Title: Zero-shot Denoising via Neural Compression: Theoretical and algorithmic framework
Venue: NeurIPS 2025
Accepted: yes
Keywords: compression-based denoising

zero-shot image denoising

neural compression
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

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 compressionbased 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 .

1 Introduction

Background and motivation Denoising is a fundamental problem in classical signal processing and has recently gained renewed attention from the machine learning community. Let x = ( x 1 , . . . , x n ) ∈ R n + denote a non-negative signal of length n , where signal x is not observable in many systems. Instead, we observe a noisy version y = ( y 1 , . . . , y n ) , where the observations are conditionally independent given x , and each entry is distributed according to a common conditional distribution:

We assume that the noise mechanism is memoryless (independent across coordinates) and homogeneous (identical across entries). The goal of a denoising algorithm is to estimate x from the noisy observations y . Given its prevalence in imaging and data acquisition systems, denoising has been a central topic in signal processing for decades. Classical denoising methods rely on explicit structural assumptions about the underlying signal x , often hand-crafted by domain experts [1, 2, 3, 4, 5, 6, 7, 8]. In contrast, recent advances in machine learning have enabled a new class of data-driven denoising

set of overlappingpatches as

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algorithms. These methods learn the optimal denoising function from data, leveraging statistical patterns directly from signal and noise distributions.

While learning-based approaches achieve state-of-the-art performance and often outperform classical methods in controlled settings, they face significant challenges in practice:

Supervision requirement: Most learning-based methods require training set of paired samples { ( y i , x i ) } i m =1 , where x i is clean signal and y i is its noisy counterpart. In practical scenarios such as medical imaging, { x i } i m =1 are unavailable or prohibitively expensive to obtain.

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 necessitates 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 to individual inputs with general purpose.

From neural compression to zero-shot denoising Denoising algorithms-ranging from classical signal processing techniques to deep learning methods-fundamentally rely on the assumption that real-world signals are highly structured. Compression-based denoising leverages this same principle, but rather than directly solving the inverse problem, it instead performs lossy compression on the

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Noisy (20.18 / 0.5408)

Restormer (31.44 / 0.9218)

JPEG2K (25.55 / 0.7728)

BM3D (27.56 / 0.8236)

DIP (26.40 / 0.7875)

DD (25.97 / 0.7902)

S2S (19.02 / 0.5867)

ZS-N2S (20.70 / 0.6776)

ZS-N2N (27.66 / 0.8231)

ZS-NCD (28.81 / 0.8615)

noisy observation y , under the hypothesis that the clean signal x lies in a 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 to noisy data, the intuition is that a lossy compressor-operating at a 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-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 on 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 are 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-without 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 in overlapping regions. This approach is illustrated in Figure 1.

Despite relying solely on 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 on inputs that lie outside the natural image distribution. We compare it against the baselines in Figure 2, ZS-NCD shows superior performance in denoising and training stability.

Paper contributions This paper introduces a zero-shot image denoising framework based on neural compression. Our main contributions are:

A zero-shot denoising algorithm using neural compression. We propose a fully unsupervised method that trains a 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 architectureagnostic and leverages only the structure present in the observed noisy image.

Theoretical results connecting denoising and compression performance. We establish finitesample upper bounds on the reconstruction error of the proposed compression-based maximum likelihood denoisers, for both Gaussian and Poisson noise models.

Extensive empirical validation. We demonstrate that our method achieves state-of-the-art performance among zero-shot denoising techniques across a range of noise models and datasets.

2 Related work

Self-supervised and zero-shot denoising Supervised learning-based denoisers such as DnCNN [25] and Restormer [19] achieve state-of-the-art 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], Noise2Void [11], Noise2Same[26] 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 recently, 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-image context, but require early stopping or under-parameterization to avoid overfitting. Self-supervised variants suffer from masking-induced information loss. Hybrid approaches, such as masked pretraining-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-based lossy compression, often referred to as neural compression, uses an autoencoder architecture combined with an entropy model to estimate and constrain the bitrate at 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 formalized by Donoho [14], who introduced the minimum Kolmogorov complexity estimator, and further refined by Weissman et al. [15], showing that, under certain conditions on both the signal and the noise, optimal lossy compression of a noisy signal-followed by suitable post-processing-can asymptotically achieve optimal denoising performance. Prior to these theoretical developments, early empirical methods such as wavelet-based 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-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 a two-step denoiser based on neural compression, trained on a single noisy image. It achieves state-of-the-art performance across both AWGN and Poisson noise models. Moreover, we contribute new theoretical results that advance the foundations of compression-based denoising.

3 Compression-based denoising: Theoretical foundations

Lossy compression Let Q ⊂ R n denote the signal class of interest, such as vectorized natural images of a fixed size. A lossy compression code for Q is defined by an encoder-decoder pair ( f, g ) , f : Q → { 1 , . . . , 2 R } , and g : { 1 , . . . , 2 R } → R n . The performance of a lossy code is characterized by: i) Rate R , indicating the number of distinct codewords; ii) Distortion δ , defined as the worst-case per-symbol mean squared error (MSE) over the signal class:

The set of reconstructions produced by the decoder forms the codebook:

Compression-based denoising We propose compression-based denoising as a structured maximum likelihood (ML) estimation. Given a noisy observation y ∼ ∏ n i =1 p ( y i | x i ) and a a lossy compression code ( f, g ) for Q , the compression-based ML denoiser solves

This formulation leverages the fact that clean signals, by virtue of their structure, are more compressible 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 case of AWGN: y = x + z , where z ∼ N ( 0 , σ 2 z I n ) , the described denoiser simplifies to:

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 y i is modeled as a Poisson random variable with mean αx i : y i ∼ Poisson( αx i ) . Under this model, the compression-based ML denoiser simplifies to

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 an alternative loss based on a normalized squared error between c and the rescaled observations:

Theoretical analysis We begin by analyzing the performance of compression-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 proofs can be found in Appendix A.

Theorem 1. Assume that x ∈ Q and let ( f, g ) denote a lossy compression for Q that operates at rate R and distortion δ . Consider y = x + z , where z ∼ N ( 0 , σ 2 z I n ) . Let ˆ x denote the output of the compression-based denoiser defined by ( f, g ) as in (1) . Then,

with a probability larger than 1 -2 -ηR +2 .

This bound decomposes the denoising error into two terms: a distortion term √ δ , which reflects the approximation quality of the compression code, and a 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 non-asymptotically and does not assume the code is optimal, only that it provides a distortionδ covering of Q . This highlights that even non-ideal compression codes can enable effective denoising, provided the rate-distortion tradeoff is well-calibrated.

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 Q n denote the set of k -sparse vectors in R n satisfying ∥ x ∥ ∞ ≤ 1 . Fix a parameter η ∈ (0 , 1) , and suppose y = x + z where z ∼ N (0 , σ 2 z I n ) . Then, there exists a family of compression codes such that, when used with the denoiser defined in (1) , with a probability larger than 1 -e -ηk log( n/k ) , the estimate ˆ x satisfies

where C = 2(1 + 2 √ η ) √ 2 ln 2 and γ n = k log 2 k 2 n + log 2 k + k (log 2 e+1) n .

Corollary 1 provides a high-probability bound on the normalized error 1 √ n ∥ ˆ x -x ∥ 2 . Squaring both sides and having ( √ a + √ b ) 2 ≤ 2( a + b ) , it follows that, with high probability

Thus, up to universal constants, the dominant term in the upper bound scales as σ 2 z k n log( n k ) . 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 2 n is an artifact of our proof or a real barrier to 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.

We next extend our analysis to signal-dependent noise model. Poisson noise is particularly relevant in imaging applications such as microscopy and astronomy, where photon counts vary with signal intensity. Unlike Gaussian noise, Poisson observations induce a 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 as in Theorem 1. Assume that for any x ∈ Q , x i ∈ ( x min , x max ) , where 0 < x min < x max < 1 . Assume that y 1 , . . . , y n are independent with y i ∼ Poisson( αx i ) . Let ˆ x denote the solution of (2) . Let C 1 = x 5 max / ( x 2 min ) and C 2 = x 2 max x 3 min β √ ( 4 ln 2 )( √ 1 + η + √ η ) . Then, with a probability larger than 1 -2 -ηR +2 ,

Theorem 3. Consider the same setup as in Theorem 2. Let ˆ x denote the solution of (3) . Let C = 4 √ ln 2( √ 1 + η + √ η +1) . Then, with a probability larger than 1 -2 -ηR +2 ,

Remark 1. Theorems 2 and 3 show that, in the case of Poisson noise, minimizing either the ML loss function or the computationally efficient MSE loss function can recover the signal. This result is also consistent with our simulations reported later in Section 5.

