Introduction

Hyperspectral imaging has the characteristics of a combination of spatial and spectral information simultaneously, which plays a crucial role in the fine recognition and extraction of materials1,2. Benefiting from the advantages of hyperspectral data with rich spectral wavelengths, which can provide chemical composition and quantitative information of surface materials, a series of imaging spectrometers have been designed and developed in deep space exploration, e.g., Mars3,4 and Moon5. In this paper, we focus on the Compact Reconnaissance Imaging Spectrometer for Mars (CRISM) aboard NASA’s Mars Reconnaissance Orbiter for acquiring a hyperspectral data cube of the Martian surface. By using the hyperspectral data acquired by CRISM, scientists can identify the main rocks and minerals on the surface of Mars, such as hydrous minerals6, etc. By comparing the spectral characteristics of different regions and features, the evolution of the chemical and mineral composition of the Martian crust can also be inferred. This information is essential for further understanding the geological history and environmental evolution of Mars, i.e., Fe–Mg-rich clay minerals were discovered in McLaughlin crater via CRISM data, which indicates a groundwater-fed lake in this region7. In addition to identifying surface components, CRISM data is also used to detect the dust and water ice aerosols in the atmosphere, which provides a valuable tool for probing atmospheric structure and improving our understanding of the Martian atmosphere8. However, the reason for the low signal-to-noise ratio (SNR) of the instrument, the presence of artifacts or calibration errors, caused mixed noise to exist widely in the CRISM data, which affects the accurate extraction and analysis of minerals.

To solve the problem, researchers have developed various techniques to reduce the impact of noise and improve the quality of the data. These techniques include applying various filters, using statistical machine learning methods, and developing more accurate deep learning-based methods to remove noise. A column on an average-based noise removal method for CRISM data was proposed in ref. 9, where the localized noise spikes are identified based on spectral derivatives and corrected using neighborhood pixel information. A denoising tool, complement to CRISM analysis toolkit (CoTCAT), provides a straightforward pipeline to reduce noise for CRISM data using sharpening-median, mobile median, and mobile average filters10. An iterative maximum log-likelihood method regularizes along-track oversampled CRISM data, which can remove the vertical striping artifacts effectively11. A hypothesis-based estimation with regularization method was proposed, which provides an improved statistical framework for reconstructing hyperspectral images from instruments like CRISM12. A new method for simultaneous atmospheric correction and denoising of CRISM data was proposed13, which significantly reduces column-dependent artifacts remaining after standard volcano scan correction14. The technique estimates the atmospheric transmission spectrum directly from each image by modeling it as an additive component in a sparse spectral unmixing framework. Large noise spikes are identified and corrected during the iterative estimation process. In addition, a hierarchical Bayesian model-based ratioing technique using bland pixel likelihoods is proposed in ref. 15, which can be used to detect subtle mineral phases even in small outcrops. In addition, the radiance data were used as a verification of the authenticity of minerals at 2.1 \(\mu m\) absorptions16.

The recent advancements in deep learning-based image processing methods have led to significant breakthroughs, owing to their exceptional speed of inversion and cutting-edge performance17. More specifically, various supervised learning methods have been designed in the image denoising field, including RGB images18, multispectral images19, and hyperspectral images20. However, to train such type of networks, we need to build a large number of noisy-clean pairs, which is obviously not feasible in the denoising of Mars, because we do not have clean images of Mars. Thanks to the development of self-supervised learning technology, we can only use noisy images for denoising, without using clean images21. Self-supervised learning is a type of deep learning that trains the model from the input data itself, without any explicit labeled data. Self-supervised learning has become increasingly popular in recent years and has been applied in the denoising field successfully21,22,23. For the denoising field, the model is trained only within the corrupted data itself, rather than being provided with paired noisy-clean data. More specifically, plenty of noisy–noisy pairs are generated within the same scene to train the network21. In practice, it is still hard to collect plenty of paired noisy–noisy pairs; hence, a neighbor sub-sampling method is proposed to generate two sub-images for network training23. By learning from these unlabeled training pairs from noisy images directly, self-supervised models can often achieve competitive performance in a denoising field. Given the unavailability of clean CRISM images in practice, the application of existing supervised deep learning-based techniques for denoising CRISM images is not feasible. Thus, we turn to self-supervised learning as an alternative approach for CRISM image denoising.

