Distance fields on unstructured grids: Stable interpolation, assumed gradients, collision detection and gap function

AbstractThis article presents a novel approach to collision detection based on distance fields. A novel interpolation ensures stability of the distances in the vicinity of complex geometries. An assumed gradient formulation is introduced leading to a C1-continuous distance function. The gap function is re-expressed allowing penalty and Lagrange multiplier formulations. The article introduces a node-to-element integration for first order elements, but also discusses signed distances, partial updates, intermediate surfaces, mortar methods and higher order elements. The algorithm is fast, simple and robust for complex geometries and self contact. The computed tractions conserve linear and angular momentum even in infeasible contact. Numerical examples illustrate the new algorithm in three dimensions

SOL-NeRF:Sunlight Modeling for Outdoor Scene Decomposition and Relighting

Outdoor scenes often involve large-scale geometry and complex unknown lighting conditions, making it difficult to decompose them into geometry, reflectance and illumination. Recently researchers made attempts to decompose outdoor scenes using Neural Radiance Fields (NeRF) and learning-based lighting and shadow representations. However, diverse lighting conditions and shadows in outdoor scenes are challenging for learning-based models. Moreover, existing methods may produce rough geometry and normal reconstruction and introduce notable shading artifacts when the scene is rendered under a novel illumination. To solve the above problems, we propose SOL-NeRF to decompose outdoor scenes with the help of a hybrid lighting representation and a signed distance field geometry reconstruction. We use a single Spherical Gaussian (SG) lobe to approximate the sun lighting, and a first-order Spherical Harmonic (SH) mixture to resemble the sky lighting. This hybrid representation is specifically designed for outdoor settings, and compactly models the outdoor lighting, ensuring robustness and efficiency. The shadow of the direct sun lighting can be obtained by casting the ray against the mesh extracted from the signed distance field, and the remaining shadow can be approximated by Ambient Occlusion (AO). Additionally, sun lighting color prior and a relaxed Manhattan-world assumption can be further applied to boost decomposition and relighting performance. When changing the lighting condition, our method can produce consistent relighting results with correct shadow effects. Experiments conducted on our hybrid lighting scheme and the entire decomposition pipeline show that our method achieves better reconstruction, decomposition, and relighting performance compared to previous methods both quantitatively and qualitatively.</p

Signed zeros of Gaussian vector fields-density, correlation functions and curvature

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.Comment: 14 pages, 1 figure, uses iopart.cls, improved presentation, to appear in J. Phys.

Riesz external field problems on the hypersphere and optimal point separation

We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed measure and $d-2 \leq s <d$, our results lead to explicit point-separation estimates for $(Q,s)$-Fekete points, which are $n$-point configurations minimizing the Riesz $s$-energy on $\mathbb{S}^d$ with external field $Q$. In the hyper-singular case $s > d$, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure

Reconstructing the Local Twist of Coronal Magnetic Fields and the Three-Dimensional Shape of the Field Lines from Coronal Loops in EUV and X-Ray Images

Non-linear force-free fields are the most general case of force-free fields, but the hardest to model as well. There are numerous methods of computing such fields by extrapolating vector magnetograms from the photosphere, but very few attempts have so far made quantitative use of coronal morphology. We present a method to make such quantitative use of X-Ray and EUV images of coronal loops. Each individual loop is fit to a field line of a linear force-free field, allowing the estimation of the field line's twist, three-dimensional geometry and the field strength along it. We assess the validity of such a reconstruction since the actual corona is probably not a linear force-free field and that the superposition of linear force-free fields is generally not itself a force-free field. To do so, we perform a series of tests on non-linear force-free fields, described in Low & Lou (1990). For model loops we project field lines onto the photosphere. We compare several results of the method with the original field, in particular the three-dimensional loop shapes, local twist (coronal alpha), distribution of twist in the model photosphere and strength of the magnetic field. We find that, (i) for these trial fields, the method reconstructs twist with mean absolute deviation of at most 15% of the range of photospheric twist, (ii) that heights of the loops are reconstructed with mean absolute deviation of at most 5% of the range of trial heights and (iii) that the magnitude of non-potential contribution to photospheric field is reconstructed with mean absolute deviation of at most 10% of the maximal value.Comment: submitted to Ap

Steiner's formula in the Heisenberg group

Steiner's tube formula states that the volume of an ∈-neighborhood of a smooth regular domain in ℝn is a polynomial of degree n in the variable ∈ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ∈-neighborhood with respect to the Heisenberg metric is an analytic function of ∈ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms

Acute silencing of hippocampal CA3 reveals a dominant role in place field responses.

Neurons in hippocampal output area CA1 are thought to exhibit redundancy across cortical and hippocampal inputs. Here we show instead that acute silencing of CA3 terminals drastically reduces place field responses for many CA1 neurons, while a smaller number are unaffected or have increased responses. These results imply that CA3 is the predominant driver of CA1 place cells under normal conditions, while also revealing heterogeneity in input dominance across cells