Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some sufficient and necessary conditions for a tensor to be centrosymmetric or skew centrosymmetric are given. We show that, a general tensor can always be expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric tensor. Some sufficient and necessary conditions for a Cauchy tensor to be centrosymmetric or skew centrosymmetric are also given. Spectral properties on H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy tensors are discussed. Some further questions on these tensors are raised

Parameterized Synthetic Image Data Set for Fisheye Lens

Based on different projection geometry, a fisheye image can be presented as a parameterized non-rectilinear image. Deep neural networks(DNN) is one of the solutions to extract parameters for fisheye image feature description. However, a large number of images are required for training a reasonable prediction model for DNN. In this paper, we propose to extend the scale of the training dataset using parameterized synthetic images. It effectively boosts the diversity of images and avoids the data scale limitation. To simulate different viewing angles and distances, we adopt controllable parameterized projection processes on transformation. The reliability of the proposed method is proved by testing images captured by our fisheye camera. The synthetic dataset is the first dataset that is able to extend to a big scale labeled fisheye image dataset. It is accessible via: http://www2.leuphana.de/misl/fisheye-data-set/.Comment: 2018 5th International Conference on Information Science and Control Engineerin

Discrete approximations to reflected Brownian motion

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains $D$ in $\mathbb{R}^n$ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on $D\cap2^{-k}\mathbb{Z}^n$ moving at the rate $2^{-2k}$ with stationary initial distribution converge weakly in the space $\mathbf{D}([0,1],\mathbb{R}^n)$, equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on $D$. We further show that the following ``myopic conditioning'' algorithm generates, in the limit, a reflected Brownian motion on any bounded domain $D$. For every integer $k\geq1$, let $\{X^k_{j2^{-k}},j=0,1,2,...\}$ be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in $D$ conditioned not to exit $D$ before time $2^{-k}$. We prove that the laws of $X^k$ converge to that of the reflected Brownian motion on $D$. These approximation schemes give not only new ways of constructing reflected Brownian motion but also implementable algorithms to simulate reflected Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/009117907000000240 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org