This study aims to develop a new machine learning denoiser for global Martian surface mineral identification to test whether the diagnostic absorption peak can be recovered from the low SNR CRISM data. A critical aspect of this research is that our network is trained solely using noisy CRISM images by subsampling two sub-images from each noisy image without necessitating any clean images. The network adaptively learns the inherent noise in the images. Our finding suggests that it is more reasonable to train the model on the CRISM target domain directly. Meanwhile, the model efficiently mitigates noise in Martian imagery, providing a viable solution for detecting low-abundance24,25 minerals on the Martian surface.

Results

E2E-CRISM principle

The self-supervised framework of E2E-CRISM is illustrated in Fig. 1. Martian spectral images exhibit redundancy in the spectral dimension. Based on subspace representation theory, the original image is projected onto orthogonal subspaces derived directly from the noisy images, resulting in eigenimages. To account for training samples, sub-images of spatially adjacent regions are sampled from the original single low-SNR eigenimage through neighbor column sampling. Each target sub-image is adjacent to the input sub-image either horizontally or vertically. Since noise among adjacent pixels is independent while signal correlation exists, one of the sub-images can be utilized as training input while the other serves as a corresponding target to optimize network parameters. Our sampling strategy comprehensively preserves spatial information compared to alternative methods. During inference, the low-SNR original Martian eigenimages will be fed into the trained E2E-CRISM model without spatial down-sampling. Additionally, since the first band of the eigenimage exhibits higher SNR, it serves as a guidance image to direct network optimization. Moreover, our network solely comprises attention-based convolutional layers and activation functions without any spatial down-sampling modules, allowing high-frequency components to pass through and avoiding loss of spatial resolution. Furthermore, the lightweight network design utilizing subspace techniques significantly reduces computational burden, enabling the proposed method to process each image in less than one minute. This is a significant improvement in speed, making the proposed method much faster and more suitable for processing the large-scale Martian data.

Fig. 1: The flowchart of E2E-CRISM hyperspectral image denoising framework.
Fig. 1: The flowchart of E2E-CRISM hyperspectral image denoising framework.The alternative text for this image may have been generated using AI.
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The method leverages Mars spectral images' spectral redundancy to generate intrinsic images via subspace projection, acquires trainingsample pairs through adjacent column sampling, and optimizes network parameters in a self-supervised manner bytreating adjacent sub-images as input-target pairs. Symbols: \([{\cal Y}]\), noisy image; \([\widetilde {\cal A}]\), eigenimage; \([{\bf{C}}]\), spectral subspace; \([\widehat{\cal X} = \widehat {\cal A}{ \times _3}{\bf{C}}]\), estimated clean image.

Statistical analysis

To verify the gap of the model trained with different noisy datasets, the proposed model trained on a simulated Interdisciplinary Computational Vision Laboratory (ICVL) dataset26 and on a real global noisy CRISM dataset in this paper are compared. We first match the most similar feature map of all layers trained with two different datasets with the Frobenius norm. Subsequently, p-value27, a probability value in statistics that measures the strength of evidence against a null hypothesis, is used to test the significance of a result or a finding in a statistical analysis to illustrate the differences of the model trained on different datasets here. To ensure a fair comparison, we kept all variables fixed in the model trained with the ICVL dataset and CRISM dataset. After training the network, we test both of them on CRISM data with IDs FRT00005f23 for comparison. The p-value shows that from layer 1 to layer 6, there is a significant difference in pixel values between the two images, which accounts for more than 99%. Based on the results of the statistical test, we reject the null hypothesis that there is no significant difference in pixel values between the two images. The phenomenon shows that there is a significant difference in the features obtained by the two different datasets.

In the last layer of the model (layer 7), there is also a significant difference in pixel values between the two images, and this phenomenon accounts for 60%. More specifically, the visualization results of feature maps in layer 7 are shown (Supplementary Figs. 1–5) with two different models. The results show that H = 0, P-value = 0.670582, and H = 0, P-value = 0.685560, in band 5 and band 6, respectively. The value H = 0 refers to the null hypothesis, which typically states that there is no effect or no difference between two images. The p-values are 0.670582 and 0.685560, which represent the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. Therefore, features obtained via two different datasets are consistent with the null hypothesis of no difference in band 5 and band 6 but have a distinct in band 2, band 3, and band 4.

