DELIMIT PyTorch enhances deep learning for diffusion imaging.
problem Applying deep learning to spherical diffusion imaging data.
method Added spherical harmonic interpolation and local convolution layers to PyTorch.
result Deep learning can now be applied conveniently to diffusion imaging data.
Spherical CNNs tackle 3D data analysis, especially spherical images.
problem Learning problems involving spherical images, like omnidirectional vision and molecular regression.
method Defined spherical cross-correlation, developed a generalized FFT for efficient computation.
result Demonstrated spherical CNNs' effectiveness in 3D model recognition and atomization energy regression.
Study spherical images of modified vector fields on a unit sphere.
problem Understanding spherical images of modified vector fields.
method Analysis of spherical images using modified orthogonal vector fields and Darboux vector.
result Characterization of spherical indicatrices with modified orthogonal frame.
A new method tracks retinal vessels more accurately than existing methods.
problem Tracking retinal vessels accurately in spherical images.
method Computing cusp-free, crossing-preserving geodesics on spherical positions and orientations.
result Crossing-preserving tracking shows clear advantages over non-crossing-preserving tracking.
In this paper, we investigate a curve whose spherical image the tangent indicatrix and binormal indicatrix is slant helix and called it as a slant helix. We obtain that the spherical images are spherical slant helices defined by [3]. This notation is a generalization of a slant helix. Furthermore, we have given some ch…
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …
In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…
Nearly spherical, positively curved surfaces are mapped from a sphere.
problem Mapping nearly spherical, positively curved surfaces from a sphere.
method Combines Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.
result Every nearly spherical, positively curved surface is the contractive image of a round sphere.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
Spherical quadrilaterals classified based on geometric properties.
problem Classifying spherical quadrilaterals up to isometry.
method Developing map and condition of genericity.
result Space of quadrilaterals with prescribed angles consists of finitely many open curves.
Simpler neural network for spherical images using Clebsch-Gordan transforms.
problem Learning spherical images rotation invariantly.
method Clebsch-Gordan transform for nonlinearity, avoiding repeated Fourier transforms.
result Improved performance compared to previous methods.
SUNLayer framework improves image denoising stability.
problem Stable denoising of images and other inverse problems.
method Introduces SUNLayer framework based on spherical harmonics to analyze generative models.
result Demonstrates stable denoising performance of SUNLayer on various activation functions.
Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.
problem Proving a formula connecting discrete Gaussian curvature to the algebraic area of Gauss images.
method Comparing winding numbers and critical point index of a normal vector to deduce the formula.
result Formula significantly limits possible shapes of Gauss images of polyhedral vertex stars.
We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agra…
Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed re…
Characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic and spherical spaces.
problem Characterizing curves in curved spaces.
method Rotation minimizing frames and exponential maps.
result Characterizes geodesic spherical curves in hyperbolic and spherical spaces through linear equations.
We consider the discrete representations of 3-manifold groups into PU(2,1) that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…
Meta Optimal Transport learns from past problems to solve similar OT problems faster.
problem Solving similar optimal transport problems repeatedly from scratch is inefficient.
method Amortized optimization to predict optimal transport maps from past solutions.
result Meta OT models can solve new problems faster than standard methods.
In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images.
We explore visual representations of tilings corresponding to Schläfli symbols. In three dimensions, we call these tilings "honeycombs". Schläfli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces in all dimensions. In three dimensions, there are only a finite number …
We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written x−1y where x and y are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type An and Bn and then show that in spherical …
Proposes a new clustering method based on expectiles for non-spherical clusters.
problem Inability of K-means to handle non-spherical clusters. method Uses expectiles to define cluster centers and searches for clusters via a greedy algorithm.
result Outperforms K-means and spectral clustering on asymmetric shaped clusters. CeCNN predicts SE and AL from UWF images, improving myopia screening.
problem Predicting axial length and spherical equivalence from UWF fundus images.
method Copula-enhanced Convolutional Neural Network (CeCNN) for multiresponse regression.
result CeCNN improves prediction of SE and AL compared to baseline CNNs.
