Implicit Function Research Articles

Improved Trajectory Reconstruction for Markerless Pose Estimation.

Markerless pose estimation allows reconstructing human movement from multiple synchronized and calibrated views, and has the potential to make movement analysis easy and quick, including gait analysis. This could enable much more frequent and quantitative characterization of gait impairments, allowing better monitoring of outcomes and responses to interventions. However, the impact of different keypoint detectors and reconstruction algorithms on markerless pose estimation accuracy has not been thoroughly evaluated. We tested these algorithmic choices on data acquired from a multicamera system from a heterogeneous sample of 53 individuals in a rehabilitation hospital. We found that using a top-down keypoint detector and reconstructing trajectories with an implicit function enabled accurate, smooth, and anatomically plausible trajectories, with a noise in the step width estimates compared to a GaitRite walkway of only 9mm.

Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above.

Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially 2pi -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by textrm{i}a_tau =(zeta -textrm{i})a-d a_{x x}-|a|^2a+textrm{i}f_0+textrm{i}f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )}. The main new feature of the problem is the specific form of the source term f_0+f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )} which describes the simultaneous pumping of two different modes with mode indices k_0=0 and k_1in mathbb {N}. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e., f_1=0, can be continued into the range f_1not =0. Our analytical findings apply both for anomalous (d>0) and normal (d<0) dispersion, and they are illustrated by numerical simulations.

Investigation of the Existence of Limit Cycles in Multi Variable Nonlinear Systems with Special Attention to 3X3 Systems

The proposed work addresses the dynamics of a general system and explores the existence of limit cycles (LC) in multi-variable Non-linear systems with special attention to 3x3 nonlinear systems. It presents a simple, systematic analytical procedure as well as a graphical technique that uses geometric tools and computer graphics for the prediction of limit cycling oscillations in three-dimensional systems having both explicit and implicit nonlinear functions. The developed graphical method uses the harmonic balance/harmonic linearization for simplicity of discussion which provides a clear and lucid understanding of the problem and considers all constraints, especially the simultaneous intersection of two straight lines &amp; one circle for determination of limit cycling conditions. The method of analysis is made simpler by assuming the whole system exhibits the limit cycling oscillations predominantly at a single frequency. The discussions made either analytically/graphically are substantiated by digital simulation by a developed program as well as by the use of the SIMULINK Toolbox of MATLAB Software.

Reconstruction and completion of high-resolution 3D cardiac shapes using anisotropic CMRI segmentations and continuous implicit neural representations

Since the onset of computer-aided diagnosis in medical imaging, voxel-based segmentation has emerged as the primary methodology for automatic analysis of left ventricle (LV) function and morphology in cardiac magnetic resonance images (CMRI). In standard clinical practice, simultaneous multi-slice 2D cine short-axis MR imaging is performed under multiple breath-holds resulting in highly anisotropic 3D images. Furthermore, sparse-view CMRI often lacks whole heart coverage caused by large slice thickness and often suffers from inter-slice misalignment induced by respiratory motion. Therefore, these volumes only provide limited information about the true 3D cardiac anatomy which may hamper highly accurate assessment of functional and anatomical abnormalities. To address this, we propose a method that learns a continuous implicit function representing 3D LV shapes by training an auto-decoder. For training, high-resolution segmentations from cardiac CT angiography are used. The ability of our approach to reconstruct and complete high-resolution shapes from manually or automatically obtained sparse-view cardiac shape information is evaluated by using paired high- and low-resolution CMRI LV segmentations. The results show that the reconstructed LV shapes have an unconstrained subvoxel resolution and appear smooth and plausible in through-plane direction. Furthermore, Bland-Altman analysis reveals that reconstructed high-resolution ventricle volumes are closer to the corresponding reference volumes than reference low-resolution volumes with bias of [limits of agreement] −3.51 [−18.87, 11.85] mL, and 12.96 [−10.01, 35.92] mL respectively. Finally, the results demonstrate that the proposed approach allows recovering missing shape information and can indirectly correct for limited motion-induced artifacts.

Dynamics of a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics.

An accurate and efficient implicit thermal network method for the steady-state temperature field

To calculate the temperature value accurately and efficiently, an implicit thermal network method (TNM) is developed in this study. The main idea of the method is the conversion of such time-varying observed variables as the thermal resistance and the heat source into latent variables to build the implicit thermal equilibrium equation. In the implicit TNM, the steady-state temperature is taken as an independent variable, then parameters related to the steady-state temperature can be expressed as the function of the independent variable. On this basis, implicit thermal equilibrium equations can be constructed. Finally, the steady-state temperature field can be obtained by Newton’s method in optimization. To validate the performance and effectiveness of the implicit TNM, case studies and the sensitivity analysis of algorithms are conducted. The result reveals that the implicit TNM outperforms conventional TNMs, the steady-state TNM and the transient TNM, in computation accuracy, computing time and allocated memory.

