Smooth Metrics Research Articles

An Attempt to Construct an Activity Cycle Catalog with Kepler Long-Cadence Light Curves

Many stars show activity cycles like the Sun. Kepler has gathered ∼200,000 light curves. Most of the Kepler stars only have long-cadence light curves, which limits their applicable methods. Some metrics, for example Sph, are effective for long-cadence light curves but require rotation periods. In order to improve the utilization of Kepler light curves, we introduce and use the smoothness metric. The smoothness metric is able to analyze stars without a measured rotation period and is applicable for long-cadence light curves. We test and validate our metric, resulting in the detection of the 11 years solar cycle and a 457 days cycle for our prototype star KIC 9017220. We analyze 92,084 Kepler long-cadence light curves, and as our main results, we detect 4455 magnetic activity cycle candidates, but about 20 percent are false cycles and 50 percent are lower limits of the real cycles, and we analyze their causes in detail. As an investigation into the performance of our method, we simulate disturbance factors and prove that the p-value test is invalid under certain circumstances.

Vanishing Theorems for Sheaves of Logarithmic Differential Forms on Compact Kähler Manifolds

Abstract In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by Hrmander’s $L^2$ estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Kähler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu’s vanishing theorem, Girbau’s vanishing theorem, Le Potier’s vanishing theorem, and a version of the Kawamata–Viehweg vanishing theorem.

Fast Joint Multi-Robot Trajectory Optimization by GPU Accelerated Batch Solution of Distributed Sub-Problems.

We present a joint multi-robot trajectory optimizer that can compute trajectories for tens of robots in aerial swarms within a small fraction of a second. The computational efficiency of our approach is built on breaking the per-iteration computation of the joint optimization into smaller, decoupled sub-problems and solving them in parallel through a custom batch optimizer. We show that each of the sub-problems can be reformulated to have a special Quadratic Programming structure, wherein the matrices are shared across all the problems and only the associated vector varies. As result, the batch solution update rule reduces to computing just large matrix vector products which can be trivially accelerated using GPUs. We validate our optimizer’s performance in difficult benchmark scenarios and compare it against existing state-of-the-art approaches. We demonstrate remarkable improvements in computation time its scaling with respect to the number of robots. Moreover, we also perform better in trajectory quality as measured by smoothness and arc-length metrics.

Noise Reduction in Human Motion-Captured Signals for Computer Animation based on B-Spline Filtering.

Motion capturing is used to record the natural movements of humans for a particular task. The motions recorded are extensively used to produce animation characters with natural movements and for virtual reality (VR) devices. The raw captured motion signals, however, contain noises introduced during the capturing process. Therefore, the signals should be effectively processed before they can be applied to animation characters. In this study, we analyzed several common methods used for smoothing signals. The smoothed signals were then compared based on the smoothness metrics defined. It was concluded that the filtering based on the B-spline-based least square method could achieve high-quality outputs with predetermined continuity and minimal parameter adjustments for a variety of motion signals.

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, {mathscr {S}}. By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric h_{ij} on a three-manifold Sigma foliated by a one-parameter family of two-spheres {mathscr {S}}_rho . It is shown first that we can always choose the shift such that the {mathscr {S}}_rho surfaces form a smooth inverse mean curvature foliation of Sigma . An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the rho -integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the {mathscr {S}}_{rho } level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.

A method for reconstructing NDVI time-series based on envelope detection and the Savitzky-Golay filter

ABSTRACT High-quality, normalized differential vegetation index (NDVI) time-series data are fundamental for environmental remote sensing applications; however, their quality is often influenced by complicated factors such as atmospheric aerosols and cloud coverage. Hence, in the current study, a robust reconstruction method based on envelope detection and the Savitzky-Golay filter (ED-SG) was developed to reduce noise in the NDVI time-series. To verify the performance of ED-SG, simulation experiments were implemented and NDVI time-series samples were selected for different land cover types derived from MOD09GQ, Sentinel-2 and Landsat 8 OLI of Yangtze River Basin, between December 2018 and December 2019. The experimental results yielded an agreement coefficient and variance of 0.9599 and 0.0006, respectively on simulated time-series, Additionally, the smoothness metrics of evergreen broadleaf forests, evergreen needleleaf forests, deciduous broadleaf forests, herbaceous, and croplands were 0.0019, 0.0017, 0.0012, 0.0012, and 0.0013, respectively. Ultimately, the reconstructed time-series metrics showed significant improvements in robustness and smoothness over conventional methods. Moreover, the simplistic mechanisms of the ED-SG model enabled it to run effectively in the Google Earth Engine over the NDVI time-series of the whole Yangtze River Basin.

