Unsupervised Hyperspectral Image Denoising via Spectral Learning Preference of Neural Networks

Highlights

What are the main findings?

• An unsupervised hyperspectral image denoising framework is proposed that exploits the inherent spectral learning preference of deep neural networks.
• An adaptive early stopping strategy is designed to align with this learning behavior, enabling the model to fit only clean spectral signals, prevent overfitting to noise and significantly improving denoising performance.

What are the implications of the main findings?

• Without requiring clean reference data, the proposed approach provides a practical solution for real-world scenarios where noise-free hyperspectral observations are unavailable.
• By leveraging intrinsic network priors rather than explicit noise modeling, the method offers strong adaptability across different noise types. This suggests a promising direction for robust and label-free hyperspectral image restoration.

Abstract

Existing hyperspectral denoising networks typically rely on large amounts of high-quality paired noisy–clean images for training, which are often unavailable. Moreover, the noise distribution in real hyperspectral images (HSIs) is complex and variable, making it challenging for existing networks to handle noise distributions not present in the training dataset, resulting in poor generalization. To address these issues, this paper proposes an unsupervised Hyperspectral image Denoising approach exploiting the spectral learning preference of neural networks with an adaptive early stopping strategy (termed HyDePre). Inspired by the Deep Image Prior, which reveals that neural networks tend to capture natural image structures before fitting noise, we observe that deep neural networks exhibit a similar learning preference in the spectral domain. Specifically, as training progresses, the network first fits smooth spectral feature curves and only later adapts to Gaussian noise and complex impulse noise. This observation provides an opportunity to use an early stopping strategy, allowing the network to fit only the clean spectral signals and thus achieve denoising. Our method does not require clean images for training, but instead optimizes network parameters to automatically learn prior spectral information from a single noisy image, modeling the intrinsic structure of the input data to uncover its underlying patterns.However, finding the optimal stopping point is challenging without access to clean images as sources of prior information. To tackle this challenge, we introduce an adaptive early stopping strategy based on the average spectral maximum variation of the reconstructed image, effectively preventing overfitting. The experimental results demonstrate that HyDePre outperforms existing methods in terms of both visual quality and quantitative metrics.

1. Introduction

Hyperspectral images (HSIs) have demonstrated enormous application potential in fields such as precision agriculture, environmental monitoring [1], and military reconnaissance, due to their unique advantage of integrating spectral and spatial information. However, limited by the complexity of imaging systems and external environments, raw HSI data inevitably suffers from various types of noise contamination, such as Gaussian noise, impulse noise, stripe noise, and deadlines. These noises not only seriously degrade the visual quality of the images but, more critically, significantly affect the accuracy of subsequent advanced tasks such as classification and unmixing. Therefore, researching efficient and robust algorithms for removing noise in hyperspectral images is of crucial theoretical and practical value.

Current hyperspectral image denoising methods can be roughly divided into two categories: model-based methods and deep learning-based methods. Model-based methods constrain the solution space by encoding the prior knowledge of HSIs in the form of regularization terms. Commonly used priors include local smoothness [2], non-local self-similarity [3], and the low-rankness in the spectral dimension [4]. Many works use total variation (TV) regularization to model local smoothness prior, such as the nonconvex correlated total variation method (NCTV [5]), enhanced 3DTV method [6], and many other methods that integrate TV regularization, and the low-rank model, such as total-variation-regularized low-rank matrix factorization (LRTV [7]), the decomposition of joint low rankness and local smoothness plus sparse matrices (DCTV-RPCA [8]), representative coefficient total variation (RCTV [9]), and low-rank tensor decomposition with total variation (LRTDTV [10]) or weighted group sparsity regularization (LRSTV [11]). To utilize non-local self-similarity prior, existing methods group these similar blocks and then perform collaborative filtering (BM3D [3], BM4D [12], VBM4D [13], and NSP [14]) or low-rank approximation (GLF [15]) on them. Due to the high correlation in the spectral dimension, HSI data cubes have strong low-rank characteristics. Exploiting this characteristic, a significant amount of Gaussian noise can be removed using techniques such as double-factor regularized low-rank tensor factorization (LRTF-DFR) [16], low-rank matrix recovery (LRMR) [4], low-rank and sparse representations [17], nonconvex regularized low-rank and sparse matrix decomposition (NonRLRS) [18], and sparse representation of the underlying HSI, e.g., Kronecker-basis-representation (KBR) [19]. As for sparse noise, LRTF-DFR [16] minimizes the ${\ell }_1$ norm to promote sparity, LRMR constrains the upper bound of the cardinality of sparse noise component, and NonRLRS [18] introduces the normalized $\epsilon$—penalty as a nonconvex regularizer. While these methods are effective in certain scenarios, they often struggle with complex noise patterns and are highly sensitive to parameter settings. Additionally, many of these methods are time-consuming due to the large sizes of HSIs and the iterative estimation procedures.

Recently, deep learning-based methods have gained popularity due to their ability to learn complex representations of noise and signals [20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Deep learning-based methods include supervised learning and self-supervised/unsupervised learning. Supervised learning methods train deep neural networks on large-scale datasets to learn an end-to-end mapping from noisy images to clean images. Notable examples include the spatial–spectral gradient network (SSGN) [21], the CNN-based HSI denoising method (HSI-DeNet [22], HSID-CNN [20], 3DADCNN [23], and QRNN3D [24]), and the transformer-based HSI denoising method (Hider [27], TRQ3DNet [28], SST [29], HSDT [30], and SERT [31]). The performance of supervised learning models heavily relies on large-scale, high-quality paired training data. However, acquiring noise-free hyperspectral images in the real world is extremely challenging. To overcome the reliance on paired data, self-supervised and unsupervised denoising approaches have been developed [34,35,36,37,38,39,40,41,42]. Representative methods include using blind spot networks [35,43], using deep image prior [34], using implicit neural representation [39], using diffusion models [38], and so on. Despite their effectiveness, most existing self-supervised/unsupervised hyperspectral denoising methods rely on specific noise statistics assumptions or handcrafted internal priors, making them prone to underfitting or overfitting when confronted with complex or non-independent noise, and often leading to unstable denoising performance and the loss of spectral or spatial details. Among them, a representative and influential line of research is the Deep Image Prior (DIP) [34,44], which reveals that neural networks themselves encode implicit image priors and tend to fit structured image content before noise, even without external training data. From this perspective, the DIP mechanism can be understood as the network implicitly imposing a form of prior during optimization. In this work, we analyze the interaction between the learning behavior of neural networks and the intrinsic spectral characteristics of hyperspectral images, and argue that extending the DIP mechanism to the spectral domain enables more effective exploitation of spectral information. Under this formulation, during network optimization, clean and smooth spectral signatures are preferentially reconstructed, while noise is fitted only at later stages of training. Such a learning behavior reflects an inherent spectral learning preference of neural networks, which can be conceptually interpreted as a deep spectral prior in the wavelength domain and which constitutes the foundation of the proposed method in this paper.

