The widely utilized physical model for explaining the formation of an image affected by light transmission hazed [7] is typically defined as follows: (1)I(x)=J(x)t(x)+A(1−t(x))where I(x) represents the observed hazy image, t(x) signifies the medium transmission map, A denotes the global atmosphere light, and J(x) corresponds to the haze-free image. Further, t(x) can be expressed as: t(x)=e−βd(x), where β represents the scattering coefficient of the atmosphere, and d(x) signifies the scene depth between the digital camera and the captured object for each pixel x in the image. During periods of dusty conditions, the varying degradation rates of red, green, and blue color components result in characteristic attributes like displacement, density, and temporal variance in the affected images. It can be formulated as: (2)A^=<AR,AG,AB>where AG=k1AR+b1, AB=k2AR+b2, A^ is the global color deviation value of the sand dust image, b is the disturbance amount, and k is the spatial distribution coefficient of the atmospheric light value of the three basic color spectrums. Based on the above formula, various degrees and types of sandy images can be synthesized. Then, with the help of these sand-dust images, we can get low-light degraded images. Specifically, for the generation of synthetic low-light images, we initially apply gamma correction to attenuate the luminance of the images. This process can be mathematically encapsulated by the following formula: (3)I(x)ll=β×(α×I(x))γwhere I(x)ll is the low-light sand-dust image, α,β and γ are the parameters of gamma correction. We randomly select these parameters to generate different levels of illumination for low-light images. α,β and γ are defined as U(0.65,0.7),U(0.65,0.7) and U(1.5,1.6), respectively. The visual examples of synthetic low-light sand-dust images are shown in Fig. 1. Furthermore, visual examples of real-world and synthetic dust images and their pixel distribution histograms are shown in Fig. 2. The top two rows represent real sand dust images and the corresponding RGB (Red, Green, Blue) distribution histograms; The bottom two rows are the synthetic degraded images and the corresponding histograms. In Fig. 2, we find the same pattern as in [8], that is, the distribution of sand dust images has obvious prior characteristics for shifting, concentration, and sequential. It can be seen that the distribution of the synthetic image satisfies all the properties of the real sandstorm image, proving the correctness and superiority of our method for synthesizing dust image data.

Li et al. [9] introduced AOD-Net, a pioneering end-to-end CNN model for image dehazing, highlighting its potential for enhancing computer vision tasks in hazy conditions. Chen et al. [10] present NAFNet, a pioneering approach that simplifies image restoration by eliminating nonlinear activations, achieving comparable state-of-the-art results with reduced computational complexity. Cui et al. [11] introduced OKNet, an efficient convolutional network for image restoration that leverages an omni-kernel module to capture multi-scale receptive fields, demonstrating state-of-the-art performance across various restoration tasks including dehazing, desnowing, and defocus deblurring. Gao et al. [12] propose a single-stage design, based on a simple U-Net architecture, with a mountain-shaped structure. The DSANet proposed by Cui and Knoll [13] enhances representation learning through spatial and frequency strip attention units, offering a significant advancement in image restoration efficiency and effectiveness. Gao et al. [14] introduced ECFNet, an efficient image restoration framework that combines spatial and frequency attention mechanisms with multi-scale blocks to adaptively handle varying degradation levels across image regions, achieving superior performance on multiple benchmark datasets. Cui et al. [15] introduced FSNet, an image restoration framework that employs frequency selection mechanisms to dynamically decompose and emphasize frequency components for effective image recovery.

Data uncertainty and model uncertainty in deep learning are key factors influencing performance, and probabilistic modeling approaches to address data and model uncertainty are gaining growing attention. Bayesian SegNet [16] leverages Monte Carlo Dropout [17] at test time to approximate posterior distributions, offering a measure of uncertainty crucial for decision-making in applications such as autonomous driving and robotic interaction. Wang et al. [18] introduce a data-uncertainty guided multi-phase learning approach for semi-supervised object detection that effectively leverages unlabeled data across varying difficulty levels. Zhou et al. [19] introduced an Uncertainty-Aware Edge Detector (UAED) that leverages the ambiguity in multiple annotations to enhance edge detection performance, representing a significant advancement in the field. Yang et al. [20] present UGTR, a pioneering approach that integrates Bayesian learning with Transformer-based [21] reasoning to enhance camouflaged object detection. Zhang et al. [22] introduced GLENet, a generative framework for modeling label uncertainty in 3D object detection using conditional variational autoencoders, which improves detection accuracy by capturing the diversity of potential bounding boxes for objects. Tang et al. [23] proposed UA-Track, an uncertainty-aware framework for 3D multi-object tracking that introduces probabilistic attention, query denoising, and uncertainty-reduced query initialization to address tracking challenges in complex scenarios. Dong et al. [24] proposed UAC, a semi-supervised learning method that combines multi-perturbation strategies and uncertainty estimation to improve medical image segmentation performance. Shao et al. [25] proposed UA-Fusion, an uncertainty-aware multimodal data fusion framework for 3-D object detection in autonomous vehicles. This framework effectively addresses challenges related to uncertainty in dynamic traffic environments through probabilistic cross-modal attention and query denoising strategies.

