Geometric compression plays a fundamental rolein virtual reality and augmented reality (VR/AR) applications. Dense meshesare re-sampled and re-tessellatedtoreduce thecomplexity. This process is called remeshing. In this work,we propose a novel remeshing algorithm based on both angle-preservingparameterization, and measurecontrollable parameterization. The conformal parameterization iscarried out by discrete surface Ricci flow method, the measurecontrollable parameterization is obtaind by an optimal mass transportation map. The sampling is performed on the measurecontrollable parameterization domain, the triangulation is computed on the conformal parameterization domain using Delaunay refinement algorithm. This method gives the user full control of sampling distribution, and produces mesheswith curvature measure convergence. The meshing resultcan emphsizethe region of interests, is curvature sensitive. Experimental results demonstrate the efficiency and efficacy of the proposed method.

As the barcode becomes more widely used, its applications and data capacity demands grow, increasing the need for barcodes with greater data density. Utilizing the quick response (QR) code–one of the many types of barcodes–we developed two algorithms. The first algorithm creates a color QR code that stores more information than a standard QR code and embeds extra data with limited access privilege. The second algorithm denoises a noisy color QR code. These algorithms consist of three techniques: (1) enlarging the data capacity of a compact QR code image by stacking multiple classical QR codes to form a color barcode, (2) embedding information into the color QR code using pseudo quantum signals in an M-band wavelet domain and selecting the discrete 4-band wavelet transforms to compress the QR images, and (3) applying Discrete M-band Wavelet Transform (DMWT) and Patch Group Prior based Denoising (PGPD) methods to denoise noisy QR code images. The peak-signal-to-noise-ratio (PSNR) summary indicates that information in a color QR code can be efficiently stored and retrieved with these methods. Moreover, it shows that our denoising algorithm effectively removes heavy noise from the noisy color QR code. Our algorithms are implemented in a flexible framework, which allows for further modifications to improve both the data capacity of a color QR code and the effectiveness of signal extraction from noisy data to meet future demands.

In this paper we first introduce a fractional form formula among a number of Euler’s formulas. We then extend the formula and with mathematical induction prove the case when the number of terms increases and the exponent is integer. Afterwards, we study the connection between Euler’s formula and Lagrange interpolating polynomial and use the latter to prove part of the extended formula. We then obtain a new formula from this connection. At last, we derive a set of new equations from the extended formula.

In [1] it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the Ryu-Takayanagi proposal must be modified for warped AdS 3 (WAdS 3) with Dirichlet boundary conditions. In this paper, we consider AdS 3 and WAdS 3 with Dirichlet-Neumann bound-ary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS 3 /WCFT and WAdS 3 /WCFT du-alities. Our bulk results are consistent with the WCFT results derived by Castro-Hofman-Iqbal using the Rindler method. Comparing with [1], we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.

The optimal mass transportation problem, proposed by Monge, is dominated by the Monge-Ampere equation. In general, as the Monge-Ampere equation is highly non-linear, this type of partial dierential equations is beyond the solving ability of a conventional nite element method. This diculty led Gu et al. alternatively to the discrete optimal mass transportation problem. They developed variational principles for this problem and reported the calculation for the optimal mapping, based on theorem that among all possible cell decompositions, with constrained measures, the trans- portation cost of the discrete mapping from cells to the corresponding discrete points is minimized by the decomposition induced by a power Voronoi diagram. Their research inspired us to consider a similar discrete optimal mass transportation problem. Here we replace the L2 Euclidean distance by the length of the shortest path connecting two points on a line network. To solve the optimal trans- portation problem on a line network, we study a type of Voronoi diagram on undirected and connected networks and propose an elegant construction algorithm. We further consider weighted distances in a network and develop a method to compute the centroidal Voronoi tessellation (CVT) for a network. By using real geographic data, the method proposed is proved ecient and eective in several practical applications, including charging station distribution in a trac network, and trash can distribution in a park.