arXiv research

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.

The paper proves a conjecture about the shape of floating bodies.

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in Rm. When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called a radial center or an illuminating center, respec…

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an inde…

In this paper, the problem of optimizing the deployment of unmanned aerial vehicles (UAVs) equipped with visible light communication (VLC) capabilities is studied. In the studied model, the UAVs can simultaneously provide communications and illumination to service ground users. Ambient illumination increases the interf…

We propose an automatic method to infer high dynamic range illumination from a single, limited field-of-view, low dynamic range photograph of an indoor scene. In contrast to previous work that relies on specialized image capture, user input, and/or simple scene models, we train an end-to-end deep neural network that di…

For a Veech surface (x,ω), we characterize subspaces of X^n, invariant under the diagonal action of the affine group of X. We prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X,ω) embedded in the…

Enhances 2D face recognition with 3D features using active illumination.

We present a deep learning framework for wide-field, content-aware estimation of absorption and scattering coefficients of tissues, called Generative Adversarial Network Prediction of Optical Properties (GANPOP). Spatial frequency domain imaging is used to obtain ground-truth optical properties from in vivo human hands…

Design optimization techniques are often used at the beginning of the design process to explore the space of possible designs. In these domains illumination algorithms, such as MAP-Elites, are promising alternatives to classic optimization algorithms because they produce diverse, high-quality solutions in a single run,…

Joint scene understanding and segmentation for automotive applications is a challenging problem in two key aspects:- (1) classifying every pixel in the entire scene and (2) performing this task under unstable weather and illumination changes (e.g. foggy weather), which results in poor outdoor scene visibility. This poo…

Visual perception is a challenging problem in part due to illumination variations. A possible solution is to first estimate an illumination invariant representation before using it for recognition. The object albedo and surface normals are examples of such representations. In this paper, we introduce a multilayer gener…

The ability to learn and adapt in real time is a central feature of biological systems. Neuromorphic architectures demonstrating such versatility can greatly enhance our ability to efficiently process information at the edge. A key challenge, however, is to understand which learning rules are best suited for specific t…

Automatic video modification to hide faces while maintaining pose, illumination, and expression.

New index characterizes non-smooth Zoll convex bodies.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

A groupoid $Ω\left( \mathcal{B} \right)$ called material groupoid is naturally associated to any simple body $\mathcal{B}$. The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the inclusion of these new objects in the theory of material bodies opens th…

Study shows volumes of complex classes can be represented by convex bodies.

The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…

Study on discrete Okounkov bodies and their applications.

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…

We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, …

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …

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…

The paper proves no multiple equichordal points exist in convex bodies.

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

When $S$ is a closed, orientable surface with genus $g(S) \geq 2$, we show that the automorphism group of the compression body graph $\mathcal{CB}(S)$ is the mapping class group. Here, vertices are compression bodies with exterior boundary $S$, and edges connect pairs of compression bodies where one contains the other.

In this paper we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…

The paper extends inequalities for projection bodies to arbitrary measures.

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

DeepBark improves tree bark re-identification accuracy.

In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with c…

Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler…

Efficiently samples arbitrary compact bodies with polynomial complexity.

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

Efficient algorithm for sampling from arbitrary compact bodies.

New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.

Functional Magnetic Resonance Imaging (fMRI) provides dynamical access into the complex functioning of the human brain, detailing the hemodynamic activity of thousands of voxels during hundreds of sequential time points. One approach towards illuminating the connection between fMRI and cognitive function is through dec…

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the co…

Analytic convex bodies' Poincaré series extended holomorphically.

Solves Christoffel-Minkowski problem for axially symmetric bodies.