Piecewise Smooth Research Articles

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted $$L_2$$ space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

Canonical Forms for Codimension One Planar Piecewise Smooth Vector Fields with Sliding Region

This paper is devoted to exhibit canonical forms for 2D codimension one piecewise smooth vector fields. All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. As consequence, 62 canonical forms were obtained.

An extrapolative approach to integration over hypersurfaces in the level set framework

We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g., curves in two dimensions that contain a finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.

Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

Plastic Deformation of Materials Under Loading Along Piecewise Smooth Trajectories with Areas of Unloading by the Elastic Law

Within the framework of a version of plasticity theory based on the concept of slip, we propose a method for the determination of plastic strains in materials with regard for the strain anisotropy in the case of loading applied along piecewise smooth trajectories (with areas of unloading according to the elastic law). The material function of plasticity Π used in this theory is determined, for a given function of hardening F , either from the tension–compression tests or from the tests with alternating torsion of thin-walled pipes.

A family of HLL-type solvers for the generalized Riemann problem

The generalized Riemann problem (GRP) is the initial value problem for a conservation law with piecewise smooth, but discontinuous initial data. We provide a new method for solving the GRP approximately, that can be used as a building block for high order finite volume or discontinuous Galerkin methods. Our new GRP solvers use the approximate states and wave speeds obtained through a HLL-type Riemann solver and use this information to build an approximation of the state in the GRP of any order. What is new about this approach compared to most previous solvers is that we no longer need to solve a classical Riemann problem exactly. We give a detailed explanation of this strategy for HLL and HLLC solvers for the Euler equations, as well as for the HLLD solver for MHD equations. We demonstrate the performance of the solvers from this new family of GRP solvers for a broad range of test problems.

T-junctions in Spline Surfaces

T-junctions occur where surface strips start or terminate. This paper develops a new way to create smooth piecewise polynomial free-form spline surfaces from quad-meshes that include T-junctions. All mesh nodes are interpreted as control points of GT-splines, that is, geometrically smoothly joined piecewise polynomials. GT-splines are akin to and compatible with B-splines and cover simple T-junctions by two polynomial pieces of degree bi-4 and more complex ones by four such patches. They complement multi-sided surface constructions in generating free-form surfaces with adaptive layout. Since GT-splines do not require a global coordination of knot intervals, GT-constructions are easy to deploy and can provide smooth surfaces with T-junctions where T-splines cannot have a smooth parameterization. GT-constructions display a uniform highlight linedistribution on input meshes where alternatives, such as Catmull-Clark subdivision, exhibit oscillations.

Adaptive Order Non-Convex Lp-norm Regularization in Image Restoration

The optimization formulations of sparse signal recovery problems use a regularizer term which encourages the solution to be sparse or piece-wise smooth. Of special interest are the ℓp norms with 0 ≤ p ≤ 1 which penalize large values and encourage the solution of the optimization problem to be sparse. In this work, we propose a new regularizer in which the exponent p, 0 ≤ p ≤ 1 of the ℓp–norm is a function of the signal element amplitude, leading to a function which is smooth and convex over non-negative values. This formulation adapts the value of the exponent p according to a sigmoid function applied on the signal values, to restrict its values to between 0 and 1. Experiments show that this formulation is more accurate in denoising, than the ℓ1 and ℓ0.5 norm regularizers, and is more accurate than the ℓ1-norm when used iteratively for deblurring or inpainting.

The performance of some implicit region-based active contours in segmenting and restoring welding radiographic images

Several domains are based on image processing and analysis. One of them is the radiographic inspection which is used in Non Destructive Testing (NDT). Active contours, snakes or deformable models are powerful techniques in image segmentation and restoration. According to the term related to the input data (image to be treated) those functionals are ranked on two categories: edge-based models and region-based models. Previous studies point out the advantages of the regionbased models over edge-based models. In this paper, we discuss and we summarize the strengths and weaknesses of four implicit region-based active contour models named: Piecewise Constant PC, Piecewise Smooth PS, Local Binary Fitting LBF and Global Local fitting energy GLF. After performing several experiments, we have concluded that all the models perform well with homogeneous images. On the contrary when images are strongly inhomogeneous, the models based on global (PC) or local (LBF) statistic intensity fail to segment such images. The PS model with its great advantage in preserving the contours has, as a drawback, the high CPU time consuming. The combination of local and global statistic image intensity gives to the GLF model the ability to better deal with such images in less CPU time.