1.1. Related Work

While traditional model-based methods remain highly effective and interpretable, deep learning–based approaches have attracted increasing attention for hyperspectral image denoising due to their ability to capture complex data characteristics. However, their efficacy typically relies on extensive datasets of paired noisy–clean training data, which are often impractical to acquire in hyperspectral scenarios. To circumvent this limitation, self-supervised and unsupervised learning approaches have emerged as promising alternatives. The core idea of self-supervised learning methods is to generate supervisory signals directly from the noisy observations and then use these signals for training. The pioneering Noise2Noise framework [45] demonstrated that denoising networks can be trained using image pairs sharing the same underlying signal but corrupted by independent noise. Yet, acquiring such paired noisy data remains a significant challange in practical HSI applications. To address this, Zhu et al. [35] developed blind-spot networks utilizing per-band 2D convolutions (N2V-2D), 3D convolutions (N2V-3D), and spatial–spectral feature separation (N2V-Sep). These methods randomly mask a subset of pixels or spectral bands during training, compelling the network to reconstruct missing values via the surrounding spatial–spectral context [43,46]. Similarly, Zhuang et al. proposed the Eigenimage2Eigenimage (E2E) framework [36], which generates training pairs from a single noisy image via random neighbor subsampling for eigenimage denoising. Nevertheless, these self-supervised approaches are often predicated on certain noise assumptions. For instance, N2V-2D, N2V-3D, and N2V-Sep presuppose that the image signal is spatially correlated while the noise is independent, whereas E2E assumes zero mean noise. Such assumptions, however, frequently fail to hold under the complex noise conditions inherent to real-world HSIs.

Deep Image Prior (DIP) [44] introduces a novel paradigm for unsupervised image denoising. The fundamental premise of DIP is that the architecture of an untrained, randomly initialized CNN acts as an implicit image prior. During the optimization process for a single noisy image, the network naturally captures low-frequency structural information before modeling high-frequency noise. Consequently, applying early stopping yields a denoised result before the network overfits the noise. The DIP paradigm requires no external training data and relies solely on the single noisy image, making it particularly advantageous for HSI denoising where clean data are scarce. Sidorov and Hardeberg were the first to apply DIP to HSI inverse problems and proposed the DIP2D and DIP3D models [34]. Then, Miao et al. integrated DIP into linear mixed models, employing U-Net and fully connected networks to represent abundance maps and endmembers, respectively [47]. Saragadam et al. further extended this approach to address general low-rank tensor and matrix factorization problems (DeepTensor, [41]), thereby establishing a unified framework for various HSI inverse tasks. However, existing DIP-based methods depend heavily on CNNs; while adept at capturing spatial locality and translation invariance, they often fail to fully exploit the high correlation and smoothness inherent in the spectral dimension. Furthermore, Chakrabarty et al. pointed out some failure cases for the traditional DIP [48]. Specifically, when an image is corrupted by periodic stripe noise with regular intervals and widths, the DIP model often struggles to distinguish true signal from artifacts. This limitation stems from the inherent inductive bias of the network architecture, which prioritizes the generation of low-frequency structures. Since such structured stripe noise exhibits low-frequency characteristics that are similar to natural spatial features, the network inevitably assimilates these artifacts during the early optimization stages, rendering standard early stopping ineffective. This poses a fundamental challenge for DIP-based methods operating solely in the spatial domain. Conversely, in the spectral dimension, the disparity between the signal and noise is significantly more pronounced: while noise typically manifests as high-frequency fluctuations, clean spectral signatures are characterized by inherent smoothness. This distinction motivates us to explore spectral correlations as a more robust prior for effective denoising. Another challange is that their training process is very time-consuming [38,44] and it is difficult to accurately determine the optimal stopping point. To solve this problem, we introduce an adaptive early stopping strategy to prevent the network from overfitting to noise, significantly improving the denoising performance.

1.2. Contributions

This study investigates whether a similar learning preference exists in our one-dimensional natural signals, i.e., the spectral vectors in HSIs. Leveraging this learning preference, we propose a straightforward yet effective method for removing mixed noise in hyperspectral images. The main contributions of this paper can be summarized as follows:

• Existing HSI denoising networks typically rely on large amounts of high-quality paired noisy–clean images, which are difficult to obtain in practice. To overcome this limitation, we propose an unsupervised denoising method that requires no clean images and optimizes network parameters to automatically learn spectral priors from a single noisy image, addressing the issue of insufficient training data.
• We observe that deep networks tend to first fit smooth spectral features before addressing the noise. Based on this observation, we introduce an adaptive early stopping strategy, which leverages the network’s prior preference to fit only clean spectral signals, preventing overfitting to noise and significantly improving denoising performance.
• Our work is addressing the limitations of traditional denoising networks in generalizing to complex noise types. By modeling the intrinsic structure of the data, our approach enhances the network’s adaptability to various noise types, such as Gaussian, stripe noise, impulse noise, and deadlines. This significantly improves the robustness and stability of the denoising process, making it more effective in real-world scenarios.

This paper is organized as follows: Section 2 provides a detailed description of our proposed denoising method HyDePre. Section 3 and Section 4 present the experimental results, including comparisons with state-of-the-art algorithms. Section 5 provides discussions of this paper. Section 6 makes a conclusion.

2. Methodology

2.1. Problem Formulation

Let $\mathbf{X}\in {\mathbb{R}}^{n_b\times mn}$ denote a hyperspectral image matrix with $m\times n$ spectral vectors (the columns of $\mathbf{X}$, equal to number of spatial pixels) of size $n_b$ (the number of bands). Under the assumption of additive noise, the observation model can be written as

$$\mathbf{Y}=\mathbf{X}+\mathbf{S}+\mathbf{N},$$

(1)

where $\mathbf{Y},\mathbf{S},\mathbf{N}\in {\mathbb{R}}^{n_b\times mn}$ represent the observed HSI data, sparse noise (e.g., impulse noise and stripes), and Gaussian noise, respectively. Traditionally, the denoising problem is formulated as

$$\underset{\mathbf{X}}{min}\mathcal{L}(\mathbf{Y},f\left(\mathbf{X}\right))+\lambda \mathcal{R}\left(\mathbf{X}\right),$$

(2)

where the loss function $\mathcal{L}$ represents the data fidelity and f models the forward physical process to obtain the observation $\mathbf{Y}$. The regularizer $\mathcal{R}$ encodes priors on $\mathbf{X}$, and $\lambda >0$ is the regularization parameter.

2.2. Network Learning Preference for HSI Denoising

Deep neural networks exhibit a well-documented learning preference toward low-frequency signal components during training, a phenomenon commonly referred to as “spectral bias” or the “frequency principle”. Numerous studies in machine learning theory have shown that neural networks tend to fit smooth and slowly varying structures at early training stages, while high-frequency components are learned progressively later [48,49,50,51,52]. This behavior has also been linked to implicit regularization effects arising from network architectures and gradient-based optimization.

In the context of hyperspectral imagery, this learning preference aligns naturally with the intrinsic properties of spectral signals. Clean hyperspectral spectra typically exhibit strong spectral correlation and smooth variation across adjacent bands, reflecting material-specific reflectance characteristics. In contrast, common noise sources, such as Gaussian noise, impulse noise, and stripe noise, are largely spectrally uncorrelated and manifest as high-frequency perturbations along the spectral dimension.

