ZS-NCD is evaluated at its final training iteration, without manual tuning or stopping criteria.
Natural images with synthetic noise where σg is the standard deviation of the Gaussian distribution, and Poisson noise defined as Poisson(az). where a is the scale factor. Note that Poisson noise is signal-dependent noise with Ely) = az. To re-scale the noisy image to the range of clean image, we followed the literature by assuming that the the scale is known, and normalize the noisy image as y/a in the experiments of this section. We evaluate on grayscale Set// [25], RGB Se113 [54] (center-cropped to 192 x 192) and Kodak24 [55] datasets. Table 2 presents the denoising performance of various methods. BM3D achieves the strongest results on grayscale images, though it relies on accurate knowledge of the noise power parameter. Existing learning-basse zero-shot denoisers, in contrast, often exhibit inconsistent
We consider two synthetic noise models, AWGN (00 o?).
8

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Table 2: Denoising performance comparison under AWGN and Poisson Noise, average PSNR(dB) and SSIM are reported. Best results are in bold, second- best are underlined.
Noise Parameter
AWGN, WWGN,N(000r" (o))
Poisson, Prissso((rar//
a or a
Method
Setl1
Set13
Kodak24
Setl
Set13
Kodak24
JPEG2K
0.7699
26.69 0.7554
27.86
0.7457
22.35/
0.569/
22.56 0.5249
BM3D
32.22/0.899
/..15/0.8008
32.37/ 0.8754
26.66 / 0.7505
25.64/0.6912
27.04/0.660
DIP
29.11/0,7990
/0.8500
31.42/ 0.8444
0.5863
05.14/0.6916
26.37 26.37/00670 /.6676
DD
28.83/0.8215
29.22 0
28.71 0.8016
24.37 / 0.6622
24.96/0.7066
25.59/ 0.6679
15
S25
26.81 00.818
20.61 0.66879
23. 23.08/ 0.7695
21.75 0.6872
19.23 0.6553
22.52//0.4418
ZS-N2S
0.849
18.18 0.5690
18.68 0.5540
25.06/ / 0.7051
21.23/ 0.6666
22.24/0.61700
ZS-N2N
30.01/ 0.8169
30.95 0088701
32.30/ 0,8650
24.04 0.5766
25.37 0.6878
26,80/ 0.6757
ZS-NCD
31.35/
31.93 0.8983
33,18/ 0.9026
25.65 0.7113
26.44/ 0.7434
27.64/ 0.7432
JPE22K
24.91 / 0.6997
24.32 0.6676
25.43 0.6550
23.03 / 0.6108
22.65 / 0..5552
23.58 / 0,5668
BM3D
29.79/ 0.8523
28.81 00.8213
008092
22.70/ D.5741
22.17 0.5992
00.5911
DIP
26.60 0.7128
27 85 0,7737
28.90/ 0.7738
24.94 / 0.6512
26. 13 0.7289
27.49 0.72243
25
DD
26.93/007553
40. 0.7832
27.62 67.62/ 0,7496
25. 0.7022
26.04/ 0.7373
26,56 0.7060
S2S
23.32 0.7306
17.95 0.5999
20.69 0.6949
23. 23.40/ 0.7355
20.18 0.6927
23.09 / 0.7674
ZS-N2S
27.30 0.7971
20.39 0.6200
80.89/ 0,6156
26.01 0.7478
21.19 0.6312
21.47 0.6277
ZS-N2N
27.18/ / 0.7173
28.36 0,8001
29.54/ 0.7798
25.40 0.6432
26.75 0.7455
28.21 0.7374
ZS-NCD
28.93 / 0.8079
29.33 0.8351
30.60/ 0.8144
27.10 0,7431
27.60 0.7827
28,77 0.7677
JPEG2K
22.05 0.5794
21.43 0.5295
22.17
0.5055
24.77 0.6811
0.6696
25.52 0.6608
BM3D
26.56/ / 0.7619
25.78 0.77134
27.06/ 0..0477
23.09 / D.5787
23.00/ 0.6281
24.49 / 0.6008
DIP
23.46 / 0.5783
24.82 0.6748
0.6494
26.30 0.7004
27.72/00788/
29. 12. 7845
50
DD
24.01 / 0.6584
24.56 0.6779
24.98 0.6413
26.87 0.7455
27.43 0.7867
27.71 / 0.7543
$2S
17.41 / 0.5200
14,21 0.3933
17.00 0,5325
25.70 0.7896
21.75 / 0.7365
23.88 0.8014
7S-N28
24.74 / 0.6883
20.62 0,5880
20.05 0.5774
27.08 0.7855
20.75 0.6033
20.25 / 0.55993
ZS-N2N
23.52 / 0.557
24.67 0.6444
25. 82 0.6151
27.26 0.7216
28.57 0.88112
30. 13 8076
ZS-NCD
25.58/
25.87 0.7269
27.89/ 0.7464
28.44 0.7914
29.09 / 0.8223
30.60 0.8235
performance across noise levels and image resolutions. For example, ZS-N2S and Se1f2Self degrade on nigh-resolution images, likely due to the limitations of training with masked pixels. ZS-N2N performs well on high-resolution images from Kodak24 but suffers on lower-resolution images in Set13 (192 x 192), as it is trained to map between two downscaled versions of the same noisy image. In comparison, ZS-NCD maintains robust performance across different noise levels and image sizes. The more realistic case of not having access to the noise parameter was discussed in Section 4. In both noise regimes, we use MSE as the loss function. However, for Poisson noise, minimizing the negative log- ligeliiel is also natural choice. We defer the results using this loss to Appendix B.3.
Table 3: First 2 rows: Denoising performance (average PSNR / SSIM of 6 images) under AWGN (0, T) on Mouse Nucle fluorescence microscopy images (image size 128 x 128). Noise levels are 10 and 20. Last row: Real camera denoising performance camera image dataset: PolyU, The images are cropped into size of 512 x 512. We report the average PSNR / SSIM of 6 random images.
JPEG2K
DIP
ZS-N2N
ZS-N25
S25
ZS-NCD
10
32.89 02824
38.65 / 0.9640
86743/08789
37.38/ /09953
36.17/
12,63 0.2966
38.23/ / D.900
20
28.57 / 0.666
34.96 / 0.9296
32.32/0.7689
33.50 0.9092
32.25 0.8532
30.11
0,8600
10.09/ 0,1559
34.71 / 0.9003
Unknown
0.8924
35.71 /0.5066
35.43/0.9408
34.83/0.9395
34.07 /0.9028
23.61 /0.8344
35.667 0.9527
35.84/
Fluorescence microscopy and real camera images To evaluate performance in low-data and domain-shift settings, we test ZS-NCD on Mouse Nuclei fluorescence microscopy images [56), which differ significantly from natural images in structure and texture. We also assess real-world denoising using the PolyU dataset [57], which contains high-resolution images captured by Canon, Nikon, and Sony cameras. Ground-tree images are obtained by averaging multiple captures, while the noisy inputs are single-shot acquisitions. Results are shown in Table 3. ZS-NCD consistently outperforms other learning-based zero-shot denoisers, demonstrating robustness unknown noise models and non-natural image distributions.
Table 4: Comparison of AWGN denoising of Conv and MLP based ZS-NCD on S.t
ZS-NCD
0 25
σ 50
Conv
28.93 / 0,8079
25.58 / 0.7144
MLP
29.52 0.8363
25.89 / 0.7306

---

Figure 4: Denoising Parrot using ZS-NCD, where only a single pixel from each overlapping patch (stride 1) is retained after compression. (AWGN, Ø= = 25.) Each heatmap value indicates the PSNR achieved when denoising is based solely on the pixel that specific location within each patch.
Robustness to overfitting. Most learning- based zero-shot methods are prone overfitting due to the lack of clean targets and the use of overparameterizee networks. In contrast, ZS-NCD, grounded in compression-bast
denoising theory, overcomes this issue given the entropy constraint. To further highlight this key aspect of ZS-NCD, we replace the convolutional encoder-decoder (* 0.4M params) with fully connected MIP (=s 2.3M params) and observe that, instead of degradation, the performance improves using the same À (see Table 4).
Effect of overlapping patch aggregation. As described Section and illustrated in Fig. ZS-NCD denoises each pixel by aggregating outputs from overlapping patches, where each patch is first compressed and then decompressed using learned neural compression model. Intuitively, one might expect the most accurate reconstruction for given pixel to come from the patch in which it lies at the center, as this location benefits from the largest available spatial context, which has been observed in [58].
This observation leads to the question: Does averaging over overlapping reconstructions improve denoising quality, or would suffice to use only the patch where pixel appears at a fixed position (e.g.. the center)? From a computational perspective, both strategies are equivalent, since in both methods every patch is processed, but in averaging scheme, each patch contributes to all the pixels it covers.
To investigate this, we conducted ablation in which, instead of averaging, each pixel (i.j) is reconstructed solely from one of the & x patches in which appears, using a fixed location in the patch (e.g.. top-left, center, etc.). The results shown in Fig. 4, where each heatmap entry reports the PSNR obtained by using only that specific location in the patch for reconstruction. As expected, performance is best when the pixel is centrally located, and degrades it moves toward the patch boundaries.
However, the key observation is that averaging across all overlapping reconstructions yields substantial performance gain. For instance, in denoising Parrot (from Setl dataset), the best single- location reconstruction achieves 25.90 dB (center), while averaging achieves 28. 14 dB, gain of over 2 dB. This highlights the denoising benefit combining multiple noisy views each pixel, consistent with principles from ensembling and variance reduction.
Conclusions
We have studied maximum likelihood compression- -aaeessioo---eeddddd denoising, and provided theoretical char- acterization of its performance under both AWGN and Poisson noise. Furthermore, we introduced ZS-NCD, a new zero-shot neural-comppession--ee based denoising and demonstrated that achieves state-of-the-art performance among zero-shot methods, in both AWGN and Poisson denoising.
The presented theoretical results derived by assuming a fixed (e.g., pre-ttrained/deffned)) compres- sion code. Extending these results to the case of zero-shot learned compression codes is an interesting direction for future research.
Acknowledgment
A.Z., X.C., S.J. were supported by NSF CCF-2237538.
10

