PCT: Point cloud transformer

The irregular domain and lack of ordering make it challenging to design deep neural networks for point cloud processing. This paper presents a novel framework named Point Cloud Transformer (PCT) for point cloud learning. PCT is based on Transformer, which achieves huge success in natural language processing and displays great potential in image processing. It is inherently permutation invariant for processing a sequence of points, making it well-suited for point cloud learning. To better capture local context within the point cloud, we enhance input embedding with the support of farthest point sampling and nearest neighbor search. Extensive experiments demonstrate that the PCT achieves the state-of-the-art performance on shape classification, part segmentation, semantic segmentation, and normal estimation tasks.

Structural studies of reelin N-terminal region provides insights into a unique structural arrangement and functional multimerization

The large, secreted glycoprotein reelin regulates embryonic brain development as well as adult brain functions. Although reelin binds to its receptors via its central part, the N-terminal region directs multimer formation and is critical for efficient signal transduction. In fact, the inhibitory antibody CR-50 interacts with the N-terminal region and prevents higher-order multimerization and signaling. Reelin is a multidomain protein in which the central part is composed of eight characteristic repeats, named reelin repeats, each of which is further divided by insertion of an EGF module into two subrepeats. In contrast, the N-terminal region shows unique “irregular” domain architecture since it comprises three consecutive subrepeats without the intervening EGF module. Here we determined the crystal structure of the murine reelin fragment named RX-R1 including the irregular region and the first reelin repeat at 2.0 Å resolution. The overall structure of RX-R1 has a branched Y-shaped form. Interestingly, two incomplete subrepeats cooperatively form one entire subrepeat structure, though an additional subrepeat is inserted between them. We further reveal that Arg335 of RX-R1 is crucial for binding CR-50. A possible self-association mechanism via the N-terminal region is proposed based on our results.

Bending of Nonconforming Thin Plates Based on the Mixed-Order Manifold Method with Background Cells for Integration

As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed-order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed-order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

Multivariate approximation of functions on irregular domains by weighted least-squares methods

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a bounded real-valued function on a general bounded domain $\varOmega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\varOmega )$, the analysis in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203) shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \ln n$. When an $L^2(\varOmega )$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203, Section 5). If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that when $\varOmega $ is an irregular domain such that the analytic form of an $L^2(\varOmega )$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \ln n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\varOmega $ and $V_n$. Numerical results validating our analysis are presented.