When a neural network is trained directly on spectral vectors, this interaction between the network’s frequency bias and the spectral characteristics of hyperspectral data causes the network to prioritize fitting smooth spectral curves before adapting to noisy fluctuations. As training continues, the network gradually begins to absorb high-frequency noise components, leading to overfitting. This observation motivates the adoption of an early stopping strategy in the spectral domain. By terminating training at an appropriate stage, the network is encouraged to capture the underlying clean spectral structure while avoiding excessive fitting to noise.

Let $G_{\Theta }$ denote a trainable deep neural network (DNN) parameterized by $\Theta$. Introducing this reparametrization into (2), we obtain

$$\underset{\Theta }{min}\mathcal{L}(\mathbf{Y},f\circ G_{\Theta }\left(\mathbf{Y}\right))+\lambda \mathcal{R}\circ G_{\Theta }\left(\mathbf{Y}\right),$$

(3)

where the symbol ∘ represents the function composition operator.

The flowchart of our proposed HyDePre is shown in Figure 1, where each pixel is treated as a sequence of spectral values, enabling the network to model the inter-band dependencies effectively. Obviously, we focus on spectral features instead of the spatial features emphasized in traditional DIP applications [34]. This is because spectral correlations in hyperspectral data are much stronger than spatial correlations [53]. Prioritizing spectral correlations enables more effective denoising (see Section 2.3) while simplifying the network, reducing parameters, and easing training. Moreover, HyDePre adopts an encoder–decoder architecture, which consists of the following three key components:

(1) Encoder: The encoder compresses the input spectral sequence for each pixel ${\mathbf{y}}_i\in {\mathbb{R}}^{n_b}$ into a low-dimensional representation ${\mathbf{y}}_i^{\mathrm{hidden}}\in {\mathbb{R}}^d$ using a stacked Long Short-Term Memory (LSTM) [54] network with three layers. This process effectively preserves the critical spectral features while mitigating redundancy and noise. The selection of an LSTM network as the backbone of the encoder framework is guided by the inherent characteristics of HSIs: (1) HSIs typically comprise hundreds of spectral bands, resulting in a long sequential spectral vector for each pixel; (2) the strong inter-band correlations within HSIs introduce complex dependencies across the spectral sequence, which the LSTM is well-suited to model and exploit. The LSTM operates on the input sequence in a layer-wise manner:

$${\mathbf{h}}_t^{\left(l\right)}={\mathrm{LSTM}}_l\left({\mathbf{h}}_t^{(l-1)},{\mathbf{c}}_t^{(l-1)}\right),l=1,2,3,t=1,\dots ,T,$$

(4)

where ${\mathbf{h}}_t^{\left(l\right)}$ and ${\mathbf{c}}_t^{\left(l\right)}$ are the hidden state and the cell state at time step t for the l-th LSTM layer, respectively. The LSTM encoder outputs a sequence of hidden states for each layer, and the final hidden state from the last layer ${\mathbf{h}}_T^{\left(3\right)}$ is taken as the final hidden state, which serves as a comprehensive representation of the entire spectral sequence:

$${\mathbf{y}}_i^{\mathrm{hidden}}={\mathbf{h}}_T^{\left(3\right)}={\mathrm{LSTM}}_{\mathrm{encoder}}\left({\mathbf{y}}_i\right).$$

(5)

(2) Latent representation: After processing the input spectral sequence ${\mathbf{y}}_i$ through the LSTM encoder, the final hidden state ${\mathbf{y}}_i^{\mathrm{hidden}}$ is then passed through a linear transformation without a bias term to project it into a lower-dimensional latent space of dimension p, (where $p\ll n_b$):

$${\tilde{\mathbf{y}}}_i={\mathbf{W}}_L{\mathbf{y}}_i^{\mathrm{hidden}},$$

(6)

where ${\mathbf{W}}_L\in {\mathbb{R}}^{p\times d}$ is the weight matrix.

The resulting latent representation ${\tilde{\mathbf{y}}}_i\in {\mathbb{R}}^p$ provides a more compact encoding of the spectral information and acts as a learned spectral prior. This latent representation encapsulates the essential spectral features of the input sequence, serving as a compressed form that retains the most critical spectral dependencies and correlations.

(3) Decoder: The decoder is responsible for reconstructing the input sequence from its latent representation. This process begins with a latent-to-hidden transformation, wherein the latent vector is projected back into the hidden space via a linear layer followed by a ReLU activation function. This transformation can be expressed as follows:

$${\mathbf{h}}_0=\mathrm{ReLU}\left({\mathbf{W}}_L^{\prime }{\tilde{\mathbf{y}}}_i\right),$$

(7)

where ${\mathbf{W}}_L^{\prime }\in {\mathbb{R}}^{d\times p}$ is the weight matrix. Next, the transformed vector ${\mathbf{h}}_0$ is reshaped into a form compatible with the decoder’s LSTM layer, denoted as ${\mathbf{h}}_0^{\left(3\right)}$. The decoder employs an LSTM structure identical to the encoder, with the same number of layers, to process the reshaped hidden vector and generate a sequence ${\mathbf{y}}_i^{\mathrm{decoded}}\in {\mathbb{R}}^d$. This process can be expressed as follows:

$${\mathbf{y}}_i^{\mathrm{decoded}}={\mathrm{LSTM}}_{\mathrm{decoder}}\left({\mathbf{h}}_0^{\left(3\right)}\right),$$

(8)

Finally, the output sequence is passed through a linear transformation (output layer) to obtain the reconstructed sequence

$${\widehat{\mathbf{x}}}_i={\mathbf{W}}_O{\mathbf{y}}_i^{\mathrm{decoded}},$$

(9)

where ${\mathbf{W}}_O\in {\mathbb{R}}^{n_b\times d}$ is the weight matrix. The ${\ell }_1$ norm between the reconstructed sequence ${\widehat{\mathbf{x}}}_i$ and the original input sequence ${\mathbf{y}}_i$ is employed as the reconstruction loss. In addition, an ${\ell }_1$-norm regularization is incorporated to promote parameter sparsity and stabilize the optimization process. Thus, the final loss function consisting of the reconstruction loss and an ${\ell }_1$ regularization term is formulated as follows:

$$\mathcal{L}(\Theta )={\displaystyle \frac{1}{n}}\sum _{i=1}^n{∥{\widehat{\mathbf{x}}}_i-{\mathbf{y}}_i∥}_1+{\lambda \parallel \Theta \parallel }_1={\displaystyle \frac{1}{n}}\sum _{i=1}^n{∥G_{\Theta }\left({\mathbf{y}}_i\right)-{\mathbf{y}}_i∥}_1+\lambda {\parallel \Theta \parallel }_1.$$

(10)

An important attribute of the deep neural network is its inherent bias towards capturing meaningful visual content more rapidly than fitting to noise during training, which leads to the phenomenon known as “early-learning-then-overfitting” [55]. This characteristic enables the framework to achieve effective denoising by stopping the training process at an optimal point before the model begins to overfit to noise.