---

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14

---

A Proofs
A.1 Auxiliary lemmas
Before stating the proofs of the mains theorems, here state some lemmas that will be used later in the proofs.
Lemma 1. Assume that 0 < am S 01, 02 S aM < oc. Then,
(mm) (02 an)²
QM (02 a1)
< (Poissoson Poisson(oz))
201
am
2011
Lemma 2. Consider independent Poisson random variables Yi,..., Yn, where Y, Poisson(as))
Consider w1..... Wn. R. Let of
Ei=1 wfa; and ww ma..e lws/. Then, for any
t [0, 30
21EM
22 2
I
Σ
W
1 +t S expp 2( wwt/3)
(12)
and
I
Σ
Y
Σ
w1:: -t) s <cxpl
(13)
2 wxt/3)
A.1.1
Proof of Lemma 1
Proof.
DRL ( 02 -i1 + 01 log
02
ai
-a log(1 + ai
Or
(14)
Using the Taylor's theorem,
log(1 +u) = a (a) u²
(15)
2
where (()) = log(1 + u) and a E (0, u). Note that
("(u) (1 x)
Letting 82-a1, 01
for (0,u),
0:M )² < (a) < - Omm )2.
(16)
Okm
aM
Combining (14), (15) and (16) yields the desired result.
A.1.2 Proof of Lemma 2
Proof. Define
n
Un
Wi Yo
willi
and
n
n
0,
w. (Y: a;))²
Σ
w ai.
15

---

Consider > O, then using the Chernoff bound, we have
I Elexp(ssu -Y a:))
P
w aiaa +t
exp(st)
i=1
exp
(ki (ai(e - 1- -8w4) st).
(17)
Note that for E (-1,1), e" - 1- 2(1 v1-v Therefore,
Assuming that S w0xr
then |sws| < 1, for all i.
(sw))
exp(
(ai - 1 ssw))) - st) S expl
015
2(1 sws/3) - -st)
i=1
(sw;)
S expl
) -s()
2(1 SWM/33
i=1
5²00
= expl
- 81).
(18)
2(1
swi /33
Evaluating this bound o7 +wttt/ since 1 SWM/3
33
o it follows that
a
0
1
st.
(19)
2(1 - SwM/3 /3)
2
2 + wxt/3)
To derive the other bound, we can follow the same steps and apply Chernoff bound as done in (17) to get
P Σ w,Y, S pn -t)
S exp(( Σ (e -w: 1 + sw)) st).
(20)
We now use the incquality for u (-1,
2² 2(1 u/3)
1 S
Assume S 1 so that s/w./ S for all Then:
(sw;)²
(e =w0, 1 sw;) <
aj 2(1 + sw, /33)
(sw;)²
4/3
1=1
s² 00
(21)
4/3
Hence,
Y, S pm -1)
52022
< exp
sl
(22)
4/3
Setting s 20 which satisfies < 1
$² 2002 F
3t
t2
st
<
(23)
4/3
2
wwt/3)
16

---

A.2 Proof of Theorem
Proof. Recall that y = x + 2. with 2 is i..d. WVO,, and
arg min ||c u1l. EEC
arg min ||c 2: 1.
cEC
Since both क. are in C,
0 v < j y11
(24)
( x) =112 < I( x) =112
(25)
| x=12 - -2(22 @
x) + || <
| x²|² 2(2, H x) + ||
(26)
|| æ/|² <
| x 2(2, x) + 2(z x)
(27)
|| ==|² W
| z1|² 2| -)) 2| 2 x)|-
(28)
e - denote the error of the compression-basee estimate of ground truth from its noisy version y, and d x denote the distortion from the compressing the ground truth x with the compression code C, then we have
||a| < |||| + 2| (z, c)1 + 2|( d)
(29)
|d| 2 + 2 (z,
|sel|
e
+ 2||||| dll
(z, ||]|| ||
d
(30)
For any possible reconstruction e E C, we define error vector e(c) = c x. Given 41, > 0, define event & and &g
(0)
cEC
E1
Sti
(31)
|lsii|
and
Eg =
22
S t2 $
(32)
||(
respectively. Conditioned on & n &g, follows from (30) that
||l| < |||| + 26, ||el| + |/2]|d]]
(33)
Therefore,
|||| 2t, ||e|| + t < ||d| + 2t2 ||d| + + (tf - 12),
(34)
|||||| t Σ / (|ld] + 12) + (1f -2),
(35)
and finally,
|cl|2 < ||d| + t tu + V -t2,
(36)
where the last line follows because Va + < Vā + Vb, for all a, > To finish the proof we need to bound P(EE, n (o) and set parameters and tz2
Note that for each
(e) |
is unit vector in R". Therefore, E 24 Tel
(c)
N((0
Hence,
IP
< 2exp
(37)
20o
Therefore, applying the union bound and noting that |C|
< 2R.
12
P(E < 2 exp
(38)
20
17

---

and
t2
P(E7) S 2 exp
(39)
2
For 17 (0, 1). set
1 = 0z /22/2222222111 + n))
and
82 02 0=V2ln2Rm
Then,
PPET 89) S nR+2
(40)
Using the selected values of i and 12 in (36) yields the desired result, i.e.,
(2 (1 + 2V7),
- 2112 S Vō + 20
where we have used the fact that 1|dd1 S 6.
A.3 Proof of Corollary
Proof. We start by designing lossy compression code for the set signals Qn. defined as
Qn fo R" ||æ||o k, ||/||>> - 1).
(41)
For k-sparse æ E Qn. let (+) E R denote the k-dimensional vector derived from the non-zero coordinates of æ. We define a lossy compression code (f.g) that achieves distortion 8. Specifically, given a k-sparse a E On the encoder operates as follows:
1. Encode the number of non-zero entries and their locations. This requires at most
T2
ne
loga l: + loga
loga A: + k logz(
k
bits.
2. Quantize the values of the non-zero coordinates such that overall distortion 5 is achieved. To achieve this goal it quantizes each non-zero coordinate of x into bits. Let [zs]o denote the b-bit quantized version of z;. Then, [x; [z.]o] < 2 2-b, Therefore, the overall & distortion can be bounded as
x
g(f(z)) 12 A2 26
Choosing
b= log. nd
ensures that | - (ete-(((x>)) g(f())))
< no.
It can be observed that overall the number of bits required for describing the signals in 2, within distortion can be bounded as
k loga 2
n
+ k logz
+ Ck. T.
where c₁ = loga + k(log, o + 1). Using the defined lossy compression code to solve (1) and applying Theorem 1, it follows that, with probability larger than 2-
x 2112 S V Vo+00 00
k logg(
+
log2
n
+
Ck
2n no'
k
-V8+0,Cy
k: logo(
+
Ct
2n
1o0
n
18

---

where C'
2n Let
n
Then,
1
x
2112 S
1
+o
h:
#log2 k
h
log2
Ck
vn
2n
n
K
+
k
1082
(T)
n
+ 77o
Vn
where
h:
logg
k
Ck
In
2n
n
Finally, since R 2 A (g),
2-
2 1
-ik log()
A.4 Proof of Theorem 2
Proof. Recall that arg mineec L(c; y). Let
a arg min
|| cl22-
cEC
Since both & and ø are in C, we have L(0; y) L(ē; y). or
Σ (ai; yi log 2.) <
(az; log z4).
(42)
Given the input signal æ E IR and e E C, let (oisson((xr) and Poisson(ac) denote the distributions corresponding to independent Poisson random variables with respective means our; and acs, Note that
DRL. DkI (Poisson((ar)) | |Poisson(ac)) (((c: xi) + cat log
(43)
i=1
Adding E,( 0.0; + oa;; log 24) the both sides of (42), it follows that
Σ
(a xi) yı log d. + oa,, log 2i) <
(a(ā; xzi yi log ã. + ax; x;), (44)
1=
or
DEL (Poissson)())) |Poisso((at)) (ooissoo(
Σ (ax; log 0;
s DKL (Poisson (oossoo(az) + Σ (az, yi) log
(45)
Given t1 > 0, define events E and Eg as
(U) - log It vVCEC
Σ =
(46)
and
Eg
(v) az; log 2 -t2 )
(47)
19

