Edge Singularity Research Articles

Automatic Grading System for Diagnosis of Breast Cancer Exploiting Co-occurrence Shearlet Transform and Histogram Features

ObjectivesBreast cancer (BC) is one of the most commonly reported health issues worldwide, especially in females. Early detection and diagnosis of BC can greatly reduce mortality rates. Samples obtained with different imaging methods such as mammography, computerized tomography, magnetic resonance, ultrasound, and biopsy are used in the diagnosis of BC. Histopathological images obtained from a biopsy contain vital information about the stage of the BC. Computer-aided systems are important tools to assist pathologists in the early detection of BC. Material and methodsIn the current study, the use of gray-level co-occurrence matrix (GLCM) of Shearlet Transform (ST) coefficients were first scrutinized as textural features. ST is an advanced decomposition-based method that can analyze images in various directions and is sensitive to edge singularities. These features make ST more robust than other decomposition methods such as Fourier and wavelet. Color channel histogram features were also utilized for a second level of evaluation in the diagnosis of the BC stage. These features are considered one of the most important building blocks that pathologists consider in the course of grading histopathological images. Then, by combining these two features, the classification results were re-assessed utilizing Support Vector Machine (SVM) as a classifier. ResultsThe assessments were performed on a BreaKHis dataset containing benign and malignant histopathological samples. The average accuracy scores were reported as being 98.2%, 97.2%, 97.8%, and 97.3% in the sub-databases with 40×, 100×, 200×, and 400× magnification factors, respectively. ConclusionsThe obtained results showed that the proposed method was quite efficient in histopathological image classification. Despite the relative simplicity of the approach, the obtained results were far superior to previously reported results.

A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

Abstract Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lame system of linear elasticity in polyhedral domains in $${\mathbb {R}}^3$$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

Extracting edge flux intensity functions along part-elliptical 3-D cracks by the quasidual function method

Part-elliptical crack fronts are common in real life, especially at free surfaces and at fastener holes. For such cases we herein extend the quasidual function method (QDFM) presented in Schapira and Yosibash (2019) to allow the extraction of edge flux intensity functions (EFIFs) from finite element solutions. We consider the Laplace operator, which provides a convenient framework for the extension of the QDFM to the elasticity operator.The QDFM post-processes the FE solution away from the singular edge. Numerical examples demonstrating the efficiency and accuracy of the proposed method are given for semi-elliptical and quarter-elliptical cracks.

Extracting edge flux intensity functions along an elliptical 3-D singular edge by the quasidual function method

Elliptical cracks are common in practice, as most cracks tend to propagate along semi-elliptical fronts. Herein we extend the quasidual function method (QDFM) for the computation of edge flux intensity functions (EFIFs) along elliptical cracks and generalized EFIFs along elliptical V-notches in 3-D domains. This QDFM is used to extract EFIFs from finite element solutions away from the singular edge and are given by a functional representation along the edge and not point-wise.The mathematical analysis of the QDFM is provided, followed by numerical examples demonstrating its efficiency and accuracy. Here we consider the Laplace operator, which provides a convenient framework for the extension of the QDFM for the elasticity operator.

The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance

The failure of frictional interfaces — the process of frictional rupture — is widely assumed to feature crack-like properties, with far-reaching implications for various disciplines, ranging from engineering tribology to earthquake physics. An important condition for the emergence of a crack-like behavior is the existence of stress drops in frictional rupture, whose basic physical origin has been recently elucidated. Here we show that for generic and realistic frictional constitutive relations, and once the necessary conditions for the emergence of an effective crack-like behavior are met, frictional rupture dynamics are approximately described by a crack-like, fracture mechanics energy balance equation. This is achieved by independently calculating the intensity of the crack-like singularity along with its associated elastic energy flux into the rupture edge region, and the frictional dissipation in the edge region. We further show that while the fracture mechanics energy balance equation provides an approximate, yet quantitative, description of frictional rupture dynamics, interesting deviations from the ordinary crack-like framework — associated with non-edge-localized dissipation — exist. Together with the recent results about the emergence of stress drops in frictional rupture, this work offers a comprehensive and basic understanding of why, how and to what extent frictional rupture might be viewed as an ordinary fracture process. Various implications are discussed.

