Visible Singularities Research Articles

Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, R, which defines the integrals of a compactly supported L2 function, f, over ellipsoids and hyperboloids with centers on a smooth connected surface, S. Our transform is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that R satisfies the Bolker condition if the support of f is contained in a connected open set that is not intersected by any plane tangent to S. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions and validate our microlocal theory.

New family of C metrics in N=2 gauged supergravity

We present a new family of charged $C$ metrics in $\mathcal{N}=2$ gauged supergravity in four dimensions. The double Wick rotation of the $C$ metric allows us to bring our solution into a different family of the $C$ metrics previously found by L\"u and V\'azquez-Poritz. In the case of zero acceleration limit, our solution with vanishing charges reduces to the scalar haired black holes in anti--de Sitter space with regular horizons. Nevertheless, it turns out that each family of neutral solutions fails to veil the curvature singularity by the event horizon, showing that neither of them represents the accelerated black holes with a scalar hair. Physical solutions without visible curvature singularities are obtained only in the case of nonvanishing charges. Causal structures of the solution are spelled out in detail. We also present conditions under which the solution preserves supersymmetry.

Smoothing of nonsmooth differential systems near regular-tangential singularities and boundary limit cycles

Understanding how tangential singularities evolve under smoothing processes was one of the first problem concerning regularization of Filippov systems. In this paper, we are interested in C n -regularizations of Filippov systems around visible regular-tangential singularities of even multiplicity. More specifically, using Fenichel theory and blow-up methods, we aim to understand how the trajectories of the regularized system transits through the region of regularization. We apply our results to investigate C n -regularizations of boundary limit cycles with even multiplicity contact with the switching manifold.

Globally visible singularity in an astrophysical setup

ABSTRACT The global visibility of a singularity as an end state of the gravitational collapse of a spherically symmetric pressureless cloud is investigated. We show the existence of a non-zero measured set of parameters: the total mass and the initial mean density of the collapsing cloud, giving rise to a physically strong globally visible singularity as the end state for a fixed velocity function. The existence of such a set indicates that such singularity is stable under small perturbation in the initial data causing its existence. This is true for marginally as well as non-marginally bound cases. The possibility of the presence of such suitable parameters in the astrophysical setup is then studied: (1) The singularities’ requirements at the centre of the M87 galaxy and at the centre of our galaxy (SgrA*) to be globally visible are discussed in terms of the initial size of the collapsing cloud forming them, presuming that such singularities are formed due to gravitational collapse. (2) The requirement for the primordial singularities formed due to a collapsing configuration after getting detached from the background universe at the time of matter-dominated era just after the time of matter-radiation equality, to be globally visible, is discussed. (3) The scenario of the collapse of a neutron star after reaching a critical mass, which is achieved by accreting the supernova ejecta expelled by its binary companion core progenitor, is considered. The primary aim of this paper is to show that globally visible singularities can form in astrophysical setups under appropriate circumstances.

Collision of a hard ball with singular points of the boundary.

Recently, physical billiards were introduced where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables, the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional); i.e., the particle may collide with these singular points. Here, we consider the collision of a hard ball with a visible singular point and demonstrate that the motion of the smooth ball after collision with a visible singular point is indeed the one that was used in the studies of physical billiards. Therefore, such collision is equivalent to the elastic reflection of hard ball's center off a sphere with the center at the singular point and the same radius as the radius of the moving particle. However, a ball could be rough, not smooth. In difference with a smooth ball, a rough ball also acquires rotation after reflection off a point of the boundary, which leads to more complicated dynamics.

Microlocal Analysis of Generalized Radon Transforms from Scattering Tomography

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier integral operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semiglobal Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the generalized Radon FIO. Our theory has specific applications of interest in Emission Compton Scattering Tomography (ECST) and Bragg Scattering Tomography (BST). We show that the ECST and BST integration curves satisfy the semiglobal Bolker assumption and provide simulated reconstructions from ECST and BST data. Additionally, we give example “sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artifacts. The observed artifacts in reconstruction are shown to align exactly with our predictions.

Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.

An image reconstruction model regularized by edge-preserving diffusion and smoothing for limited-angle computed tomography

Limited-angle computed tomography is a very challenging problem in applications. Due to a high degree of ill-posedness, conventional reconstruction algorithms will introduce blurring along the directions perpendicular to the missing projection lines, as well as streak artifacts when applied on limited-angle data. Various models and algorithms have been proposed to improve the reconstruction quality by incorporating priors, among which the total variation, i.e. l1 norm of gradient, and l0 norm of the gradient are the most popular ones. These models and algorithms partially solve the blurring problem under certain situations. However, the fundamental difficulty remains. In this paper, we propose a reconstruction model for limited-angle computed tomography, which incorporates two regularization terms that play the role of edge-preserving diffusion and smoothing along the x-direction and y -direction respectively. Then, an alternating minimization algorithm is proposed to solve the model approximately. The proposed model is inspired by the theory of visible and invisible singularities of limited-angle data, developed by Quinto et al. By incorporating visible singularities as priors into an iterative procedure, the proposed algorithm could produce promising results and outperforms state-of-the-art algorithms for certain limited-angle computed tomography applications. Extensive experiments on both simulated data and real data are provided to validate our model and algorithm.

The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for k-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of 2-parameter families, , of planar Filippov systems assuming that Z0,0 presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.

