Linear Singularities Research Articles

Asymptotic Rules of Equilibrium Desingularization

A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).

Engineering phase and polarization singularity sheets

Optical phase singularities are zeros of a scalar light field. The most systematically studied class of singular fields is vortices: beams with helical wavefronts and a linear (1D) singularity along the optical axis. Beyond these common and stable 1D topologies, we show that a broader family of zero-dimensional (point) and two-dimensional (sheet) singularities can be engineered. We realize sheet singularities by maximizing the field phase gradient at the desired positions. These sheets, owning to their precise alignment requirements, would otherwise only be observed in rare scenarios with high symmetry. Furthermore, by applying an analogous procedure to the full vectorial electric field, we can engineer paraxial transverse polarization singularity sheets. As validation, we experimentally realize phase and polarization singularity sheets with heart-shaped cross-sections using metasurfaces. Singularity engineering of the dark enables new degrees of freedom for light-matter interaction and can inspire similar field topologies beyond optics, from electron beams to acoustics.

Generation and Absorption of Periodic Waves Traveling on a Uniform Current in a Fully Nonlinear BEM-based Numerical Wave Tank

Accurate and efficient numerical wave generation and absorption of two-dimensional nonlinear periodic waves traveling on a steady, uniform current were carried out in a potential, fully nonlinear numerical wave tank. The solver is based on the Βoundary Εlement Μethod (ΒΕΜ) with linear singularity distributions and plane elements and on the mixed Eulerian–Lagrangian formulation of the free surface equations. Wave generation is implemented along the inflow boundary by imposing the stream function wave solution, while wave absorption at both end-boundaries is effectively treated by introducing absorbing layers. On the absorbing beach side, the outflow boundary condition is modified to ensure that the solution accurately satisfies the dispersion relation of the generated waves. The modification involves a free-parameter that depends on the mass flux through the domain and is determined through a feedback error-correction loop. The developed method provides accurate time domain wave solutions for shallow, intermediate, and deep water depths of high wave steepness (wave heights up to 80% of the maximum value) that remain stable for 150 wave periods. This also holds in case a coplanar or opposing uniform current of velocity up to 20% of the wave celerity interacts with the wave.

Deep geometric convolutional network for automatic modulation classification

A recent trend of automatic modulation classification is to automatically learn high-level abstraction of signals, instead of manually designing features for further classification. In this paper, we propose a new deep geometric convolutional network (DGCN) to hierarchically extract discriminative features from Wigner–Ville distribution map of signals. A group of geometric filters are constructed from a least square support vector machine, to capture the linear singularity existed in maps. The filters are cascaded to construct a deep network for extracting discriminative features and classifying signals with different modulation types. Some experiments are taken to investigate the performance of DGCN, and the results show that it can achieve high accuracy in classifying 15 types of modulation signals.

Dynamical quantum phase transitions of quantum spin chains with a Loschmidt-rate critical exponent equal to 12

We describe a new universality class of dynamical quantum phase transitions of the Loschmidt amplitude of quantum spin chains off equilibrium criticality. We demonstrate that in many cases it is possible to change the conventional linear singularity of the Loschmidt rate function into a smooth peak by tuning one parameter of the quench protocol. Exactly at the point when this change-over occurs, the singularity of the Loschmidt rate function persists, with a critical exponent equal to $\frac{1}{2}$ . The non-equilibrium renormalization group fixed-point controlling this universality class is described. An asymptotically exact renormalization group recursion relation is derived around this fixed-point to obtain the critical exponent.

Block-Extraction and Haar Transform Based Linear Singularity Representation for Image Enhancement

In this paper, we develop a novel linear singularity representation method using spatial K-neighbor block-extraction and Haar transform (BEH). Block-extraction provides a group of image blocks with similar (generally smooth) backgrounds but different image edge locations. An interblock Haar transform is then used to represent these differences, thus achieving a linear singularity representation. Next, we magnify the weak detailed coefficients of BEH to allow for image enhancement. Experimental results show that the proposed method achieves better image enhancement, compared to block-matching and 3D filtering (BM3D), nonsubsampled contourlet transform (NSCT), and guided image filtering.

