Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Density Fingerprint of a Periodic Point Set
71
0
0.0
(
0
)
Added by
Teresa Heiss
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.
rate research
Read More
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.
The Hausdorff distance, the Gromov-Hausdorff, the Frechet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $inf_rho F(rho)$ where $F$ is a suitable functional and $rho$ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space $mathcal{K}$, in such a way that the composition in $mathcal{K}$ (extending the composition of homeomorphisms) passes to the limit and, at the same time, $mathcal{K}$ is compact.
The present work proposes a solution to the challenging problem of registering two partial point sets of the same object with very limited overlap. We leverage the fact that most objects found in man-made environments contain a plane of symmetry. By reflecting the points of each set with respect to the plane of symmetry, we can largely increase the overlap between the sets and therefore boost the registration process. However, prior knowledge about the plane of symmetry is generally unavailable or at least very hard to find, especially with limited partial views, and finding this plane could strongly benefit from a prior alignment of the partial point sets. We solve this chicken-and-egg problem by jointly optimizing the relative pose and symmetry plane parameters, and notably do so under global optimality by employing the branch-and-bound (BnB) paradigm. Our results demonstrate a great improvement over the current state-of-the-art in globally optimal point set registration for common objects. We furthermore show an interesting application of our method to dense 3D reconstruction of scenes with repetitive objects.
In our previous two papers, we studied (positive) 3D gadgets in origami extrusions which create a top face parallel to the ambient paper and two side faces sharing a ridge with two simple outgoing pleats. Then a natural problem comes up whether it is possible to construct a `negative 3D gadget from any positive one having the same net without changing the outgoing pleats, that is, to sink the top and two side faces of any positive 3D gadget to the reverse side without changing the outgoing pleats. Of course, simply sinking the faces causes a tear of the paper, and thus we have to modify the crease pattern. There are two known constructions of negative 3D gadgets before ours, but they do not solve this problem because their outgoing pleats are different from positive ones. In the present paper we give an affirmative solution to the above problem. For this purpose, we present three constructions of negative 3D gadgets with a supporting triangle on the back side, which are based on our previous ones of positive 3D gadgets. The first two are an extension of those presented in our previous paper, and the third is new. We prove that our first and third constructions solve the problem. Our solutions enable us to deal with positive and negative 3D gadgets on the same basis, so that we can construct from an origami extrusion constructed with 3D gadgets its negative using the same pleats if there are no interferences among the 3D gadgets. We also treat repetition/division of negative 3D gadgets under certain conditions, which reduces their interferences with others.
We consider the number of distinct distances between two finite sets of points in ${bf R}^k$, for any constant dimension $kge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Omegaleft(minleft{ n^{2/3}m^{2/3},; frac{n^{10/11}m^{4/11}}{log^{2/11}m},; n^2,; m^2right}right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