We propose UAKAN for the removal of sand-dust in a single-image under low-light conditions, as shown in Fig. 3. To expand the application of Kolmogorov-Arnold Networks (KANs) in low-level vision, we propose novel Tokenized KAN modules, namely the Downsampling Tokenized KAN (DTKAN) and Upsampling Tokenized KAN (UTKAN) modules. These modules leverage Patch Embedding and Patch Unembedding techniques for tokenization, followed by the application of KAN layers for further embedding, thereby enhancing the representational capabilities of the network. Furthermore, to tackle the aleatoric uncertainty inherent in the data dimensions of the concurrent task of single-image low-light enhancement and sand-dust removal, we introduce a module that couples uncertainty awareness with deterministic representation termed the Uncertainty Coupling Certainty Module (UCCM). The UCCM achieves the integration of uncertainty and determinism by feature coupling between a distribution modeling representation module and a residual learning module, thereby characterizing the fusion of uncertainty and determinism. Finally, addressing uncertainties of the epistemic dimension, we introduce the Uncertainty-aware Distributional Spatial Modulator (UDSM) and integrate it into the backbone of our framework, which we term the Uncertainty-aware Feature Modeling Block (UFMBlock). The incorporation of UFMBlock equips our framework with the capability to model comprehensive uncertainty, subsequently augmenting its ability to encapsulate the intrinsic uncertainties associated with the dual tasks of low-light image enhancement and image sand-dust removal. Our network adopts the same MIMOUNet [26] architecture, which is a multi-input multi-output U-shaped encoder-decoder structure. The total structure of our proposed UAKAN is shown in Fig. 3.

Analogous to a Multilayer Perceptron (MLP), a KAN with K layers can be described as a composition of successive KAN layers, this process can be formulated as following: (4)KAN(I)=(ΦN−1⊙ΦN−2⊙⋯⊙Φ1⊙Φ0)where I is the input feature; Φi denote the i-th layer of the KAN framework. Each KAN-Layer, with nin-dimensional input and nout-dimensional output, Φ compries nin×nout learnable activation functions ϕ: (5)Φ={ϕq,p},p=1,2,⋯,nin,q=1,2,⋯,noutwhere ϕq,p is the parameter that can be learned, and the computing process of the i-layer to the i+1-layer can be formulated in matrix from Ii+1=ΦiIi, where: (6)Φi=(ϕk,1,1(⋅)ϕk,1,2(⋅)…ϕk,1,nk(⋅)ϕk,2,1(⋅)ϕk,2,2(⋅)…ϕk,2,nk(⋅)⋮⋮⋮ϕk,nk+1,1(⋅)ϕk,nk+1,2(⋅)…ϕk,nk+1,nk(⋅))by employing edge-learned activation functions and parameterized activation functions in place of weights, KAN distinguishes itself from traditional MLP, thereby eliminating the need for linear weight matrices. The architectural design bolsters the interpretability of KAN while maintaining high performance, rendering them apt for a broad spectrum of applications. The detailed framework of KAN layers is shown in Fig. 3d.

To expand the application of KAN in low-level vision, we introduce a novel approach that integrates KANs with the sampling layers within the U-Net architecture, specifically the Down-sampling Tokenized KAN (DTKAN) and Up-sampling Tokenized KAN (UTKAN) modules. These modules utilize Patch Embedding (PE) and Patch Unembedding (PU) techniques for tokenization, followed by the application of KAN layers for advanced embedding processes, thereby enhancing the network’s capability to handle complex visual tasks. The detailed structure of DTKAN and UTKAN is shown in Fig. 3b,c. For DTKAN, this process can be formulated as: (7)Iout=KAN(PE(Iin))meanwhile, for UTKAN, it can be expressed as: (8)Iout=KAN(PU(Iin))where Iin and Iout are the input and output features, respectively.

