On the Stability Problem of Equilibrium Discrete Planar Curves

We study planar polygonal curves with the variational methods. We show a unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.

Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

Optimization Analysis of Smart Steel-Plastic Geogrid Support for Tunnel

With the concept of smart geogrid coming out, many scholars have built optical fiber into the geogrid to form a kind of smart geogrid material with self-sensing function of structural deformation. It can not only reinforce the parts with potential safety hazards, but also have the functions of safety monitoring, intelligent prevention, and control of engineering disasters, which is of great significance for ensuring the safety of tunnel construction and improving the tunnel monitoring methods. Based on predecessors’ research on smart geogrid tensile calibration experiment and sensor method simulation and experimental verification, this paper analyzes the smart geogrid and the tunnel surrounding rock as a whole, to study the deformation coordination mechanism between the geogrid material and the tunnel surrounding rock. Referring to the relevant engineering practice case, through finite element numerical simulation, the optimal layout of smart geogrid material was explored, and the principle of discrete curvature reconstruction curve sensing of smart geogrid was optimized by simulating the working conditions of different construction methods and supporting conditions, in order to provide a theoretical basis for the application of smart geogrid material in practical tunnel engineering.

Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

We study the volume growth of metric balls as a function of the radius in discrete spaces and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the 1st author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, $\lambda _2$ of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the 2nd eigenvalue (i.e. the 1st nonzero eigenvalue). We also describe a method for proving Buser’s Inequality in graphs, particularly under a lower bound assumption on curvature.

A Discrete Curvature Estimation Based Low-Distortion Adaptive Savitzky–Golay Filter for ECG Denoising

Electrocardiogram (ECG) sensing is an important application for the diagnosis of cardiovascular diseases. Recently, driven by the emerging technology of wearable electronics, massive wearable ECG sensors are developed, which however brings additional sources of noise contamination on ECG signals from these wearable ECG sensors. In this paper, we propose a new low-distortion adaptive Savitzky-Golay (LDASG) filtering method for ECG denoising based on discrete curvature estimation, which demonstrates better performance than the state of the art of ECG denoising. The standard Savitzky-Golay (SG) filter has a remarkable performance of data smoothing. However, it lacks adaptability to signal variations and thus often induces signal distortion for high-variation signals such as ECG. In our method, the discrete curvature estimation is adapted to represent the signal variation for the purpose of mitigating signal distortion. By adaptively designing the proper SG filter according to the discrete curvature for each data sample, the proposed method still retains the intrinsic advantage of SG filters of excellent data smoothing and further tackles the challenge of denoising high signal variations with low signal distortion. In our experiment, we compared our method with the EMD-wavelet based method and the non-local means (NLM) denoising method in the performance of both noise elimination and signal distortion reduction. Particularly, for the signal distortion reduction, our method decreases in MSE by 33.33% when compared to EMD-wavelet and by 50% when compared to NLM, and decreases in PRD by 18.25% when compared to EMD-wavelet and by 25.24% when compared to NLM. Our method shows high potential and feasibility in wide applications of ECG denoising for both clinical use and consumer electronics.