Bifurcation of Quasiperiodic Orbit in a 3D Piecewise Linear Map

Earlier investigations have demonstrated how a quasiperiodic orbit in a three-dimensional smooth map can bifurcate into a quasiperiodic orbit with two disjoint loops or into a quasiperiodic orbit of double the length in the shape of a Möbius strip. Using a three-dimensional piecewise smooth (PWS) normal form map, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of creation of tori with multiple loops may result from border collision bifurcations. We also illustrate the occurrence of multiple attractor bifurcations due to the interplay between the stable and the unstable manifolds. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in PWS systems, but fail in some cases — especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by change in eigenvalues of the unstable fixed points.

Bifurcation of Nonhyperbolic Limit Cycles in Piecewise Smooth Planar Systems with Finitely Many Zones

Like for smooth systems, it is very important to discuss the stability and bifurcation of limit cycles in a piecewise smooth planar system. Most of the previous works focus only on hyperbolic limit cycles. Few works have considered nonhyperbolic limit cycles. In fact, to date, no concrete examples of piecewise smooth planar system with nonhyperbolic limit cycles have been given in literature. In this paper, we consider for the first time the bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with discontinuities on finitely many straight lines intersecting at the origin. We present a method of Melnikov type to derive two quantities which can be used to determine the stability and the number of limit cycles that can bifurcate from a nonhyperbolic limit cycle of a piecewise smooth planar system. As applications, we present two examples of piecewise smooth systems with two and three zones respectively whose unperturbed system has a nonhyperbolic limit cycle.

Bifurcation of Limit Cycles from a Class of Piecewise Smooth Systems with Two Vertical Straight Lines of Singularity

Perturbing the following systems [Formula: see text] having a linear center with two vertical straight lines ([Formula: see text]) of singularity inside the class of all piecewise polynomials of degree [Formula: see text], we give the estimate of the maximum number [Formula: see text] of limit cycles bifurcating from the period annulus around the center based on the argument principle and the first order averaging theory. It shows that [Formula: see text] is at least [Formula: see text] bigger than the maximum number of limit cycles bifurcating from the period annulus around the linear center with two parallel straight lines of singularity in [Li & Liu, 2015].

Reconstruction and visualization of large-scale volumetric models of neocortical circuits for physically-plausible in silico optical studies

BackgroundWe present a software workflow capable of building large scale, highly detailed and realistic volumetric models of neocortical circuits from the morphological skeletons of their digitally reconstructed neurons. The limitations of the existing approaches for creating those models are explained, and then, a multi-stage pipeline is discussed to overcome those limitations. Starting from the neuronal morphologies, we create smooth piecewise watertight polygonal models that can be efficiently utilized to synthesize continuous and plausible volumetric models of the neurons with solid voxelization. The somata of the neurons are reconstructed on a physically-plausible basis relying on the physics engine in Blender.ResultsOur pipeline is applied to create 55 exemplar neurons representing the various morphological types that are reconstructed from the somatsensory cortex of a juvenile rat. The pipeline is then used to reconstruct a volumetric slice of a cortical circuit model that contains ∼210,000 neurons. The applicability of our pipeline to create highly realistic volumetric models of neocortical circuits is demonstrated with an in silico imaging experiment that simulates tissue visualization with brightfield microscopy. The results were evaluated with a group of domain experts to address their demands and also to extend the workflow based on their feedback.ConclusionA systematic workflow is presented to create large scale synthetic tissue models of the neocortical circuitry. This workflow is fundamental to enlarge the scale of in silico neuroscientific optical experiments from several tens of cubic micrometers to a few cubic millimeters.AMS Subject ClassificationModelling and Simulation

Stability and Perturbations of Generalized Heteroclinic Loops in Piecewise Smooth Systems

We investigate a class of planar piecewise smooth systems with a generalized heteroclinic loop (a closed curve composed of hyperbolic saddle points, generalized singular points and regular orbits). We give conditions for the stability of the generalized heteroclinic loop and provide some sufficient conditions for the maximum number of limit cycles that bifurcate from the heteroclinic connection. The discussions rely on the approximation of the Poincare map, which is constructed near the generalized heteroclinic loop. To obtain it, we introduce the Dulac map and use Melnikov method. By analyzing the fixed point of the Poincare map, we get the number of limit cycles, which can be produced from the generalized heteroclinic loop. As applications to our theories, we give an example to show that two limit cycles can appear.

Improvement of the basis for the solution of the Dirac equation in Cassini coordinates

We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in Ref. [1]. For the calculations in Ref. [1], we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-splines only. Therefore, we include basis functions which are defined using functions with step-like behaviour instead of B-splines. Thereby, we achieve a significant increase of accuracy of results as compared to Ref. [1].