Real-Time Multi-Convex Model Predictive Control for Occlusion-Free Target Tracking With Quadrotors

This paper proposes a Model Predictive Control (MPC) algorithm for target tracking amongst static and dynamic obstacles. Our main contribution lies in improving the computational tractability and reliability of the underlying non-convex trajectory optimization. The result is an MPC algorithm that runs real-time on laptops and embedded hardware devices such as Jetson TX2. Our approach relies on novel reformulations for the tracking, collision, and occlusion constraints that induce a multi-convex structure in the resulting trajectory optimization. We exploit these mathematical structures using the split Bregman Iteration technique, eventually reducing our MPC to a series of convex Quadratic Programs solvable in a few milliseconds. The fast re-planning of our MPC allows for occlusion and collision-free tracking in complex environments even while considering a simple constant-velocity prediction for the target trajectory and dynamic obstacles. We perform extensive bench-marking in a realistic physics engine and show that our MPC outperforms the state-of-the-art algorithms in visibility, smoothness, and computation-time metrics.

Positivity of direct images with a Poincaré type twist

Abstract We consider a holomorphic family $f:\mathscr {X} \to S$ of compact complex manifolds and a line bundle $\mathscr {L}\to \mathscr {X}$ . Given that $\mathscr {L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathscr {D}$ , the direct image $f_*(K_{\mathscr {X}/S}\otimes \mathscr {D} \otimes \mathscr {L})$ carries a smooth hermitian metric. If $\mathscr {L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Păun in a special situation of a big line bundle.

Measuring Movement Quality of the Stroke-Impaired Upper Extremity with a Wearable Sensor: Toward a Smoothness Metric for Home Rehabilitation Exercise Programs.

Remote patient monitoring systems show promise for assisting stroke patients in home exercise programs. While these systems typically measure exercise repetitions in order to monitor compliance, a key goal of therapists is to also monitor movement quality. Here we develop a measure of movement quality - Peak Intensity - that is a measure of movement smoothness that is implementable with a wrist-worn inertial measurement unit (IMU) in the context of performing repetitions of an upper extremity exercise. To calculate Peak Intensity, we assume we have an accurate count of the number of exercise repetitions in an exercise set, then calculate Peak Intensity as the total number of movement peaks from the continuous stream of IMU data generated across the set, divided by the number of repetitions. Using wrist-worn IMU measurements from 19 participants with chronic stroke performing a sample exercise in which they picked up and moved blocks across a divider (i.e. the Box and Blocks Test) we show that Peak Intensity is moderately correlated with a widely used measure of movement quality, the Quality of Movement score of the Motor Activity Log. Peak Intensity is also strongly correlated with a measure of hand function (the BBT score itself), but is more sensitive at greater levels of impairment. Finally, we show Peak Intensity can be validly derived from either wrist acceleration or angular velocity. These results suggest Peak Intensity could serve as an indicator of movement exercise quality for therapists monitoring home rehabilitation, and, potentially, as a means to provide augmented feedback to patients about their exercise quality.

Smoothness metrics for reaching performance after stroke. Part 1: which one to choose?

BackgroundSmoothness is commonly used for measuring movement quality of the upper paretic limb during reaching tasks after stroke. Many different smoothness metrics have been used in stroke research, but a ‘valid’ metric has not been identified. A systematic review and subsequent rigorous analysis of smoothness metrics used in stroke research, in terms of their mathematical definitions and response to simulated perturbations, is needed to conclude whether they are valid for measuring smoothness. Our objective was to provide a recommendation for metrics that reflect smoothness after stroke based on: (1) a systematic review of smoothness metrics for reaching used in stroke research, (2) the mathematical description of the metrics, and (3) the response of metrics to simulated changes associated with smoothness deficits in the reaching profile.MethodsThe systematic review was performed by screening electronic databases using combined keyword groups Stroke, Reaching and Smoothness. Subsequently, each metric identified was assessed with mathematical criteria regarding smoothness: (a) being dimensionless, (b) being reproducible, (c) being based on rate of change of position, and (d) not being a linear transform of other smoothness metrics. The resulting metrics were tested for their response to simulated changes in reaching using models of velocity profiles with varying reaching distances and durations, harmonic disturbances, noise, and sub-movements. Two reaching tasks were simulated; reach-to-point and reach-to-grasp. The metrics that responded as expected in all simulation analyses were considered to be valid.ResultsThe systematic review identified 32 different smoothness metrics, 17 of which were excluded based on mathematical criteria, and 13 more as they did not respond as expected in all simulation analyses. Eventually, we found that, for reach-to-point and reach-to-grasp movements, only Spectral Arc Length (SPARC) was found to be a valid metric.ConclusionsBased on this systematic review and simulation analyses, we recommend the use of SPARC as a valid smoothness metric in both reach-to-point and reach-to-grasp tasks of the upper limb after stroke. However, further research is needed to understand the time course of smoothness measured with SPARC for the upper limb early post stroke, preferably in longitudinal studies.