---

respectively. Conditioned on & n E2, DRL [Pooisso((az)|lPoosssooo and consequently from Lemma 1,
DKL (Poisson(oar) (aapaannooooPPPosoooo((( (Poissnn((zt) + t 12
(48)
2
T min
alla x 011 -
max
a:|
2112 + (t1 + t2).
(49)
min
To finish the proof, we bound P(Ei n E2) and set 4 and (2-
To bound (P(E), we apply Lemma 2, where for each c, we set (c) = log 1. Then,
o" (c) (log
jaa; S aB²
Zi
i=1 G
1=1
and
WM max [
B
where
3 = log
Tmin
Therefore, using the union bound, it follows that
t7
P(E))= < R exp(
2(a.2 Σn 23 + 311/3)
17
s R exp(
(50)
(ina32mmar + Bt, /3)
To bound (PEE)) we again apply Lemma 2, with w; = log x. and derive
22
P(E)) S exp(
2(a33 21 Tj + 3B2233)
t2
exp( 2(no B2 xmax + /11/3)
(51)
Setting ( and ls such that they are both smaller than 3na33, and noting that Tmax 1, we have
P(E n ( < 2n exp(
+ exp( 4na
(52)
4no;32
Choosing 1 3/ 2 nR(1 a) and to BB 4 nRma, it follows that
P(E n E5) 2
(53)
A.5
Proof of Theorem
Proof. Recall that
arg min ||c y/al3 eEC
a arg min ||c ar/
CEC
Following the similar setup as in Section A.2, we get
c y/all
< || y/al2
c x + x a/a < || - x + x - y/al2
[ - x0|2 2(y/aa xr, Φ x) + y/a m|| < || - x - 2/y (a x. x)) + 1ly/a - a:|2
||
æ1| <
| - x|2 2 2-222/y/ 2, x) + 2 x, à x)
||
x |
<
| x 2 22// à x) 2 x, 5 x)
20

---

Defining e, d and e c E C, as done in the proof of Theorem 1, we have
||| < |||| 2 (y/a z,e)| 2 x. d)/.
(54)
Define events
v - ax)
Sti == 1, I,..., 2
(55)
and
&
W ex (3))
S $12e.
(56)
Conditioned on & n &g, it follows that
|||l
|d| + 2(1 t2)/a,
(57)
Using Lemma 2 with y/ Poisson(ax;;) and e it follows that
t
Us
ax (e)
21
S 2exp
(58)
2( + HMM133)
1=1
where
2
W 04
(c) (1)))22; S naa
(59)
i=1
and
WM
[el S Tmax.
iE(L....,
Using the union bound and noting that C
28 we have
t
P(Ef) S R+1 exp
(60)
(nor max a + t maxx /3)
and
P(ES) < 2exp
(61)
2(n max t max /3)
Setting ' and to such that they are both smaller than 3nx and noting that Tmax < 1, we have
t 4na
t
P( n Eg) 2 exp(
+ + exxp
(62)
4na
For 1) E (0, 1), set
4 = 2/ nlnn 2) + n)RR
and
1/ = 2 mmn(l 2)aR.
Then,
P(E U E3) 2 R2+2 -
(63)
Using the selected values of t and to in (57) yields the desired result, i.c.,
|ell < ||d]| + 4/ n(In 21 + n) R + mm
< ||dll + V nR (4vIn2)
(VI+ vn+1).
21

---

(.)
g6g (.)(.)
log P(-)
(()))(
Conv FC
GDN RoLU
Conv FC
GDN/ ReLU
Conv / FC
Conv/FC
/DN ReLU
Conv/ FC
GDN ReLU
Conv FC
Entropy Modal
encoder
decoder
Figure 5: Neural compression network used for denoising. Conv and FC denote the convolutiona and fully connected layer, respectively. GDN and ReI.U are activation functions.
B Additional experiments and experimental settings
In this section, we provide the details of the networks structures and experimental settings. We also present more experiments for Poisson denoising using MLE and MSE loss functions with unknown noise level.
B.1 Network structure
For our experiments we used 3 convolutional layers in the encoder with 128 number of channels for the first two layers in the encoder (and the last two layers of decoder), see Figure 5. For color images choose the number of channels in the last encoder (and first decoder) layer equals to 32, and for grayscale images equals to 16. The MLP-based network of the ablation study in Section has 3 fully connected layers the encoder with 1024 hidden units for the first two layers in the encoder (and the last two layers of decoder). The number of hidden units in the last encoder (and first decoder) layer equals 16. As activation function we use GDN [32] for Conv network and ReLU for MLP.
B.2 More recent improvements on Zero-shot Noise2Noise
Zero-shot Noise2Noise (ZS-N2N) [24] learns mapping between two downsampled noisy images extracted from the original image. This strategy generating noisy samples was extended two recent concurrent works. Dual-sampling Noise2Noise (DS-N2N) [29] identifies key sub-optimality in ZS-N2N: the network learns to denoise at a lower resolution but is then directly applied to the larger, original resolution. To address this, the authors proposed apply an additional bicubic upsampling to the downsampled images and train the network on both the low-resolution and the upsampled pairs, which boosts the performance of ZS-N2N.
In Pixel2Pixel [30]. the authors identified the local sampling ZS-N2N insufficient to break the spatial correlation of real-world noise. To address this issue, Pixel2Pixel first finds "pixel bank" for each pixel based on the non-local similar patches. Then, by randomly choosing pixels from the bank, it generates multiple noisy pairs to be used pseudo-training dataset for zero-shot denoising.
We have compared the performance of both methods under AWGN and Poisson noise in Tables 5 and 6, respectively. Each paper's 's official code was used to report the numbers in the tables. In AWGN, compression-basee denoiser ZS-NCD outperforms both methods while achieving the second-best performance in Poisson denoising. Pixel2Pixel performs well in Poisson denoising.
Table 5: Average PSNR(dB)/SSI) denoising performance on Kodak24 dataset under AWGN
(O o?). Best results are in bold, second-best are underlined.
15
25
0= 50
ZS-N2N N2N (2023)
32.30 0.8650
29.54/ 0.7798
25.82 / 0.6151
DS-N2N (2025)
32.31 0.8803
29.64/ / 0.8044
25.42 / 0.6378
Pixe12Pixel (2025)
31.31 0.8707
29.89/
0.8098
26.55 / 0.6873
Zs-NCD
33.18 0.9026
30.60
0.8144
27.89 / 0.7464
22

---

Table Average PSNR(dB)SSSM denoising performance on Kodak24 dataset under Poison noise, Poisson(ax)/oo Best results are in bold, second second-best are underlined.
15
25
50
ZS-N2N (2023)
26.80 0.6757
28.21
0.7374
30. 13 / 0.8076
DS-N2N (2025)
27.29 / 0.7016
28.50/ / 0.7540
30.27 / 0.8250
Pixel2Pixel (2025)
28.22/ / 0.7390
29.26 / 0.7891
30.41 / 0.8372
ZS-NCD
27.64 0.7432
28.77 0.7677
30.60 / 0.8235
B.3 MSE and likelihood estimation under Poisson noise without knowing true a
We compare the MSE and MLE distortion for Poisson denoising in Table 7 using the estimated & as explained in Section 4.
Table Minimizing Poisson negative log- likelikl (NLL) vs. MSE with estimated & for Camera- man image in Setl 1. PSNR SSIM are reported here.
Q
MSE (with estimated â) NLL (with estimated à)
15
23.41 0.7554
23.13 0.7567
50
25.22 / 0.7961
24.88 / 0.7460
B.4 Study on factors in patch-wise compression affecting denoising
In this section, we explain the intuition behind why learning compression networks and denoising on overlapped patches is feasible. The centered pixels in the patches are better compressed as empirically observed in [58], thus they can provide better denoising performance. To study the contribution of each patch containing the single pixel be denoised we design the experiment that, in the denoising phase, we denoise the overlapped patches, but only a single pixel at the same location from each patch used to construct the final denoised image, instead of averaging all of them as in (9). We show the denoising performance of each pixel location in Figure 6. The PSNR at each pixel denotes the denoising performance of only using the specific pixel of each overlapped patches with stride 1, We can find that the boundary pixels give lower PSNR, which is consistent with previous research findings that the centered pixels are better compressed. Next, we analyze the effect of patch size in both learning and denoising phases. Given that scaler quantization is applied and the entropy model is learned on latent code of the patches, the compression performance on the latent code is affected by both the patch size and the number of downsampling operations in CNN-basee encoder. We design the experiment that 3 downsampling operators are applied to patch size and 16, where the latent code sizes are 1 x X n6 and 2 x 2 x n2, respectively, where the denoising performance at each pixel location is in Figure 6 (Left) and (Middle), and if we increase the downsampling to for patch size 16, which results in the latent code size to be 1 x 76, the denoising performance is in Figure 6 (Right). We find that spatial size of the latent code to be quantized matters given the scaler quantization limitation, the reconstructed output by the decoder will be restricted by the only correlated latent code as we can observe. Motivated from this, we perform the learning and denoising phases both patch-wise with proper networks structure, all pixels in each patch are used and the overlapped areas are averaged properly to reduce the variance of the compression-basee based estimates.
C Additional numerical results
In this section, we provide the full denoising numerical results of the denoisers on all the test images. All the experiments were run on Nvidia RTX 6000 Ada with 48 GB memory. It takes 40 minutes to denoise a grayscale image of size 256 256, and 50 minutes for an RGB image of size 512 x 768. Adam optimizer is used for training the networks over 20K steps, with initial learning rate of 5 x 10-3 decreased to 5 10-4 after 16K steps for the Conv-based network. The learning rate for MI.P-based networks is 1 x 10-3
23

