This paper introduces a novel class of neural networks called Incommensurate Fractional Recurrent Neural Networks (FRNNs) , inspired by incommensurate fractional difference systems. Unlike classical RNNs, FRNNs incorporate learnable fractional orders to capture heterogeneous temporal dependencies. To address the failure of back-propagation caused by cross-layer and cross-time coupling, we propose a forward-forward recursive gradient computation method combined with mini-batch SGD and ADAM optimization. Extensive experiments on multiple time-series datasets, including synthetic sinusoidal sequences, chaotic sequences, and real-world financial and energy data, demonstrate that FRNN achieves lower prediction error compared to RNN. These results confirm the effectiveness of fractional memory dynamics for sequential data processing.

Images taken in low-light conditions are frequently affected by limited visibility, diminished contrast and severe noise, adversely impacting the performance of various computer vision tasks. Most variational-based Retinex decomposition methods mainly depend on integer norms to constrain the illumination and reflectance components. However, this strategy may fail to achieve the ideal Retinex decomposition. In this paper, we propose a Retinex-based variational model that incorporates flexible constraints for both illumination and reflectance. Specifically, we impose the Lpnorm constraints with varying values of p to ensure the piece-wise smoothness of the illumination and promote the presence of abundant textures in the reflectance. Moreover, we develop two effective pixel-wise weight matrices that consider variance and gradients of the input image respectively, with the objective of preserving the structural edges of the illumination and retaining more details in the reflectance. In addition, we use an L2norm to estimate the overall noise level and avoid noise amplification. Through incorporating these above constraints, our proposed variational model can obtain a structure-aware illumination and a detail-revealed reflectance. Qualitative and quantitative comparisons on real-world and synthetic datasets indicate that our approach yields results with superior visual quality and outperforms several state-of-the-art algorithms on objective metrics. Besides, our algorithm can also address similar low-level computer vision challenges, such as image dehazing and underwater image enhancement. The source code is available at https://github.com/Enping-Hu/dual weighted lp.

Images captured under haze weather conditions usually suffer from visual quality degradations, such as blurred details, faded colors, and decreased saturation. Existing physicsbased dehazing methods mainly have two drawbacks: 1) the atmospheric light is treated as a constant for the entire image, and 2) pixel-or patch-based strategies are employed to estimate the model parameters, resulting in inaccurate haze density estimations. Therefore, these methods may lead to over-dehazing or under-dehazing due to insufficient utilization of features from regions with similar haze densities. To address these issues, a novel single image dehazing framework based on fuzzy region segmentation and haze density decomposition is proposed. Specifically, a region-based physical model that considers the non-uniform atmospheric light is first constructed based on the classic atmospheric scattering model. Then, a fuzzy segmentation algorithm is improved to divide the input hazy image into several separate regions. Subsequently, we formulate a simple linear relationship between the atmospheric light and brightness to estimate region-based atmospheric light. On the other hand, we develop a novel haze density decomposition algorithm based on boundary constraints to separate the atmospheric veil into two components: thin part and dense part. Three haze-related features, contrast, gradient and clarity, are extracted from the input hazy image to construct weight maps and a multi-scale fusion is further exploited to combine weight maps and boundary veils to acquire the refined atmospheric veil. Finally, the model inversion is performed to acquire the haze-free result. Experiments on six diverse hazy datasets demonstrate that the proposed algorithm outperforms several state-of-the-art dehazing methods in both visual quality and objective evaluation.

The visibility of images plays a crucial role in Intelligent Transportation Systems (ITS). However, images captured under hazy environments can degrade visual quality, significantly reducing the working performance of ITS. Although existing dehazing methods have achieved remarkable performance for daytime hazy images, they struggle to overcome the unique degradations under nighttime haze conditions such as glows, weak illumination, hidden noise, and color distortions. To simultaneously address these degradations, we propose VNDHR, a novel Variational Nighttime Dehazing framework using Hybrid Regularization focusing on enhancing the perceptual visibility of nighttime hazy scenarios. Specifically, a new physical model that accounts for multiple degradations under nighttime haze conditions is first constructed. Then, a novel hybrid variational model comprising an$\ell _{p}$norm, a weighted$\ell _{2}$norm, and a total variation regularization is developed to obtain a structure-aware illumination and a noise-free reflectance, simultaneously. To remove the nonhomogeneous haze in the illumination, we employ the dark channel prior to estimate parameters in each grid patch. Furthermore, a simple but effective nonlinear stretching function is designed to enhance the texture in the decomposed reflectance component. Finally, the dehazed illumination and the stretched reflectance are combined to generate a haze-free result. Experiments performed on synthetic and real-world nighttime hazy images prove that our VNDHR framework achieves state-of-the-art dehazing performance, providing results with clear details and less noise. Besides, our VNDHR can also handle various types of degraded images well, such as low-light images, daytime hazy images, sandstorm images, and underwater images.

A neural network method for solving fractional diffusion equations is presented in this letter. An adaptive gradient descent method is proposed to minimize energy functions. Due to the memory effects of the fractional calculus, the gradient of energy function becomes much more complicated, and we suggest a simplified method. Numerical examples with one-layer and two-layer neurons show the effectiveness of the method.

The fractional logistic map holds rich dynamics and is adopted to generate chaotic series. A watermark image is then encrypted and inserted into the original images. Since the encryption image takes the fractional order within (0, 1], it increases the key space and becomes difficult to attack. This study provides a robust watermark method in the protection of the copyright of hardware, images, and other electronic files.