3D point cloud lossy compression using quadric surfaces

The presence of 3D sensors in hand-held or head-mounted smart devices has motivated many researchers around the globe to devise algorithms to manage 3D point cloud data efficiently and economically. This paper presents a novel lossy compression technique to compress and decompress 3D point cloud data that will save storage space on smart devices as well as minimize the use of bandwidth when transferred over the network. The idea presented in this research exploits geometric information of the scene by using quadric surface representation of the point cloud. A region of a point cloud can be represented by the coefficients of quadric surface when the boundary conditions are known. Thus, a set of quadric surface coefficients and their associated boundary conditions are stored as a compressed point cloud and used to decompress. An added advantage of proposed technique is its flexibility to decompress the cloud as a dense or a course cloud. We compared our technique with state-of-the-art 3D lossless and lossy compression techniques on a number of standard publicly available datasets with varying the structure complexities.

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Maximal families of nodal varieties with defect

In this paper we prove that a nodal hypersurface in $$\mathbf {P}^4$$ P 4 with positive defect has at least $$(d-1)^2$$ ( d - 1 ) 2 nodes, and if it has at most $$2(d-2)(d-1)$$ 2 ( d - 2 ) ( d - 1 ) nodes and $$d\ge 7$$ d ≥ 7 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of $$\mathbf {P}^3$$ P 3 ramified along a surface of degree 2d with positive defect has at least $$d(2d-1)$$ d ( 2 d - 1 ) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in $$\mathbf {P}^{2n+2}$$ P 2 n + 2 with positive defect for d sufficiently large.

ACOUSTIC SIMULATION OF THE CENTRAL HALL IN PALAU GÜELL BY GAUDÍ

Introduction: Quadric surfaces are commonly used in buildings due to their geometric ability to distribute and focus sound waves. The Central Hall in Palau Güell — a UNESCO World Heritage Site — is topped by an ellipsoidal dome. Antoni Gaudí envisaged this room as a concert hall where the organ and the dome play a lead role. Methods: The two previously mentioned elements are the main subject of our paper, which serves two purposes: 1) determining the values of the acoustic parameters of the hall through onsite measurement and also through simulation, and 2) using the geometric parameters of the quadric surface, which best fits the dome, in order to check whether it is possible to improve the acoustics of the hall by placing a new emission source at the focus of the dome’s ellipsoid. Results and Discussion: Contrary to the authors’ expectations, due to the focal reflection properties of the quadric surface, some acoustic parameters on the listening plane do not improve significantly. Therefore, we conclude that Gaudí took the acoustical impact into account when designing this hall.

Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.