---

Figure 6: Denoising AWGN (02 = =25) of image Parrot by only compressing a single pixel in each overlapped patches with stride 1. The PSNR at each pixel denotes the final denoising performance by only compressing the pixel at that specific location in each patch. Left: patch size x 8, with downsampling factor equals 8 in fo.; Middle: patch size 16 x 16, with downsampling factor equals 8 in fo; Right: patch size 16 x 16, with downsampling factor equals 16 in fo.-
C.1 Set11 Dataset
For noise levels (15, 25, 50) set À (300, 850, 3000). Similar to Kodak and other experiments we set training epochs to have 20K steps of gradient back propagation. For Poisson denoising = (15, 25, 50) the À= (3000, 1500, 1000). We report the detailed results of AWGN denoising in Table 8, and Poisson noise denoising in Table 9.
Table 8: Setll Denoising performance comparison under AWGN (N o
Mediol
Hoara
Peppers
Startia
Moravch
Turfar
Tauns
Pizee
Coople
KM60
31 2014
3570
53.00-9115
JSNUM90
$1.000 5455
31 1153RLM05
31.511 9632
3504R19055
32 12.10000 8634
31.34018726
32. 1000 8755
3656017664
26
27,700
271228/7411
27.4082000
2734607741 7417
3041 8111
200995.27 8213
2840023000 8387
13330 3344
37.50-258
24 1746
22
2685953485
3V005730
500003 295
26 4400222
98.3.334 6447
26 4630
23 254 1955
1m0 3335
25.5009215
25 N15
2841 6942
28
28
596
3400890
37.77 6538
76 75008551
35 4540 51
20 6311
25-MCP
3085622544
45515535
31.2953844
11.00339955
3153855522
42 5454
31 410400
33.000833 8279
3135012200
2S-NCD (MLP))
86 46.7
52-199 9665
13 6761
11-5918536
RM60
501305505
28 5645
26 5300344
04040 5551
25 9039
25.1498835
3440017266
2591807136
24:3.0
24
36
8841
35 22.521.550 6347
90 4853 9225
2525
2435007120
21:240 185
24
27 17200 115
21.1597333 11:11
25 N35
363461511 TM4
24 2330 4771
55 115000
X 1117777
15 233 1860
1171653113 7017
149053477
154053405
5510554444
55.14 14.45 B05T
22 MASSIE
28.50 1444
2054100000
25,51001545
ZS-NCD (MLP)
17 6575
76-77 8441
8111
7811
1000 7456
KM60
7517
29 5000000
26550 1911
550005933
2528819135
16.000 1873
20:30.285 285
T00 2013
25.110.650 6977
217304607
21 21LLMACI
6015
21-5145777
51100005
25
33
31 11 5061
21 21400220
227599-221 1215
25
22 22100002
2121/2 2411
22 221311200
24.000 6530
24.10155922
2401/1044
25 N35
36 33
26.66 7281
213000 7608
21,000 1514
23 11009966
25 160 2813
7130816:0 6147
267/004883
1139003183
Z5-MCP
23.866361
25 3382517795
23.202333
1385
245587255
13.310 1236
23
20/2210001
21. 23.10918510
(M-L)) (MLP))
74007757
36 950
1745
7546
1587766
76 764
6648
75 (1668550
7104
Table 9: Set1l Denoising performance comparison under Poisson noise (
512 512
Mothod
Czras
House
Peppoo
Staifall
Monncch
Akplacc
Pank
Bashara
Boaua
Paskc
Ooogle
IPR02K
31 99209933
24 4115
23. 130 K213
21 115330 5567
21 2413 6499
30.77 4
23 23.0206575
21 16960 5985
23 (96 5346
75750 5510
25 25.04.053 5337
22.300 5883
THP
23 5385
26 33
24 2300 6666
25,
6875
23.0008895 8938
23.818.5755
23 23.50-5915
24
24558555 5611
5363
23
15:
24 5011
25. 41448888
24060355 6016
23 130 6530
25 23.9000000 6841
25. 5565
338580.50
23 050033955
24 404000 Suns
34 810 5664
25 949 5355
3464
N25
226-2317
11157453 1431
4335 7457
7058
233770-9887
20,
73723 R665
22.0445339
23 7925
22
78133
16
30.270075044
22
21-75815772
23-NCD
24 S
26
24
23, 9544 7700
54.21810558
25 33.50007266
24 34.11044622
25 4844 ASTO
9344 6700
25 2514 6052
2RANNLYESS
230811991 1094017755
22.79551225
37 CUCAAA
7547 6178
454 6011
11 1AA 5644
57 000 9606
75 11115 9605
3043 5601
21 240 1966
DPP
52 2000
4340
T967
21 SE0KAM
6A17
2491445538
6511
2404N05512
23-82N
11
22 1405025-
24
115
24 2251
24 S0N0
24 24100-ATTS
25 2000 A01S
11110 2054
23
25
211560.114
2444119956
21
33
210000055
26
26.1123100
4001
27-2011
25 2ASANIC
25
74 00000MM
21 21.200 A000
5e
5160
S1 TISS
2300n 1150
23 6001
2 SA CTEC
21 1240
20 20 1114
ZASTONCITY
23 100000001
25 6157
24
24
XX 10N
25 2-34418544
51
21 7A66
57 1214
SX. 2USTNESE55
50
25-228
23 12.0001111
21505115 1434
26 24354 3535
1551
27-3541733 7723
25 50500055
21.310 1834
27:48 T002
21.20172215
$25
27 10052165
5550035500
5500
204413 1126
11990.567
11.18850011
27 94577594577
21 140 Trton
38.110
7754
27 22.440 TE55
2S-NCH
28
K261
76
11600 (501
74 164 797%
74
110 T166
1156
24

---

C.2
Set13 Dataset
All images are center-cropped at size of 192 x 192. For this of images we set À (100, 200, 800) and for noise levels dg = (15, 25, 50) and for Poisson denoising we have À = (900, 500, 200) for noise levels a = (15, 25,50), We report the detailed results of AWGN denoising in Table 10, and Poisson noise denoising in Table 11.
Table 10: S Denoising performance comparison under AWGN (0)
2331402574
25788 5135
24 80008817
21.30 26308883
20.33 106
25 RT0URR3
25436300 364
28318 7141
28 99080
35.30433
36688 7543
27348 901
40004101
8711
32
315433314 1141
385403348
31258
160 9m
10.08 1331
11946-306
30563K
2414010001
9011
25771 2003
25006-106
132507701
31588-230
24335 6676
150
14110000
20-323
12-10-294
1330000
10
13 135087611
22450 5T55
955
11 000051
11.00 104
11 kd
1000 1161
31440 135
DO
2
24
24858
36-16769
547608779
11250
809
127762315
15.0/ 2713
12
(6548 540)
32
14:30 2015
25A00
25708-122
21.5607356
25.0633117
Table 11: S Denoising performance comparison under Poisson noise Poisson (ax)) //a
Matted
Balwan
Barbess
Cronpeerd
Cinicc
Fioe
Floeco
Forzus
Him
Foperrs
F973
Avtrae
18
2.4013341
24
5055
T0nn and
54.0 0135
54340 0136
54448
2LA8 944
13
56588 7741
25 25.00AR6
2500
25-NC0
11 405
23 137777991
21300
242902200
19
26589998032
25:500890
36469
23.400710
21-55155400
25446 3424
355/2X
15133
243900631
22008
22
2902
77000 0535
253234
34 66552803
6667771
27419 6236
114000013
34000
34730 3-55
25-110
37461983
23 239102211
21216399
27329
27580 500
27400 7627
01500
1001
1345
25338
2400 mm
511
34008
15 S5aauEs
2:440
31,50 6/6
1120038
25323
13336161
264111
2613 136
25 600LY
3 6.501.04
24538 (D)
21
2011
21 333408
11510 1112
2:333 145
11300000
600
165
36368165
20310
365167755
34.1806311
20060
C.3 Kodak24 Dataset
For Gaussian denoising A = (75, 150, 750) for noise levels 02 = (15, 25, 50) and for Poisson de- noising À (750, 300, 150) for a (15, 25, 50). For BM3D Poisson denoising of = (15, 50) we set ØBM3D = (50, 25, 15). We report the detailed results of AWGN denoising in Table 12, and Poisson noise denoising in Table 13.
Table 12: Kodak24 Denoising performance comparison under AWGN denoising II.
25