In this work, we further investigate how this learning preference manifests in hyperspectral images and argue that transferring the DIP paradigm from the spatial domain to the spectral (wavelength) dimension allows the network to more effectively exploit intrinsic spectral correlations. The rationale and empirical evidence supporting the effectiveness of spectral domain DIP are discussed in the following subsection (Section 2.3). Based on this spectral learning behavior, effective denoising can be achieved by stopping the optimization process before the network starts fitting noise. However, the critical challenge lies in identifying this optimal stopping point without access to the ground truth clean image. To address this challenge, we introduce an adaptive early stopping strategy, requiring no clean images, which balances the trade-off between noise suppression and overfitting. The details of this strategy are elaborated in the subsequent subsection (Section 2.4).

2.3. Why Deep Image Prior Works Better in the Spectral Domain

Structured noise, such as stripe noise and deadline noise, is a common degradation in hyperspectral imagery and poses a challenge for DIP-based denoising methods. When such noise is spatially coherent, spatial domain modeling may inadvertently treat it as valid image structure and thus fit it during optimization. In this subsection, we provide an illustrative analysis to explain why modeling the deep prior from the spectral viewpoint is more effective in handling structured noise.

We consider Case 1 on the Washington DC Mall dataset, which simulated Gaussian noise and stripe noise. Figure 2a shows the clean spectral curve, the noisy observation, and the spectral vector recovered by the proposed HyDePre at the pixel point (20, 199). The clean image exhibits smooth variation along the spectral dimension, whereas the structured noise introduces sharp, localized perturbations at specific bands. During optimization, the proposed method based on spectral learning preference prioritizes fitting the smooth spectral trend while suppressing these abrupt fluctuations, resulting in a reconstructed spectrum that closely follows the clean signal without reproducing noise-induced spikes.

To further analyze how structured noise is perceived under different modeling viewpoints, we select a representative band (Band 32) where the spectral fluctuation is most pronounced. Figure 2b,c show the noisy and clean images of this band, respectively. The noise appears as strong spatially structured artifacts, which are highly coherent in the spatial domain. When a conventional spatial DIP [44] is applied directly to this single band, as shown in Figure 2d, the structured noise is partially preserved, indicating that the spatial model tends to interpret such patterns as image structures. This phenomenon is aligned with the failure case reported by Chakrabarty et al. in [48], which pointed out that when an image is corrupted by periodic stripe noise with regular intervals and widths, the DIP model often struggles to distinguish true signal from artifacts.

In contrast, Figure 2e shows the Band 32 image extracted from the denoised hyperspectral cube produced by the proposed HyDePre. Since the optimization is performed on pixel-wise spectral vectors rather than spatial patches, the structured spatial noise is implicitly transformed into localized spectral irregularities. These irregularities correspond to high-frequency components along the spectral dimension and are therefore less favored during the early stages of neural network optimization. As a result, the structured noise is effectively suppressed without being explicitly modeled or detected.

This analysis demonstrates that the advantage of spectral domain modeling lies in its ability to reshape the representation of structured noise in a way that aligns with the learning preference of neural networks. By transferring the deep prior from the spatial domain to the spectral domain, structured spatial noise becomes less dominant during optimization, leading to more robust denoising behavior.

2.4. Adaptive Early Stopping Strategy

By exploiting the fact that the spectral curves of clean HSI pixels are smooth, while noisy spectral curves exhibit irregularities, we design an early stopping strategy based on the average spectral maximum variation (ES-ASMV). The approach involves stopping the training process when significant variability is detected in the reconstructed spectral curves, signaling that the network has started to learn noise rather than the desired visual content.

We use superscripts to denote the number of training iterations, where ${\Theta }^t$ represents the network parameters after the t-th training iteration, and ${\widehat{\mathbf{X}}}^t=[{\widehat{\mathbf{x}}}_1^t,\dots ,{\widehat{\mathbf{x}}}_n^t]=G_{{\Theta }^t}\left(\mathbf{Y}\right)$ denotes the reconstructed image produced by the network at iteration t. For the i-th pixel of the t-th reconstruction image, its spectral maximum variation (SMV) is defined as the maximum absolute value of the variations computed along the spectral dimension

$${v^t}_i={∥\mathrm{D}{\widehat{\mathbf{x}}}_i^t∥}_{\infty }={∥{\left[{\widehat{\mathbf{x}}}_i^t\right]}_{2:n_b}-{\left[{\widehat{\mathbf{x}}}_i^t\right]}_{1:n_b-1}∥}_{\infty },$$

(11)

where ${∥\mathbf{x}∥}_{\infty }={max}_j\left|{\mathbf{x}}_j\right|$ is the infinity norm of vector $\mathbf{x}$, and the subscript ${\left[\mathbf{x}\right]}_{j_1:j_2}$ represents the subvector of $\mathbf{x}$ consisting of elements from the $j_1$-th to the $j_2$-th element.

As shown in Figure 3, the SMV values of the clean image are much smaller than those of the noisy image. This indicates that the average value of SMV is an effective indicator for distinguishing between clean images and noisy images. Based on this, the average spectral maximum variation (ASMV) of the entire reconstructed HSI at the t-th iteration is defined as the mean SMV of all pixels.

$${\mathrm{ASMV}}^t=\mathrm{mean}\left({v^t}_i\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{v}_i.$$

(12)

Given ${\mathrm{ASMV}}^t$, the rate of change in ASMV at the t-th iteration is defined as follows:

$$\Delta {\mathrm{ASMV}}^t=({\mathrm{ASMV}}^t-{\mathrm{ASMV}}^{t-1})/{\mathrm{ASMV}}^{t-1}.$$

(13)

When this rate ($\Delta \mathrm{ASMV}$) exceeds a predefined threshold $\gamma$, it indicates that the network begins to learn high-frequency noise components, and training should be terminated to prevent the network from overfitting. However, considering that the network parameters update drastically and the rate of change in ASMV is unstable in the early stages of training, the fact that $\Delta \mathrm{ASMV}$ exceeds the threshold in the early stages cannot truly reflect the overfitting trend of the network. Therefore, we introduce a Warm-up Period. Specifically, a starting epoch $E_S$ is defined, during which the early stopping mechanism remains inactive. Once the training epoch surpasses $E_S$, early stopping is triggered if the $\Delta \mathrm{ASMV}$ exceeds the threshold $\gamma$. This strategy can not only effectively avoid misjudgments in the initial training stage but also ensures that training halts when significant spectral variability is observed, thereby improving computational efficiency while preserving the quality of the reconstruction results. The proposed HyDePre algorithm for HSI mixed denoising is summarized in Algorithm 1.

Algorithm 1 HyDePre: Hyperspectral Image Denoising via Spectral Learning Preference

Require: Observation $\mathbf{Y}$; randomly initialized parameters ${\Theta }^0$; iteration counter $t=0$; ${\mathrm{ASMV}}^0=0$; threshold $\gamma$; and starting epoch $E_S$

Ensure: Reconstruction $\widehat{\mathbf{X}}$

1: while not stopped do

2:    Update $\Theta$ via (3) to obtain ${\Theta }^{t+1}$.

3:    Reconstruct ${\widehat{\mathbf{X}}}^{t+1}=G_{{\Theta }^{t+1}}\left(\mathbf{Y}\right)$.