Extraction of Single Diode Model Parameters of Solar Cells and PV Modules by Combining an Intelligent Optimization Algorithm with Simplified Explicit Equation Based on Lambert W Function

In this paper, the explicit equation of the single diode model (SDM) expressed by the Lambert W function was reduced to its simplified form through variable replacement; then the simplified explicit equation was combined with an intelligent optimization algorithm to estimate the SDM parameters of solar cells and PV modules. To evaluate the parameter extraction performance of the new method, eight typical intelligent optimization algorithms were combined with the implicit, explicit, and simplified explicit equation to extract the SDM parameters of a solar cell and three types of PV modules. The results show that the new method not only improves the accuracy of parameter extraction but also enhances the robustness and convergence speed. Most importantly, the new method can nearly improve the parameter extraction accuracy of a poor-performing algorithm in traditional methods to the level of other well-performing algorithms without enhancing the algorithm itself. In a word, this study offers a new choice for a more accurate and reliable extraction of SDM parameters from both solar cells and PV modules.

Analytical prediction of the temperature and the lifetime of an evaporating spherical droplet

In this paper, we propose to predict analytically the temperature of an evaporating spherical droplet. To do so, we first review, from data in the literature, the effect of temperature on the physical parameters involved in cooling-induced evaporation, namely the saturating vapor pressure, the diffusion coefficient of vapor in air, the liquid density, the enthalpy of vaporization and the thermal conductivity of air. These data support a series of approximations that allows us to derive an implicit equation for the liquid temperature. We propose a quadratic approximation of the variation of the saturating vapor concentration with temperature to obtain an explicit prediction of the drop temperature. As a result, an analytical prediction of the droplet lifetime including the cooling effect is proposed.

Reasonable thickness determination for implicit porous sheet structure using persistent homology

Porous structures are widely used in various industries because of their excellent properties. Porous surfaces have no thickness and should be thickened to sheet structures for further fabrication. For porous surfaces with implicit representation, sheet structures can be generated by extracting the region between two iso-surfaces. However, too small sheet thickness leads to extra small holes emerging on the sheet structure, and too large sheet thickness makes the pores closed. In this study, we propose a novel method for determining the sheet thickness for porous surfaces represented by implicit functions. Specifically, after calculating the persistent diagram (PD) of the scalar field dictated by the implicit function, the points in the PD are clustering into three clusters. Using these clusters, a topology control method is developed, which allows users to determine how many pores can be closed. Moreover, a reasonable thickness can be determined, by which all of the pores keep open, and the extra small holes are eliminated. Experimental results show the effectiveness of the developed method.

Stability of Nonlinear Implicit Differential Equations with Caputo–Katugampola Fractional Derivative

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, the existence of the solution of the implicit equation is proved, and the relevant conclusions about the stability of Ulam–Hyers are obtained. Finally, the correctness of the conclusions is verified by an example.

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

Study on the microwave heating temperature uniformity of an intelligent synergistic mode stirrer

Aiming at the intercoupling of multiple physics and the time-varying characteristics of boundary conditions in the microwave heating process, how to optimize the temperature uniformity of materials with a cooperative-mode stirrer is focused in this paper. First, the formation of standing waves can be effectively avoided by introducing the mode stirrer, thereby improving the heating efficiency of the material. Description of pattern stirrer motion processes using implicit functions and level set methods. Second, in order to optimize the temperature uniformity of the material, the stirrer is made intelligent, and the electromagnetic field distribution can be optimized by using different stirrer positions and directions, and the temperature uniformity is improved through intelligent coordination of stirrers in different positions. Finally, the finite element method is used to perform numerical simulation calculations on integer and continuous variables. The results show that the proposed method can improve the temperature uniformity of the conventional microwave heating model by 28.2%–81.44% and 23.2%∼63.91% at each level and lead hammer section respectively, and can improve the heating efficiency by 14.8%∼36.4%. According to its performance when heating different shapes and different materials, the proposed method is efficient and feasible.

Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.

Diversity Enabled Low-Latency Wireless Communications With Hard Delay Constraints

The emerging next generation Ultra-Reliable and Low-Latency Communications (xURLLC) is expected to play a central role in supporting mission-critical mobile applications because it holds the promise of improving the Quality-of-Service (QoS) substantially. However, it is quite challenging to satisfy the hard delay constraint in harsh wireless environments due to sporadic deep fades, especially when the average power is strictly limited. In this paper, we aim at assuring hard delay constraints with the aid of frequency or spatial diversity techniques. To this end, we focus on both parallel and multiple-input-multiple-output (MIMO) fading channels, in which time domain power adaptation is exploited to provide just-in-time services (JITS). It is shown that the hard delay constraint can be satisfied with a finite average power when the frequency or spatial diversity gains are no less than two. By adopting the implicit function theorem, we reveal the relationship between the required average power, the delay constrained throughput, and the outage probability without power adaptation. Furthermore, by adopting Ferrari’s solution to fourth order algebraic equations, we show that hard delay constrained transmission is feasible even when the sub-channels in the frequency and spatial domains are highly but not fully correlated.