---

Table 13: Kodak24 Denoising performance performance under Poisson noise Poisson (aason((aa)) (a2)/aa
C.4 Microscopy Mouse Nuclei Dataset
For these images with noise level a₂ = (10, 20) we set (200, 600), we train the networks for 20K steps to obtain the results. We report the detailed denoising performance in Table 14 and 15 respectively.
Table 14: Denoising performance under AWGN (NO, 2T) )) on fluorescence microscopy dataset: Mouse Nuclei. Images cropped into 128 x 128. Noise level 0=
10.
JPECIK
BM3D
DIP
DD
ZS-N2N
ZS-N2S
S2S
ZS-NCD
22.90007954
38.88A199331
37.31/0.8973
37.73/0.9464
36.37/0.9356
34.70/0.9410
10.88/0.1687
39.03X1.9536
2
32.32/0.8300
37.53/0.9613
$5,93/0.8909
36.46/0.9560
35,26/0.9345
28.78/0.8504
13.08/0.4000
96.83/099546
32.97/0
38,43/0.
16.17/0.8482
37.03/0.9631
35.86/0.9405
31.53/0.9307
12.76/0.3574
37.81/0.966 9634
4
$2.57/0.8418
38.0570.9605 9605
35.82/0.9107
36.7000.9478
14.86/0.9066
120.
14.42/0.3639
37.51/0.9303
$4.54/0.6646 7646
41,53/0 9596
38. 38.09/0.80 8268
40.02/0.9438
39,30/0.9252
29.75/0.7976
10,22/0.
40,93/0, 9420
$2.02/088800
37 37.49/0.9703
89.2770.8999
36.05/0.9628
35.38/0.9491
30.630.89899
14.42/.3931
17.26600.958
Average
32.89/0.8294
38.65/0.9640
36.43/0.8789
37.33009533
36.17/0.9319
31.26/0.8812
12.63/0.2966
38. 38.23/0.9508
Table 15: Denoising performance under AWGN ((0, o2!) on fluorescence microscopy dataset:
Mouse Nuclei. Images are cropped into 128 x 128. Noise level 0g 20.
JPEGIK
BM3D
DIP
DD
ZS-N2N
ZS-N2S
S2S
ZS-NCD
28.37/0.3337
35.10/0.9211
33.09/0. 8485
33.70/0.8938
32.32/0. 8609
32.59/0.8763
9.30/0.0240
34.98/0.884
2
27.97/0.7255
33.800199400
31.41/0.7986
32.39/0.9239
8421
28.42/0.8328
10.73/0.2833
33.75/0.9172
3
28.42/0.7121
34.45019352
31.47/0.7642
32.64/0.9096
31.63/0.866 8660
31.08/0.9003
10.12/0.1807
34.25/0.9216
4
29.31/0.7557
34.3000.9255
31.02/0.5998
32.71/0.9000
31.12018168
30.6000.8551
11.377.1947
33.87/0.8947
5
29.62/0.5932
58.500.9158
35.45/0.7763
37.18/0.9068
35.90008858
32.89/0.86410 8640
8,23/0.0650
37.700.9137
27.71/0.7113
33.6100.3399
31.48/0.7860
32.41/0.9206
31.4340.7700
26.87/0.8312
10.83/0.2328
33,72/0.9245
Average
28.57/0.6986
34.96/0.9296
32.32/0.7889
33.500.9092
32.250085332
30.41/0.8600
10.09/0.1599
34.7180.9993
C,5 Real Camera Noise Dataset PolyU
For these images with unknown noise model/level À 25. Also for BM3D the best peroformance was achieved with setting 0BM3D 15. We report the detailed denoising performance Table 16.
26

---

Table 16: Real camera denoising performance on camera image dataset: PolyU. The dataset includes photos taken from 3 brands of cameras. Randomly selected images are cropped into 512 x 512.
Models
C.plug11
C.bike10
N.flowerl
N.plant10
S.plant13
S.doer10
Average
JPEG2K
36.26/ 0.9615
04.23/0.937)
33.55 /09199
36.74/ 1/44/0.00 0091157
30.39 / 009001
00.9022
34.33 / 0.9225
BM3D
37.15 0.9955
34.85 0.9615
35.81/0.955 /0.9504
38.40/ 0..4100
31.65/ / 0,9466
36.43/ 0.9285
35.71 0.9506
DIP
37.62/ 0.9724
34.85 0.9534
34,93 / 0.9396
37.64 0.99556
31.50/ / 0.9396
36.02/ 0.9145
35.43 / 0,9408
DD
36.79 0..99/0.9722
34.73 0.9566
34.85/ 0.9936
07.84/0.9327
30.91 009305
33.88 0.9084
34.83/ 0.9395
ZS-N2N
36.30/ 0.9621
33.18/ 0.8883
33.28 0,8974
36.21 0088622
30.57 / 009052
34,89/ 0.8804
34.07 / 0.9028
7S N2S
22.76 0.9119 0.9119
20.36 0066/0..00 0.8113
25.20 0.86670
33.63 / 0089900
21.33 0.8256
18.39 0.6966
23.61 / 0.8344
37.75 / 0.9768
33.56/ 0.9545
0.9537
31.93 0,9483
0.9527
ZS-NCD
36,99 / 0.9763
34.79/ 0.9586
35.43/ 0,9489
38.65 0.9449
31.79 0,9464
37.42 0.9451
35.84/ 0.9534
27

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D Visual Comparisons
In this section, we provide more visualization comparison of the zero-shot denoisers. The reconstruc- tion PSNR and SSIM are above the images.
D.1 Kodak24
Grount Truth (x, 1.0)
Noisy (20.17 /0.4078)
BM3D (28.20) / 0.7966)
DD (25.12/ 0.7364)
S25 (17.67 / 0.7054)
ZS-N25 (18.52 / 0.5813)
ZS-N2N (27.99 0.7707)
ZS-NCD (29.21 0.8189)
Figure 7: Kodim24 under additve white Gaussian noise (a, = 25).
28

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Ground Truth (w, 1.0)
Noisy (17.80 / 00.3365)
BMBD (23.97 0.6702
DD (24.57 0.7007)
S2S (18.05 0.7221)
ZS ZS-N25 (14.82 0.5742)
ZS-N2N (26.39 0.7239)
ZS-NCD (27.82 0.8012)
Figure 8: K4dim24 under Poisson noise (a = 25).
29

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D.2
Mouse Nuclei
Ground Truth (s/ / 1.0)
Noisy (22.1 0.3839)
BM3D (35.10/ / 0.9211)
DD (33.70 0.8938)
S2S (9.30/ 0.0240)
ZS-N2S / 0.8763)
ZS-N2N (32.32/ / 0.8609)
ZS-NCD (34.98/ / 0.8843)
Figure 9: Mouse nuclei reconstruction comparison under additive white Gaussian noise (a, = 20).
30