Analysis of nascent silicon phase-change gratings induced by femtosecond laser irradiation in vacuum

The formation of periodic surface structures is a general effect of femtosecond laser irradiation of solid targets showing promising interest in material science and technology. However, the experiments are typically carried out in air, a condition in which the target surface becomes densely decorated with nanoparticles that can influence the formation of the surface structures in the early stage of the irradiation process. Here we report an investigation of structures generation on a silicon surface irradiated in vacuum (10−5 mbar) with a low number of laser pulses (N ≤ 10) that exploits several microscopy techniques (optical, atomic force, electron and Raman). Our analyses allow identifying the creation of silicon phase-change gratings consisting of alternating amorphous and crystalline periodic lines, with almost no material removal, located at the periphery of a shallow ablation crater. These gratings originate from two different kinds of defects: (i) the first is characterized by a peculiar lobed shape that is produced by the first few laser pulses; (ii) the second is provided by the one-dimensional, linear singularity defined by the ablation edge of the nascent crater. Both kind of defects lead to grating structures extending outwards the amorphous central area of the crater along the direction of the laser polarization. Comparative analysis with the surface formed in air, in the same experimental conditions, evidences the important role played by nanoparticles densely decorating the target in air and the striking variation occurring in vacuum.

Geometric Separation in $$\mathbb {R}^3$$ R 3

The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution containing a non-trivial superposition of point and curvilinear singularities into its distinct geometric constituents. The solution proposed in Donoho and Kutyniok (2013) considers expansions with respect to a combined wavelet-curvelet dictionary and applies an $$\ell ^1$$ -norm minimization over the expansion coefficients to achieve separation asymptotically at fine scales. However, the original proof of this result uses a heavy machinery relying on sparse representations of Fourier integral operators which does not extend directly to the 3D setting. In this paper, we extend the geometric separation result to the 3D setting using a novel and simpler argument which relies in part on techniques developed by the authors for the shearlet-based analysis of curvilinear edges. Our new result also yields a significantly simpler proof of the original 2D geometric separation problem and extends a prior result by the authors which was limited to piecewise linear singularities.

Sub-Compartments Within Orientation Columns of Primary Visual Cortex: A Proposal for a Contour Building Architecture

PURPOSE: In the mammalian visual system, early stages of visual form perception begin with orientation selective neurons in primary visual cortex (V1). In many species (including humans, monkeys, tree shrews, cats, and ferrets), these neurons are organized in beautifully arrayed orientation columns, which shift in orientation preference across V1 and are highlighted by orientation pinwheels. However, to date, the relationship of orientation architecture to the encoding of elemental aspects of visual contours is still unknown. METHODS: Using a novel highly accurate method of targeting electrode position, combining with optical image and single-unit recording, we report for the first time the presence of three functionally distinct zones within single orientation domains. RESULTS: We found evidence for three concentric sub-regions centered on the orientation-pinwheel. The central-most region contains neurons with small receptive fields and strong suppressive surrounds, while the outermost region contains neurons with larger receptive fields and weak suppressive surrounds. CONCLUSIONS: We suggest that these zones subseries computation of distinct aspects of visual contours (linear orientation, local curvature, and contour singularities). The orientation domain thus embodies a three stage visual contour processor in V1.

Appearance of fractional vortex dipoles by using spiral linear zone plate

We present a novel technique based on linear zone plate (LZP) by which linear singularities regarded as fractional vortex dipoles are efficiently generated. Our approach requires applying spiral phase to a LZP, so fractional vortex dipoles are then acquired. By also implementing transverse phase shift to the spiral LZP, we are able to equally divide the number of the produced dark beams carrying mixed phase dislocations into two transverse separate parts. As a result, by varying a so called the controlling parameter the lateral positions of the focused fractional vortex dipoles are changed. Since we are able to generate dark beams in different and specific linear positions, therefore we suggest these features will be of great interest in one-dimensional optical trapping. All results are completely verified by experimental works.