4 Zero-shot compression-based denoiser

We refer to a 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 a low-complexity reconstruction from the output of a 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 or designed using classical methods, and applied independently of the noisy input. This setup is not zero-shot, as it relies on prior training or 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 is trained directly on patches extracted from a single noisy image, without access to clean targets or external data. This section describes the ZS-NCD architecture and optimization procedure in detail.

Proposed zero-shot denoiser: ZS-NCD Let P ( i,j ) : R h × w → R k × k denote the patch extraction operator, which returns a k × k patch whose top-left corner is at pixel ( i, j ) . Let f θ 1 and g θ 2 denote the encoder and decoder networks, parameterized by weights θ 1 and θ 2 , respectively. Define I as the set of all coordinates ( i, j ) ∈ { 1 , . . . , h -k +1 } × { 1 , . . . , w -k +1 } from which a valid k × k patch can be extracted.

Given a single noisy image y , the ZS-NCD is trained to minimize the following patchwise objective:

where P ( f θ 1 ( P ( i,j ) ( y )) ) denotes the likelihood (or entropy model) of the latent code produced by the encoder, and λ > 0 is a hyperparameter controlling the trade-off between fidelity and compressibility.

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In (8), K = k 2 , and the function L K : R K × R K → R + is a distortion loss determined by the noise model, as defined in Section 3. For example, in the AWGN case, L K corresponds to the squared ℓ 2 norm of the distance between a noisy patch and its neural compression reconstruction. Note that f θ 1 maps the input into a discrete latent space, which is non-differentiable and thus incompatible with standard gradient-based optimization. To address this, we follow the neural compression framework of [32], using a continuous relaxation during training (e.g., uniform noise injection) and applying actual discretization only at test time. The entropy term P is modeled using a factorized, non-parametric density [34].

After training, the denoised image is obtained by applying the encoder and decoder to each patch and averaging the overlapping outputs. For each pixel ( i, j ) , let I ( i,j ) ⊂ I denote the set of patch locations such that P ( i ′ ,j ′ ) includes the pixel ( i, j ) . The final estimate at location ( i, j ) is given by

where |I ( i,j ) | denotes the number of patches covering pixel ( i, j ) , and ·| ( a,b ) denotes the ( a, b ) -th pixel of the patch output. For interior pixels away from the boundary, |I ( i,j ) | = K . As shown later in Section 5, this aggregating of reconstructed patches significantly enhances denoising performance.

Setting the hyperparameter λ The ZS-NCD objective in (8) includes a hyperparameter, λ , which balances reconstruction fidelity and compressibility. Interpreted through the lens of lossy compression, varying λ 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 λ be selected to optimize denoising performance? In the following, we explain our approach for setting λ under both AWGN model and Poisson noise model.

Case I: Gaussian noise. Let ˆ x denote the output of the ZS-NCD denoiser, and consider the AWGN model y = x + z with z ∼ N (0 , σ 2 z I n ) . Then,

the first term is the noise variance, and the second is the true denoising error. While z and ˆ x are not fully independent, they are intuitively weakly correlated in successful denoising regimes, where the estimate ˆ x depends only indirectly on the noise. Thus, the cross term is expected to be small: 1 n E [ z ⊤ ( x -ˆ x ) ] ≈ 0 . This approximation suggests that, ideally, under low-noise regimes, 1 n ∥ y -ˆ x ∥ 2 2 is expected to be close to σ 2 z . Based on this insight, we propose a simple and effective heuristic for choosing λ : select λ such that 1 n ∥ y -ˆ x ∥ 2 2 is closest to the known noise variance σ 2 z . This procedure can be implemented efficiently via a tree-based search strategy, as described in Algorithm 1. To apply Algorithm 1, one needs an estimate of the noise power σ 2 z . This is 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 of wavelet coefficients.

Finally, we observe that the performance of ZS-NCD is relatively robust to the choice of λ . For instance, on the Nuclei dataset (Figure 3), ZS-NCD outperforms the state-of-the-art zero-shot learning-

based denoiser, ZS-Noise2Noise, across a wide range of λ values, for both σ z = 10 and σ z = 20 . A similar approach can also be applied to the case of Poisson noise, as well.

Case II: Poisson noise In the case of Poisson noise, in addition to estimating λ , we need to estimate α , which is used to normalize the measurements, in both the MSE-based and the MLE-based methods. Note that in the case of Poisson noise, E [ y i ] = αx i , and therefore, with high probability, 1 n ∑ n i =1 y i ≈ α 1 n ∑ n i =1 x i . Using this observation, and assuming that 1 n ∑ n i =1 x i ≈ 0 . 5 , we estimate α as ˆ α = 2 n ∑ n i =1 y i . We then use the estimated noise level ˆ α to normalize both MSE and MLE based optimization for denoising Poisson noise. See Table 1 for the result of this noise parameter estimation on a sample image. Having an estimate of noise parameter α , we then follow a similar procedure we used in the case of AWGN to set the parameter λ . Specifically, we write the MSE between normalized y and the denoised image ˆ x as

Again, assuming that we are in a low-noise regime, i.e., the second and the third terms in (11) are close to zero, and using 1 n ∑ n i =1 x i ≈ 0 . 5 approximation, it follows that 1 n E [ ∥ y /α -ˆ x ∥ 2 2 ] ≈ 1 2 α . This implies that, in the case of Poisson noise, we can still use Algorithm 1 to find λ , after updating ∥ ˆ x ( k ) -y ∥ 2 > nσ 2 z to ∥ ˆ x ( k ) -y ∥ 2 > 1 2 α . We empirically observe that MSE( y /α , ˆ x ) in training our networks is close to 1 2 α as reported in Table 1, which indicates that 1 2 α is a good approximation of the MSE that can be used as a threshold for selecting λ . When α is not known, we obtain the estimate ˆ α , and use 1 2ˆ α as a valid threshold to set λ .

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 learning-based methods. All baselines are dataset-free, i.e., they operate solely on the noisy image to be denoised. For non-learning methods, we include JPEG-2K and BM3D. Although rarely used as a denoising baseline, JPEG-2K provides a useful point of comparison from the perspective of compression-based denoising, as it represents a fixed, pre-defined compression code. For learning-based methods, we evaluate Deep Image Prior (DIP) [21], Deep Decoder (DD) [22], Zero-Shot 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 We consider two synthetic noise models, AWGN N (0 , σ 2 z ) , where σ z is the standard deviation of the Gaussian distribution, and Poisson noise defined as Poisson( αx ) , where α is the scale factor. Note that Poisson noise is signal-dependent noise with E [ y ] = α x . To re-scale the noisy image to the range of clean image, we followed the literature by assuming that the the scale α is known, and normalize the noisy image as y /α in the experiments of this section. We evaluate on grayscale Set11 [25], RGB Set13 [54] (center-cropped to 192 × 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-based zero-shot denoisers, in contrast, often exhibit inconsistent

performance across noise levels and image resolutions. For example, ZS-N2S and Self2Self degrade on high-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 × 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 α was discussed in Section 4. In both noise regimes, we use MSE as the loss function. However, for Poisson noise, minimizing the negative log-likelihood is also a natural choice. We defer the results using this loss to Appendix B.3.

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-truth 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 to unknown noise models and non-natural image distributions.

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24.35

24.67

24.82

24.74

24.64

24.48

23.09

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25.45

25.71

25.60

25.80

25.60

25.36

24.43

24.61

25.37

25.71

25.82

25.90

25.65

25.42

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24.60

25.45

25.75

25.90

25.80

25.64

25.48

24.72

24.70

25.50

25.80

25.78

25.79

25.78

25.67

24.74

24.53

25.52

25.77

25.80

25.84

25.71

25.59

24.73

24.32

25.26

25.55

25.62

25.66

25.61

25.29

24.47

23.11

24.33

24.71

24.73

24.77

24.78

24.49

23.31

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Robustness to overfitting. Most learning-based zero-shot methods are prone to overfitting due to the lack of clean targets and the use of overparameterized networks. In contrast, ZS-NCD, grounded in compression-based 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 . 4 M params) with a fully connected MLP ( ≈ 2 . 3 Mparams) and observe that, instead of degradation, the performance improves using the same λ (see Table 4).