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We claim that our proposed method is zero-shot, the details how to crop patches from the single given noisy image can be found in Figure 1 and Section 4.
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Reviews:
[{"review_id": "quzJDCk2rk", "reviewer": "Reviewer_t45e", "summary": "**Focus Issue**\n\n\nZero-shot denoising: Denoising observations without training samples or clean reference images.\n\n\n**Contributions**\n\n\nThe paper proposes the Zero-Shot Neural Compression Denoiser (ZS-NCD):1.It optimizes on patches extracted from a single noisy image. The final reconstruction is obtained by aggregating outputs from the trained model over overlapping patches.\n2.It provides novel finite-sample theoretical results describing the achievable upper bound on reconstruction error for maximum likelihood-based compression denoisers, further establishing the theoretical foundation of compression-based denoising.\n\n\n**Performance**\n\nZS-NCD naturally avoids overfitting and eliminates the need for manual regularization or early stopping.\nZS-NCD achieves state-of-the-art performance among zero-shot denoisers for both Gaussian and Poisson noise removal tasks.", "questions": "In addition to the doubts raised in the \"Strengths And Weaknesses\" section, the article does not present the performance of the proposed method on completely clean images. Specifically, ZS-NCD achieves the desired results by overlapping and training patches from the same image. This raises the following questions:\n\nWill it significantly degrade the quality of clean images?\n\n\nIs the computational cost acceptable since a model has to be trained for each individual image?\n\n\nDoes a model trained on a specific image possess generalizability? If not, would it be impossible to complete the denoising task for a large number of images within a reasonable timeframe?\n\n\nIf the authors can address these questions, I would consider raising the score.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Quality and Clarity**\n\n\nThe presentation of key issues and solutions is relatively clear, and the figures and tables are displayed in a neat and concise manner. Therefore, I tend to give a positive rating in terms of Quality and Clarity.\n\n\n**Significance and Originality**\n\n\nThe proposed problem, \"Zero-shot denoising via neural compression,\" is valuable, and its innovativeness is also recognizable. However, the baseline models used for comparison are somewhat outdated, with most dating back to 2020 or earlier. Additionally, I do not see a significant leading edge in the performance metrics. Therefore, the claim that \"S-NCD achieves state-of-the-art performance among zero-shot denoisers for both Gaussian and Poisson noise removal tasks\" fails to convince me. Furthermore, the experimental discussion regarding \"naturally avoiding overfitting and eliminating the need for manual regularization or early stopping\" is overly simplistic and almost glossed over. In summary, I tend to give a relatively conservative rating.", "was_revised": true, "final_justification": "We still have some unresolved concerns, and the authors have not provided further clarification. Nevertheless, we believe that this method represents the current state-of-the-art (SOTA), and some concerns are inherent issues within the field. Therefore, we have decided to give a score of 4.", "rebuttal": "We thank the reviewer for their valuable feedback. We first address the specific questions raised, followed by a response to the points noted in the weaknesses section.\n\n>Q1. Will it significantly degrade the quality of clean images?\n\nZS-NCD learns a lossy compression code using the patches extracted from the input noisy image. When the input is a noisy image, enforcing a rate-distortion trade-off via a penalty  on the rate  effectively suppresses noise and yield a denoised image. The denoising performance depends on the rate constraint. To achieve the best performance, one needs to set the rate penalty such that the operating rate-distortion point is consistent with the noise level.  If the input image is a clean noise-free image, the output may or may not be distorted depending on the rate penalty. In the case of no rate penalty ($\\lambda = 0$), ZS-NCD is able to recover an almost lossless reconstruction of the clean image, provided the network bottleneck has sufficient capacity for the required entropy (as distortion goes to zero, required rate grows without bound).\n\nThis contrasts with methods like ZS-N2N, ZS-N2S and S2S, which  inherently cannot  recover a clean image due to disjoint-pixel training strategies or masking.\n\nTo further clarify this point, we empirically evaluated different methods on a clean image (*kodim05* from Kodak24 dataset). The results, reported as PSNR (dB)/SSIM, are shown in the following table. $n_b$ denotes the dimension of latent code in the bottleneck of ZS-NCD.\n\n|| kodim05 ($\\sigma=0$)|\n|:----------:|:------:|\n|DIP  | 42.02 / 0.9913|\n|DD  |28.14 / 0.8736|\n|ZS-N2N  | 33.46 / 0.9845|\n|ZS-NCD ($n_b=32$) | 35.52 / 0.9797\n|ZS-NCD ($n_b=128$) | 54.57 / 0.9997 \n\n\n>Q2. Is the computational cost acceptable since a model has to be trained for each individual image?\n\nTraining a model per noisy image is an inherent aspect of zero-shot learning-based denoisers, which assume no access to training datasets or noisy/clean image pairs. This is true for all zero-shot learning-based denoisers, and not just ZS-NCD.  This implies that compared to semi-supervised and supervised methods, such algorithms require a higher computational complexity during test time. However, the trade-off is justified in cases where access to data is very limited. This is true in various biomedical imaging  and scientific imaging applications.\n\nWe provide timing and hardware details  in Appendix C (lines 638–643). Note that ZS-NCD has not yet been optimized for speed. We believe that there is considerable room for reducing the training time through more efficient architecture designs, weight reuse and adaptive training. The goal of this work is to both empirically and theoretically demonstrate the promise of compression-based zero-shot learning-based denoising, as a novel framework.\n\n>Q3. Does a model trained on a specific image possess generalizability? If not, would it be impossible to complete the denoising task for a large number of images within a reasonable timeframe?\n\nThis is an interesting question. The notion of generalizability is different in  zero-shot learning-based denoisers  compared to  supervised methods. In zero-shot settings, the model is trained and evaluated on a single noisy image. Therefore, generalization beyond the observed image is not a goal. However, exploring generalization across images is an intriguing question. Zero-shot denoising methods, such as DIP and deep decoder, are image-dependent and fail completely when applied to an unseen noisy image. On the other hand, ZS-NCD, due to its compression-based structure and the fact that it is trained on image patches,  shows promising generalizability  when trained on a noisy image and applied to another unseen noisy image. \n\nTo empirically verify this point, in the following two tables, we report the performance of ZS-NCD trained on a noisy image and used to denoise another image, for two different noise levels ($\\sigma=15$ and $\\sigma=50$). In each case, we also report the performance achieved when ZS-NCD is used in its original form, i.e., using the same training and test image. It can be observed that, despite expected performance drop, still ZS-NCD shows reasonable and promising generalization. *kodim01* and *kodim05* are chosen from Kodak24 dataset.\n\n|AWGN ($\\sigma=15$) | Test on kodim01 | Test on kodim05|\n|:----------:|:------:|:------:|\n|**Train on kodim01** | 31.28 / 0.9059 | 26.71 / 0.8944|\n|**Train on kodim05** | 30.29 / 0.8874 | 31.63 / 0.9286|\n\n|AWGN ($\\sigma=50$) | Test on kodim01 | Test on kodim05|\n|:----------:|:------:|:------:|\n|**Train on kodim01** | 26.01 / 0.7245 | 23.59 / 0.7300|\n|**Train on kodim05** | 25.19 / 0.6966 | 25.60 / 0.7608|\n\n\n>W1. More recent baselines comparison.\n\nThe most recent state-of-the-art learning-based zero-shot denoiser we were aware of at the time of submission was ZS-N2N [Mansour2023], published in 2023. ZS-N2N is included in our comparisons as reported for example in Table 1 of the manuscript. We have since become aware of two concurrent zero-shot denoising methods:  DS-N2N [Bai2025] and Pixel2Pixel [Ma2025], pointed to us by Reviewer FvQP. Both algorithms are variations of ZS-N2N [Mansour2023].\n\nIn the revised version of our manuscript we will cite both papers in the Related Work section. \nAdditionally, we have tested DS-N2N and Pixel2Pixel on the Kodak24 dataset using the public code provided by the authors on our own machines, in line with our policy of reproducing all denoisers performances under the same condition for fairness. \nIn the table below, we compare the performance of our proposed ZS-NCD under AWGN. The table reports the average PSNR (dB) and SSIM over the 24 images in Kodak24 dataset.\n\n|| $\\sigma=15$ | $\\sigma=25$ | $\\sigma=50$|\n|:---|:---|:---|:---|\n|ZS-N2N (2023) |  32.30 / 0.8650 | 29.54 / 0.7798  | 25.82 / 0.6151|\n|DS-N2N (2025) | 32.31 / 0.8803 | 29.64 / 0.8044  | 25.42 / 0.6378|\n|Pixel2Pixel (2025) | 31.31 / 0.8707  | 29.89 / 0.8098  | 26.55 / 0.6873|\n|ZS-NCD (ours) | **33.18 / 0.9026**  | **30.60 / 0.8144** | **27.89 / 0.7464** |\n\n[Mansour2023] **ZS-N2N**: Mansour, Y., \\& Heckel, R. (2023). Zero-shot noise2noise: Efficient image denoising without any data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 14018-14027).\n\n[Bai2025] **DS-N2N**: Bai, J., Zhu, D., \\& Chen, M. (2025). Dual-sampling noise2noise: Efficient single image denoising. IEEE Transactions on Instrumentation and Measurement.\n\n[Ma2025] **Pixel2Pixel**: Ma, Q., Jiang, J., Zhou, X., Liang, P., Liu, X., \\& Ma, J. (2025). Pixel2Pixel: A Pixelwise Approach for Zero-Shot Single Image Denoising. IEEE Transactions on Pattern Analysis and Machine Intelligence.