SoRoForge: End-to-End Soft Actuator Design

We present SoRoForge, a design tool that addresses several limitations in the existing design and fabrication workflow for pneumatic soft actuators. We use implicit geometry functions to specify geometry and material distribution, a GUI-based software tool for interactive exploration of computational network representations of these implicit functions, and an automated tool for generating and displaying rapid simulation results of candidate designs. We prioritize seamless connectivity between all stages of the design and fabrication process, and elimination of steps that require human intervention. The workflow presented here integrates with existing capabilities for multimaterial additive manufacturing, and is also forward-compatible with emerging automated design techniques. The data gathered from human-interaction with this tool will be used by future automation tools to enable automated soft actuator design based on high-level specifications. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In this paper we present SoRoForge, a software tool that integrates major stages of soft actuator design and fabrication, which are ordinarily distinct and require deep domain expertise to investigate.This work promotes cross-discipline collaboration, allowing experts in other fields to map soft robotics solutions onto their design problems.Our key contribution is the automation of normally tedious or complex aspects of soft actuator design and fabrication: the specification of design geometry, the preparation of numerical simulation studies, and the physical realization of finished designs.Applications for soft actuators designed using our tool include soft manipulators on automated assembly lines, patient-specific biomedical implements, and food handling equipment for easily bruised produce.

A note on the existence of stationary vortex patches for the SQG equation in bounded domain

By studying the contour dynamics equation and using the implicit function theorem, we prove the existence of stationary vortex patches with fixed vorticity and total flux for each patch for the surface quasi-geostrophic equation in a bounded domain near non-degenerate critical points of the Kirchhoff-Routh function.

The nonlinear effects of spinning on the dynamics of a pitching cantilever

We investigate the structural dynamics of a pitching cantilever as it spins about a fixed axis, emulating the motion of the rotor blades of wind turbines or rotorcraft to some extent. From first principles, we derive a system of equations consisting of a nonlinear partial differential equation (PDE) coupled with a nonlinear ordinary differential equation (ODE) that describes the motion of the system in bending and pitch as it spins according to a prescribed rotor velocity and acceleration. The equations of motion consider the interaction between dynamics of a flexible body in bending, rigid body motion in pitch, and the nonlinear effects of spinning, concurrently. This simplified model of a rotor blade permits an intuitive understanding of the various nonlinear terms that arise in the equations of motion, and their relative influence on the system dynamics. In a series of numerical experiments, we study the effects of the coupling between the bending and pitch degrees of freedom, the influence of rotor velocity, and rotor acceleration on the system dynamics. We compare the full nonlinear simulation results to those of the underlying linear system, obtained by imposing a small angle assumption in pitch and excluding all remaining nonlinear terms. The numerical experiments are performed by non-dimensionalizing the equations, discretizing the non-dimensionalized equations in space by applying the Galerkin method, discretizing the resulting system of ODEs in time using the Houbolt method and finally solving the resulting system of implicit algebraic equations using Newton’s method.

Occupancy Planes for Single-View RGB-D Human Reconstruction

Single-view RGB-D human reconstruction with implicit functions is often formulated as per-point classification. Specifically, a set of 3D locations within the view-frustum of the camera are first projected independently onto the image and a corresponding feature is subsequently extracted for each 3D location. The feature of each 3D location is then used to classify independently whether the corresponding 3D point is inside or outside the observed object. This procedure leads to sub-optimal results because correlations between predictions for neighboring locations are only taken into account implicitly via the extracted features. For more accurate results we propose the occupancy planes (OPlanes) representation, which enables to formulate single-view RGB-D human reconstruction as occupancy prediction on planes which slice through the camera's view frustum. Such a representation provides more flexibility than voxel grids and enables to better leverage correlations than per-point classification. On the challenging S3D data we observe a simple classifier based on the OPlanes representation to yield compelling results, especially in difficult situations with partial occlusions due to other objects and partial visibility, which haven't been addressed by prior work.

NeAF: Learning Neural Angle Fields for Point Normal Estimation

Normal estimation for unstructured point clouds is an important task in 3D computer vision. Current methods achieve encouraging results by mapping local patches to normal vectors or learning local surface fitting using neural networks. However, these methods are not generalized well to unseen scenarios and are sensitive to parameter settings. To resolve these issues, we propose an implicit function to learn an angle field around the normal of each point in the spherical coordinate system, which is dubbed as Neural Angle Fields (NeAF). Instead of directly predicting the normal of an input point, we predict the angle offset between the ground truth normal and a randomly sampled query normal. This strategy pushes the network to observe more diverse samples, which leads to higher prediction accuracy in a more robust manner. To predict normals from the learned angle fields at inference time, we randomly sample query vectors in a unit spherical space and take the vectors with minimal angle values as the predicted normals. To further leverage the prior learned by NeAF, we propose to refine the predicted normal vectors by minimizing the angle offsets. The experimental results with synthetic data and real scans show significant improvements over the state-of-the-art under widely used benchmarks. Project page: https://lisj575.github.io/NeAF/.