Effect of overlapping patch aggregation. As described in Section 4 and illustrated in Fig. 1, ZS-NCD denoises each pixel by aggregating outputs from overlapping patches, where each patch is first compressed and then decompressed using a learned neural compression model. Intuitively, one might expect the most accurate reconstruction for a 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 it 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 an ablation in which, instead of averaging, each pixel ( i, j ) is reconstructed solely from one of the k × k patches in which it appears, using a fixed location in the patch (e.g., top-left, center, etc.). The results are 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 as it moves toward the patch boundaries.

However, the key observation is that averaging across all overlapping reconstructions yields a substantial performance gain. For instance, in denoising Parrot (from Set11 dataset), the best singlelocation reconstruction achieves 25.90 dB (center), while averaging achieves 28.14 dB, a gain of over 2 dB. This highlights the denoising benefit of combining multiple noisy views of each pixel, consistent with principles from ensembling and variance reduction.

6 Conclusions

We have studied maximum likelihood compression-based denoising, and provided theoretical characterization of its performance under both AWGN and Poisson noise. Furthermore, we introduced ZS-NCD, a new zero-shot neural-compression-based denoising and demonstrated that it achieves state-of-the-art performance among zero-shot methods, in both AWGN and Poisson denoising.

The presented theoretical results are derived by assuming a fixed (e.g., pre-trained/defined) compression 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.

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A Proofs

A.1 Auxiliary lemmas

Before stating the proofs of the mains theorems, here we state some lemmas that will be used later in the proofs.

Lemma 1. Assume that 0 < α m ≤ α 1 , α 2 ≤ α M < ∞ . Then,

Lemma 2. Consider independent Poisson random variables Y 1 , . . . , Y n , where Y i ∼ Poisson( α i ) . Consider w 1 , . . . , w n ∈ R . Let σ 2 n = ∑ n i =1 w 2 i α i and w M ≜ max i ∈{ 1 ,...,n } | w i | . Then, for any t ∈ [0 , 3 σ 2 n 2 w M ] ,

and

A.1.1 Proof of Lemma 1

Proof.

Using the Taylor's theorem,

Letting u = α 2 -α 1 α 1 , for α ∈ (0 , u ) ,

Combining (14), (15) and (16) yields the desired result.

A.1.2 Proof of Lemma 2

Proof. Define and

where f ( u ) = log(1 + u ) and α ∈ (0 , u ) . Note that

Consider s > 0 , then using the Chernoff bound, we have

Note that for u ∈ ( -1 , 1) , e u -1 -u ≤ u 2 2(1 -u/ 3) . Assuming that s ≤ 1 w M , then | sw i | < 1 , for all i . Therefore,

Evaluating this bound at s = t σ 2 n + w M t/ 3 , since 1 -sw M / 3 = σ 2 n σ 2 n + w M t/ 3 , it follows that

To derive the other bound, we can follow the same steps and apply Chernoff bound as done in (17) to get

We now use the inequality for u ∈ ( -1 , 1) :

Assume s ≤ 1 w M so that s | w i | ≤ 1 for all i . Then:

Setting s = 3 t 2 σ 2 n , which satisfies s ≤ 1 w M ,

Hence,

A.2 Proof of Theorem 1

Proof. Recall that y = x + z , with z is i.i.d. N (0 , σ 2 z ) , and

Since both ˆ x , ˜ x are in C , and

Therefore, and finally,

Let e = ˆ x -x denote the error of the compression-based estimate of ground truth x from its noisy version y , and d = ˜ x -x denote the distortion from the compressing the ground truth x with the compression code C , then we have

For any possible reconstruction c ∈ C , we define error vector e ( c ) = c -x . Given t 1 , t 2 > 0 , define event E 1 and E 2 as

respectively. Conditioned on E 1 ∩ E 2 , it follows from (30) that

where the last line follows because √ a + b ≤ √ a + √ b , for all a, b > 0 . To finish the proof we need to bound P (( E 1 ∩ E 2 ) c ) and set parameters t 1 and t 2 .

Note that for each c , e ( c ) ∥ e ( c ) ∥ is a unit vector in R n . Therefore, ∑ n i =1 z i e ( c ) i ∥ e ( c ) ∥ ∼ N (0 , σ 2 z ) . Hence,

Therefore, applying the union bound and noting that |C| ≤ 2 R ,

and

For η ∈ (0 , 1) , set and

Then, bits.

Quantize the values of the non-zero coordinates such that overall distortion δ is achieved. To achieve this goal it quantizes each non-zero coordinate of x into b bits. Let [ x i ] b denote the b -bit quantized version of x i . Then, | x i -[ x i ] b | ≤ 2 -b . Therefore, the overall ℓ 2 distortion can be bounded as

Choosing

Using the selected values of t 1 and t 2 in (36) yields the desired result, i.e.,

where we have used the fact that 1 n ∥ d ∥ 2 2 ≤ δ .

A.3 Proof of Corollary 1

Proof. We start by designing a lossy compression code for the set of signals in Q n , defined as

For a k -sparse x ∈ Q n , let x ( k ) ∈ R k denote the k -dimensional vector derived from the non-zero coordinates of x . We define a lossy compression code ( f, g ) that achieves distortion δ . Specifically, given a k -sparse x ∈ Q n , the encoder operates as follows:

Encode the number of non-zero entries and their locations. This requires at most

It can be observed that overall the number of bits required for describing the signals in Q n within distortion δ can be bounded as

where c k = log 2 k + k (log 2 e + 1) . Using the defined lossy compression code to solve (1) and applying Theorem 1, it follows that, with a probability larger than 1 -2 -ηR +2 ,

where C = 2(1 + 2 √ η ) √ 2 ln 2 . Let

Then, where

Finally, since R ≥ k log 2 ( n k ) , and

A.4 Proof of Theorem 2

Proof. Recall that arg min c ∈C L ( c ; y ) . Let

Since both ˆ x and ˜ x are in C , we have L (ˆ x ; y ) ≤ L (˜ x ; y ) , or

Given the input signal x ∈ R n and c ∈ C , let Poisson( α x ) and Poisson( α c ) denote the distributions corresponding to independent Poisson random variables with respective means αx i and αc i . Note that

Adding ∑ i ( -αx i + αx i log x i ) to the both sides of (42), it follows that

Given t 1 , t 2 > 0 , define events E 1 and E 2 as

respectively. Conditioned on E 1 ∩ E 2 ,

D KL (Poisson( α x ) ∥ Poisson( α ˆ x )) ≤ D KL (Poisson( α x ) ∥ Poisson( α ˜ x )) + t 1 + t 2 , (48) and consequently from Lemma 1,

To finish the proof, we bound P ( E 1 ∩ E 2 ) and set t 1 and t 2 .

To bound P ( E c 1 ) , we apply Lemma 2, where for each c , we set w i ( c ) = log 1 c i . Then,

and where

Therefore, using the union bound, it follows that

To bound P ( E c 2 ) , we again apply Lemma 2, with w i = log 1 ˜ x i , and derive

Setting t 1 and t 2 such that they are both smaller than 3 nαβ , and noting that x max < 1 , we have

Choosing t 1 = β √ 4 ln 2 nR (1 + η ) α and t 2 = β √ 4 ln 2 nRηα , it follows that

A.5 Proof of Theorem 3

Proof. Recall that

Following the similar setup as in Section A.2, we get

Defining e , d and e ( c ) , c ∈ C , as done in the proof of Theorem 1, we have

Conditioned on E 1 ∩ E 2 , it follows that

Using Lemma 2 with y i ∼ Poisson( αx i ) and w i = e ( c ) i , it follows that

where and

Define events and

and

For η ∈ (0 , 1) , set and

Then,

Using the union bound and noting that |C| ≤ 2 R , we have

Setting t 1 and t 2 such that they are both smaller than 3 nx 2 max α , and noting that x max < 1 , we have

Using the selected values of t 1 and t 2 in (57) yields the desired result, i.e.,

Entropy Model

Conv / FC

GDN / ReLU

Conv / FC

GDN / ReLU

Conv / FC

Conv / FC

GDN / ReLU

Conv / FC

GDN / ReLU

Conv / FC

encoder

decoder

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 we 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 5 has 3 fully connected layers in 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 to 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 a mapping between two downsampled noisy images extracted from the original image. This strategy of generating noisy samples was extended in two recent concurrent works. Dual-sampling Noise2Noise (DS-N2N) [29] identifies a 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 to 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 in ZS-N2N insufficient to break the spatial correlation of real-world noise . To address this issue, Pixel2Pixel first finds a '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 as a 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 official code was used to report the numbers in the tables. In AWGN, our compression-based denoiser ZS-NCD outperforms both methods while achieving the second-best performance in Poisson denoising. Pixel2Pixel performs well in Poisson denoising.