\n\n>W2. Clarification on performance claims  of ZS-NCD.\n\nThe claim refers to ZS-NCD's performance  among **learning-based zero-shot denoisers**. To ensure that the wording of the claim is clear, in the revised version, we will emphasize that the comparison is within learning-based zero-shot denoising methods. Specifically, we will update that sentence as follows: **ZS-NCD achieves state-of-the performance among learning-based zero-shot denoisers for both Gaussian and Poisson noise**, as supported by the empirical  results reported in Table 1 and Table 2 of the manuscript. In both tables, we have included the performance of BM3D as a strong classical non-learning-based baseline to provide broader context.\n\n>W3. Clarification on overfitting claims.\n\nThe statement in the abstract is further elaborated  in Section 5, ``*Robustness to Overfitting*'' (lines 282–290) and is empirically supported by Figure 2. Specifically, ZS-NCD leverages the entropy constraint of the  compression-based architecture, which   inherently limits amount of information that can pass through the bottleneck. As a result, the model avoids overfitting, despite being trained on all noisy pixels.  This is a key advantage of zero-shot compression-based denoising compared to other zero-shot approaches. \n\nThis claim is further verified in Figure 2, which shows  the PSNR of the denoised image over training iterations. It can be observed that the PSNR of ZS-NCD improves steadily and does not show degradation that is typical to overfitting. In contrast, in other zero-shot denoising methods overfitting is a known issue and is typically mitigated manually. For example,\n\n- DIP and Deep Decoder rely on manual early stopping or under-parameterization.\n- ZS-N2S, S2S and ZS-N2N modify the loss function to exclude some pixels, which is another form of manual regularization.  \n\nIn contrast, ZS-NCD does not require any of these manual heuristics and architectural constrains. In our approach, overfitting is prevented naturally by the compression-based  bottleneck that does not allow the model to reproduce the high-entropy noisy input image. We believe that this is a key advantage of our method.\n\n>Paper formatting concerns.\n\nWe will update the manuscript to address all the formatting issues and have detailed explanations for the figures in the appendix."}, {"review_id": "wPDzJOW8gc", "reviewer": "Reviewer_FvQP", "summary": "This paper is composed of two parts. The first part focuses on the relationship between compression and denoising, and presents several theoretical results that give upper bounds to the denoising performance of a compression-based denoiser. The second part applies compression-based denoising to the \"zero-shot\" setting, where only a single noisy image is available. The original high-resolution image is broken up in small patches that are used to train a neural compressor. Numerical experiments demonstrate the method performs comparably to other zero-shot baselines.", "questions": "- Do the authors expect the dependence of their bounds in rate $R$ and distortion $\\delta$ to be tight?\n- The authors propose to select the RD tradeoff parameter $\\lambda$ such that the distortion of the compressed noisy patches becomes comparable to the noise level. Another natural choice would be to optimize the upper bounds given in the Theorems along the RD curve. For instance, Theorem 1 suggests to set $\\lambda = (8\\ln 2)(1 + 2\\sqrt \\eta)^2\\frac{\\sigma_z^2}{n}$. How do these two criteria compare?\n- More broadly, I am surprised that \"compression performs better at denoising than denoising\". That is, why does compressing 8x8 patches would outperform an approach tailored to denoising such as SURE or any other mentioned in the paper? The authors mention an increased robustness to overfitting, but it is not intuitive to me why a compressor would overfit less than a denoiser (e.g., in SURE, the divergence of the denoiser is explicitly regularized).", "limitations": "yes", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Compression-based denoising\n\nThe paper makes novel contributions to compression-based denoising through Theorems 1 to 3, which show that better compression performance (as measured by rate and distortion) leads to an improved upper bound on denoising performance. However, it is in my opinion difficult to evaluate these results due to the choice of setting. Indeed, denoising performance is here measured in the worst case over a deterministic class of signals, rather than average MSE over a signal distribution as is more common. In particular, this is the metric that is used in the numerical comparisons of Table 1. Thus, it is unclear if additional factors compared to optimal MMSE performance (such as $\\log_2 n$ in Corollary 1) are due to this choice of setting, looseness of the bounds, or a fundamental underperformance of compression-based denoising.\n\nRelated to the above point, the authors propose to select the codeword $c$ via a maximum a posteriori procedure, rather than minimum mean squared error (eqs. (2) vs (3)). Although it is stated that the former is always the best choice, it seems to me that the upper bound in Theorem 3 becomes more advantageous to that of Theorem 2 in the regime of small rates and large distortions.\n\n## Application to zero-shot denoising\n\nA strength of the paper is its experimental results, which show superior zero-shot denoising performance to other approaches. I note that there are other very recent concurrent works, which build on ZS-N2N and may be relevant to this paper:\n\n> Bai, J., Zhu, D., & Chen, M. (2025). Dual-sampling noise2noise: Efficient single image denoising. IEEE Transactions on Instrumentation and Measurement.\n>\n> Ma, Q., Jiang, J., Zhou, X., Liang, P., Liu, X., & Ma, J. (2025). Pixel2Pixel: A Pixelwise Approach for Zero-Shot Single Image Denoising. IEEE Transactions on Pattern Analysis and Machine Intelligence.\n\nAs defined by NeurIPS policy for concurrent works, these works do not qualify as prior work, but could be of interest to the authors.\n\nA critical component of the method is the chosen patch size. I could not find it mentioned in the paper (though inspection of the figure indicates that it might be $8 \\times 8$?). How was it selected? It seems to me that different choices of patch sizes would greatly affect the results. For instance, a small patch size limits the range of spatial dependencies in the signal that can be exploited by the denoiser, effectively limiting it to small noise levels. However, a large patch size leads to a small number of patches available to train the compressor.", "was_revised": true, "final_justification": "I have no remaining concerns.", "rebuttal": "We thank the reviewer for their valuable feedback. We first address the specific questions raised, followed by a response to the points noted in the weaknesses section. In our response to Question 1, we also address some of the comments regarding the choice of settings in the theoretical results, as mentioned in the Strengths And Weaknesses.\n\n>Q1. Tightness of bounds in Section 3.\n\nOur analysis is framed in a deterministic, high-probability setting: we provide uniform high-probability guarantees on reconstruction error for signals in a fixed class, without assuming any prior distribution. This framework is common in the information-theoretic literature on compression-based inference (e.g., compression-based compressed sensing), and it offers practical advantages, particularly in the zero-shot denoising scenario where the underlying data distribution may be unknown or difficult to model. The deterministic nature of our bounds ensures they hold uniformly over the signal class, rather than in expectation.\n\nRegarding the tightness of the bounds, in the discussion following Corollary 1, we have compared our result with the optimal Bayesian performance (in terms of information dimension). In the revised version, we will also include a comparison to known minimax rates for $k$-sparse signal recovery under Gaussian noise. For the case $k=k_n$, such that $k_n/n\\to 0$, as $n$ grows without bound,  the minimax  risk in known to scale as ${\\sigma_z^2 k \\log(n/k) \\over n} $ [Johnstone2019] [Donoho1994] [Donoho1992]. This further confirms the tightness of our bound in the worst-case setting, and potential optimality of compression-based compressed sensing. \n\nFinally, while in Table 1 we report average denoising performance, our method is designed to perform well on each individual instance, consistent with the worst-case theoretical guarantees we provide. This aligns with the goals of zero-shot denoising, where robustness to the specific realization is more critical than expected-case optimality.\n\n[Johnstone2019] Johnstone, I. M. Gaussian Estimation: Sequence and Wavelet Models.\n\n[Donoho1994] Donoho, D. L., et al. Minimax risk over $\\ell_p$ balls for $\\ell_q$ losses.\n\n[Donoho1992] Donoho, D. L., et al. Maximum entropy and the nearly black object.\n\n>Q2. Selecting $\\lambda$ based on theoretical upper bounds.\n\nTheorem 1  provides a non-asymptotic upper bound on the reconstruction error, assuming that a compression code with a given rate $R$ and distortion $\\delta$ is available. The constants appearing in the bound (such as $(8 \\ln 2)(1 + 2\\sqrt{\\eta})^2$) characterize the relationship between rate and MSE, but they are not intended to prescribe a specific choice of the Lagrange multiplier $\\lambda$ used in training.\n\nIndeed, in principle, one could consider minimizing the upper bound in Theorem 1 as a function of $\\delta$ (or $R$), which in turn could suggest an “optimal” rate-distortion operating point for denoising. However, solving such an optimization is highly non-trivial. Specifically, note that the upper bound in Theorem 1 can be written as\n$$\nu(\\delta) = \\sqrt{\\delta} + \\sigma_z \\zeta \\sqrt{R(\\delta)},\n$$where $\\zeta = 2 \\sqrt{\\frac{2 \\ln 2}{n}}(1 + 2\\sqrt{\\eta})$, and $R(\\delta)$ is the rate corresponding to distortion $\\delta$. Differentiating and setting $u'(\\delta) = 0$ gives:\n$$\n\\frac{1}{2\\sqrt{\\delta}} + \\sigma_z \\zeta \\cdot \\frac{R'(\\delta)}{2\\sqrt{R(\\delta)}} = 0.