Identification of Martian surface minerals based on E2E-CRISM

Electromagnetic spectral characteristics ranging from visible to short-wave infrared are commonly employed for identifying rocks, soils, and minerals on the Martian surface. Each type of mineral exhibits absorption spectral features that can offer valuable insights into their chemical composition, structure, and paragenetic information, thereby facilitating geological environment inference. The significance of Mars as a primary target for scientific exploration, in terms of determining habitability and the potential presence of life, prompts our specific focus on identifying hydrated clay minerals. Here, we assessed the result in mineral identification performance following the improvement of Martian image SNR. We first focused on the mineral’s interpretation in the Eridania basin. Before applying the proposed method, photometric angles and atmospheric correction are corrected using the CAT 7.4 software. Fig. 2 displays the noisy band 83 of the CRISM images for datasets with IDs 7f7e and 865f, along with their corresponding denoising results obtained through the proposed E2E-CRISM method, and the histogram of noise intensity for the corresponding noisy image estimated by HySime28. The spectral band 83 recovered by the proposed E2E-CRISM method exhibits remarkable outcomes in comparison to the corrected version solely utilizing CAT. The proposed method demonstrates clear effectiveness in Gaussian noise removal. Moreover, it exhibits the capability to partially eliminate striping noise.

Fig. 2: CRISM HSI, 7f7e, and 865f, of Band 83 in the experiments.
Fig. 2: CRISM HSI, 7f7e, and 865f, of Band 83 in the experiments.The alternative text for this image may have been generated using AI.
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a CRISM data (7f7e) processing using CAT 7.4. b CRISM data (7f7e) processing using CAT 7.4 and the proposed E2E-CRISM. c Histogram of noise intensity (estimated by HySime) for (a), fitted by a Gaussian distribution. d CRISM data (865f) processing using CAT 7.4. e CRISM data (865f) processing using CAT 7.4 and the proposed E2E-CRISM. f Histogram of noise intensity (estimated by HySime) for (d), fitted by a Gaussian distribution.

Furthermore, the spectral interpretation results indicate a significant presence of alteration minerals in this region, characterized by prominent Mg–OH absorptions at approximately 2.31 μm and OH overtones ranging from 1.39 to 1.4 μm, along with H–OH absorptions near 1.91 μm associated with adsorbed water. More specifically, the mineralogy results in the colles deposits are presented in Fig. 3. Figure 3a illustrates the spectral curves with four images, wherein the presence of severe spectral noise poses challenges in identifying the effective absorption peaks of minerals. Conversely, the denoised curves (depicted in Fig. 3b) exhibit distinct absorption features at the positions of 1.4146 μm, 1.9281 μm, 2.2054 μm, 2.3112 μm, 2.3972 μm, and 2.5097 μm, respectively. These diagnostic features make it easier to identify different types of minerals; for instance, the presence of IDs, caf3, can be determined by observing absorptions at 1.1416 μm, 1.9281 μm, and 2.2054 μm, (as shown in Fig. 3b), indicating the presence of opal mineral.

Fig. 3: Mineralogy in the colles deposits.
Fig. 3: Mineralogy in the colles deposits.The alternative text for this image may have been generated using AI.
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a CRISM data processing using CAT 7.4. b CRISM data processing using CAT 7.4 and the proposed E2E-CRISM.

The proposed E2E-CRISM method was further applied to the McLaughlin crater, which is one of the highly rated candidate landing sites for China’s first Mars sample return mission, known as “Tianwen-3”, to validate its efficacy in identifying hydrous clay minerals. Specifically, our focus is on detecting the presence of serpentine, an important indicator of mineral for potential life existence that is rare and sparsely distributed on Mars. The false color image before and after denoising is presented in Fig. 4, along with a comparative result of the serpentine spectral curves pre- and post-denoising, as well as a comparison with laboratory spectral curves. It can be clearly seen that after denoising, the CoTCAT method results in oversmoothed spectra, leading to the erroneous removal of some diagnostic absorption peaks, such as at 2.3244 μm. In contrast, the proposed E2E-CRISM method achieves a good balance between preserving spectral information and maintaining absorption peak characteristics, thereby demonstrating its superiority. We can distinctly identify the combination of four absorption peaks of serpentine, including 1.3883 μm, 2.1063 μm, 2.3244 μm, and 2.5097 μm. Although the presence of serpentine was mentioned in previous studies conducted in this area, no direct evidence supporting its existence through spectral curve analysis has been provided29,30. Based on the proposed method, we directly obtain the spectral curves of serpentine mineral from the images. The discovery of this mineral, which is often associated with the presence of water and potentially indicative of past habitable environments, provides valuable information and reference for our scientific questions regarding the search for traces of life in future Mars sample return missions. By using the implementation of the proposed E2E-CRISM methodology, we have successfully identified talc for the first time within this study area, as illustrated in Fig. 4. It is evident that we can distinctly discern the presence of three absorption peaks associated with talc at wavelengths of 1.3883 μm, 2.3112 μm, and 2.3972 μm.