B.3 MSE and likelihood estimation under Poisson noise without knowing true α

We compare the MSE and MLE distortion for Poisson denoising in Table 7 using the estimated ˆ α as explained in Section 4.

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 to 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 is 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-based encoder. We design the experiment that 3 downsampling operators are applied to patch size 8 and 16, where the latent code sizes are 1 × 1 × n b and 2 × 2 × n b respectively, where the denoising performance at each pixel location is in Figure 6 ( Left ) and ( Middle ), and if we increase the downsampling to 4 for patch size 16, which results in the latent code size to be 1 × 1 × n b , 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-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 × 768 . Adam optimizer is used for training the networks over 20K steps, with initial learning rate of 5 × 10 -3 decreased to 5 × 10 -4 after 16K steps for the Conv-based network. The learning rate for MLP-based networks is 1 × 10 -3 .

1

2

3

4

5

6

7

8

23.32

24.35

24.67

24.82

24.74

24.64

24.48

23.09

24.64

25.45

25.71

25.60

25.80

25.60

25.36

24.43

24.61

25.37

25.71

25.82

25.90

25.65

25.42

24.68

24.60

25.45

25.75

25.90

25.80

25.64

25.48

24.72

24.70

25.50

25.80

25.78

25.79

25.78

25.67

24.74

24.53

25.52

25.77

25.80

25.84

25.71

25.59

24.73

24.32

25.26

25.55

25.62

25.66

25.61

25.29

24.47

23.11

24.33

24.71

24.73

24.77

24.78

24.49

23.31

23.5

24.0

24.5

25.0

25.5

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

23.83

24.74

25.01

25.09

24.99

25.06

25.01

24.85

23.95

24.72

24.93

24.93

24.73

24.73

24.40

22.90

24.85

25.41

25.70

25.78

25.88

25.76

25.66

25.53

24.95

25.46

25.68

25.67

25.65

25.61

25.30

24.18

24.86

25.61

25.80

25.81

26.05

25.90

25.79

25.64

24.95

25.65

25.75

25.77

25.98

25.74

25.49

24.42

24.99

25.62

25.87

25.95

26.12

25.89

25.81

25.64

25.03

25.61

25.83

25.84

25.96

25.65

25.42

24.46

25.05

25.77

25.95

26.05

26.10

26.09

25.93

25.80

25.12

25.77

25.92

26.03

25.99

25.90

25.60

24.63

25.09

25.79

25.91

25.91

26.03

25.99

25.89

25.71

25.23

25.77

25.85

25.86

25.91

25.87

25.62

24.67

24.97

25.65

25.80

25.77

25.92

25.83

25.71

25.63

25.03

25.62

25.78

25.74

25.87

25.70

25.49

24.63

24.83

25.39

25.53

25.68

25.80

25.64

25.58

25.43

24.89

25.38

25.54

25.55

25.65

25.48

25.27

24.29

23.94

24.82

25.03

25.19

25.05

25.10

25.02

24.92

24.05

24.81

24.95

25.03

24.85

24.84

24.49

23.00

24.76

25.42

25.66

25.70

25.84

25.70

25.62

25.50

24.89

25.42

25.62

25.65

25.61

25.55

25.27

24.09

24.77

25.59

25.78

25.78

25.97

25.85

25.80

25.56

24.82

25.55

25.71

25.71

25.86

25.63

25.41

24.27

24.78

25.49

25.77

25.91

26.01

25.77

25.73

25.56

24.82

25.49

25.73

25.82

25.86

25.49

25.31

24.26

24.83

25.66

25.85

25.93

25.98

25.97

25.79

25.71

24.86

25.61

25.75

25.87

25.84

25.71

25.44

24.39

24.85

25.63

25.77

25.75

25.92

25.86

25.75

25.51

24.92

25.57

25.68

25.66

25.81

25.59

25.43

24.35

24.51

25.38

25.52

25.53

25.66

25.59

25.47

25.35

24.51

25.27

25.43

25.38

25.55

25.38

25.07

24.09

22.91

24.18

24.49

24.54

24.59

24.66

24.68

24.37

22.95

24.09

24.36

24.31

24.37

24.23

23.97

22.60

23.0

23.5

24.0

24.5

25.0

25.5

26.0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

24.04

24.98

25.37

25.56

25.61

25.61

25.53

25.50

25.31

25.34

25.30

25.32

25.24

25.08

24.81

23.77

25.03

25.77

25.99

26.05

26.11

26.06

26.03

26.06

26.06

26.06

25.96

25.90

25.99

25.86

25.69

25.02

25.36

25.94

26.15

26.11

26.13

26.16

26.18

26.18

26.11

26.12

26.09

26.07

26.08

26.03

25.88

25.36

25.45

26.03

26.13

26.15

26.22

26.22

26.24

26.19

26.10

26.14

26.14

26.05

26.06

26.03

25.72

25.28

25.40

26.01

26.21

26.22

26.32

26.27

26.29

26.23

26.21

26.19

26.16

26.15

26.10

26.11

25.94

25.39

25.46

26.08

26.28

26.28

26.39

26.30

26.30

26.33

26.26

26.31

26.23

26.14

26.22

26.18

25.93

25.46

25.43

26.03

26.17

26.26

26.32

26.28

26.24

26.32

26.22

26.26

26.15

26.16

26.16

26.20

25.93

25.53

25.42

26.04

26.22

26.21

26.34

26.32

26.25

26.25

26.16

26.20

26.18

26.16

26.15

26.04

25.90

25.47

25.21

26.01

26.21

26.27

26.17

26.35

26.33

26.24

26.03

26.10

26.05

26.09

26.01

25.97

25.79

25.19

25.40

26.11

26.21

26.30

26.33

26.36

26.32

26.22

26.15

26.17

26.21

26.13

26.16

26.07

25.89

25.35

25.44

26.06

26.18

26.25

26.26

26.23

26.24

26.11

26.05

26.09

26.22

26.16

26.12

26.09

25.86

25.35

25.42

26.08

26.16

26.17

26.21

26.16

26.20

26.17

26.04

26.04

26.12

26.09

26.04

26.06

25.79

25.24

25.17

25.95

26.12

26.18

26.10

26.15

26.20

26.19

26.00

26.02

25.96

25.96

25.84

25.88

25.80

25.16

25.20

25.97

26.11

26.17

26.17

26.11

26.11

26.19

26.05

26.12

25.96

25.96

25.91

25.95

25.78

25.19

24.87

25.69

25.88

26.00

25.94

25.87

25.97

25.94

25.76

25.77

25.81

25.84

25.81

25.68

25.51

24.88

23.77

24.97

25.34

25.47

25.50

25.51

25.42

25.44

25.16

25.32

25.31

25.25

25.17

25.09

24.83

23.93

24.0

24.5

25.0

25.5

26.0

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

C.1 Set11 Dataset

For noise levels (15 , 25 , 50) we set λ = (300 , 850 , 3000) . Similar to Kodak and other experiments we set training epochs to have 20K steps of gradient back propagation. For Poisson denoising α = (15 , 25 , 50) the λ = (3000 , 1500 , 1000) . We report the detailed results of AWGN denoising in Table 8, and Poisson noise denoising in Table 9.

C.2 Set13 Dataset

All images are center-cropped at size of 192 × 192 . For this set of images we set λ = (100 , 200 , 800) and for noise levels σ z = (15 , 25 , 50) and for Poisson denoising we have λ = (900 , 500 , 200) for noise levels α = (15 , 25 , 50) . We report the detailed results of AWGN denoising in Table 10, and Poisson noise denoising in Table 11.