To address the aleatoric uncertainty inherent in the data dimensions of the intricate visual task encompassing joint low-light image enhancement and sand-dust removal, we introduce the Uncertainty Coupling Certainty Module (UCCM). This module is designed to manage the stochastic elements present in these challenging image-processing tasks, thereby enhancing the robustness and reliability of the visual system. The detailed structure of our UCCM is shown in Fig. 4. In the UCCM, we employ a coupling mechanism between an uncertainty modeling branch and a deterministic representation branch to estimate the stochastic uncertainty present in the data. For the deterministic branch, we utilize a residual block from the ResNet [27] architecture for certain representations. As a quintessential component within convolutional neural networks, residual blocks possess formidable learning capabilities. The detailed process of the residual block can be articulated through the following formula:(9)Icertainty=Iin+Conv(ReLU(Conv(Iin)))meanwhile, for the uncertainty branch, we utilize a distribution modeling module for learning the uncertainty inside data-wise, it can be expressed as: (10)Iuncertainty=S(fμ(Iin)+ϵfσ(Iin))where S is the sampling operator from the Gaussian distribution by the parameterization trick [19], Icertainty and Iuncertainty are the output features of the certainty branch and the uncertainty branch, respectively, and ϵ∼N(0,I). The pseudo-code of the uncertainty-aware sampling processing S is shown in Algorithm 1. We utilize a learnable distribution in the feature space to introduce uncertainty modeling into the sand-dust removal task. We use two independent convolutional layers fμ and fσ to achieve the feature distribution, and we model the distribution of Iin at each pixel as Gaussian following [20]. The results are denoted as the mean μ and the variance σ of these two layers. Unlike other ordinary convolutional layers, these two convolutional layers learn a distribution N(μ,σ) parameterized by mean μ and variance σ. Meanwhile, the representation of each sample is not a deterministic point embedding but a stochastic embedding sampled from N(μ,σ). Finally, we employ a convolutional layer to reduce the dimensionality of the concatenated features from these two sections, a process we refer to as “coupling”. This coupling process of our proposed UCCM can be shown as: (11)Iout=Conv([Icertainty,Iuncertainty])

In order to make a proper response to the epistemic uncertainty in neural networks, that is, the model uncertainty, we embed an uncertainty-aware feature selective distribution learning module. This module is called the Uncertainty-aware Distributional Spatial Module (UDSM). We incorporate a distribution representation learning module, enabling our model to learn both a mean feature and a standard deviation feature. These two features are then combined to form a distribution from which we sample to obtain a single-channel selective map. This selective feature map is subsequently used to refine the expression of the input features selectively, yielding the final feature map for low-light enhancement and simultaneous sand-dust removal. The detailed structure of our proposed UDSM is shown in Fig. 4, and the whole process of UDSM can be described in Eq. (12): (12)UDSM(Iin)=Iin×S(fμ(Iin)+ϵfσ(x))

Furthermore, we embed UDSM in the backbone of our UAKAN and call the backbone Uncertainty-aware Feature Modeling Block (UFMBlock). Inspired by the architecture of Transformers [21], our backbone structure can be divided into two analogous stages, each initiated with a layer normalization. Drawing from the design of residual blocks [27], we incorporate two convolutional layers within each stage, with an activation function interposed between them. Influenced by the NAFBlock [10], we introduce channel attention [28] and depth-wise convolution in the first stage to enhance the network’s fitting and learning capabilities. Consequently, we integrate the UDSM into the second stage of our backbone to balance the computational load between the two stages. The specific process of our backbone can be delineated by the following formula:(13)Imid=f1×1(CA(SG(DWConv(f1×1(LN(Iin))))))+Iin,Iout=f1×1(UDSM(SG(f1×1(LN(Imid))))))+Imidwhere CA,SG,DWConv, f1×1, LN, and Imid denote channel attention [28], simple gate activation function, depth-wise convolution, convolutional layer with kernel 1, layer normalization [29], and the middle features, respectively.