Similarity maps and field-guided T-splines

A variety of techniques were proposed to model smooth surfaces based on tensor product splines (e.g. subdivision surfaces, free-form splines, T-splines). Conversion of an input surface into such a representation is commonly achieved by constructing a global seamless parametrization, possibly aligned to a guiding cross-field (e.g. of principal curvature directions), and using this parametrization as domain to construct the spline-based surface. One major fundamental difficulty in designing robust algorithms for this task is the fact that for common types, e.g. subdivision surfaces (requiring a conforming domain mesh) or T-spline surfaces (requiring a globally consistent knot interval assignment) reliably obtaining a suitable parametrization that has the same topological structure as the guiding field poses a major challenge. Even worse, not all fields do admit suitable parametrizations, and no concise conditions are known as to which fields do. We present a class of surface constructions (T-splines with halfedge knots ) and a class of parametrizations ( seamless similarity maps ) that are, in a sense, a perfect match for the task: for any given guiding field structure, a compatible parametrization of this kind exists and a smooth piecewise rational surface with exactly the same structure as the input field can be constructed from it. As a byproduct, this enables full control over extraordinary points. The construction is backward compatible with classical NURBS. We present efficient algorithms for building discrete conformal similarity maps and associated T-meshes and T-spline surfaces.

Total Variation Regularized Reweighted Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing

Blind hyperspectral unmixing (HU), which includes the estimation of endmembers and their corresponding fractional abundances, is an important task for hyperspectral analysis. Recently, nonnegative matrix factorization (NMF) and its extensions have been widely used in HU. Unfortunately, most of the NMF-based methods can easily lead to an unsuitable solution, due to the nonconvexity of the NMF model and the influence of noise. To overcome this limitation, we make the best use of the structure of the abundance maps, and propose a new blind HU method named total variation regularized reweighted sparse NMF (TV-RSNMF). First, the abundance matrix is assumed to be sparse, and a weighted sparse regularizer is incorporated into the NMF model. The weights of the weighted sparse regularizer are adaptively updated related to the abundance matrix. Second, the abundance map corresponding to a single fixed endmember should be piecewise smooth. Therefore, the TV regularizer is adopted to capture the piecewise smooth structure of each abundance map. In our multiplicative iterative solution to the proposed TV-RSNMF model, the TV regularizer can be regarded as an abundance map denoising procedure, which improves the robustness of TV-RSNMF to noise. A number of experiments were conducted in both simulated and real-data conditions to illustrate the advantage of the proposed TV-RSNMF method for blind HU.

Effect of stochastically moving border on basins of attraction in a class of piecewise smooth maps

Determination of the basin of attraction of an attractor plays an important role in understanding the dynamics of a system. In all the existing literature, the basin of attraction of any attractor has been described to be deterministic. In this paper we show the existence of a non-deterministic basin of attraction of an attractor. We have considered a piecewise smooth (PWS) one-dimensional map, having stochastically varying border, which is allowed to move in a small bounded region of the phase space while retaining the deterministic dynamics on each compartment of the phase space. In case of this type of systems there exists a region in the phase space with the property that orbits starting from a single point lying inside this region do not display the same property of convergence or divergence, i.e., one may converge while another may diverge. In other words, the convergence or divergence of an orbit starting from a point inside this region is a probabilistic event, and the probabilities of convergence and divergence are both non-zero. We also derive the upper and lower bounds of the corresponding probability curves. Since all physical systems contain noise, the occurrence of such non-deterministic basin of attraction is a definite possibility, if the noise affects the position of the border. This may lead to dangerous consequences, as a region of the basin of attraction of an attractor may become non-deterministic, with a non-zero probability of divergence of orbits starting inside it.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

Sensitivity analysis of primary resonances and bifurcations of a controlled piecewise-smooth system with negative stiffness

In this paper, the combination of the cubic nonlinearity and time delay is proposed to improve the performance of a piecewise-smooth (PWS) system with negative stiffness. Dynamical properties, feedback control performance and symmetry-breaking bifurcation are mainly considered for a PWS system with negative stiffness under nonlinear position and velocity feedback control. For the free vibration system, the homoclinic-like orbits are firstly derived. Then, the amplitude-frequency response of the controlled system is obtained analytically in aspect of the Lindstedt–Poincaré method and the method of multiple scales, which is also verified through the numerical results. In this regard, a softening-type behavior, which directly leads to the multi-valued responses, is illustrated over the negative position feedback. Especially, the five-valued responses in which three branches of them are stable are found. And complex multi-valued characteristics are also observed in the force-amplitude responses. Furthermore, for explaining the effectiveness of feedback control, the equivalent damping and stiffness are also introduced. Sensitivity of the system response to the feedback gain and time delay is comprehensively considered and interesting dynamical properties are found. Relatively, from the perspective of suppressing the maximum amplitude and controlling the resonance stability, the selection of the feedback parameters is discussed. Finally, the symmetry-breaking bifurcation and chaotic motion are considered.