Related papers: Improving Frenet's Frame Using Bishop's Frame
This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel…
In the plane, we study the transform $R_\gamma f$ of integrating a unknown function $f$ over circles centered at a given curve $\gamma$. This is a simplified model of SAR, when the radar is not directed but has other applications, like…
One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods…
Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse…
Bayesian neural networks (BNNs) estimate the posterior distribution of model parameters and utilize posterior samples for Bayesian Model Averaging (BMA) in prediction. However, despite the crucial role of flatness in the loss landscape in…
Normalization Equivariance (NE) is a structural prior that improves robustness to distribution shift in image-to-image tasks. A function $f$ is normalization equivariant iff $f(a y + b\mathbf{1}) = a f(y) + b\mathbf{1}$ for all $a>0$ and…
Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of…
Convolutional neural network (CNN) slides a kernel over the whole image to produce an output map. This kernel scheme reduces the number of parameters with respect to a fully connected neural network (NN). While CNN has proven to be an…
Existing video recognition algorithms always conduct different training pipelines for inputs with different frame numbers, which requires repetitive training operations and multiplying storage costs. If we evaluate the model using other…
In this paper we study the fusion frame potential, that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. The structure of local and global minimizers of this potential…
Applications of machine learning techniques for materials modeling typically involve functions known to be equivariant or invariant to specific symmetries. While graph neural networks (GNNs) have proven successful in such tasks, they…
A set of equations is developed to describe a curve in space given the curvature $\kappa$ and the angle of rotation $\theta$ of the osculating plane. The set of equations has a solution (in terms of $\kappa$ and $\theta$) that indirectly…
Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an…
One may associate several frames to a given polytope, such as its collection of vertices, edges, or facet normal vectors. In this note, we use these frames to generate geometric inequalities for the simplex in $\mathbb{R}^d$ and polytopes…
Parsing urban scene images benefits many applications, especially self-driving. Most of the current solutions employ generic image parsing models that treat all scales and locations in the images equally and do not consider the geometry…
This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full…
This paper presents a survey of the Fermat principle within the framework of general relativity, tracing its evolution from classical optics to its modern variational formulation in Lorentzian geometry. In particular, we provide its proof…
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a…