Disk Bundle Research Articles

Fast detection of the main anatomical structures in digital retinal images based on intra- and inter-structure relational knowledge

Background and objective: The anatomical structure detection in retinal images is an open problem. However, most of the works in the related literature are oriented to the detection of each structure individually or assume the previous detection of a structure which is used as a reference. The objective of this paper is to obtain simultaneous detection of the main retinal structures (optic disc, macula, network of vessels and vascular bundle) in a fast and robust way.Methods: We propose a new methodology oriented to accomplish the mentioned objective. It consists of two stages. In an initial stage, a set of operators is applied to the retinal image. Each operator uses intra-structure relational knowledge in order to produce a set of candidate blobs that belongs to the desired structure. In a second stage, a set of tuples is created, each of which contains a different combination of the candidate blobs. Next, filtering operators, using inter-structure relational knowledge, are used in order to find the winner tuple. A method using template matching and mathematical morphology is implemented following the proposed methodology.Results: A success is achieved if the distance between the automatically detected blob center and the actual structure center is less than or equal to one optic disc radius. The success rates obtained in the different public databases analyzed were: MESSIDOR (99.33%, 98.58%, 97.92%), DIARETDB1 (96.63%, 100%, 97.75%), DRIONS (100%, n/a, 100%) and ONHSD (100%, 98.85%, 97.70%) for optic disc (OD), macula (M) and vascular bundle (VB), respectively. Finally, the overall success rate obtained in this study for each structure was: 99.26% (OD), 98.69% (M) and 98.95% (VB). The average time of processing per image was 4.16 ± 0.72 s.Conclusions: The main advantage of the use of inter-structure relational knowledge was the reduction of the number of false positives in the detection process. The implemented method is able to simultaneously detect four structures. It is fast, robust and its detection results are competitive in relation to other methods of the recent literature.

Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.

A remark about weak fillings

Let L be a closed manifold of dimension n≥2 which admits a totally real embedding into Cn. Let ST*L be the space of rays of the cotangent bundle T*L of L, and let DT*L be the unit disk bundle of T*L defined by any Riemannian metric on L. We observe that ST*L endowed with its standard contact structure admits weak symplectic fillings W which are diffeomorphic to DT*L and for which any closed Lagrangian submanifold N⊂W has the property that the map H1(N,R)→H1(W,R) has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.

Ricci curvature of double manifolds via isoparametric foliations

Given a closed manifold M and a vector bundle ξ of rank n over M, by gluing two copies of the disc bundle of ξ, we can obtain a closed manifold D(ξ,M), the so-called double manifold.In this paper, we firstly prove that each sphere bundle Sr(ξ) of radius r>0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r>0 small enough, the induced metric of Sr(ξ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n≥3.As an application, if M admits a metric with positive Ricci curvature and n≥2, then we construct a metric with positive Ricci curvature on D(ξ,M). Moreover, under the same metric, D(ξ,M) admits a natural isoparametric foliation.For a compact minimal isoparametric hypersurface Yn in Sn+1(1), which separates Sn+1(1) into S+n+1 and S−n+1, one can get double manifolds D(S+n+1) and D(S−n+1). Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations (cf. [25]), we study Ricci curvature of them with isoparametric foliations in the last part.

Finiteness of $$\pi _1$$ π 1 -sensitive Hofer–Zehnder capacity and equivariant loop space homology

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer–Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.

Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $\pi_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $\pi_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.

Aggregation behavior of cyclic rod-coil diblock copolymers in selective solvents

The aggregation behavior of cyclic rod-coil (RC) diblock copolymers in dilute solutions is investigated through dissipative particle dynamics simulation. By varying the rod length and coil length, cyclic RC copolymers in selective solvents exhibit various morphologies, including spherical micelle, vesicle, bilayer disc, and ribbon bundle structure. Compared with the equivalent linear RC copolymer, only spherical micelle and barrel bundle phase are observed. Rod length is the major factor that controls the liquid-crystalline behavior of RC copolymer systems, while the coil length has a secondary effect on the aggregate morphology. The size of rod bundle varies with the coil length, especially for the end-toend ribbon bundle and side-by-side barrel bundle, which are assembled by cyclic and linear RC copolymer solutions. This finding indicates that the ribbon bundle or nanofiber-like structure in cyclic RC copolymers can be obtained by controlling the rod length and coil length, and thus the optical and electrical properties of RC copolymer would be further controlled and optimized. Results illustrate that cyclization of a linear RC copolymer induces remarkable differences in the rod arrangement and aggregation behavior, thereby indicating the competition between interfacial energy, rod orientational entropy, coil stretching entropy, and packing constraints.