Fig. 4: Mineral imagery and spectral analysis of McLaughlin crater.
Fig. 4: Mineral imagery and spectral analysis of McLaughlin crater.The alternative text for this image may have been generated using AI.
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a CRISM data (a5aa) processing using CAT 7.4, followed by the proposed E2E-CRISM. b CRISM spectral curve analysis matched with serpentine mineral via USGS spectral library. c CRISM data (c42e) processing using CAT 7.4, followed by the proposed E2E-CRISM. d CRISM spectral curve analysis matched with talc mineral via Mica and USGS spectral library.

Discussion

The identification of clay minerals on the Martian surface serves as a foundational element for analyzing the planet’s early climate, hydrological activity, geological history, and habitability. While current methodologies have successfully identified high-abundance minerals on Mars, several challenges remain unresolved: can we effectively explore low-abundance minerals? What advantages does enhanced image quality offer for mineral identification? This paper introduces an innovative machine learning approach that leverages global noisy image data from Mars to develop the first comprehensive model for enhancing mineral data quality. This model efficiently mitigates noise in Martian imagery, providing a viable solution for detecting low-abundance minerals on the Martian surface.

To elucidate the differences between models trained on Martian imagery and those developed using terrestrial simulations, we initially performed a statistical analysis, which revealed a 60% disparity in the feature maps generated by the two models. Hence, it is more reasonable to train the self-supervised network on the target domain directly, CRISM data, rather than other images, i.e., ICVL. Hence, compared with the model in ref. 26, in this paper, we trained a self-supervised model only with noisy CRISM HSI data with the neighbor column sampling, meanwhile, a total variation (TV) loss is considered to get a smooth result of the CRISM.

The absorption features of clay minerals are observed at ~1.4, 1.9, and 2.1–2.5 µm, respectively. Specifically, the band at 1.4 µm corresponds to OH and H2O molecules, while the band at 1.9 µm is attributed solely to H2O molecules. Additionally, the bands in the range of 2.1–2.5 µm correspond to various molecular species, including Fe–OH, Mg–OH, and Al–OH. From the analysis of the original noisy spectral curve, it was found that complex noise seriously affects the identification of effective absorption peaks of minerals, causing significant interference and many incorrectly identified minerals. In contrast, the Mars images after E2E-CRISM denoising clearly revealed the distinct absorption peak positions of different minerals (shown in Fig. 3b). Previously, these peaks were unrecognizable due to positioning errors and noise (shown in Fig. 3a). Subsequently, we conducted a comparative analysis by aligning the identified minerals, serpentine and talc, using two spectral libraries. We carefully examined their spectral shapes and absorption peak positions. Notably, denoising techniques significantly enhanced mineral detection results, revealing a remarkable consistency between the detected minerals and those in the Martian MICA mineral spectral library31 and the USGS spectral library32 (shown in Fig. 4). Serpentine, opal, and talc minerals suggest the potential for a habitable environment and the possible existence of life during the Noachian period.

Methods

Notations

The concepts introduced in this subsection are initially presented. Vectors are represented by lowercase bold letters (e.g., x), where \({\bf{x}}\in {{\mathbb{R}}}^{n}\) indicates a vector of dimension \(n\). Matrices are denoted by bold capital letters (e.g., \({\bf{X}}\in {{\mathbb{R}}}^{{I}_{1}\times {I}_{2}}\)) and represent two-dimensional arrays. Tensors are indicated using Euler-script letters; for instance, the notation \({\mathscr{Y}}\in {{\mathbb{R}}}^{{I}_{1}\times {I}_{2}\times {I}_{3}}\) denotes a third-order tensor with three dimensions.