4:    Compute ${\mathrm{ASMV}}^{t+1}$ for ${\widehat{\mathbf{X}}}^{t+1}$ via (12).

5:    Compute $\Delta {\mathrm{ASMV}}^{t+1}$ for ${\widehat{\mathbf{X}}}^{t+1}$ via (13).

6:    if $\Delta {\mathrm{ASMV}}^{t+1}>\gamma$ and $t>E_S$ then

7:         return ${\widehat{\mathbf{X}}}^t$.

8:    end if

9:    $t\leftarrow t+1$

10: end while

2.5. Comparative Analysis of Deep 1D Spectral Prior and DIP

The core finding of this study corroborates the hypothesis that the “Deep Image Prior” (DIP) effect, originally characterized in spatial domains by Ulyanov et al. [44], is also intrinsic to one-dimensional spectral vectors in hyperspectral images (HSIs). Our experiments demonstrate that neural networks exhibit a distinct high-frequency impedance in the spectral domain, fitting smooth spectral signatures significantly faster than noise components. This work is best contextualized against two related approaches: DHP [34] and DS2DP [47]. DHP represents a direct extension of DIP to the spatial domain of HSIs using CNNs; however, it incurs substantial computational burdens due to the requirement for large network capacity and excessive iteration overhead. To address efficiency, DS2DP employs a linear mixture model framework, utilizing a U-Net and fully connected networks to model abundance maps and endmembers, respectively. While DS2DP shares a conceptual similarity with our work in applying deep priors to 1D vectors (specifically endmembers), our approach diverges fundamentally by targeting the raw spectral vectors directly. The dual-network architecture of DS2DP, which jointly optimizes abundance and endmembers in an unsupervised manner, is susceptible to error propagation, potentially compromising global reconstruction quality. In contrast, our method directly exploits the network’s inherent learning preference for 1D spectral vectors. This paradigm avoids the complexities of unmixing-based decomposition, offering a straightforward yet highly effective solution for HSI denoising.

3. Experiments on Simulated Images

3.1. Noise Simulation

To assess the performance of the proposed method in comparison with state-of-the-art denoising algorithms, experiments were conducted on simulated data. A subimage from the Washington DC Mall dataset (https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html, accessed on 1 September 2024) (of size $150\times 200\times 100$) and a subimage from the Pavia University dataset (https://www.ehu.eus/ccwintco/index.php?title=Hyperspectral_Remote_Sensing_Scenes, accessed on 1 September 2024) (of size $310\times 250\times 87$) were selected for evaluation. Both subimages are of high quality and are commonly regarded as clean reference images in the literature [16,53]. To simulate noisy images, we introduced five types of additive noise into the data, as follows: Case 1∼Case 3 include a combination of Gaussian noise and stripe noise, impulse noise, and deadlines, respectively. Case 4 combines Gaussian noise, stripe noise, and impulse noise. Case 5 includes Gaussian noise, stripe noise, impulse noise, and deadlines simultaneously. These configurations allow for the assessment of the denoising performance under increasingly challenging and realistic mixed noise scenarios.

3.2. Network Architecture and Training Configuration

The proposed HyDePre framework adopts an encoder–latent–decoder architecture to model pixel-wise spectral vectors in an unsupervised manner. Both the encoder and decoder are implemented using stacked LSTM networks with three layers. The encoder maps the input spectral sequence into a compact latent representation, which serves as a bottleneck to suppress noise while preserving essential spectral structures. The decoder then reconstructs the spectral signal from this latent representation. All experiments share the same network architecture and training configuration. The detailed architectural and optimization parameters are summarized in

Table 1. Network architecture and training configuration of HyDePre.

d	p	Optimizer	Learning Rate	Batch Size	$\mathit{\lambda }$
128	8	Adam	0.001	64	${10}^{-7}$

d: hidden state dimensionality of each LSTM layer in both encoder and decoder; p: dimensionality of the latent representation; and $\lambda$: regularization coefficient used in the loss function.

. In addition, the Warm-up Period $E_S$ and the threshold $\gamma$ are set to {(6, 0.07), (10, 0.03), (6, 0.1), (6, 0.1), (6, 0.1)} for the Washington DC Mall dataset under five cases and {(25, 0.06), (25, 0.06), (45, 0.06), (10, 0.07), and (30, 0.06)} for the Pavia University dataset.

3.3. Comparisons and Evaluation Metrics

The proposed HyDePre is compared with eight state-of-the-art methods including FastHyMix [53] (https://github.com/LinaZhuang, accessed on 10 September 2024), KBR [19] (https://github.com/XieQi2015/KBR-TC-and-RPCA, accessed on 10 September 2024), FallHyDe [56] (https://github.com/ChenYong1993/FallHyDe, accessed on 15 December 2025), NonRLRS [18], DDS2M [38] (https://github.com/miaoyuchun/DDS2M, accessed on 1 December 2025), HLRTF [37] (https://github.com/YisiLuo/HLRTF, accessed on 1 December 2025), LRTFR [39] (https://github.com/YisiLuo/Continuous-Tensor-Toolbox, accessed on 1 December 2025), and EGD-Net. Among them, KBR, FallHyDe, and NonRLRS are model-based methods; DDS2M, FastHyMix, HLRTF, LRTFR, and EGD-Net are deep-learning-based or hybrid methods. NonRLRS utilizes the idea of low-rank matrix recovery by introducing a normalized $\epsilon$-penalty term to constrain the low-rank potential clean image and sparse noise components. KBR utilizes the idea of sparse representation, measuring the sparsity of a tensor through Kronecker-basis-representation. DDS2M is based on DIP and linear mixed models, using U-Net and fully connected networks to model abundance maps and endmembers, respectively. HLRTF and LRTFR both use low-rank tensor fractorization. HLRTF embeds a deep neural network into tensor singular value decomposition, while LRTFR proposes a low-rank tensor function representation which can continuously represent data beyond meshgrid with infinite resolution. FastHyMix is a two-stage hybrid method, involving signal subspace estimation followed by subspace coefficient regularization. It uses a Gaussian mixture model and a plug-and-play technique. Similarly, FallHyDe is also based on low-rank subspace representation. Its primary innovation lies in a non-iterative estimation of spatial representation coefficients, which leverages the high signal-to-noise ratio bands inherent in real HSIs. This design renders FallHyDe exceptionally fast and makes it highly suitable for large-scale HSI denoising tasks. EGD-Net is an eigenimage guided diffusion network for hyperspectral mixed noise removal. In the experiments, the hyperparameters of all algorithms were adjusted to the optimal values according to the authors’ recommendations in their papers through manual tuning. The experiments were conducted on a laptop equipped with a 12th Gen Intel(R) Core(TM) i7-12700H processor (2.30 GHz) and 16 GB of RAM. All the model-based methods were implemented in MATLAB R2022b. All the deep learning-based methods were implemented using the PyTorch 2.4.1 framework, accelerated by an NVIDIA GeForce RTX 3060 GPU.