Proposes a new latent variable model for hyperspherical latent spaces.
problem Efficiently modeling heavy-tailed distributions in hyperspherical latent spaces.
method Introduces spherical Cauchy (spCauchy) latent variables and applies Möbius transformations.
result Shows spCauchy recovers vMF geometry in high-concentration limits and avoids complex evaluations.
FHDMs achieve optimal convergence in spherically supported data.
problem Statistical convergence properties of FHDMs for spherical data.
method FHDMs leverage random generation time and Doob's h-transform to optimize convergence rate.
result Achieve minimax optimal convergence rate in total variation for spherically supported Sobolev smooth data.
Study of light function singularities on surfaces.
problem Characterizing singularities of the slant function on surfaces.
method Analyzing the differential geometry of the parabolic set and its spherical image under the Gauss map.
result The type of singularities of the slant function is determined by the geometry of the parabolic set and its spherical image.
We describe a model for capturing the statistical structure of local amplitude and local spatial phase in natural images. The model is based on a recently developed, factorized third-order Boltzmann machine that was shown to be effective at capturing higher-order structure in images by modeling dependencies among squar…
In this study, we give the dual characterizations of Mannheim offsets of the ruled surface in terms of their integral invariants and the new characterization of the Mannheim offsets of developable surface. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets …
New autoencoder improves latent space learning by optimizing sliced Gromov-Wasserstein discrepancies.
problem Improving inner discrepancy between prior and posterior distributions in autoencoders.
method Proposed spherical sliced fused Gromov Wasserstein (SSFG) and variants (MSSFG, PSSFG) to find important directions.
result New autoencoders achieve favorable performance in latent manifold learning, image generation, and reconstruction.
Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…
Generative Latent Implicit Conditional Optimization (GLICO) learns from small samples.
problem Learning from small labeled datasets.
method Generative Latent Implicit Conditional Optimization (GLICO) learns a latent space and generator from small labeled data.
result GLICO synthesizes new samples for every class using as few as 10 examples per class.
The paper lifts spherical Morse functions to immersions and embeddings.
problem Lifting spherical Morse functions to other maps.
method New methods to lift to special generic maps with non-positive codimensions.
result Constructs most lifts to special generic maps.
Extends curvature properties to arbitrary closed sets in R^n.
problem Establishing curvature properties for arbitrary closed sets.
method Analyzes eigenvalues of spherical image maps and introduces second fundamental form.
result Integrates support measure and generalizes curvature properties.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value on the real…
Authors compute Weingarten map and curvatures for SL(n, R).
problem Computing curvatures of the special linear group.
method Elementary calculations to derive Weingarten map and curvatures.
result Explicit computation of curvatures at identity of SL(n, R).
The study classifies singularities of spherical orthotomic curves.
problem Classifying singularities of spherical orthotomic curves.
method Defining spherical orthotomic curves and classifying their singularities.
result Singularities of spherical orthotomic curves are classified.
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
We present an algorithm for converting an indoor spherical panorama into a photograph with a simulated overhead view. The resulting image will have an extremely wide field of view covering up to 4π steradians of the spherical panorama. We argue that our method complements the stereographic projection commonly used in t…
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space S12. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…
In this paper, we obtain the characterizations of Mannheim offsets of the timelike ruled surface with spacelike rulings in dual Lorentzian space. We give the relations between terms of their integral invariants and also we give the new characterization of the Mannheim offsets of developable timelike ruled surface. More…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
The paper proves rigidity theorems for hypersurfaces in spherical space forms.
problem Proving rigidity of hypersurfaces in spherical space forms under certain curvature conditions.
method Topological and homotopical methods, including diffeomorphisms and weak homotopy equivalences.
result The universal cover of the hypersurface is diffeomorphic to the n-sphere and fundamental group bounds.
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.