σ

Method

Baboon

Bridge

Coastguard

Comic

Face

Flowers

Man

Monarch

Peppers

Zebra

Average

15

BM3D

28.56/0.7797

33.07/0.9151

30.39/0.8723

30.18/0.8799

28.74/0.9289

30.28/0.7665

29.62/0.9040

35.83/0.9369

29.88/0.8323

31.13/0.9361

31.77/0.8199

34.49/0.9588

31.02/0.9198

31.15/0.8808

25.12/0.6539

27.01/0.7674

26.27/0.7426

25.79/0.7323

24.86/0.8017

27.35/0.6482

25.52/0.7986

30.86/0.8397

25.97/0.6983

25.63/0.7649

28.27/0.7161

28.00/0.8183

26.33/0.8243

26.69/0.7543

DIP

27.25/0.7498

30.92/0.8403

30.18/0.8692

30.79/0.9036

28.25/0.9091

29.62/0.7611

28.85/0.8817

33.81/0.8947

29.99/0.8333

31.18/0.9085

30.30/0.7782

32.22/0.8908

30.63/0.9203

30.31/0.8570

DD

26.35/0.7029

24.27/0.7066

29.16/0.8508

28.80/0.8421

26.44/0.8932

29.59/0.7398

27.34/0.8634

34.87/0.9274

28.68/0.8073

30.02/0.9151

31.07/0.8003

33.09/0.9277

30.22/0.9052

29.22/0.8371

ZS-N2N

28.63/0.7992

28.45/0.7803

32.08/0.9041

31.54/0.9141

28.70/0.9018

30.64/0.8048

29.67/0.8955

34.02/0.8817

31.63/0.8801

31.65/0.9120

31.28/0.8089

32.84/0.9009

31.27/0.9277

30.95/0.8701

ZS-N2S

20.92/0.5844

21.14/0.5730

21.37/0.6468

20.78/0.4893

15.80/0.4987

22.03/0.5641

17.49/0.5154

8.43/0.3637

22.69/0.6725

12.21/0.6124

21.60/0.6438

11.58/0.4462

20.37/0.7871

18.18/0.5690

S2S

22.36/0.5810

30.39/0.8769

22.74/0.7485

22.72/0.7108

17.44/0.7015

17.23/0.4383

21.14/0.7121

16.78/0.8102

15.65/0.4463

26.92/0.8838

24.20/0.7398

15.48/0.7528

14.84/0.5400

20.61/0.6879

ZS-NCD

28.10/0.7831

33.85/0.9208

31.49/0.9051

32.65/0.9345

29.23/0.9355

30.51/0.7891

29.69/0.9077

35.85/0.9381

31.49/0.8867

33.21/0.9445

31.60/0.8281

35.07/0.9601

32.37/0.9448

31.93/0.8983

25

BM3D

26.56/0.6892

30.70/0.8824

28.06/0.7955

27.59/0.7664

25.90/0.8698

28.87/0.6978

26.93/0.8396

33.51/0.9087

27.35/0.7428

28.72/0.9056

30.11/0.7823

31.66/0.9324

28.61/0.8643

28.81/0.8213

23.98/0.5711

24.14/0.6801

24.14/0.6400

23.53/0.5801

21.36/0.7014

26.17/0.5692

22.77/0.6809

28.30/0.7960

23.51/0.5667

23.03/0.7101

26.47/0.6729

25.23/0.7707

23.58/0.7389

24.32/0.6676

DIP

25.70/0.6734

27.27/0.7021

27.79/0.7999

27.86/0.8126

25.11/0.8382

28.16/0.6599

26.03/0.8164

31.17/0.8384

27.27/0.7446

29.02/0.8785

28.82/0.7313

29.20/0.8106

28.65/0.8820

27.85/0.7837

DD

25.56/0.6589

23.56/0.6676

26.94/0.7758

26.77/0.7542

24.76/0.8397

28.37/0.6819

25.72/0.8058

32.22/0.8849

27.09/0.7366

27.68/0.8566

29.42/0.7615

29.94/0.8937

28.11/0.8641

27.40/0.7832

ZS-N2N

26.85/0.7287

26.73/0.7136

29.00/0.8282

28.30/0.8312

25.95/0.8462

28.71/0.7250

26.67/0.8248

31.31/0.8156

28.88/0.7940

29.03/0.8683

28.95/0.7282

29.78/0.8205

28.54/0.8768

28.36/0.8001

ZS-N2S

19.22/0.4963

15.44/0.3973

22.05/0.5859

21.55/0.5581

16.64/0.5734

25.76/0.6423

17.98/0.6324

20.90/0.6792

22.30/0.6170

21.29/0.7778

22.45/0.6535

17.04/0.6502

22.41/0.7964

20.39/0.6200

S2S

18.66/0.4947

24.97/0.7969

19.43/0.6054

20.69/0.5890

16.30/0.6168

15.08/0.3846

17.57/0.5267

15.53/0.7605

14.11/0.3892

21.34/0.7849

22.40/0.6795

13.60/0.6805

13.69/0.4890

17.95/0.5998

ZS-NCD

26.54/0.7128

30.65/0.8388

28.78/0.8332

29.46/0.8718

26.83/0.8847

29.14/0.7367

27.05/0.8393

32.99/0.8870

28.83/0.8027

30.18/0.8917

29.62/0.7656

31.56/0.8898

29.63/0.9023

29.33/0.8351

50

BM3D

24.66/0.5953

26.99/0.7854

25.31/0.6606

24.45/0.5245

22.30/0.7299

27.19/0.6140

23.52/0.7068

29.84/0.8400

24.77/0.6272

25.62/0.8492

27.34/0.7139

27.51/0.8657

25.59/0.7613

25.78/0.7134

21.99/0.4807

21.81/0.5541

21.55/0.4898

21.41/0.3641

18.50/0.5254

23.43/0.4603

19.92/0.5254

24.51/0.6873

20.74/0.4169

19.95/0.5819

23.66/0.5810

21.26/0.6490

19.81/0.5670

21.43/0.5295

DIP

24.07/0.5778

23.06/0.5943

25.05/0.6773

24.10/0.6179

21.64/0.7142

26.07/0.5804

22.84/0.6871

28.34/0.7754

24.60/0.6173

25.68/0.7922

26.06/0.6398

25.60/0.7060

25.55/0.7926

24.82/0.6748

DD

23.87/0.5745

23.06/0.6436

24.01/0.6402

23.43/0.4732

21.37/0.6956

26.83/0.6127

22.73/0.6819

29.12/0.8434

24.08/0.6011

24.09/0.7616

26.82/0.7055

25.40/0.8137

24.41/0.7659

24.56/0.6779

ZS-N2N

24.41/0.5999

23.99/0.5573

25.19/0.6726

24.45/0.6513

21.87/0.7014

25.90/0.5732

22.72/0.6619

27.10/0.6419

24.91/0.6111

25.11/0.7571

25.35/0.5654

25.14/0.6243

24.62/0.7594

24.67/0.6444

ZS-N2S

22.32/0.5704

16.65/0.5507

22.29/0.6475

21.75/0.4859

15.72/0.3989

25.49/0.6110

18.56/0.6121

24.16/0.7812

22.50/0.5681

19.83/0.6154

19.54/0.5440

16.26/0.4869

23.00/0.7724

20.62/0.5880

S2S

14.08/0.3567

17.56/0.5046

14.95/0.3650

17.25/0.3204

13.55/0.3265

12.73/0.2916

13.31/0.2712

13.82/0.5715

12.16/0.2727

14.59/0.5444

17.16/0.4959

11.42/0.5186

12.16/0.2807

14.21/0.3938

ZS-NCD

24.32/0.6035

26.84/0.7333

25.70/0.7172

25.76/0.7385

23.14/0.7768

26.90/0.6401

23.74/0.7212

28.46/0.7716

25.46/0.6710

26.29/0.8075

26.40/0.6722

27.25/0.7856

26.10/0.8113

25.87/0.7269

α

Method

Baboon

Bridge

Coastguard

Comic

Face

Flowers

Man

Monarch

Peppers

Zebra

Average

15

BM3D

24.46/0.5651

26.63/0.7712

25.18/0.6462

24.45/0.5186

22.07/0.7101