Asymptotic solution of the elasticity equations in the vicinity of an elliptical crack front

The asymptotic solution of the elasticity equations in the vicinity of an elliptical singular edge is derived and presented explicitly for an elliptical crack with traction free BCs. It is composed of edge stress intensity functions (ESIFs), which are functions along the singular edge, eigenfunctions and their shadows. The recursive equations to derive the eigenfunctions and their shadows are provided.The explicit presentation of the series is required to identify the different quantities of interest as the ESIFs and the T-stresses and allows the extension of the quasi dual function method for the extraction of the ESIFs along elliptical singular edge. We also present the dual eigenfunctions needed to extract KI in the case of an elliptical crack with traction free BCs.

Edge singularities in the one-dimensional Bariev model

Bariev's model for correlated hopping in one dimension is integrable and soluble within the framework of two nested Bethe Ansätze. For repulsive interaction and finite magnetization the low energy excitation spectrum is a two-component Luttinger liquid with conformal charges c=1. In zero magnetic field the spectrum of spin excitations is gapped. For higher energy excitations the single-hole (single-particle) spectral functions display edge singularities that deviate from those expected from a Luttinger liquid. The system can be described as a mobile impurity coupled to the Luttinger liquid yielding an effective x-ray edge type model. Using the Bethe Ansatz solution for repulsive interaction we obtain expressions for the critical exponents for the single-particle and single-hole Green's functions in zero magnetic field.

Fractal x-ray edge problem at the critical point of the Aubry-André model

We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the "core-hole" Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power laws and dynamical exponents characterizing the x-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.

NUFFT for the Efficient Spectral Domain MoM Analysis of a Wide Variety of Multilayered Periodic Structures

In this paper the Method of Moments (MoM) in the spectral domain is used for the analysis of multilayered structures containing periodic arrays of either patches or apertures. The patches and apertures may have many different geometries including complex surfaces limited by two parallel lines and two arbitrary curves, circular and elliptic rings, circular and elliptic arcs, and circular and elliptic sectors. Basis functions accounting for edge singularities are used in the approximation of the electric/magnetic current density on the patches/apertures, which enables a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, these Fourier transforms are efficiently computed by means of the nonuniform fast Fourier transform (NUFFT) algorithm. Results have been obtained for frequency-selective surfaces (FSSs), and for the elements used in the design of both reflectarray and metasurface antennas. The results obtained indicate that the software based on the NUFFT is only 15% slower than the standard spectral domain MoM software used for structures in which the 2-D Fourier transform of the basis functions is analytical, and between 50 and 80 times faster than CST.

Scaling of the Berry Phase in the Yang-Lee Edge Singularity

We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.

Dimensional reduction by conformal bootstrap

Abstract Dimensional reductions in the branched polymer model and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. Small minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to the $D'=D-2$D Yang–Lee edge singularity and to the pure $D'=D-2$D Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$ and, for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.

A Fragile Watermark Scheme for Image Recovery Based on Singular Value Decomposition, Edge Detection and Median Filter

Many fragile watermark methods have been proposed for image recovery and their performance has been greatly improved. However, jagged edges and confusion still exist in the restored areas and these problems need to be solved to achieve a better visual effect. In this paper, a method for improving recovery quality is proposed that adopts singular value decomposition (SVD) and edge detection for tamper detection and then uses a median filter for image recovery. Variable watermark information can be generated that corresponds to block classifications. With mapping and neighborhood adjustment, the area that has been tampered can be correctly detected. Subsequently, we adopt a filtering operation for the restored image obtained after the inverse watermark embedding process. During the filtering operation, a median filter is used to smooth and remove noise, followed by minimum, maximum and threshold operations to balance the image intensity. Finally, the corresponding pixels of the restored image are replaced with the filtered results. The experimental results of six different tampering attacks conducted on eight test images show that tamper detection method with the edge detection can identify the tampered region correctly but has a higher false alarm rate than other methods. In addition, compared with the other three similar methods previously, using a median filter during image recovery not only improves the visual effect of the restored image but also enhances its quality objectively under most tampering attack conditions.

Revisiting the field-driven edge transition of the tricritical two-dimensional Blume-Capel model.

We reconsider the tricritical Blume-Capel model on the square lattice with a magnetic field acting on the open boundaries in one direction. Periodic boundary conditions are applied in the other direction. We apply three types of Monte Carlo algorithms, local Metropolis updates, and cluster algorithms of the Wolff and geometric type, adapted to the symmetry properties of the model. Statistical analyses of the bulk magnetization, the bulk Binder ratio, the edge magnetization, and the connected product of the edge and bulk magnetizations lead to new results confirming the presence of a singular edge transition at H_{sc}≈0.68, as we reported earlier [Phys. Rev. E 71, 026109 (2005)PLEEE81539-375510.1103/PhysRevE.71.026109]. We provide a plausible answer concerning a discrepancy between the behavior of the edge Binder ratio reported in that work and our new results.

