arXiv research

Convolutional Neural Networks (CNNs) have become the method of choice for learning problems involving 2D planar images. However, a number of problems of recent interest have created a demand for models that can analyze spherical images. Examples include omnidirectional vision for drones, robots, and autonomous cars, mo…

DeepSphere improves spherical CNNs by balancing efficiency and rotation equivariance.

Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…

CeCNN predicts SE and AL from UWF images, improving myopia screening.

The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be exte…

The Hausdorff Distance (HD) is widely used in evaluating medical image segmentation methods. However, existing segmentation methods do not attempt to reduce HD directly. In this paper, we present novel loss functions for training convolutional neural network (CNN)-based segmentation methods with the goal of reducing HD…

Group equivariant neural networks simplify complex tasks with group representation theory.

Characterization of lung nodules as benign or malignant is one of the most important tasks in lung cancer diagnosis, staging and treatment planning. While the variation in the appearance of the nodules remains large, there is a need for a fast and robust computer aided system. In this work, we propose an end-to-end tra…

Coordinate-independent convolutions on manifolds avoid reference frame ambiguity.

Unified geometric scattering model for measure spaces.

Paper introduces spherical knot mosaics for knot and link invariants.

Study geodesics on spherical polyhedra, estimating their number.

Study predicts lens performance using neural networks.

The abstract proves spherical surface decompositions with conical singularities.

For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.

Study spherical curves with curvature dependent on distance to a great circle.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

New findings show fundamental group is not audible in spherical space forms.

Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

Study on spherical CR manifolds with non-trivial Chern classes.

PGF kernels analyze spherical data using generalized RBF kernels.

New approximative kernels improve PDE-G-CNNs for geometric deep learning.

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

In recent years, deep learning poses a deep technical revolution in almost every field and attracts great attentions from industry and academia. Especially, the convolutional neural network (CNN), one representative model of deep learning, achieves great successes in computer vision and natural language processing. How…

Classifies involutions on spherical 3-manifolds.

The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.

We attempt to interpret how adversarially trained convolutional neural networks (AT-CNNs) recognize objects. We design systematic approaches to interpret AT-CNNs in both qualitative and quantitative ways and compare them with normally trained models. Surprisingly, we find that adversarial training alleviates the textur…

Paper develops invariants for spherical curves using chord diagrams.

In this paper we consider the spherical slant helices in $R^3$. More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.

The paper characterizes spherically symmetric metrics with scalar curvature.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

New formula for spherical polygon area via prequantization.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…

Convolutional Neural Networks (CNNs) have revolutionized performances in several machine learning tasks such as image classification, object tracking, and keyword spotting. However, given that they contain a large number of parameters, their direct applicability into low resource tasks is not straightforward. In this w…

2D CNNs approximate Korobov functions with near-optimal rates.

Recently we initiated the study of spherical T-duality for spacetimes that are principal SU(2)-bundles. In this paper, we extend spherical T-duality to spacetimes that are oriented non-principal SU(2)-bundles. There are several interesting new examples in this case and a new phenomenon appearing in the non-principal ca…

In image classification, visual separability between different object categories is highly uneven, and some categories are more difficult to distinguish than others. Such difficult categories demand more dedicated classifiers. However, existing deep convolutional neural networks (CNN) are trained as flat N-way classifi…

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

Proposes a fixed smooth convolutional layer to reduce checkerboard artifacts in CNNs.

In this paper, we address the issue of how to enhance the generalization performance of convolutional neural networks (CNN) in the early learning stage for image classification. This is motivated by real-time applications that require the generalization performance of CNN to be satisfactory within limited training time…

Spherical Plateau problem studies minimal surfaces in quotients of spheres.