Artifacts and Visible Singularities in Limited Data X-Ray Tomography

We describe a principle to determine which features of an object will be easy to reconstruct from limited X-ray CT data and which will be difficult. The principle depends on the geometry of the data set, and it applies to any limited data set. We also describe a characterization of Frikel and the author explaining artifacts that can be added to limited angle reconstructions, and we provide an easy-to-implement method to decrease them. These ideas are justified using microlocal analysis, deep mathematics that involves Fourier theory. Reconstructions from simulated and real limited data are given to illustrate our ideas.

Thermo- and Photoacoustic Tomography with Variable Speed and Planar Detectors

We analyze the mathematical model of multiwave tomography with a variable speed with integrating measurements on planes tangent to a sphere surrounding the source. We prove sharp uniqueness and stability estimates with full and partial data and propose a time reversal algorithm which recovers the visible singularities.

On The Strength Of Streak Artifacts In Filtered Back-projection Reconstructions For Limited Angle Weighted X-Ray Transform

In this article, we study the limited angle problem for the weighted X-ray transform. We consider two approximate reconstructions by applying filtered back-projection formulas to the limited data. We prove that each resulted operator can be decomposed into the sum of three Fourier integral operators whose symbols are of types \((\varrho ,\,\delta ) \ne (1,\,0).\) The first operator, being a pseudo-differential operator, is responsible for the reconstruction of visible singularities. The other two are responsible for the generation of the artifacts. The theory of Fourier integral operators then implies, in particular, the continuity of the reconstruction operator and geometry of the artifacts. We then extend the technique developed by the author in [Inverse Problems 31 (2015) 055003] to obtain more refined microlocal estimates for the strength of the artifacts.

Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction. &nbsp Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.

Limited Data Problems for the Generalized Radon Transform in ${{\mathbb R}^n}$

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in ${{\mathbb R}^n}$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in ${{\mathbb R}^n}$ and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in ${{{\mathbb R}^2}}$ and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradig...

Electrical resistance of copper under shock compression: Experimental data

The electrical resistance of copper foil under shock compression is measured. The electrical resistance and electrical conductivity are plotted as functions of the shock pressure in the interval up to 20 GPa. These dependences are monotonic and have no visible inflections or singularities. A qualitative dependence of the electrical resistance of the metal on the shock impedance of the material of the block containing the sample is found. A comparison of the data obtained in this study with results of other authors shows that it is important to take into account the block material, the shape and thickness of the sample, and the procedure of determining the state of the sample.

Slow–fast n-dimensional piecewise linear differential systems

In this article we analyze n-dimensional slow–fast systems in a piecewise linear framework. In particular, we prove a Fenichel's-like Theorem where we give an explicit expression for the invariant slow manifold, that leads to the proof of the existence and location of maximal canards orbits. We show that these orbits perturb from singular orbits through contact points, of order greater than or equal to two, between the reduced flow and the fold manifold. In the particular case n=3, we show that the unique contact point is a visible two-fold singularity.

Numerical Recovery of Source Singularities via the Radiative Transfer Equation with Partial Data

The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator $X_{V}^{*}X_{V}$, where $X_{V}$ is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by Monard and Bal [J. Comput. Phys., 229 (2010), pp. 4952--4979]. We will see that one can detect quite well the visible singularities of an internal optical source $f$ for generic anisotropic $k$ and $\sigma$, with or without noise added to the accessible data $X_{V}f$. In particular, we use a truncated Neumann series to estimate $X_{V}$ and $X_{V}^{*}$, which provides a good approximation of $X_{V}^{*}X_{V}$ with an error of higher Sobolev regularity. This paper provides a visual demonstration of the author's previous work [M. Hubenthal, Inverse Problems, 27 (2011), 125009] in recovering the microlocally visible singularities of an unknown source from partial data. We also g...

Is a Curved Flight Path in SAR Better than a Straight One?

In the plane, we study the transform $R_\gamma f$ of integrating an unknown function $f$ over circles centered at a given curve $\gamma$. This is a simplified model of synthetic aperture radar (SAR), when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set $WF(f)$. If the visible singularities of $f$ hit $\gamma$ once, we show that $WF(f)$ cannot be recovered; i.e., the artifacts cannot be resolved. If $\gamma=\partial\Omega$ is the boundary of a strictly convex domain $\Omega$, we show that this is still true. On the other hand, in the latter case, if $f$ is known a priori to have singularities in a compact set, then we show that one can recover $WF(f|_\Omega)$, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.

On the genericity of spacetime singularities

We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse. The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.

On the dT=0 toy model in reggeon field theory

Reggeon field theory with zero transverse dimensions is critically reanalyzed in the Hamiltonian formulation for both a sub- and a supercritical pomeron. Different mathematical aspects of the model, starting from the scalar products in the space of quantum states, are discussed. The probabilistic picture is addressed in the absence of pomeron merging. The issue of the large loop approximation is discussed in terms of the Hamiltonian evolution and its relation to the probabilistic picture recalled. A perturbative treatment, based on the PT symmetry of the model, is proposed that may be useful for more realistic models. Finally, we present numerical calculations for the various parameters of the models, α(0)-1=μ and the triple-pomeron coupling constant λ, which help one to understand some mathematical aspects and the different approximation regimes. They show that the triple-pomeron interaction always makes amplitudes fall with rapidity, irrespective of the value of the intercept. The smaller the values of the ratio λ/μ, the higher are the rapidities y at which this fall starts, so that at small values of λ it begins at asymptotically high rapidities (for λ/μ<1/4 the fall is noticeable only at μy>100). No visible singularity is seen for the critical pomeron.