To evaluate the denoising performance of different algorithms, in addition to visual effects, we use the Mean Peak Signal-to-Noise Ratio (MPSNR), Mean Structural Similarity Index (MSSIM), and Mean Feature Similarity Index (MFSIM) as quantitative metrics. These metrics are obtained by calculating the corresponding PSNR, SSIM [57], and FSIM [58] band by band and then taking the average. A higher PSNR, SSIM, and FSIM value indicates better denoising performance. Let $\mathcal{X}\in {\mathbb{R}}^{m\times n\times n_b}$ denote the clean hyperspectral image and $\widehat{\mathcal{X}}\in {\mathbb{R}}^{m\times n\times n_b}$ be the denoised result, where $n_b$ represents the number of spectral bands. These metrics are calculated by averaging the values obtained from each band b as follows:

$$\mathrm{MPSNR}={\displaystyle \frac{1}{n_b}}\sum _{b=1}^{n_b}10{log}_{10}\left({\displaystyle \frac{{\mathrm{MAX}}_b^2}{{\mathrm{MSE}}_b}}\right),$$

(14)

where ${\mathrm{MAX}}_b$ is the maximum pixel intensity in the b-th band, and ${\mathrm{MSE}}_b$ represents the mean squared error between the reference and denoised bands.

$$\mathrm{MSSIM}={\displaystyle \frac{1}{n_b}}\sum _{b=1}^{n_b}{\displaystyle \frac{(2{\mu }_{{\mathcal{X}}_b}{\mu }_{{\widehat{\mathcal{X}}}_b}+C_1)(2{\sigma }_{{\mathcal{X}}_b{\widehat{\mathcal{X}}}_b}+C_2)}{({\mu }_{{\mathcal{X}}_b}^2+{\mu }_{{\widehat{\mathcal{X}}}_b}^2+C_1)({\sigma }_{{\mathcal{X}}_b}^2+{\sigma }_{{\widehat{\mathcal{X}}}_b}^2+C_2)}},$$

(15)

where $\mu$ and ${\sigma }^2$ denote the mean and variance of the image intensity, ${\sigma }_{{\mathcal{X}}_b{\widehat{\mathcal{X}}}_b}$ is the covariance, and $C_1$ and $C_2$ are constants to ensure stability. ${\mathcal{X}}_b$ and $\widehat{{\mathcal{X}}_b}$ indicate the b-th band of the clean and reconstructed HSI.

$$\mathrm{MFSIM}={\displaystyle \frac{1}{n_b}}\sum _{b=1}^{n_b}\mathrm{FSIM}({\mathcal{X}}_b,{\widehat{\mathcal{X}}}_b).$$

(16)

where the calculation method of FSIM is defined in [58], which utilizes phase congruency and gradient magnitude to capture the preservation of image features.

3.4. Experimental Results

We present the quantitative assessment of the simulated experimental results in

Table 2. Quantitative assessment of the proposed and comparison methods on Washington DC Mall Dataset in different cases.

Case	Metric	Noisy	FastHyMix (TNNLS, 2021) [53]	KBR (TPAMI, 2017) [19]	FallHyDe (TGRS, 2024) [56]	NonRLRS (TIP, 2019) [18]	DDS2M (ICCV, 2023) [38]	HLRTF (CVPR, 2022) [37]	LRTFR (TPAMI, 2024) [39]	EGD-Net (JSTARS, 2025) [59]	HyDePre* (Proposed)	HyDePre (Proposed)
Case 1	MPSNR	20.10	32.72	33.70	30.63	32.08	32.06	34.03	31.78	33.55	34.74	34.74
	MSSIM	0.5673	0.9377	0.9600	0.8915	0.9529	0.8670	0.9642	0.8493	0.9572	0.9737	0.9737
	MFSIM	0.7913	0.9645	0.9791	0.9535	0.9750	0.9277	0.9836	0.9212	0.9753	0.9865	0.9865
	Time	-	5	63	0.2	202	1194	21	64	7	89	89
Case 2	MPSNR	16.38	30.56	29.56	25.53	28.58	29.27	29.03	30.26	30.65	30.39	30.39
	MSSIM	0.3705	0.9280	0.9103	0.7448	0.8980	0.7687	0.9152	0.8064	0.9258	0.9343	0.9343
	MFSIM	0.6846	0.9667	0.9557	0.8893	0.9525	0.8730	0.9613	0.9006	0.9663	0.9702	0.9702
	Time	-	3	47	0.2	97	2507	22	60	7	183	183
Case 3	MPSNR	17.68	31.83	31.72	27.33	29.86	29.81	32.22	30.59	31.69	32.53	32.44
	MSSIM	0.4637	0.9182	0.9403	0.8237	0.9290	0.7842	0.9482	0.8026	0.9497	0.9583	0.9601
	MFSIM	0.7395	0.9585	0.9701	0.9255	0.9638	0.8823	0.9763	0.8960	0.9774	0.9803	0.9801
	Time	-	3	76	0.2	52	1202	20	61	7	68	51
Case 4	MPSNR	17.28	28.04	30.36	26.74	29.62	28.56	30.13	30.11	28.13	30.93	30.62
	MSSIM	0.4420	0.8795	0.9299	0.8075	0.9253	0.7992	0.9421	0.8185	0.8771	0.9509	0.9488
	MFSIM	0.7106	0.9380	0.9696	0.9196	0.9647	0.8961	0.9762	0.9058	0.9388	0.9782	0.9776
	Time	-	3	90	0.1	90	2389	20	61	7	198	214
Case 5	MPSNR	17.25	29.29	32.08	27.58	31.12	29.13	31.57	31.29	28.66	32.53	32.53
	MSSIM	0.4538	0.9164	0.9500	0.8625	0.9447	0.8074	0.9539	0.8380	0.9106	0.9615	0.9615
	MFSIM	0.7219	0.9577	0.9754	0.9438	0.9725	0.9000	0.9805	0.9157	0.9548	0.9808	0.9808
	Time	-	3	87	0.2	45	1203	37	60	7	59	59
Mean	MPSNR	17.74	30.49	31.48	27.56	30.25	29.77	31.40	30.81	30.54	32.22	32.14
	MSSIM	0.4595	0.9160	0.9381	0.8260	0.9300	0.8053	0.9447	0.8230	0.9241	0.9557	0.9557
	MFSIM	0.7296	0.9571	0.9700	0.9257	0.9657	0.8958	0.9756	0.9079	0.9625	0.9792	0.9790
	Time	-	3	73	0.2	97	1699	24	61	7	119	119

Note: The best and the second best results are highlighted in bold and underline, respectively.

and

Table 3. Quantitative assessment of the proposed and comparison methods on Pavia University dataset in different cases.