27.46/0.6342

23.51/0.7017

29.05/0.7747

24.98/0.6405

25.60/0.8101

27.53/0.7041

26.16/0.7288

26.19/0.7802

25.64/0.6912

22.01/0.4873

21.82/0.5515

21.77/0.5163

21.69/0.3727

18.47/0.5237

24.97/0.5289

20.42/0.5841

23.83/0.6996

21.69/0.4723

20.21/0.5809

24.17/0.6012

20.73/0.5858

21.04/0.6381

21.76/0.5494

DIP

24.30/0.5907

23.05/0.6010

25.46/0.7069

24.21/0.5996

21.79/0.7282

26.99/0.6150

23.41/0.7456

28.03/0.7686

25.10/0.6444

26.34/0.8136

26.26/0.6510

25.44/0.7088

26.37/0.8180

25.14/0.6916

DD

23.89/0.5750

23.16/0.6479

24.83/0.6873

23.80/0.5544

21.52/0.7107

27.22/0.6387

23.41/0.7339

29.25/0.8522

25.03/0.6514

24.67/0.7745

27.09/0.7156

25.12/0.7640

25.43/0.8020

24.96/0.7006

ZS-N2N

24.82/0.6330

24.11/0.5768

25.82/0.7281

25.35/0.6946

21.99/0.7055

27.11/0.6779

23.67/0.7490

27.20/0.6491

26.60/0.7290

25.99/0.7740

26.01/0.6047

25.02/0.6004

26.13/0.8190

25.37/0.6878

ZS-N2S

21.39/0.5390

17.46/0.4084

22.23/0.6428

21.83/0.5714

17.53/0.5771

25.14/0.6103

17.97/0.5082

24.33/0.7854

22.94/0.5959

21.15/0.7157

21.53/0.5467

19.43/0.6122

23.08/0.7725

21.23/0.6066

S2S

16.58/0.5042

21.66/0.6523

18.07/0.6269

22.14/0.6206

15.01/0.5319

24.18/0.6548

18.87/0.6736

14.80/0.7562

24.78/0.7188

17.73/0.7807

21.51/0.6626

12.61/0.5630

22.09/0.7728

19.23/0.6553

ZS-NCD

24.66/0.5807

27.63/0.7844

26.43/0.7443

26.40/0.7436

23.17/0.7781

27.53/0.6510

24.30/0.7680

29.58/00.8302

26.15/0.6997

26.46/0.7991

27.34/0.7196

26.68/0.7217

27.45/0.8440

26.44/0.7434

25

BM3D

20.61/0.4799

21.75/0.5241

22.93/0.6748

22.04/0.5620

19.94/0.6637

24.88/0.6358

22.35/0.7387

22.34/0.5167

23.11/0.5790

21.99/0.6799

22.75/0.5533

20.01/0.4239

23.48/0.7582

22.17/0.5992

22.59/0.5060

22.38/0.5732

22.81/0.5904

22.12/0.4455

19.59/0.6087

25.30/0.5533

21.51/0.6444

25.19/0.7343

22.10/0.4841

21.52/0.6226

24.78/0.6181

22.07/0.6515

22.44/0.7057

22.65/0.5952

DIP

24.85/0.6243

24.02/0.5860

26.63/0.7646

25.41/0.6907

22.75/0.7640

27.42/0.6386

24.61/0.7935

26.07/0.7119

27.02/0.8225

27.36/0.6839

26.86/0.7348

27.25/0.8467

26.13/0.7289

DD

24.51/0.6017

23.30/0.6625

25.78/0.7389

24.96/0.6226

22.80/0.7680

28.18/0.6808

24.65/0.7872

30.34/0.8723

25.91/0.6824

26.31/0.8306

28.07/0.7242

26.92/0.7823

26.80/0.8309

26.04/0.7373

ZS-N2N

25.81/0.6876

25.11/0.6387

27.17/0.7874

26.65/0.7562

23.45/0.7635

28.28/0.7277

24.95/0.8033

28.68/0.7031

28.10/0.7878

27.67/0.8276

27.47/0.6713

26.90/0.6864

27.43/0.8511

26.75/0.7455

ZS-N2S

19.29/0.4409

21.36/0.5393

21.65/0.6879

20.44/0.4223

16.56/0.4988

25.41/0.6417

15.88/0.5870

25.29/0.8037

19.35/0.6591

21.09/0.6962

24.71/0.6801

22.47/0.7677

22.00/0.7806

21.19/0.6312

S2S

17.40/0.5115

24.01/0.7441

18.56/0.6458

22.99/0.6967

15.48/0.5679

25.07/0.6826

19.74/0.7193

15.06/0.7385

26.12/0.7550

18.86/0.8094

23.26/0.6959

12.85/0.6302

22.89/0.8084

20.18/0.6927

ZS-NCD

25.23/0.6145

29.09/0.8189

27.60/0.7969

27.66/0.8053

24.50/0.8223

28.31/0.6908

25.65/0.8231

30.99/0.8571

27.21/0.7491

27.82/0.8309

28.40/0.7464

27.98/0.7490

28.42/0.8712

27.60/0.7827

50

BM3D

21.78/0.5516

22.44/0.5350

23.85/0.7114

22.60/0.6103

21.36/0.7094

25.06/0.6670

23.51/0.7921

23.08/0.4721

23.86/0.6477

22.82/0.6871

23.24/0.5494

21.35/0.4511

24.07/0.7814

23.00/0.6281

23.83/0.5611

23.92/0.6612

24.26/0.6565

23.53/0.5775

21.09/0.6866

26.46/0.6041

22.88/0.7170

27.56/0.7817

23.77/0.5931

23.00/0.7123

26.46/0.6628

24.44/0.7320

24.00/0.7584

24.25/0.6696

DIP

25.67/0.6702

26.62/0.6890

28.12/0.8173

27.79/0.8136

24.29/0.8131

28.63/0.7149

26.30/0.8449

30.97/0.8395

27.29/0.7409

29.10/0.8779

28.24/0.7107

28.64/0.7849

28.74/0.8816

27.72/0.7845

DD

25.39/0.6439

23.57/0.6720

27.23/0.7930

26.69/0.7441

24.40/0.8285

28.69/0.7060

26.08/0.8358

32.04/0.8838

27.45/0.7575

27.94/0.8613

29.45/0.7641

29.37/0.8636

28.33/0.8740

27.43/0.7867

ZS-N2N

27.06/0.7422

26.44/0.7151

29.33/0.8535

28.72/0.8361

25.47/0.8308

29.57/0.7822

26.98/0.8566

30.77/0.7918

30.02/0.8491

29.50/0.8730

29.03/0.7330

29.21/0.7853

29.34/0.8968

28.57/0.8112

ZS-N2S

19.68/0.5011

19.80/0.4843

22.87/0.6603

19.48/0.3374

16.44/0.4783

18.21/0.4958

18.06/0.5203

25.22/0.8178

22.22/0.6118

21.95/0.7717

22.69/0.6847

19.84/0.7204

23.22/0.7586

20.75/0.6033

S2S

19.15/0.5499

27.76/0.8445

19.57/0.6641

23.63/0.7330

16.26/0.6303

26.79/0.7083

20.99/0.7695

15.34/0.7890

27.87/0.8113

21.54/0.8363

26.51/0.7211

13.43/0.6657

23.93/0.8512

21.75/0.7365

ZS-NCD

26.05/0.6622

30.15/0.8172

29.25/0.8571

29.83/0.8761

25.99/0.8641

29.36/0.7458

27.52/0.8790

32.51/0.8739

29.05/0.8147

29.27/0.8500

29.64/0.7647

29.53/0.7775

30.03/0.9081

29.09/0.8223

C.3 Kodak24 Dataset

For Gaussian denoising λ = (75 , 150 , 750) for noise levels σ z = (15 , 25 , 50) and for Poisson denoising λ = (750 , 300 , 150) for α = (15 , 25 , 50) . For BM3D Poisson denoising of α = (15 , 25 , 50) we set σ BM3D = (50 , 25 , 15) . We report the detailed results of AWGN denoising in Table 12, and Poisson noise denoising in Table 13.

