Efficient Rock Mass Point Cloud Registration Based on Local Invariants

Point cloud registration is one of the basic research hotspots in the field of 3D reconstruction. Although many previous studies have made great progress, the registration of rock point clouds remains an ongoing challenge, due to the complex surface, arbitrary shape, and high resolution of rock masses. To overcome these challenges, a novel registration method for rock point clouds, based on local invariants, is proposed in this paper. First, to handle the massive point clouds, a point of interest filtering method based on a sum vector is adopted to reduce the number of points. Second, the remaining points of interest are divided into several cluster point sets and the centroid of each cluster is calculated. Then, we determine the correspondence between the original point cloud and the target point cloud by proving the inherent similarity (using the trace of the covariance matrix) of the remaining point sets. Finally, the rotation matrix and translation vector are calculated, according to the corresponding centroids, and a correction method is used to adjust the positions of the centroids. To illustrate the superiority of our method, in terms of accuracy and efficiency, we conducted experiments on multiple datasets. The experimental results show that the method has higher accuracy (about ten times) and efficiency than similar existing methods.

The influence of the history of folk music development on Russian and Chinese folk dances

The authors consider and analyze the peculiarities of means of dance and plastic expression: pantomime, gestures, choreographic lexics, choreographic pattern, rhythmics, remarks and exclamations. The idea of a combination of a choreographic image and music of Russian and Chinese dances, declared by the authors, is a multi aspect complicated issue which is of a significant scientific interest. The purpose of the research is to reveal the contents of a folk dance stage adaptation which is conveyed with the help of improved means of expression and is an effective tool for the expression of national peculiarities of Russian and Chinese choreography. The authors study the folk dance stage adaptation as a key means of expression of folk music in Russia and China. The scientific novelty of the research consists in a comprehensive analysis and substantiation of the need for preservation of folk dance traditions in modern China as an important component of a traditional training of a future choreography teacher. The authors prove the presence of definitive features of local invariants of Russian folk dances and dance canon and its differences from the canon of folk dances of other regions.  The research actualizes the problem of music training of future choreography teachers in the pedagogical theory and artistic education practice.  The consideration of this topic is determined by the solution of an urgent problem of preservation of a unique cultural phenomenon of a nation in the aspect of continuous assimilation of culture both among regions and in interstate realms (border areas of countries with unique cultural codes).  

ISOTROPIC MOTIVES

In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$ -coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with $\mathbb{F} _p$ -coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.

The Geometry of Marked Contact Engel Structures

A contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$ ( M , C , γ ) is a 5-dimensional manifold $${\mathcal M}$$ M together with a contact distribution $$\mathcal {C}$$ C and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$ γ ⊂ P ( C ) compatible with the conformal symplectic form on $$\mathcal {C}$$ C . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$ G 2 . In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$ σ : M → γ , which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$ γ x . We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$ ( M , C , γ , σ ) , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$ M by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$ γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$ ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.