Case	Metric	Noisy	FastHyMix (TNNLS, 2021) [53]	KBR (TPAMI, 2017) [19]	FallHyDe (TGRS, 2024) [56]	NonRLRS (TIP, 2019) [18]	DDS2M (ICCV, 2023) [38]	HLRTF (CVPR, 2022) [37]	LRTFR (TPAMI, 2024) [39]	EGD-Net (JSTARS, 2025) [59]	HyDePre* (Proposed)	HyDePre (Proposed)
Case 1	MPSNR	24.30	34.93	35.70	27.14	34.09	30.85	33.64	30.38	35.37	35.56	35.44
	MSSIM	0.4448	0.9291	0.9327	0.6432	0.9302	0.8345	0.9158	0.8270	0.9411	0.9357	0.9358
	MFSIM	0.7357	0.9617	0.9716	0.8043	0.9680	0.8846	0.9617	0.8914	0.9686	0.9735	0.9736
	Time	-	7	161	0.3	56	2009	55	161	8	349	334
Case 2	MPSNR	22.95	34.16	33.86	24.41	33.32	28.23	32.15	29.97	33.96	33.98	33.95
	MSSIM	0.3992	0.9091	0.9166	0.5417	0.9005	0.7692	0.9048	0.8150	0.9018	0.9219	0.9220
	MFSIM	0.6963	0.9673	0.9672	0.7445	0.9580	0.8596	0.9592	0.8841	0.9655	0.9729	0.9730
	Time	-	7	162	0.4	74	2117	55	158	8	374	358
Case 3	MPSNR	26.44	40.81	35.78	28.88	36.37	29.56	37.28	31.55	37.19	39.02	38.99
	MSSIM	0.6031	0.9786	0.9330	0.6735	0.9520	0.7907	0.9564	0.8621	0.9504	0.9680	0.9678
	MFSIM	0.8281	0.9895	0.9673	0.8357	0.9756	0.8687	0.9799	0.9097	0.9788	0.9883	0.9881
	Time	-	10	163	0.3	76	2027	55	157	8	646	661
Case 4	MPSNR	21.43	29.40	30.54	23.33	30.13	26.74	29.00	27.62	29.69	30.94	30.78
	MSSIM	0.3752	0.8195	0.8154	0.4999	0.8294	0.7460	0.7687	0.7481	0.8185	0.8716	0.8753
	MFSIM	0.6549	0.9298	0.9169	0.7142	0.9260	0.8695	0.8827	0.8525	0.9295	0.9480	0.9560
	Time	-	7	158	0.5	71	2037	55	156	8	219	168
Case 5	MPSNR	22.14	32.53	36.50	29.22	35.85	28.20	35.27	31.02	31.32	36.76	36.76
	MSSIM	0.4428	0.8888	0.9581	0.7984	0.9535	0.8061	0.9501	0.8485	0.8910	0.9534	0.9534
	MFSIM	0.7037	0.9606	0.9833	0.9243	0.9790	0.9054	0.9802	0.9031	0.9652	0.9837	0.9837
	Time	-	21	245	0.4	63	2056	55	157	8	486	486
Mean	MPSNR	23.45	34.37	34.48	26.60	33.95	28.72	33.47	30.11	33.51	35.25	35.18
	MSSIM	0.4530	0.9050	0.9112	0.6313	0.9131	0.7893	0.8992	0.8201	0.9006	0.9301	0.9309
	MFSIM	0.7237	0.9618	0.9613	0.8046	0.9613	0.8776	0.9527	0.8882	0.9615	0.9733	0.9749
	Time	-	10	178	0.4	68	2049	55	158	8	415	401

Note: The best and the second best results are highlighted in bold and underline, respectively.

. To rigorously assess the effectiveness of our proposed adaptive early stopping strategy, we also include a variant denoted as HyDePre*, where the stopping iteration was manually selected to maximize the MPSNR for each specific case (serving as the idea performance that our framework can achieve). As shown in Table 2, our proposed HyDePre and HyDePre* outperform the compared algorithms in nearly all cases on the Washington DC Mall Dataset. Although FastHyMix surpasses our HyDePre and HyDePre* in terms of MPSNR in Case 2, it shows a significant performance decline in Case 4 and Case 5, indicating that it cannot handle complex scenarios where images simultaneously contain multiple types of noise (including Gaussian noise, stripe noise, impulse noise, and deadlines).

As shown in Table 3, our HyDePre and HyDePre* rank first or second in nearly all individual cases on Pavia University dataset. While FastHyMix demonstrates exceptional performance in Case 3 (Gaussian noise and deadlines), its efficacy drops noticeably in complex mixed-noise scenarios (e.g., Case 5, which contains Gaussian noise, stripe noise, impulse noise, and deadlines), whereas HyDePre maintains robust performance across all degradation types. Compared to the DIP-based method DDS2M, our proposed approach achieves superior denoising performance with significantly less computational cost. This validates the efficacy of our design principle: prioritizing spectral correlation over spatial features enables a straightforward yet more powerful deep image prior for hyperspectral image denoising. Figure 4 and Figure 5 show the visual effects of different denoising algorithms. For the visual comparison of the Washington DC Mall dataset, we present the results from the five top-performing algorithms to ensure clarity within the paper’s space constraints. The selection was based on the highest mean MPSNR achieved across all cases. These algorithms are KBR, HLRTF, LRTFR, EGD-Net, and the proposed HyDePre with adaptive early stopping strategy. Additionally, for the Pavia University dataset, we present the results of all algorithms except FallHyDe. As we can see, with the exception of the proposed HyDePre, all comparing methods encountered significant challenges when handling the complex mixed noise scenarios in Case 4 and Case 5.

The comparison between HyDePre (with the proposed adaptive early stopping) and HyDePre* (with manual stopping) indicates that the performance of HyDePre is remarkably close to that of HyDePre*. For instance, in Cases 1∼4 on the Pavia University dataset, the difference in MPSNR is negligible (<0.2 dB), and in Case 5, the metrics are identical. This strongly validates that our proposed adaptive early stopping criterion can accurately identify the optimal trade-off point between noise removal and signal preservation without requiring ground truth references. In summary, HyDePre offers a superior balance of high-quality restoration, robust generalization across varying noise intensities, and practical automation via the adaptive stopping mechanism.

3.5. Effectiveness of the Proposed Early Stopping Strategy

We proposed an adaptive early stopping strategy which aims to stop training at an optimal point to avoid overfitting to noise. To validate the effectiveness of this strategy, experiments were conducted on two simulated datasets, the Washington DC Mall dataset and the Pavia University dataset. The effectiveness of the proposed early stopping strategy can be observed in Figure 6. The figure depicts the relationship between the MPSNR and $\Delta$ASMV across training epochs for two cases: Case 1 (Washington DC Mall dataset) and Case 4 (Pavia University dataset). In these experiments, the Warm-up Period was configured to ten epochs with a threshold $\gamma =0.07$. Notably, the $\Delta$ASMV value surpasses the predefined threshold $\gamma$ after the Warm-up Period, coinciding with the point where the MPSNR value peaks and subsequently begins to decline. This result validates that the proposed early stopping strategy can effectively identify the optimal stopping point for training.

3.6. Sensitivity Analysis of the Regularization Coefficient $\lambda$

The proposed loss function contains a regularization term weighted by a coefficient $\lambda$, which controls the strength of the prior imposed during optimization. In all experiments, $\lambda$ is set to ${10}^{-7}$, and this value is used consistently across different datasets and noise settings. To examine the sensitivity of the proposed method to this hyperparameter, we conduct a simple sensitivity analysis on simulated data, specifically Case 1 of the Washington DC Mall dataset. We evaluate three representative values, $\lambda \in \{{10}^{-6}, {10}^{-7}, {10}^{-8}\}$. Figure 7 reports the corresponding training loss and MPSNR curves during training. As shown in Figure 7, when $\lambda ={10}^{-6}$, the training loss is high and the MPSNR is significantly low, indicating that excessive regularization leads to constant degradation; when $\lambda ={10}^{-8}$, the training loss is the lowest but the MPSNR decreases as training progresses, suggesting that the network overfits to noise. In contrast, $\lambda ={10}^{-7}$ achieves a favorable trade-off between training stability and denoising performance, yielding the highest MPSNR and a more stable optimization trajectory. Based on this analysis, $\lambda ={10}^{-7}$ is adopted in all experiments, as it effectively balances bias and variance under the current model architecture and data scale, avoids extreme degradation, and leads to more reliable denoising results.