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

We look at lattices in Iso+(H2R)Iso+(HR2), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP2CP2 via the natural embedding of SO+(2,1)↪SU(2,1)⊂SL(3,C)SO+(2,1)↪SU(2,1)⊂SL(3,C). We use the Hermitian cross product in C2,1C2,1 introduced by Bill Goldman, to determine the topology of the Kulkarni limit set ΛKulΛKul of these lattices, and show that in all cases its complement ΩKulΩKul has three connected components, each being a disc bundle over H2RHR2. We get that ΩKulΩKul coincides with the equicontinuity region for the action on CP2CP2. Also, it is the largest set in CP2CP2 where the action is properly discontinuous and it is a complete Kobayashi hyperbolic space. As a byproduct we get that these lattices provide the first known examples of discrete subgroups of SL(3,C)SL(3,C) whose Kulkarni region of discontinuity in CP2CP2 has exactly three connected components, a fact that does not appear in complex dimension 11 (where it is known that the region of discontinuity of a Kleinian group acting on CP1CP1 has 00, 11, 22 or infinitely many connected components).

Free-floating vitreous cyst

Free-floating intraocular cysts can be divided into those occurring in the anterior chamber, those occurring in the retrolental space, and those occurring in the vitreous cavity. The origin of vitreous cysts may be associated with remnants of the hyaloid system (congenital), or can result from trauma or ocular disease. The aim of this study was to assess the impact of free-floating vitreous cyst on the macula using optical coherence tomography, ultrasonography, and fluorescein angiography. A 58-year-old female patient presented for eye examination complaining of annoying movable floaters with glare and difficultly in close works. Ophthalmoscopic examination through the dilated fundus revealed posterior vitreous detachment, unremarkable peripheral retinal degenerations with no abnormalities at the optic disc, the macula or papillomacular bundle, and the anterior segment and pupillary reaction were free. This case report represents a case of free-floating vitreous cyst that has been advised for follow-up until it becomes visually significant, requiring interference either with argon laser cystotomy or aspiration through pars plana vitrectomy.

On the Szegö kernel of Cartan–Hartogs domains

Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szeg\"o kernel of the disk bundle over a noncompact K\"ahler manifold. In particular we compute the Szeg\"o kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szeg\"o kernel of the disk bundle over an Herimitian symmetric space of compact type.

Rationally convex domains and singular Lagrangian surfaces in $$\mathbb {C}^2$$ C 2

We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.

Stability of pseudoconvexity of disc bundles over compact Riemann surfaces and application to a family of Galois coverings

It is proved that Galois coverings of smooth families of compact Riemann surfaces over Stein manifolds are holomorphically convex if the covering transformation groups are isomorphic to discrete subgroups of the automorphism group of the unit disc. The proof is based on an extension of the fact that disc bundles over compact Kähler manifolds are weakly 1-complete.

Differentiable classification of 4-manifolds with singular Riemannian foliations

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $$\mathbb {S}^4$$ , $$\mathbb {CP}^2$$ , $$\mathbb {S}^2\times \mathbb {S}^2$$ , or $$\mathbb {CP}^2$$ # $$\pm \mathbb {CP}^2$$ . As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard $$\mathbb {S}^4$$ , $$\pm \mathbb {CP}^2$$ and $$\mathbb {S}^2\times \mathbb {S}^2$$ . A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds.

Algebras of semiclassical pseudodifferential operators associated with Zoll-type domains in cotangent bundles

We consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr–Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols. The Schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow-out of the null foliation of the boundary of the domain. We develop a symbolic calculus, prove the existence of projectors (under a mild additional assumption) whose range can be thought of as quantizing the domain, give a symbolic proof of a Szegö limit theorem, and study associated propagators.

Double plumbings of disk bundles over spheres

We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold $X$ and a suitable pair of classes in $H_{2}(X)$, we investigate when this pair of classes may be represented by a configuration of surfaces in $X$ whose regular neighbourhood is a double plumbing of disk bundles over spheres. Using similar methods we study single plumbings of two disk bundles over spheres inside $X$.

Exceptional Holonomy on Vector Bundles with Two-Dimensional Fibers

An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N 6 can be defined as a pair of a 2-form ω and a 3-form ρ. We prove that any analytic SU(3)- or SU(1,2)-structure on N 6 with dω∧ω=0 can be extended to a parallel $\operatorname{Spin}(7)$ - or $\operatorname{Spin}_{0}(3,4)$ -structure Φ that is defined on the trivial disc bundle N 6×B ϵ (0) for a sufficiently small ϵ>0. Furthermore, we show by an example that Φ is not uniquely determined by (ω,ρ) and discuss if our result can be generalized to non-trivial bundles.

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kahler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Englis (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.

Transnormal functions on a Riemannian manifold

We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge–Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function.

On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

Given a holomorphic line bundle over the complex affine quadric $Q^2$, we investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant biholomorphism, these are all contained in a maximal one, say $\Omega_{max}$. By removing the zero section to $\Omega_{max}$ one obtains the unique Stein, SU(2)-equivariant, punctured disc bundle over $Q^2$ which contains entire curves. All other such punctured disc bundles are shown to be Kobayashi hyperbolic.