Effective action study of the 𝒫𝒯-symmetric (iϕ3)6−𝜖 theory and the Yang–Lee edge singularity

We use the effective potential method to study the [Formula: see text]-symmetric [Formula: see text] field theory in [Formula: see text] space–time dimensions. For [Formula: see text], we obtained the first two energy levels which are real as well as reflecting the stability of the spectrum. [Formula: see text]-symmetry breaking occurs at [Formula: see text] where the two levels merge and beyond this critical point they have complex values. Since there exist no results in the literature to compare with, we extracted the critical exponents of the theory to test the accuracy of our calculations where we find them agree with exact results from the literature. We showed that the critical point is in fact a Yang–Lee edge singularity which is the first time to link [Formula: see text]-symmetry breaking to the existence of a Yang–Lee edge singularity. For [Formula: see text], the fixed point is nontrivial and exists for negative [Formula: see text] values as expected from Yang–Lee theory for ferromagnetic systems.

Characterizing singularities of a surface in Lie sphere geometry

The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.

Grand-canonical polymers under confinement: Dense solutions.

We theoretically study dense polymer solutions under open (capillary and slit) and closed (box) confinement. The theory is formulated for grand-canonical polymers and corrections to the self-consistent mean-field results are discussed. In contrast to the mean-field prediction, we found that the partition function of a labeled chain is affected by confinement even under neutral von Neumann boundary conditions and the chain length distribution is biased to short chains. As the container size increases, the contribution of the transverse excited states to the free energy of a labeled chain is found to approach its bulk value nonmonotonically (through an extremum) for the box and the capillary confinement but not for the slit. So does the confinement free energy of a labeled chain. The confinement energy of the solution is well behaved for open confinement but formally diverges for a closed box in the limit that the average chain length goes to infinity. Counted per chain, the confinement energy of the dense solution is qualitatively weaker than for a single ideal chain under similarly strong confinement (by one power in transverse container size). The container boundary contributes a surface tension to the free energy, which makes the effective monomer-wall affinity more repulsive. This correction increases with the average chain length. If present, edge or vertex singularities also contribute to the grand potential of the solution.

Dual Sheet Meshing: An Interactive Approach to Robust Hexahedralization

AbstractThe combinatorial dual of a hex mesh induces a collection of mutually intersecting surfaces (dual sheets). Inspired by Campen et al.'s work on quad meshing [CBK12, CK14], we propose to directly generate such dual sheets so that, as long as the volume is properly partitioned by the dual sheets, we are guaranteed to arrive at a valid all‐hex mesh topology. Since automatically generating dual sheets seems much harder than the 2D counterpart, we chose to leave the task to the user; our system is equipped with a few simple 3D modeling tools for interactively designing dual sheets. Dual sheets are represented as implicit surfaces in our approach, greatly simplifying many of the computational steps such as finding intersections and analyzing topology. We also propose a simple algorithm for primalizing the dual graph where each dual cell, often enclosing singular edges, gets mapped onto a reference polyhedron via harmonic parameterization. Preservation of sharp features is simply achieved by modifying the boundary conditions. We demonstrate the feasibility of our approach through various modeling examples.

Long-time existence of the edge Yamabe flow

This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov–Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.

A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes

A virtual element method is introduced for the mixed approximation of a simple model problem for the Laplace operator on a polyhedron. The method is fully analysed when the meshes are made up of triangular right prisms, pyramids and tetrahedra. The local discrete spaces coincide with the lowest order Raviart–Thomas spaces on tetrahedral and triangular right prismatic elements, and extend them to pyramidal elements. The discrete scheme is well posed and optimal interpolation error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the discrete element space are optimal and anisotropic on anisotropic right prisms. Furthermore, a discretization of the model problem in the presence of edge and vertex singularities is analysed for the proposed method on a family of suitably designed graded meshes, and optimal estimates for the approximation error are obtained, extending in this way the results of Farhloul et al. (ESAIM Math Model Numer Anal 35:907–920, 2001) where cylindrical domains with edge singularities were considered.