Spherical Embeddings Research Articles

Orientation Determination of Cryo-EM Projection Images Using Reliable Common Lines and Spherical Embeddings.

Three-dimensional (3D) reconstruction in single-particle cryo-electron microscopy (cryo-EM) is a critical technique for recovering and studying the fine 3D structure of proteins and other biological macromolecules, where the primary issue is to determine the orientations of projection images with high levels of noise. This paper proposes a method to determine the orientations of cryo-EM projection images using reliable common lines and spherical embeddings. First, the reliability of common lines between projection images is evaluated using a weighted voting algorithm based on an iterative improvement technique and binarized weighting. Then, the reliable common lines are used to calculate the normal vectors and local X-axis vectors of projection images after two spherical embeddings. Finally, the orientations of projection images are determined by aligning the results of the two spherical embeddings using an orthogonal constraint. Experimental results on both synthetic and real cryo-EM projection image datasets demonstrate that the proposed method can achieve higher accuracy in estimating the orientations of projection images and higher resolution in reconstructing preliminary 3D structures than some common line-based methods, indicating that the proposed method is effective in single-particle cryo-EM 3D reconstruction.

Deep Clustering Analysis via Dual Variational Autoencoder With Spherical Latent Embeddings.

In recent years, clustering methods based on deep generative models have received great attention in various unsupervised applications, due to their capabilities for learning promising latent embeddings from original data. This article proposes a novel clustering method based on variational autoencoder (VAE) with spherical latent embeddings. The merits of our clustering method can be summarized as follows. First, instead of considering the Gaussian mixture model (GMM) as the prior over latent space as in a variety of existing VAE-based deep clustering methods, the von Mises-Fisher mixture model prior is deployed in our method, leading to spherical latent embeddings that can explicitly control the balance between the capacity of decoder and the utilization of latent embedding in a principled way. Second, a dual VAE structure is leveraged to impose the reconstruction constraint for the latent embedding and its corresponding noise counterpart, which embeds the input data into a hyperspherical latent space for clustering. Third, an augmented loss function is proposed to enhance the robustness of our model, which results in a self-supervised manner through the mutual guidance between the original data and the augmented ones. The effectiveness of the proposed deep generative clustering method is validated through comparisons with state-of-the-art deep clustering methods on benchmark datasets. The source code of the proposed model is available at https://github.com/fwt-team/DSVAE.

3D reconstruction from cryo-EM projection images using two spherical embeddings

Single-particle analysis (SPA) in cryo-electron microscopy has become a powerful tool for determining and studying the macromolecular structure at an atomic level. However, since the SPA problem is a non-convex optimization problem with enormous search space and there is high level of noise in the input images, the existing methods may produce biased or even wrong final models. In this work, to deal with the problem, consistent constraints from the input data are explored in an embedding space, a 3D spherical surface. More specifically, the orientation of a projection image is represented by two intersection points of the normal vector and the local X-axis vector of the projection image on the unit spherical surface. To determine the orientations of the projection images, the global consistency constraints of the relative orientations of all the projection images are satisfied by two spherical embeddings which estimate the normal vectors and the local X-axis vectors of the projection images respectively. Compared to the traditional methods, the proposed method is shown to be able to rectify the initial computation errors and produce a more accurate estimation of the projection angles, which results in a better final model reconstruction from the noisy image data.

Global spherical tropicalization via toric embeddings

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.

Spherical Spaces

The notion of a spherical space over an arbitrary base scheme is introduced as a generalization of a spherical variety over an algebraically closed field. It is studied how the sphericity condition behaves in families. In particular it is shown that sphericity of subgroup schemes is an open and closed condition over arbitrary base schemes generalizing a result by Knop and Roehrle. Moreover spherical embeddings are classified over arbitrary fields generalizing and simplifying results by Huruguen.

Spherical orbifold tutte embeddings

This work presents an algorithm for injectively parameterizing surfaces into spherical target domains called spherical orbifolds. Spherical orbifolds are cone surfaces that are generated from symmetry groups of the sphere. The surface is mapped the spherical orbifold via an extension of Tutte's embedding. This embedding is proven to be bijective under mild additional assumptions, which hold in all experiments performed. This work also completes the adaptation of Tutte's embedding to orbifolds of the three classic geometries - Euclidean, hyperbolic and spherical - where the first two were recently addressed. The spherical orbifold embeddings approximate conformal maps and require relatively low computational times. The constant positive curvature of the spherical orbifolds, along with the flexibility of their cone angles, enables producing embeddings with lower isometric distortion compared to their Euclidean counterparts, a fact that makes spherical orbifolds a natural candidate for surface parameterization.