Image degradation model

The observed CRISM hyperspectral image cube can be expressed as a third-order tensor \({\mathscr{Y}}\in {{\mathbb{R}}}^{{n}_{row}\times {n}_{col}\times L}\) with \(N(N={n}_{row}\times {n}_{col})\) pixels, and \(L\) spectral bands. Assuming that noise is additive with a mean of zero, the image degradation model of CRISM hyperspectral data can be written as

$${\mathscr{Y}}={\mathscr{X}}+{\mathscr{N}},$$
(1)

where \(\{{\mathscr{X}},{\mathscr{N}}\}\in {{\mathbb{R}}}^{{n}_{row}\times {n}_{col}\times L}\) denote a clean CRISM HSI image and corresponding noise, respectively.

The spectral vectors of a hyperspectral image tend to lie in a low-dimensional subspace because of the strong correlation between spectral bands33,34. The subspace representation theory has been widely used to solve inverse problems in hyperspectral imaging and has shown promising results35,36. Based on the subspace theory, the clean image \({\mathscr{X}}\) can be expressed as follows:

$${\mathscr{X}}={\mathscr{A}}{\times }_{3}{\bf{C}},$$
(2)

where columns of \({\bf{C}}\in {{\mathbb{R}}}^{L\times p}(p\,\ll \,L)\) hold an orthogonal basis for the spectral subspace, and elements of \({\mathscr{A}}\in {{\mathbb{R}}}^{{n}_{row}\times {n}_{col}\times p}\) are representation coefficients of \({\mathscr{X}}\) with respect to \({\bf{C}}\). Hereafter, mode-3 slices of \({\mathscr{A}}\) are termed eigenimages. Hence, the image degradation model (1) can be rewritten as follows:

$${\mathscr{Y}}={\mathscr{A}}{\times }_{3}{\bf{C}}+{\mathscr{N}}.$$
(3)

E2E for CRISM denoising

The proposed CRISM hyperspectral image denoising farmwork can be divided into three parts, including subspace representation of CRISM data, self-supervised training phase via noisy eigenimages, and inference phase.

Subspace representation of CRISM data

Given the spectral correlation matrix of the observed data, \({{\mathscr{Y}}}_{(3)}{{\mathscr{Y}}}_{(3)}^{T}/N\), the subspace representation is obtained by performing eigen-decomposition, expressed as

$${\bf{U}}{{\bf{S}}}^{2}{{\bf{U}}}^{T}={{\mathscr{Y}}}_{(3)}{{\mathscr{Y}}}_{(3)}^{T}/N$$
(4)

and

$${\bf{C}}={\bf{U}}(:,{\bf{1}}:p),$$
(5)

where \({\bf{U}}\) is an orthogonal matrix. The eigenvalues in the diagonal matrix \({\bf{S}}\) are ordered by decreasing magnitude. The basis matrix \({\bf{C}}\) can be obtained from \({\bf{U}}\) with the first \(p\) columns. In addition, \(N\) is the total number of image pixels. Therefore, given \({\bf{C}}\), the CRISM denoising problem can be reformulated as an eigenimages denoising with the following formula:

$$\widehat{{\mathscr{A}}}={f}_{\theta }({\mathscr{Y}}{\times }_{3}{{\bf{C}}}^{T}),$$
(6)

where \({f}_{\theta }(\cdot )\) is the E2E-CRISM denoising network introduced in the next section.

Training phase

In this section, a self-supervised training framework with only noisy CRISM eigenimages is introduced. E2E framework for CRISM data brings two benefits as follows:

  • We generate two sub-images from a single noisy eigenimage via a neighbor column sub-sampler, inspired by the neighbor2neighbor23. This sub-image generation strategy enables us to train a network without clean images.

  • The first eigenimage is of high quality, which can help guide the feature extraction of other eigenimage bands.

This paper focuses on the utilization of full resolution targeted observations (FRT) image cubes acquired by the \(L\) detector with 231 bands ranging from 1.1127 \(\mu m\) to 2.6285\(\mu m\), among various types of CRISM Mars images. The training of our model utilizes all global CRISM FRT images (a total of 10,406 images, excluding any invalid ones). It is worth mentioning that prior to network training, only photometric angles and atmospheric correction have been rectified using CAT software. The training phase of the E2E-CRISM framework is depicted in Fig. 1 with a blue arrow flow. The modified lightweight RRG network37 is employed as the underlying architecture of our denoising framework, comprising a total of 7 layers. More specifically, the RRG network comprises 2 RRG modules, and each RRG module contains 4 dual attention blocks that utilize channel attention and spatial attention mechanisms. Accordingly, the number of kernels and their size were set to 64 and 3, respectively.

