On k-circulant matrices with the generalized third-order Pell numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

Critical Overview of Visual Tracking with Kernel Correlation Filter

With the development of new methodologies for faster training on datasets, there is a need to provide an in-depth explanation of the workings of such methods. This paper attempts to provide an understanding for one such correlation filter-based tracking technology, Kernelized Correlation Filter (KCF), which uses implicit properties of tracked images (circulant matrices) for training and tracking in real-time. It is unlike deep learning, which is data intensive. KCF uses implicit dynamic properties of the scene and movements of image patches to form an efficient representation based on the circulant structure for further processing, using properties such as diagonalizing in the Fourier domain. The computational efficiency of KCF, which makes it ideal for low-power heterogeneous computational processing technologies, lies in its ability to compute data in high-dimensional feature space without explicitly invoking the computation on this space. Despite its strong practical potential in visual tracking, there is a need for an in-depth critical understanding of the method and its performance, which this paper aims to provide. Here we present a survey of KCF and its method along with an experimental study that highlights its novel approach and some of the future challenges associated with this method through observations on standard performance metrics in an effort to make the algorithm easy to investigate. It further compares the method against the current public benchmarks such as SOTA on OTB-50, VOT-2015, and VOT-2019. We observe that KCF is a simple-to-understand tracking algorithm that does well on popular benchmarks and has potential for further improvement. The paper aims to provide researchers a base for understanding and comparing KCF with other tracking technologies to explore the possibility of an improved KCF tracker.

A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

Estimates for solutions of systems of linear equations with circulant matrices

In the present paper we consider the problem to estimate a solution of the system of equations with a circulant matrix in uniform norm. We give the estimate for circulant matrices with diagonal dominance. The estimate is sharp. Based on this result and an idea of decomposition of the matrix into a product of matrices associated with factorization of the characteristic polynomial, we propose an estimate for any circulant matrix.