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Mathematical Foundations of Computing

November 2019 , Volume 2 , Issue 4

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On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory
Jianqin Zhou, Wanquan Liu, Xifeng Wang and Guanglu Zhou
2019, 2(4): 279-297 doi: 10.3934/mfc.2019018 +[Abstract](2041) +[HTML](477) +[PDF](382.41KB)

The linear complexity and the \begin{document}$ k $\end{document}-error linear complexity of a binary sequence are important security measures for the security of the key stream. By studying binary sequences with the minimum Hamming weight, a new tool, named as the hypercube theory, is developed for \begin{document}$ p^n $\end{document}-periodic binary sequences. In fact, the hypercube theory is based on a typical sequence decomposition and it is a very important tool for investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a \begin{document}$ p^n $\end{document}-periodic binary sequence. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the \begin{document}$ m $\end{document}-hypercubes with the same linear complexity.

Big Map R-CNN for object detection in large-scale remote sensing images
Linfei Wang, Dapeng Tao, Ruonan Wang, Ruxin Wang and Hao Li
2019, 2(4): 299-314 doi: 10.3934/mfc.2019019 +[Abstract](3512) +[HTML](586) +[PDF](6332.53KB)

Detecting sparse and multi-sized objects in very high resolution (VHR) remote sensing images remains a significant challenge in satellite imagery applications and analytics. Difficulties include broad geographical scene distributions and high pixel counts in each image: a large-scale satellite image contains tens to hundreds of millions of pixels and dozens of complex backgrounds. Furthermore, the scale of the same category object can vary widely (e.g., ships can measure from several to thousands of pixels). To address these issues, here we propose the Big Map R-CNN method to improve object detection in VHR satellite imagery. Big Map R-CNN introduces mean shift clustering for quadric detecting based on the existing Mask R-CNN architecture. Big Map R-CNN considers four main aspects: 1) big map cropping to generate small size sub-images; 2) detecting these sub-images using the typical Mask R-CNN network; 3) screening out fragmented low-confidence targets and collecting uncertain image regions by clustering; 4) quadric detecting to generate prediction boxes. We also introduce a new large-scale and VHR remote sensing imagery dataset containing two categories (RSI LS-VHR-2) for detection performance verification. Comprehensive evaluations on RSI LS-VHR-2 dataset demonstrate the effectiveness of the proposed Big Map R-CNN algorithm for object detection in large-scale remote sensing images.

A Sim2real method based on DDQN for training a self-driving scale car
Qi Zhang, Tao Du and Changzheng Tian
2019, 2(4): 315-331 doi: 10.3934/mfc.2019020 +[Abstract](3957) +[HTML](1127) +[PDF](5616.33KB)

The self-driving based on deep reinforcement learning, as the most important application of artificial intelligence, has become a popular topic. Most of the current self-driving methods focus on how to directly learn end-to-end self-driving control strategy from the raw sensory data. Essentially, this control strategy can be considered as a mapping between images and driving behavior, which usually faces a problem of low generalization ability. To improve the generalization ability for the driving behavior, the reinforcement learning method requires extrinsic reward from the real environment, which may damage the car. In order to obtain a good generalization ability in safety, a virtual simulation environment that can be constructed different driving scene is designed by Unity. A theoretical model is established and analyzed in the virtual simulation environment, and it is trained by double Deep Q-network. Then, the trained model is migrated to a scale car in real world. This process is also called a sim2real method. The sim2real training method efficiently handles these two problems. The simulations and experiments are carried out to evaluate the performance and effectiveness of the proposed algorithm. Finally, it is demonstrated that the scale car in real world obtains the capability for autonomous driving.

A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance
Juanjuan Huang, Yan Zhou, Xuerong Shi and Zuolei Wang
2019, 2(4): 333-346 doi: 10.3934/mfc.2019021 +[Abstract](2265) +[HTML](551) +[PDF](1445.06KB)

In this paper, dynamical behaviors of three-dimensional chaotic system with time-delay and external periodic disturbance are investigated. When the periodic perturbation term and time-delay are added, the system presents more abundant dynamic behaviors, which can be switched between periodic state and chaotic state. Based on Lyapunov stability theory, a sufficient condition for finite-time synchronization is given. A single controller is proposed to realize finite-time synchronization of time-delay chaotic system with external periodic disturbance. The addressed scheme is provided in the form of linear inequality which is simple and easy to be realized. At the same time, it also displays that when delay term \begin{document}$ \tau $\end{document} takes different values, the time of synchronization shows certain difference. The feasibility and effectiveness of the finite-time synchronization method is verified by theoretical analysis and numerical simulation.


Erratum: The affiliations of all four authors have been corrected to School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, China. NSF has been corrected to NSFC under Fund Project. We apologize for any inconvenience this may cause.

Image enhancement algorithm using adaptive fractional differential mask technique
Xuefeng Zhang and Hui Yan
2019, 2(4): 347-359 doi: 10.3934/mfc.2019022 +[Abstract](3560) +[HTML](810) +[PDF](1909.94KB)

This paper addresses a novel adaptive fractional order image enhancement method. Firstly, an image segmentation algorithm is proposed, it combines Otsu algorithm and rough entropy to segment image accurately into the objet and the background. On the basis of image segmentation and the knowledge of fractional order differential, an image enhancement model is established. The rough characteristics of each average gray value are obtained by image segmentation method, through these features, we can determine the optimal fractional order of image enhancement. Then image will be enhanced using fractional order differential mask, from which fractional order is obtained adaptively. Several images are used for experiments, the proposed model is compared with other models, and the results of comparison exhibit the superiority of our algorithm in terms of image quality measures.

2021 CiteScore: 0.2



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