To measure the eigenimage denoising performance, several cost functions have been investigated in the image denoising field, such as \({l}_{1}\) loss, \({l}_{2}\) loss, etc. We adopt \({l}_{1}\) loss to measure the reconstruction accuracy, and TV loss to impose spatial smoothness. Let \(\tilde{{\mathscr{A}}}={\mathscr{Y}}{\times }_{3}{{\bf{C}}}^{T}\) represents noisy eigenimage, hence the total loss function can be expressed as follows:

$${\mathcal L} ={\Vert \,{f}_{\theta }({\tilde{\tilde{{\mathscr{A}}}}}_{i})-{g}_{2}({\tilde{{\mathscr{A}}}}_{i})\Vert }_{1}+\alpha { {\mathcal L} }_{TV}+\gamma {\Vert \,{f}_{\theta }({\tilde{\tilde{{\mathscr{A}}}}}_{i})-{g}_{2}({\tilde{{\mathscr{A}}}}_{i})-({g}_{1}(\,{f}_{\theta }({\bar{A}}_{i}))-\,{g}_{2}(\,{f}_{\theta }({\bar{A}}_{i})))\Vert }_{1},$$
(7)

where \({f}_{\theta }\) is a network parameterized by \(\theta\), and the modified lightweight RRG network31 is adopted in E2E framework. For simplify, let \(\tilde{{{\mathscr{A}}}_{i}}=\tilde{{\mathscr{A}}}(:,:,i)\), \({\tilde{\tilde{{\mathscr{A}}}}}_{i}=cat({g}_{1}(\tilde{{\mathscr{A}}}(:,:,1)),{g}_{1}(\tilde{{\mathscr{A}}}(:,:,i))\), and \({\overline{{\mathscr{A}}}}_{i}=cat(\tilde{{\mathscr{A}}}(:,:,1),\tilde{{\mathscr{A}}}(:,:,i))\). The function of \(cat({\bf{A}},{\bf{B}})\) defines the concatenation of matrix \({\bf{A}}\) and \({\bf{B}}\) along the frequency dimension, and \(\tilde{{\mathscr{A}}}(:,:,1)\) is the first noisy eigenimage with high quality, which is used as a guide image to guide the rest of eigenimages. The \(\alpha\), and \(\gamma\) are two hyperparameters for network training. In addition, \({ {\mathcal L} }_{TV}\) is the TV loss, which can be expressed as follows:

$$\begin{array}{c}{ {\mathcal L} }_{TV}={\Vert {\nabla }_{h}{f}_{\theta }({\tilde{\tilde{{\mathscr{A}}}}}_{i})\Vert }_{1}+{\Vert {\nabla }_{v}{f}_{\theta }({\tilde{\tilde{{\mathscr{A}}}}}_{i})\Vert }_{1},\end{array}$$
(8)

where \({\nabla }_{h}\) and \({\nabla }_{v}\) are the horizontal and vertical gradients of the image \({f}_{\theta }({\tilde{\tilde{{\mathscr{A}}}}}_{i})\), respectively.

Denoising phase

Given that the eigenimage denoising network has been well-trained, the proposed denoising framework contains three steps:

Step 1: Subspace projection of the noisy CRISM HSI \({\mathscr{Y}}\) onto an orthogonal subspace: \({\tilde{{\mathscr{A}}}}_{i}={\mathscr{Y}}{\times }_{3}{{\bf{C}}}^{T}\)

Step 2: Denoise observed eigenimages using the proposed self-supervised network:

$${\widehat{{\mathscr{A}}}}_{i}={f}_{\theta }(cat({\tilde{{\mathscr{A}}}}_{1},{\tilde{{\mathscr{A}}}}_{i}),(i=1,2,\cdots ,p),$$
(9)

where \({f}_{\theta }\) is the well-trained denoising network, and \({\widehat{{\mathscr{A}}}}_{i}\) is the \(i\) th denoised eigenimage.

Step 3: The denoised CRISM HSI data can be multiplied by the estimated eigenimage \(\widehat{{\mathscr{A}}}\) and the spectral subspace bias \({\bf{C}}\), i.e., \(\widehat{{\mathscr{X}}}=\widehat{{\mathscr{A}}}{\times }_{3}{\bf{C}}.\)