Method (

σ

)

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Average

25.20/0.7120

29.23/0.6996

29.81/0.7678

29.09/0.7299

25.55/0.7728

26.71/0.7343

28.97/0.7880

25.24/0.7730

29.29/0.7756

29.27/0.7540

27.34/0.7033

29.28/0.7219

24.73/0.7477

26.65/0.7137

29.23/0.7490

28.29/0.7205

28.86/0.7709

26.54/0.7195

27.76/0.7400

29.28/0.8055

27.21/0.7749

27.81/0.6983

30.82/0.8039

26.38/0.7215

27.86/0.7457

BM3D (15)

29.46/0.8549

33.06/0.8266

35.19/0.9096

33.30/0.8583

30.37/0.9023

31.00/0.8664

34.59/0.9384

30.37/0.9019

34.59/0.9083

34.38/0.8952

31.70/0.8475

34.12/0.8611

27.97/0.8295

30.60/0.8425

33.84/0.8781

32.69/0.8647

33.46/0.8898

30.48/0.8558

32.18/0.8568

33.84/0.8936

31.46/0.8932

31.60/0.8355

35.90/0.9205

30.83/0.8800

32.37/0.8754

DIP (15)

29.91/0.8724

31.98/0.7986

33.53/0.8588

32.07/0.8180

30.00/0.8875

30.85/0.8605

33.21/0.8851

29.54/0.8838

33.00/0.8551

32.71/0.8348

30.58/0.8144

32.79/0.8203

28.16/0.8619

30.37/0.8531

32.22/0.8231

32.26/0.8486

31.99/0.8492

29.78/0.8235

30.96/0.8111

32.66/0.8676

30.77/0.8396

30.82/0.8107

34.09/0.8716

29.94/0.8403

31.42/0.8454

DD (15)

26.10/0.7586

29.56/0.7297

30.98/0.8323

30.11/0.8023

26.17/0.8114

27.28/0.7715

30.92/0.8905

24.65/0.7766

31.34/0.8565

31.57/0.8480

28.45/0.7793

32.09/0.8177

23.42/0.6831

27.77/0.7891

29.31/0.7730

29.31/0.7730

30.93/0.8688

27.14/0.7826

28.25/0.7786

28.87/0.8549

28.03/0.8158

29.23/0.7820

31.88/0.8775

25.72/0.7853

28.71/0.8016

ZS-N2N (15)

31.13/0.8951

33.19/0.8418

34.04/0.8582

32.93/0.8463

30.38/0.8890

31.96/0.8826

33.51/0.8872

30.77/0.9045

33.75/0.8532

33.01/0.8367

32.15/0.8571

33.48/0.8290

28.98/0.8885

31.14/0.8681

32.96/0.8401

33.23/0.8687

33.21/0.8743

30.44/0.8587

32.65/0.8736

33.86/0.8856

32.51/0.8849

31.83/0.8416

33.85/0.8395

30.29/0.8548

32.30/0.8650

ZS-N2S (15)

18.03/0.4389

25.76/0.6821

17.06/0.6117

25.30/0.6915

18.35/0.5545

10.16/0.3461

22.51/0.7272

17.54/0.5125

15.42/0.6011

19.79/0.6228

23.15/0.6176

8.06/0.2929

19.24/0.3889

22.86/0.6371

23.91/0.7191

23.05/0.6010

16.88/0.5174

21.44/0.5637

11.59/0.5257

5.47/0.0930

22.77/0.6687

20.22/0.5615

22.33/0.7443

17.36/0.5761

18.68/0.5540

S2S (15)

25.73/0.7941

25.01/0.7193

23.30/0.8177

27.41/0.8106

22.73/0.7274

19.32/0.7144

29.75/0.9184

19.82/0.7318

29.36/0.8927

25.72/0.8723

22.71/0.7244

22.15/0.7985

19.75/0.5547

24.55/0.7461

17.09/0.7066

27.15/0.8337

21.30/0.7254

24.15/0.7016

28.58/0.7811

10.63/0.6711

23.49/0.8304

23.60/0.7693

22.17/0.8469

18.37/0.7790

23.08/0.7695

ZS-NCD (15)

31.28/0.9059

33.93/0.8669

35.61/0.9215

34.19/0.8876

31.63/0.9286

32.07/0.9009

35.34/0.9382

31.27/0.9222

35.03/0.9088

34.71/0.9097

32.40/0.8895

34.89/0.8939

28.61/0.8931

31.95/0.8978

34.37/0.8951

34.04/0.9004

34.04/0.9089

31.25/0.8860

33.26/0.8956

34.13/0.9096

32.45/0.8973

32.52/0.8762

35.83/0.9236

31.48/0.9046

33.18/0.9026

22.70/0.5639

27.54/0.6312

27.86/0.7130

27.10/0.6704

25.55/0.7728

24.21/0.6104

26.01/0.6863

22.11/0.6519

26.88/0.7440

26.57/0.7022

24.92/0.5912

27.20/0.6589

21.73/0.5638

24.45/0.5960

26.89/0.7150

26.60/0.6330

26.37/0.6632

24.10/0.5928

25.43/0.6575

26.43/0.7676

24.76/0.7002

25.40/0.5919

28.75/0.7779

23.70/0.6015

25.43/0.6550

BM3D (25)

26.98/0.7554

31.29/0.7717

32.74/0.8618

31.23/0.7994

27.56/0.8236

28.42/0.7789

31.87/0.9026

27.74/0.8497

32.20/0.8715

31.92/0.8480

29.29/0.7690

32.07/0.8068

25.21/0.6973

28.19/0.7513

31.75/0.8337

30.33/0.7813

31.03/0.8382

27.82/0.7616

30.14/0.7936

31.72/0.8522

28.89/0.8366

29.37/0.7491

33.59/0.8899

28.09/0.7966

29.98/0.8092

DIP (25)

26.90/0.7759

29.87/0.7204

30.91/0.7859

30.03/0.7510

26.40/0.7875

27.91/0.7697

30.77/0.8496

26.51/0.8059

30.41/0.7959

30.28/0.7784

27.77/0.7115

30.79/0.7586

25.52/0.7763

27.74/0.7625

30.16/0.7568

29.96/0.7803

29.72/0.7923

26.80/0.7322

28.77/0.7591

30.54/0.8216

28.41/0.7963

28.38/0.7238

31.72/0.8261

27.00/0.7388

28.90/0.7738

DD (25)

25.26/0.7193

29.52/0.7298

29.58/0.7641

28.73/0.7327

25.97/0.7902

26.42/0.7245

29.27/0.8238

23.99/0.7493

29.50/0.7825

29.41/0.7732

27.37/0.7223

30.11/0.7407

22.97/0.6570

26.79/0.7397

29.39/0.7752

28.17/0.7014

29.63/0.8146

26.21/0.7263

26.99/0.7077

28.03/0.8136

27.01/0.7501

28.00/0.7173

30.10/0.8084

25.12/0.7364

27.62/0.7496

ZS-N2N (25)

28.30/0.8249

30.67/0.7618

31.10/0.7676

30.28/0.7562

27.66/0.8231

29.01/0.7975

30.48/0.8057

30.81/0.7545

30.24/0.7380

29.31/0.7645

30.69/0.7276

26.66/0.8113

28.32/0.7801

30.39/0.7501

30.46/0.7826

30.62/0.8000

28.07/0.7713

29.76/0.7902

30.95/0.8091

29.45/0.8093

29.09/0.7451

30.98/0.7461

27.99/0.7707

29.54/0.7798

ZS-N2S (25)

17.06/0.4493

26.46/0.6886

26.94/0.7692

24.96/0.7041

20.70/0.6776

21.31/0.5206

20.89/0.6756

16.54/0.5188

20.46/0.6864

22.14/0.7248

23.98/0.6349

10.42/0.4087

17.29/0.4015

22.67/0.6045

25.05/0.7175

25.18/0.6550

24.09/0.7115

21.99/0.5551

20.16/0.6362

5.76/0.3971

21.54/0.6619

22.72/0.5972

24.98/0.7995

18.52/0.5813

20.89/0.6156

S2S (25)

23.60/0.7118

20.04/0.6040

21.16/0.7692

23.86/0.7267

19.02/0.5867

17.49/0.7154

23.04/0.8244

17.92/0.6895

27.46/0.8618

24.39/0.8275

19.65/0.6347

21.04/0.7575

18.72/0.4512

22.24/0.5982

15.36/0.6307

25.01/0.7566

17.56/0.6245

20.93/0.5421

25.52/0.7214

10.09/0.6607

22.32/0.7695

22.60/0.6994

20.52/0.8077

17.67/0.7054

20.72/0.6949

ZS-NCD (25)