Vortex polarity switching in magnets with surface anisotropy

Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow- or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction.

Geometric Separation of Singularities Using Combined Multiscale Dictionaries

Several empirical results appeared in the literature during the last decade have shown that it is often possible to separate images and other multidimensional data into geometrically distinct constituents. A rigorous mathematical analysis of the geometric separation problem in the two-dimensional setting was recently introduced by Donoho and Kutyniok (Comm Pure Appl Math 66:1–47, 2013), who proposed a mathematical framework to separate point and smooth curve singularities in 2D images using a combined dictionary consisting of curvelets and wavelets. In this paper, we adapt their approach and introduce a novel argument to extend geometric separation to the three-dimensional setting. We show that it is possible to separate point and piecewise linear singularities in 3D using a combined dictionary consisting of shearlets and wavelets. Our new approach takes advantage of the microlocal properties of the shearlet transform and has the ability to handle singularities containing vertices and corner points, which cannot be handled using the original arguments.

The director field distribution with the strongly pinned alignment in nematic structures at the polymer surface

We investigate orientational nematic liquid crystal structures that form at the polymer surface due to adsorption, their stability against the schlieren texture and features of point and linear singularities in them. We demonstrate that surface disclination lines, which also occur due to molecular adsorption, can lead to azimuthal anchoring of a bulk layer and make the singularities with the charge m = ±1 stable. We consider the director field distribution in a nematic layer as a result of the combination of two aligning factors, which are the linear disclination that tend to impose uniform planar ordering and point defects that impart the radial configuration to the director. With the use of the calculation, we analyse the energies of elastic distortions of the director field for the radial–planar, radial–homeotropic and radial–radial configurations, depending on the nematic layer thickness.

Comparison of stress singularities of kinked carbon and glass fibres weakening compressed unidirectional composites: a three-dimensional trimaterial junction stress singularity analysis

A three-dimensional eigenfunction expansion technique, based in part on separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion in conjunction with the Eshelby–Stroh formalism, is used to compute the local stress singularity, in the vicinity of a kinked fibre/matrix trimaterial junction front, representing a measure of the degree of inherent flaw sensitivity of unidirectional carbon-fibre-reinforced composites under compression. Micro-kinking, more prominent in the misaligned regions of carbon fibres, is caused by crystallite disorientations, as detected by the Raman and X-ray measurements, as well as dislocation glide in crystallites and intra/intercrystallite disorders. The present analysis explains the test results relating to propagation of failure from such discontinuities in a unidirectional composite under compression. Numerical results presented include the effect of fibre wedge aperture angle on the strengths of the mode I and mode II singularities. Of special practical interest is the comparison of the inherent flaw sensitivity of carbon/epoxy and glass/epoxy composites, because improvement of the compressive strength and kink toughness was earlier accomplished through commingling of highly anisotropic (and crystalline) carbon and isotropic (and amorphous) glass fibres at the tow level. Compression fracture of these composites can be fully explained and quantified by the present three-dimensional linear elastic stress singularity analysis-based method. Finally, numerical results, pertaining to the through-thickness variation of ‘stress intensity factor’ for symmetric uniform load and its skew-symmetric counterpart that also satisfies the boundary conditions on the top and bottom surfaces of a compressed composite monolayer, in the vicinity of a trimaterial junction front, are also presented.

Electric and magnetic polarization singularities of first-order Laguerre–Gaussian beams diffracted at a half-plane screen

Based on the vector Fresnel diffraction integrals, analytical expressions for the electric and magnetic components of first-order Laguerre-Gaussian beams diffracted at a half-plane screen are derived and used to study the electric and magnetic polarization singularities in the diffraction field for both two- and three-dimensional (2D and 3D) cases. It is shown that there exist 2D and 3D electric and magnetic polarization singularities in the diffraction field, which do not coincide each other in general. By suitably varying the waist width ratio, off-axis displacement parameter, amplitude ratio, or propagation distance, the motion, pair-creation, and annihilation of circular polarization singularities, and the motion of linear polarization singularities take place in 2D and 3D electric and magnetic fields. The V point, at which two circular polarization singularities with the same topological charge but opposite handedness collide, appears in the 2D electric field under certain conditions in the diffraction field and free-space propagation. A comparison with the free-space propagation is also made.