3.7. Ablation Experiments

In addition to the reconstruction loss, our network incorporates an ${\ell }_1$ regularization term, which facilitates implicit feature selection by promoting parameter sparsity and guiding the network toward targeted learning. To evaluate the effectiveness of this regularization term, we conducted ablation experiments by excluding the ${\ell }_1$ regularization term while retaining all other model configurations. The experimental results are reported in

Table 4. Performance comparison of the proposed HyDePre* with and without ${\ell }_1$ regularization in different cases.

	HyDePre* w/o ${\mathit{\ell }}_1$ Regularization			HyDePre*
Washington DC Mall Dataset
	MPSNR	MSSIM	MFSIM	MPSNR	MSSIM	MFSIM
Case 1	34.58	0.9742	0.9860	34.74	0.9737	0.9865
Case 2	30.07	0.9353	0.9708	30.39	0.9343	0.9702
Case 3	32.30	0.9582	0.9797	32.53	0.9583	0.9803
Case 4	30.78	0.9418	0.9728	30.93	0.9509	0.9782
Case 5	32.21	0.9602	0.9807	32.53	0.9615	0.9808
Mean	31.99	0.9539	0.9780	32.22	0.9557	0.9792
Pavia University dataset
Case 1	35.10	0.9182	0.9688	35.56	0.9357	0.9735
Case 2	33.56	0.9188	0.9723	33.98	0.9219	0.9729
Case 3	38.70	0.9672	0.9880	39.02	0.9680	0.9883
Case 4	30.69	0.8739	0.9537	30.94	0.8716	0.9480
Case 5	36.18	0.9501	0.9830	36.76	0.9534	0.9837
Mean	34.85	0.9256	0.9732	35.25	0.9301	0.9733

Note: The best results are highlighted in bold.

. As shown, removing the ${\ell }_1$ regularization degrades performance across nearly all scenarios, indicating its positive contribution to the overall effectiveness of our method.

4. Experiments on Real Images

4.1. Mars Image

The first real image we used is the Mars image, which was captured over the McLaughlin Crater region on Mars by the CRISM sensor on 10 March 2008. We selected a sub-region of it with a spatial size of $120\times 120$ and 418 spectral bands for the experiment. Figure 8 presents the restoration results of four selected bands (bands 2, 262, 320, and 350). As observed in the comparison, obvious residual vertical stripes remain visible in the restored images of FastHyMix, KBR, FallHyDe, NonRLRS, and HLRTF (especially in bands 262 and 320). DDS2M effectively suppresses the stripe noise but suffers from severe over-smoothing, causing the significant loss of spatial details and blurring the sharp edges of the crater. LRTFR introduces noticeable blocking artifacts. In contrast, the proposed HyDePre method and EGD-Net method achieve the most favorable visual quality, striking a superior balance between noise removal and detail preservation.

4.2. Hyperion Cuprite Image

The second real image we used is the Hyperion Cuprite image, which was captured by the Hyperion sensor in the cuprite area of Nevada, USA. It contains 242 bands, with a 10 nm spectral resolution and a 30 m spatial resolution. We selected a sub-region of it with a spatial size of $200\times 160$ and 177 spectral bands for experiment. Figure 9 presents the restoration results of 4 selected bands (bands 52, 98, 166, and 177). As illustrated in Figure 9, the reconstruction results of DDS2M suffer from excessive smoothing and blurring, while LRTFR introduces distinct blocking artifacts. KBR, FallHyDe, NonRLRS, and EGD-Net fail to fully eliminate stripe noise, with noticeable stripes remaining in bands 52, 166, and 177. While FastHyMix and HLRTF effectively suppress most noise, they compromise fine structural details within the regions highlighted by red dashed ovals (bands 166 and 177). Collectively, this visual evidence validates the robustness of HyDePre in handling complex, real-world hyperspectral degradation.

5. Discussion

Analysis of the Adaptive Early Stopping Strategy

The success of the proposed unsupervised learning framework hinges on the precise termination of the optimization process. To achieve this without ground truth references, we introduce the rate of change in average spectral maximum variation ($\Delta$ASMV) as a pseudo-metric to monitor the noise level of the generated HSI (Section 2.4). The ASMV is defined as the mean of the maximum absolute gradients along the spectral dimension for all pixels.

The rationale behind using $\Delta$ASMV lies in the intrinsic difference between the spectral properties of natural signals and mixed noise. Natural spectral signatures generally exhibit high autocorrelation and continuity, meaning their first-order differences (gradients) are relatively small and bounded. Conversely, mixed noise—particularly impulse noise and stripes—manifests as high-frequency singularities in the spectral domain. By calculating the maximum variation rather than the average variation per pixel, the ASMV becomes highly sensitive to these anomalous spikes, effectively acting as a detector for high-frequency spectral artifacts. The effectiveness of using $\Delta$ASMV can be confirmed by Figure 10.

6. Conclusions

In this paper, we propose a new denoising method for HSIs, HyDePre, which takes advantage of the intrinsic learning preference of neural networks for spectral vectors. At its core, HyDePre utilizes an encoder–decoder architecture to learn the deep 1D spectral prior directly from the noisy data. The elegance of this approach lies in its conceptual simplicity: it requires no pre-trained models or complex parameter tuning, yet demonstrates remarkable effectiveness in separating signal from noise. To validate the effectiveness of our method, experiments on simulated and real datasets were conducted. The experimental results demonstrate the effectiveness and robustness of our proposed method, achieving significant improvements in HSI denoising performance. Beyond denoising, the core principle HyDePre offers a promising paradigm for a broader range of HSI inverse problems such as inpainting, destriping, and super resolution.

By focusing on spectral vectors, HyDePre establishes a concise and fully unsupervised denoising framework that effectively exploits the abundance of spectral samples in hyperspectral data. This design choice leads to a transparent and reproducible methodology with stable optimization behavior and denoising performance that is competitive with most state-of-the-art approaches. As HyDePre is formulated as a single-image optimization problem, the inference stage involves iterative updates for each test image. From an engineering perspective, the resulting computational cost may be mitigated by incorporating meta-learning or transfer learning strategies to provide favorable initialization, while preserving the core architecture and underlying principle of the proposed method. Furthermore, although the present work deliberately emphasizes 1D spectral modeling to maintain conceptual simplicity and robustness, hyperspectral images inherently exhibit rich spatial–spectral structures. Extending HyDePre to integrate spatial context in a controlled and principled manner constitutes a natural direction for future research, which is expected to further enhance denoising performance at the expense of increased model complexity and computational overhead.