All the experiments are performed on the Ubuntu 20.04 operating system with a NVIDIA RTX4090 GPU (24 GB) and an Intel i7-13700k CPU processor. The network training and testing processes are implemented by using Python 3.10 language environment and encompass torch 2.3.1+cu118 in conjunction with Anaconda. We train all the models for 200 epochs with the AdamW optimizer. The initial learning rate is set to 1×10−4, and we employ CosineAnnealingLR to adjust the learning rate. For data augmentation, we apply horizontal flipping and randomly rotate the image to 0, 90, 180, and 270 degrees. The batch size is set as 4 with a patch size of 128 × 128. Flops are computed on a patch size of 128 × 128.

For synthetic images used to train neural networks, we use a dataset named SLLIE6K that contains 6000 images for training and validation, and 2397 images for testing. We chose the Pascal VOC 2012 [30] dataset as clear images. For the training set, we picked 799 sharp images from the Pascal VOC 2012 dataset. Then, according to different Atmospheric Light, we synthesized three types of degraded images: dust, sand and sandstorm, and each type of image contained five degraded images with different degrees according to the Scattering Coefficient. As a result, 11,985 degraded images were obtained. Finally, in order to reduce the training data set and remove some training data with similar training effects, we selected 6000 images from all degraded images as training data. Then, for the test set, we also selected 799 clear images from the Pascal VOC 2012 dataset, and then, according to different Atmospheric Light, we synthesized three types of degraded images: Dust, Sand and Sandstorm. Therefore, the test data totals 2397 image pairs.

We use the Char loss [31] as the loss function for training all the neural networks. The Char loss [31] can be formulated as:(14)ℓ(I′,I^)=‖I′−I^‖2+ϵ2,where I′ is the restored image, I^ is the ground-truth image, and ϵ=10−3 is a constant in all the experiments.

In the comparative experiments with other state-of-the-art methods, we compare our proposed method with various cutting-edge approaches and compute PSNR, SSIM, MAE, and LPIPS scores using the RGB channel following [1]. In addition, to compare the computational complexity of these learning-based methods, we have also listed relevant measurement metrics, including Params, FLOPs, inference tims, and memory cost. As shown in Table 1, our method achieves significant performance gains that are more than the state-of-the-art methods. Our method outperforms all comparison methods in all indicators. It outperforms the second-best FSNet by 0.11 dB in PSNR, and shows significant superiority in MAE and LPIPS. For the running time and memory consumption, we use a server equipped with an RTX4090 graphics card for inference time evaluation, and the image size is 128 × 128 during inference. Although our method achieves SOTA, our method is obviously more computationally demanding than Sandformer [1], SANet [32], and OKNet [11], which is a disadvantage of our method. However, compared to the second-best FSNet, our method has less than half of the computational cost, whether in terms of Flops(G) or Params(M). Furthermore, the visual results shown in Fig. 5 are in close alignment with the quantitative findings, substantiating the superior image restoration capabilities of our proposed method. Given that all comparison methods yield similar results for real images, we employ error maps [33] to highlight the differences from the original clear images. As shown in the error maps, AODNet [9], NAFNet [10], SANet [32], OKNet [11], and ECFNet [14] deliver sub-par image restoration, with large-scale errors. In contrast, DSANet [13] and FSNet [15] produce smaller errors, yet struggle with details in texture-lacking areas like the sky or walls. Our method, however, stands out by restoring the most visually pleasing images, highlighting its superior performance. Therefore, Fig. 5 shows the superiority of our proposed method in the task of simultaneously removing the dust attachment from the image and performing low-light image enhancement.

Table 2 presents ablation experimental metrics for our proposed DTKAN, UTKAN, UCCM, and UDSM. We conducted detailed ablation experiments on the components proposed in this paper. For the convenience and efficiency of the experiment, only 60% of the 6000 images used for training and validation in the SLLIE6K dataset were used for model training in the ablation experiment section, and the remaining images were used for testing. As Table 2 shown, it demonstrates that our proposed component enhances the performance of the model, underscoring the rationality of the model. For example, replacing the pooling convolution or strided convolution with the tokeniz KAN can enhance the network’s ability of image restoration. Embedding the UCCM module can increase PSNR by 1.22 dB. Our proposed UDSM module outperforms the spatial attention module and plain convolution layers by 0.21 and 0.30 dB, respectively. Therefore, Table 2 also indicates the superiority of our proposed individual modules.