Image Noise Reduction via Geometric Multiscale Ridgelet Support Vector Transform and Dictionary Learning

Advances in machine learning technology have made efficient image denoising possible. In this paper, we propose a new ridgelet support vector machine (RSVM) for image noise reduction. Multiscale ridgelet support vector filter (MRSVF) is first deduced from RSVM, to produce a multiscale, multidirection, undecimated, dyadic, aliasing, and shift-invariant geometric multiscale ridgelet support vector transform (GMRSVT). Then, multiscale dictionaries are learned from examples to reduce noises existed in GMRSVT coefficients. Compared with the available approaches, the proposed method has the following characteristics. The proposed MRSVF can extract the salient features associated with the linear singularities of images. Consequently, GMRSVT can well approximate edges, contours and textures in images, and avoid ringing effects suffered from sampling in the multiscale decomposition of images. Sparse coding is explored for noise reduction via the learned multiscale and overcomplete dictionaries. Some experiments are taken on natural images, and the results show the efficiency of the proposed method.

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely l 1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that--provided the size of the missing part is asymptotic to the size of the analyzing functions--asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.

Structural singularities of the Eu2+-doped spatially coherent [KBr0.097I0.903](0.348):[KBr0.459Cl0.511I0.030](0.652) composite as seen under the epifluorescence optical microscopy

Large crystal growths of the Eu2+-doped spatially coherent [KBr0.097I0.903](0.348):[KBr0.459Cl0.511I0.030](0.652) composite were characterized by X-ray diffraction and then studied by epifluorescence optical microscopy. Doping Eu2+ ions were observed to prefer sites located at certain linear structural singularities of the composite matrix to be segregated at. These singularities (2.1 × 107 singularities cm−2), identified as crystal lattice dislocations, were found to be distributed within the composite matrix so that they form periodic arrays of linear structural singularities (1.8 × 104 singularities cm−1). These arrays, identified as grain sub-boundaries, were found to envelope individual structural domains (1–5 µm in size) of either KBr0.459Cl0.511I0.030:Eu2+ or KBr0.097I0.903:Eu2+. These domains were found to aggregate among themselves to form the whole composite building. Small misfit angles (e.g. 7′ ± 1′ and 10′ ± 1′) characterize homo-phase structural domains while large misfit angles are characteristic of hetero-phase structural domains. Crystal lattice dislocations, forming the grain sub-boundaries, were found to present, as structural features, kinks and bifurcation points. The spatial configurations adopted by two of these features are carefully described.

A Line Enhancement Algorithm for Infrared Image Based on Ridgelet

In order to well detect target’s line character for infrared image, we propose improving line character in image through using ridgelet transform which is a kind of multiscale geometric analysis tool and it is especially suitable for describing the 2-D signals which have linear or super-plane singularities. This paper briefs the principle of ridgelet transform and explains our enhancement algorithm in some detail. Its core consists of the method of tensile scale of coefficient in Radon transform domain, filtering method in processing of one dimensional wavelet transform and the algorithm steps of improving line character. It reconstructs images and compares with the other traditional method as presented and analyzes scale of coefficient and time complexity. Our enhancement method is proved effective preliminarily by experimental results.

The Finite Ridgelet Transform for Defeat Detection of Quill

A defeat detection method for quill using the finite ridgelet transform is mentioned. This method reduce the average gray values of quill and analytic statistics methods are used in ridgelet transform according to points of quill. After ridgelet transform the linear singularity is further highlighted and the quantization errors caused by cutting image of quill is overcomed. With this method the differences between background and quill are eliminated too and automaticly adjusting the sub-image size for transformation according to the width of quill is realized. The experiment results show that this method can effectively detect smaller linear defects and hence satisfy the industrial production needs.