Cooperative People Tracking by Distributed Cameras Network

In the application of video surveillance, reliable people detection and tracking are always challenging tasks. The conventional single-camera surveillance system may encounter difficulties such as narrow-angle of view and dead space. In this paper, we proposed multi-cameras network architecture with an inter-camera hand-off protocol for cooperative people tracking. We use the YOLO model to detect multiple people in the video scene and incorporate the particle swarm optimization algorithm to track the person movement. When a person leaves the area covered by a camera and enters an area covered by another camera, these cameras can exchange relevant information for uninterrupted tracking. The motion smoothness (MS) metrics is proposed for evaluating the tracking quality of multi-camera networking system. We used a three-camera system for two persons tracking in overlapping scene for experimental evaluation. Most tracking person offsets at different frames were lower than 30 pixels. Only 0.15% of the frames showed abrupt increases in offsets pixel. The experiment results reveal that our multi-camera system achieves robust, smooth tracking performance.

The classification of flat Riemannian metrics on the plane

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally occurring, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane—as a subject—is new to the literature. Much of our research focuses on conformal metrics of the form \(e^{2\varphi }g_0\), where \(\varphi : {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a harmonic function and \(g_0\) is the standard Euclidean metric on \({\mathbb {R}}^2\). We find that all such metrics, which we call “harmonic,” arise from Riemann surfaces.

Motion Smoothness Metrics for Cannulation Skill Assessment: What Factors Matter?

Medical training simulators have the potential to provide remote and automated assessment of skill vital for medical training. Consequently, there is a need to develop “smart” training devices with robust metrics that can quantify clinical skills for effective training and self-assessment. Recently, metrics that quantify motion smoothness such as log dimensionless jerk (LDLJ) and spectral arc length (SPARC) are increasingly being applied in medical simulators. However, two key questions remain about the efficacy of such metrics: how do these metrics relate to clinical skill, and how to best compute these metrics from sensor data and relate them with similar metrics? This study addresses these questions in the context of hemodialysis cannulation by enrolling 52 clinicians who performed cannulation in a simulated arteriovenous (AV) fistula. For clinical skill, results demonstrate that the objective outcome metric flash ratio (FR), developed to measure the quality of task completion, outperformed traditional skill indicator metrics (years of experience and global rating sheet scores). For computing motion smoothness metrics for skill assessment, we observed that the lowest amount of smoothing could result in unreliable metrics. Furthermore, the relative efficacy of motion smoothness metrics when compared with other process metrics in correlating with skill was similar for FR, the most accurate measure of skill. These results provide guidance for the computation and use of motion-based metrics for clinical skill assessment, including utilizing objective outcome metrics as ideal measures for quantifying skill.

Private life of the Liouville field that causes new anomalies in the Nambu-Goto string

I consider higher-order terms of the Seeley expansion of the heat kernel, which for smooth metrics are suppressed as inverse powers of the UV cutoff Λ, and demonstrate how they result in an anomalous contribution to the string effective action after doing uncertainties Λ−2×Λ2. For the Polyakov string these anomalies precisely reproduce at one loop the result of KPZ-DDK obtained for the Liouville theory by the conformal field theory technique. For the Nambu-Goto string I find a deviation from this result which shows that the two string formulations may differ.

REALIZING METRICS OF CURVATURE ON CLOSED SURFACES IN FUCHSIAN ANTI-DE SITTER MANIFOLDS

AbstractWe prove that any metric with curvature less than or equal to $-1$ (in the sense of A. D. Alexandrov) on a closed surface of genus greater than $1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$ . The proof is by approximation, using a result about isometric immersion of smooth metrics by Labourie and Schlenker.

Bergman–Weil expansion for holomorphic functions

Using a modified Cauchy–Weil representation formula in a Weil polyhedron $${\varvec{D}}_f\subset U\subset \mathbb {C}^n$$ , we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by $$V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap {\varvec{D}}_f$$ , when $$m>n$$ and $$\{f_1,\ldots ,f_m\}$$ is minimal as a defining system. Thus the set $$V_{{\varvec{D}}_f}(f)$$ fails to be a complete intersection. We present our result as an averaged version of the classic Lagrange interpolation formula in the case $$m=n$$ . We invoke to that purpose Crofton’s formula, which plays a key role in the construction of Vogel generalized cycles as proposed in Andersson et al. (J Reine Angew Math 728: 105–136, 2017; Math Ann, 2020. https://doi.org/10.1007/s00208-020-01973-y). This leads us naturally to the construction of Bochner–Martinelli kernels. We also introduce $$f^{-1}(\{0\})$$ -Lagrange interpolators (at any order) subordinate to the choice of a smooth hermitian metric on the trivial m-bundle $$\mathbb {C}_U^m=U\times \mathbb {C}^m$$ , while the mapping $$f = (f_1,\ldots ,f_m)$$ is considered as its section.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

Higher order obstructions to the desingularization of Einstein metrics

We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kahler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kahler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that almost all flat orbifold metrics on $\mathbb{T}^4/\mathbb{Z}_2$ should not be limit of Ricci-flat metrics with generic holonomy while bubbling out Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation, we also identify a $14$-dimensional family of desingularizations satisfying all of our $84$ obstructions.

New error bounds for Laplace approximationviaStein’s method

We use Stein’s method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [ALEA Lat. Am. J. Probab. Math. Stat.11(2014) 571–587] for Stein’s method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.