\n$$However, since the function $R(\\delta)$ is generally not known in closed form for real-world data distributions or learned compressors, this optimization cannot be carried out analytically or even tractably in practice.\n\nIn contrast, during training, $\\lambda$ is used as a Lagrange multiplier in a rate-distortion objective to implicitly select an operating point on the empirical RD curve of the single image. This RD curve is not analytically accessible and varies from image to image, making the mapping from $\\lambda$ to $(R, \\delta)$ nontrivial and data-dependent.\n\nOur proposed heuristic, choosing $\\lambda$ such that the distortion roughly matches the noise level, is based on signal recovery intuition and performs well empirically, as demonstrated in our experiments and in Section 4, \"*Setting the hyperparameter $\\lambda$*\".\n\n>Q3. Seemingly surprising effectiveness of compression at denoising.\n\nThis is an insightful and intuitive question. While it may seem surprising that a compressor, which is not explicitly trained to denoise, can outperform methods designed specifically for denoising, the idea of using compression codes to solve inverse problems such as denoising and compressed sensing has a long history in information theory and signal processing. Prior work has shown that lossy compression codes can be used as a powerful implicit prior for solving inverse problems (see, e.g., [Donoho2002, Weissman2005, Jalali2016, Rezagah2017, Chang1997, Chang2000, Natarajan2002]). \n\nIn the zero-shot setting, where only a single noisy observation is available and no clean data is accessible, most learning-based denoisers rely on indirect forms of regularization to prevent overfitting. These include masking strategies (ZS-N2S), architectural bias (DIP), or underparameterized networks (Deep Decoder). While useful, these techniques are often heuristic and do not provide a principled way to control information flow from input to output.\n\nIn contrast, compression-based denoising  imposes a fundamental information-theoretic constraint via the rate-distortion trade-off. This restricts the number of bits that can pass through the model's bottleneck, acting as a principled regularizer that prevents the model from memorizing noise. The model is therefore implicitly biased toward compressible (structured) reconstructions, which aligns well with the nature of clean signals.\n\nAlthough SURE provides an unbiased estimate of the MSE under Gaussian noise with known variance, its effectiveness depends on accurate noise modeling and stable estimation of the divergence term. In practice, especially in the zero-shot setting with a single noisy image and no access to clean data, these conditions may not hold. Moreover, SURE implementations often rely on Monte Carlo approximations of the divergence, which can introduce significant variance, particularly in deep neural networks.\n\nThus, while traditional denoising methods are explicitly designed for the task, compression-based approaches like ZS-NCD offer a robust and principled alternative in data-scarce regimes, combining inductive bias with effective optimization.\n\n[Chang1997] Chang, S. G., et al. Image denoising via lossy compression and wavelet thresholding.\n\n[Chang2000] Chang, S. G., et al. Adaptive wavelet thresholding for image denoising and compression.\n\n[Donoho2002] Donoho, D. L. The kolmogorov sampler.\n\n[Natarajan2002] Natarajan, B. K. Filtering random noise from deterministic signals via data compression.\n\n[Weissman2005] Weissman, T., et al. The empirical distribution of rate-constrained source codes.\n\n[Jalali2016] Jalali, S., et al. From compression to compressed sensing.\n\n[Rezagah2017] Rezagah, F. E., et al. Compression-based compressed sensing. \n\n[Soltanayev2018] Soltanayev, S., et al. Training deep learning based denoisers without ground truth data.\n\n[Metzler2018] Metzler, C. A., et al. Unsupervised learning with stein's unbiased risk estimator.\n\n>W1. Better bound in Theorem 3 over Theorem 2.\n\nNote that while the bound in Theorem 2 is on MSE, the bound in Theorem 3 is on RMSE. To compare  the results of the two theorems, we can square the bound in Theorem 3, which shows that while the MSE in Theorem 2 scales as $\\sqrt{R/n}$, the MSE in Theorem 3 scales as $R^2/n$. This suggests  that, unlike in Theorem 2, for the bound in Theorem 3 to be small,  the rate $R$ cannot scale linearly with $n$. Therefore, minimizing the log-likelihood over a compression code should, in principle, yield better performance under Poisson noise. However, due to the highly non-convex nature of the log-likelihood loss, minimizing the MSE loss may lead to better performance in practice. A brief discussion of this observation, along with empirical results, is provided in Appendix B.2.\n\n>W2. Two concurrent zero-shot denoisers.\n\nWe thank the reviewer for bringing these results to our attention.  The recent works advance the ZS-N2N framework. We will cite both references in the Related Work Section (Section 2) of our manuscript. Additionally, we have tested DS-N2N and Pixel2Pixel on the Kodak24 dataset under AWGN using the public code provided by the authors. Average PSNR/SSIM over the 24 images are reported in the table below. \n\n||$\\sigma=15$| $\\sigma=25$| $\\sigma=50$|\n|:-:|:-:|:-:|:-:|\n|DS-N2N|32.31 / 0.8803|29.64 / 0.8044|25.42 / 0.6378|\n|Pixel2Pixel|31.31 / 0.8707| 29.89 / 0.8098|26.55 / 0.6873|\n|ZS-NCD (ours)|**33.18 / 0.9026**|**30.60 / 0.8144**|**27.89 / 0.7464**|\n\n[Bai2025] **DS-N2N**: Bai, J., et al. Dual-sampling noise2noise: Efficient single image denoising.\n\n[Ma2025] **Pixel2Pixel**: Ma, Q., et al. Pixel2Pixel: A Pixelwise Approach for Zero-Shot Single Image Denoising.\n\n>W3. Patch size in ZS-NCD.\n\nWe thank the reviewer for pointing this out. We indeed use a patch size of $8 \\times 8$ in all experiments and will clearly state this in the revised version of the paper.\n\nEmpirically, we observe that ZS-NCD is fairly robust to patch size, as long as the size is within a reasonable range, i.e., large enough to contain meaningful structure, but small enough to ensure enough training data per image.\n\nTo support this, we include below denoising results of ZS-NCD for different patch sizes, reported as PSNR (dB)/SSIM, on the *kodim05* image from the Kodak24 dataset under AWGN.\n\n|$\\sigma$| 4x4 | 8x8 | 16x16 | 32x32|\n|:-:|:-:|:-:|:-:|:-:|\n|15|31.76 / 0.9246|31.63 / 0.9286|31.66 / 0.9285| 31.70 / 0.9280|\n|25|28.46 / 0.8439|28.81 / 0.8615|29.01 / 0.8647|28.92 / 0.8621|\n|50|24.79 / 0.7264|25.60 / 0.7608| 25.58 / 0.7608|25.60 / 0.7613|"}, {"review_id": "q5bHHzWuSc", "reviewer": "Reviewer_TJvX", "summary": "The authors propose a novel method for zero-shot image denoising via neural compression in the single-image setting. Specifically, given a single noisy image, an encoder-decoder neural network with a rate constraint is trained to compress and decompress overlapping patches extracted from the image. Once trained, the denoised image is reconstructed by applying the compression-decompression process to all patches and aggregating the outputs. The approach builds on information-theoretic principles, assuming that clean images are more structured and thus more compressible than noisy ones. The rate constraint naturally biases the model toward reconstructing structured (clean) components while suppressing unstructured (noisy) variations.", "questions": "I do not have questions for the authors.", "limitations": "The authors address the limitations in section 6.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "The proposed method is well-justified theoretically and backed by finite-sample performance guarantees for both AWGN and Poisson noise models. Extensive experiments are conducted on diverse image domains (natural, microscopy, real-world camera images) and across multiple noise types. The results are consistently strong and demonstrate state-of-the-art performance among zero-shot methods. The architecture and optimization procedure are well-detailed, and the authors provide reproducibility information.", "was_revised": true, "final_justification": "I have no concerns about this paper.", "rebuttal": "We appreciate the reviewer’s positive feedback."}]

Author rebuttal:

Meta-review: This paper proposes a new approach for "zero-shot" denoising (i.e., image denoising with machine learning models without requiring any prior training data) based neural compression. The authors provide a finite-sample theoretical guarantee in terms of an upper bound to the reconstruction error for their proposed approach for Gaussian and Poisson noise models. Experimental evaluation demonstrates the performance of their method, showing state of the art results.

**Strengths**
- Nice balance between theoretical analysis and strong-performing method.
- Intuitively simply approach based on information-theoretic notions.
- State of the art results for zero-shot denoising.

**Weaknesses**
- Somewhat limited and outdated baselines in their experiments.
- Some gaps between their theoretical analysis and the reported metrics in their results.
- Some parts of the presentation can be improved (e.g. in images in the appendix).

**Discussions and conclusion**

This paper received 3 reviews, and reviewers were mildly supportive of this paper initially. However, the discussion period was productive, and the authors have provided (i) further clarifications on their theoretical analysis, (ii) further experimental results on other datasets. This further increased the support. There's consensus that this paper represents an interesting and strong contribution to the field of image denoising in general, and it's worthy of publication - and I concur.
Decision: Accept (spotlight)