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2. In Section 3, we used some vague statements to affirm the training process of the neural network, which cannot support others to reproduce the results of the paper. In addition, this section does not show the difference between this…

In this paper, we propose a novel formulation to extend CNNs to two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific…

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic…

The abundance of data has given machine learning considerable momentum in natural sciences and engineering, though modeling of physical processes is often difficult. A particularly tough problem is the efficient representation of geometric…

Cortical surface registration plays a crucial role in aligning cortical functional and anatomical features across individuals. However, conventional registration algorithms are computationally inefficient. Recently, learning-based…

We introduce an inference-time scene optimization algorithm utilizing triangle soup, a collection of disconnected translucent triangle primitives, as the representation for the geometry and appearance of a scene. Unlike full-rank Gaussian…

3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a…

We present the Julia package Manifolds$.$jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric…

To endow machines with the ability to perceive the real-world in a three dimensional representation as we do as humans is a fundamental and long-standing topic in Artificial Intelligence. Given different types of visual inputs such as…

Understanding the 3-dimensional structure of the world is a core challenge in computer vision and robotics. Neural rendering approaches learn an implicit 3D model by predicting what a camera would see from an arbitrary viewpoint. We extend…

Homogenization is a technique commonly used in multiscale computational science and engineering for predicting collective response of heterogeneous materials and extracting effective mechanical properties. In this paper, a three-dimensional…

3D scene graphs have recently emerged as a powerful high-level representation of 3D environments. A 3D scene graph describes the environment as a layered graph where nodes represent spatial concepts at multiple levels of abstraction and…

The increase of existing computational capabilities has made simulation emerge as a third discipline of Science, lying midway between experimental and purely theoretical branches [1, 2]. Simulation enables the evaluation of quantities which…

Spatial perception is a key task in several machine intelligence applications such as robotics and computer vision. In general, it involves the nonlinear estimation of hidden variables that represent the system's state. However, in the…

Reconstructing an object's geometry and appearance from multiple images, also known as inverse rendering, is a fundamental problem in computer graphics and vision. Inverse rendering is inherently ill-posed because the captured image is an…

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

Polygonal meshes provide an efficient representation for 3D shapes. They explicitly capture both shape surface and topology, and leverage non-uniformity to represent large flat regions as well as sharp, intricate features. This…

We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of…

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer…

We describe in this paper Hydra, an ensemble of convolutional neural networks (CNN) for geospatial land classification. The idea behind Hydra is to create an initial CNN that is coarsely optimized but provides a good starting pointing for…