Related papers: The Gibbs Representation of 3D Rotations
Robots in dynamic environments need fast, accurate models of how objects move in their environments to support agile planning. In sports such as ping pong, analytical models often struggle to accurately predict ball trajectories with spins…
Researchers have now achieved great success on dealing with 2D images using deep learning. In recent years, 3D computer vision and Geometry Deep Learning gain more and more attention. Many advanced techniques for 3D shapes have been…
Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…
Recent advances in computer vision have predominantly relied on data-driven approaches that leverage deep learning and large-scale datasets. Deep neural networks have achieved remarkable success in tasks such as stereo matching and…
We introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By…
Standard Vision Transformers flatten 2D images into 1D sequences, disrupting the natural spatial topology. While Rotary Positional Embedding (RoPE) excels in 1D, it inherits this limitation, often treating spatially distant patches (e.g.,…
Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the…
A rotation in a Euclidean space V is an orthogonal map on V which acts locally as a plane rotation with some fixed angle. We give a classification of all pairs of rotations in finite-dimensional Euclidean space, up to simultaneous…
Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework…
The problem of 3D reconstruction from posed images is undergoing a fundamental transformation, driven by continuous advances in 3D Gaussian Splatting (3DGS). By modeling scenes explicitly as collections of 3D Gaussians, 3DGS enables…
We present fast and accurate ways to normalize two and three dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the…
Starting from the Hamiltonian representation of the dynamics in \cite{rosengren2015chaos,colombo2019long}, this work proposes an innovative procedure to design fully-analytical maneuvers for post-mission disposal of HEOs satellites,…
A novel single-frame quaternion estimator processing two vector observations is introduced. The singular cases are examined, and appropriate rotational solutions are provided. Additionally, an alternative method involving sequential…
We present DirectTriGS, a novel framework designed for 3D object generation with Gaussian Splatting (GS). GS-based rendering for 3D content has gained considerable attention recently. However, there has been limited exploration in directly…
We review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For 3D point data,…
Representations of quantum computations are almost always based on a tensor product $\otimes$-structure. This coincides with what we are able to execute in our experiments, as well as what we observe in Nature, but it makes certain familiar…
3D Gaussian splatting (3DGS) has recently emerged as an alternative representation that leverages a 3D Gaussian-based representation and introduces an approximated volumetric rendering, achieving very fast rendering speed and promising…
A combination $\mathbf{a}+\mathrm{i}\mathbf{b}$ where ${\mathrm i}^2=-1$ and $\mathbf{a}, \, \mathbf{b}$ are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He…
Integral transforms are invaluable mathematical tools to map functions into spaces where they are easier to characterize. We introduce the hyperdimensional transform as a new kind of integral transform. It converts square-integrable…
Searching for a unified scene representation remains a research challenge in computer graphics. Traditional mesh-based representations are unsuitable for dense, fuzzy elements, and introduce additional complexity for filtering and…