Applications of ordinary voltage graph theory to graph embeddability

AbstractLet p be a prime greater than 5. We show that, while the generalized Petersen graphs of the form have cellular toroidal embeddings, they have no such embeddings having the additional property that a free action of a group on the graph extends to a cellular automorphism of the torus. Such an embedding is called a derived embedding. We also show that does have a derived embedding in the torus, and we show that for any odd q, each generalized Petersen graph of the form has a derived embedding in the Klein bottle, which has the same Euler characteristic as the torus. We close with some comments that frame these results in the light of Abrams and Slilaty's recent work on graphs featuring group actions that extend to spherical embeddings of those graphs.

On which groups can arise as the canonical group of a spherical latin bitrade

We address a question of Cavenagh and Wanless asking: which finite abelian groups arise as the canonical group of a spherical latin bitrade? We prove the existence of an infinite family of finite abelian groups that do not arise as canonical groups of spherical latin bitrades. Using a connection between abelian sandpile groups of digraphs underlying directed Eulerian spherical embeddings, we go on to provide several, general, families of finite abelian groups that do arise as canonical groups. These families include:any abelian group in which each component of the Smith Normal Form has composite order;any abelian group with Smith Normal Form Zpn ⊕ (⨁ i = 1kZpai), where 1 ≤ k, 2 ≤ a1, a2, …, ak, p and n ≤ 1 + 2∑ i = 1k(ai − 1); andwith two exceptions and two potential exceptions any abelian group of rank two.

Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

Let $\mathcal{G}$ be a properly face $2$-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group.The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face 2-coloured spherical triangulations with $t$ triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.

Spherical and Hyperbolic Embeddings of Data.

Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space.

Spherical Stochastic Neighbor Embedding of Hyperspectral Data

In hyperspectral imagery, low-dimensional representations are sought in order to explain well the nonlinear characteristics that are hidden in high-dimensional spectral channels. While many algorithms have been proposed for dimension reduction and manifold learning in Euclidean spaces, very few attempts have focused on non-Euclidean spaces. Here, we propose a novel approach that embeds hyperspectral data, transformed into bilateral probability similarities, onto a nonlinear unit norm coordinate system. By seeking a unit <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm nonlinear manifold, we encode similarity representations onto a space in which important regularities in data are easily captured. In its general application, the technique addresses problems related to dimension reduction and visualization of hyperspectral images. Unlike methods such as multidimensional scaling and spherical embeddings, which are based on the notion of pairwise distance computations, our approach is based on a stochastic objective function of spherical coordinates. This allows the use of an Exit probability distribution to discover the nonlinear characteristics that are inherent in hyperspectral data. In addition, the method directly learns the probability distribution over neighboring pixel maps while computing for the optimal embedding coordinates. As part of evaluation, classification experiments were conducted on the manifold spaces for hyperspectral data acquired by multiple sensors at various spatial resolutions over different types of land cover. Various visualization and classification comparisons to five existing techniques demonstrated the strength of the proposed approach while its algorithmic nature is guaranteed to converge to meaningful factors underlying the data.

Toric varieties and spherical embeddings over an arbitrary field

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field k. Then we provide some situations where toric varieties over k are classified by Galois-stable fans, and spherical embeddings over k by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over k whose fan is Galois-stable but which admits no k-form. In the spherical setting, we offer an example of a spherical homogeneous space X 0 over R of rank 2 under the action of SU ( 2 , 1 ) and a smooth embedding of X 0 whose fan is Galois-stable but which admits no R -form.

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry.

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry.

Progressive mesh metamorphosis

AbstractThis paper describes a new integrated scheme for metamorphosis between two closed manifold genus‐0 polyhedral models. Spherical parameterizations of the source and target models are created first. To control the morphing, any number of feature vertex pairs is specified and a fold‐over free warping method is used to align two spherical embeddings. Our method does not create a merged meta‐mesh or execute re‐meshing to construct a common connectivity for morphs. Alternatively, a scheme for the progressive connectivity transformation of two spherical parameterizations is employed to generate the intermediate meshes. A novel semi‐overlay with a geomorph scheme is proposed to reduce the popping effects caused by the connectivity transformation. We demonstrate several examples of aesthetically pleasing morphing sequences using the proposed scheme. Copyright © 2005 John Wiley &amp; Sons, Ltd.

A note on the spherical embeddings of strongly regular graphs

A primitive symmetric association scheme of class 2 is naturally embedded as a two-distance set in the unit sphere of Euclidean space, with respect to the primitive idempotent E1 of the Bose–Mesner algebra of the association scheme. Then it is shown that the ratio of the two distances of the two-distance set is instantly read from the character table (i.e., the first eigen matrix P) of the association scheme.

Spherical varieties with the $A_k$-property

An algebraic variety is said to have the $A_k$-property if any $k$ points are contained in some common affine open neighbourhood. A theorem of W{\l}odarczyk states that a normal variety has the $A_2$-property if and only if it admits a closed embedding into a toric variety. Spherical varieties can be regarded as a generalization of toric varieties, but they do not have the $A_2$-property in general. We provide a combinatorial criterion for the $A_k$-property of spherical varieties by combining the theory of bunched rings with the Luna-Vust theory of spherical embeddings.