28.88/0.8364

31.87/0.7865

32.78/0.8226

30.38/0.7368

28.81/0.8615

29.58/0.8146

32.01/0.8524

28.87/0.8661

32.17/0.8171

32.18/0.8286

30.03/0.7961

32.64/0.8248

26.88/0.8315

29.50/0.8210

31.75/0.7962

31.14/0.7939

31.33/0.8108

29.02/0.8012

30.71/0.8008

31.77/0.8283

29.79/0.7852

30.14/0.7875

33.06/0.8351

29.21/0.8189

30.60/0.8144

20.49/0.3922

24.11/0.5988

24.31/0.6804

23.21/0.5842

19.99/0.4471

21.63/0.4500

22.79/0.5132

19.11/0.4891

22.96/0.4615

22.86/0.4371

21.92/0.4196

23.57/0.6529

19.47/0.3673

21.92/0.4560

22.37/0.4966

24.41/0.5511

22.58/0.4844

21.55/0.4266

22.13/0.4597

21.65/0.6122

21.96/0.4778

22.64/0.4896

23.40/0.7331

21.11/0.4517

22.17/0.5055

BM3D (50)

24.13/0.5866

29.05/0.6999

30.00/0.7943

28.58/0.7121

24.05/0.6675

25.50/0.6364

28.28/0.8224

24.25/0.7340

29.04/0.7971

28.61/0.7567

26.42/0.6604

29.58/0.7402

22.18/0.5110

25.41/0.6189

29.16/0.7708

27.76/0.6630

27.83/0.7463

24.82/0.6048

27.70/0.7210

28.92/0.7997

25.88/0.7341

26.97/0.6380

30.42/0.8371

24.92/0.6603

27.06/0.7047

DIP (50)

23.92/0.6204

27.50/0.6301

28.26/0.6983

27.23/0.6295

23.31/0.6552

24.92/0.6190

27.26/0.7310

22.73/0.6684

27.57/0.6856

28.32/0.7120

25.16/0.5884

27.95/0.6415

22.05/0.5876

24.69/0.6148

27.49/0.6667

26.78/0.6076

26.58/0.6644

24.01/0.5880

25.56/0.6271

27.48/0.7482

25.27/0.6876

25.56/0.5805

28.28/0.7283

23.67/0.6063

25.90/0.6494

DD (50)

22.80/0.5694

26.73/0.6471

27.70/0.7500

26.03/0.6491

22.99/0.6464

23.74/0.5735

25.83/0.6793

22.06/0.6528

26.30/0.7166

25.49/0.6800

24.86/0.5904

25.87/0.6003

21.31/0.5467

24.32/0.6093

26.48/0.6855

26.30/0.6032

26.06/0.7191

23.95/0.5820

24.49/0.5819

26.19/0.7493

24.27/0.6003

25.44/0.5967

27.24/0.7664

22.98/0.5957

24.98/0.6413

ZS-N2N (50)

24.35/0.6495

27.75/0.6501

27.35/0.5999

26.63/0.5774

23.65/0.6601

25.18/0.6166

26.54/0.6457

23.63/0.6966

27.08/0.5835

26.60/0.5708

25.61/0.5946

27.12/0.5673

23.06/0.6429

24.71/0.6124

27.01/0.6080

26.54/0.5792

26.80/0.6368

24.47/0.5828

25.95/0.6185

27.29/0.6791

25.51/0.6433

25.61/0.5731

26.87/0.5712

24.44/0.6037

25.82/0.6151

ZS-N2S (50)

19.72/0.5883

24.42/0.4968

26.65/0.7305

15.11/0.4906

15.88/0.5239

22.01/0.5405

25.43/0.7356

17.24/0.5660

16.06/0.5159

17.70/0.5480

23.87/0.6170

11.75/0.4637

14.02/0.3956

22.95/0.6008

25.16/0.6968

26.24/0.6367

25.18/0.7260

21.98/0.5817

14.07/0.5146

24.56/0.7294

12.48/0.3895

23.78/0.6031

17.04/0.6914

17.89/0.4751

20.05/0.5774

S2S (50)

19.95/0.4689

21.03/0.6290

17.13/0.6781

17.53/0.5853

14.73/0.3175

15.47/0.5126

20.89/0.6709

14.94/0.3918

21.78/0.7174

20.36/0.6616

15.85/0.4717

17.66/0.6762

15.68/0.2853

15.75/0.3939

12.24/0.5437

19.63/0.5559

13.91/0.4744

14.55/0.3322

18.09/0.5861

9.24/0.5644

18.79/0.5944

20.52/0.5298

16.35/0.6779

15.94/0.4615

17.00/0.5325

ZS-NCD (50)

26.01/0.7245

29.30/0.7205

30.20/0.7927

28.87/0.7287

25.60/0.7608

26.72/0.7133

29.05/0.8173

25.47/0.7785

29.75/0.7946

29.40/0.7790

27.18/0.7023

29.80/0.7531

23.91/0.6930

26.58/0.7086

29.38/0.7633

28.67/0.7191

28.76/0.7668

26.05/0.6914

28.22/0.7414

29.07/0.7850

27.17/0.7449

27.62/0.6922

30.48/0.8184

26.18/0.7240

27.89/0.7464

C.4 Microscopy Mouse Nuclei Dataset

For these images with noise level σ z = (10 , 20) we set λ = (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.

C.5 Real Camera Noise Dataset PolyU

For these images with unknown noise model/level λ = 25 . Also for BM3D the best peroformance was achieved with setting σ BM3D = 15 . We report the detailed denoising performance in Table 16.

D Visual Comparisons

In this section, we provide more visualization comparison of the zero-shot denoisers. The reconstruction PSNR and SSIM are above the images.

D.1 Kodak24

Grount Truth (

, 1.0)

Noisy (20.17 / 0.4078)

BM3D (28.20 / 0.7966)

DD (25.12 / 0.7364)

S2S (17.67 / 0.7054)

ZS-N2S (18.52 / 0.5813)

ZS-N2N (27.99 / 0.7707)

ZS-NCD (29.21 / 0.8189)

Ground Truth (

, 1.0)

D.2 Mouse Nuclei

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification:

We claim that our proposed method is zero-shot, the details of how to crop patches from the single given noisy image can be found in Figure 1 and Section 4.

We provide the theory connects compression and denoising, which can be found in Section 3, we characterize the performance for both AWGN and Poisson noise.

We claim that the proposed ZS-NCD is the state-of-the-art zero-shot denoiser over both natural and non-natural images, this can be found empirically in Table 2 and 3.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We have a section 6 which includes the discussion on the limitation of our work.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification:

We assume that the compression code for Q is defined by an encoder-decoder pair ( f, g ) characterized by R and δ , these appear in the theoretical results we provide.

The proof of all the proposed theorems and corollary are provided in Appendix A.

Guidelines:

The answer NA means that the paper does not include theoretical results.

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4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification:

The neural networks structures we used are provided in Appendix B. The learning parameters including learning rate and optimizer are also included. The hyperparameter λ for each noise level are introduced in Appendix C.

The code of our proposed method is included in the supplementary.

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If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

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(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

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5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification:

We provide the implementation code of our method in ZS-NCD. We provide the instruction and comments in how to run the code. We also provide the requirements of the environments to run the code.

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The answer NA means that paper does not include experiments requiring code.

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6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We provide the networks structures, hyperparameter λ , learning rate and optimizer in training the neural networks.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No] .

Justification: Our experiments follow standard practice in image denoising, where evaluation is performed on fixed noisy images and results are reported using metrics such as PSNR.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

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It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We provide the information of computing resources and the running time in Appendix C.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: Our research on zero-shot neural compression based denoising conducted in the paper conform with the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: The paper proposes a novel way of denoising. Zero-shot denoiser has great potential to be used for imaging systems without ground truth and large amount of noisy images, such as microcopy and medical imaging. Providing the good denoised version of the acquired noisy images is beneficial to the downstream tasks.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

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The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

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Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We have cited properly about the datasets we used in the paper.

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The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

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Justification: Our paper does not release new assets.

Guidelines:

The answer NA means that the paper does not release new assets.

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14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

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Justification: Our paper does not involve crowdsourcing nor research with human subjects.

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Justification: Our paper does not involve crowdsourcing nor research with human subjects.

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The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

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Peer reviews:
comments: 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.

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.

QUESTIONS:
I do not have questions for the authors.

LIMITATIONS:
The authors address the limitations in section 6.
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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.\n\nQUESTIONS:\n- 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).\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n**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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\nIn 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.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}