Convolution plays a crucial role in various applications in signal and image processing, analysis and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processing point cloud data. However, one of the major challenges is to define a proper way to "sweep" filters through the point cloud as a natural generalization of the planar convolution and to reflect the point cloud's geometry at the same time. In this paper, we consider generalizing convolution by adapting parallel transport on the point cloud. Inspired by a triangulated surface based method [Stefan C. Schonsheck, Bin Dong, and Rongjie Lai, arXiv:1805.07857.], we propose the Narrow-Band Parallel Transport Convolution (NPTC) using a specifically defined connection on a voxelized narrow-band approximation of point cloud data. With that, we further propose a deep convolutional neural network based on NPTC (called NPTC-net) for point cloud classification and segmentation. Comprehensive experiments show that the proposed NPTC-net achieves similar or better results than current state-of-the-art methods on point clouds classification and segmentation.
are more intuitive and similar to the Euclidean case, and this is one of the reasons why most of the existing works fall into this category. The philosophy behind these methods is that the tangent plane of a @math -dimensional manifold is embedded to a @math -dimensional Euclidean domain where convolution can be easily defined. In this paper, we make the first attempt to interpret some of the existing mesh-based methods in a unified framework. We claim that most of the spatial mesh-based methods can be formulated as Here, @math is a convolution kernel and @math with @math being the size of the kernel. The mapping @math is defined as where @math , @math . For simplicity, we will denote @math . Most of the designs of the existing manifold convolutions focused on the designs of @math . We remark that possible singularities will lead to no convolution operation at those points. These are isolated points on a closed manifold and do not effect experiment results. In addition, singularities from a given vector field can be overcome using several pairs of vector fields and pooling @cite_15 .
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Abstract This paper mainly focuses on dimensional reduction of fused dataset of holistic and geometrical face features vectors by solving singularity problem of linear discriminant analysis and maximizing the Fisher ratio in nonlinear subspace region with the preservation of local discriminative features. The combinational feature vector space is projected into low dimensional subspace using proposed Kernel Locality Preserving Symmetrical Weighted Fisher Discriminant Analysis (KLSWFDA) method. Matching score level fusion technique has been applied on projected subspace and combinational entire Gabor subspace is framed. Euclidean distance metric (L2) and support vector machine (SVM) classifier has been implemented to recognize and classify the expressions. Performance of proposed approach is evaluated and compared with state of art approaches. Experimental results on JAFFE, YALE and FD expression database demonstrate the effectiveness of the proposed approach. A convenient way of analysing Riemannian manifolds is to embed them in Euclidean spaces, with the embedding typically obtained by flattening the manifold via tangent spaces. This general approach is not free of drawbacks. For example, only distances between points to the tangent pole are equal to true geodesic distances. This is restrictive and may lead to inaccurate modelling. Instead of using tangent spaces, we propose embedding into the Reproducing Kernel Hilbert Space by introducing a Riemannian pseudo kernel. We furthermore propose to recast a locality preserving projection technique from Euclidean spaces to Riemannian manifolds, in order to demonstrate the benefits of the embedding. Experiments on several visual classification tasks (gesture recognition, person re-identification and texture classification) show that in comparison to tangent-based processing and state-of-the-art methods (such as tensor canonical correlation analysis), the proposed approach obtains considerable improvements in discrimination accuracy. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. This work addresses the problem of approximating a manifold by a simplicial mesh, and the related problem of building triangulations for the purpose of piecewise-linear approximation of functions. It has long been understood that the vertices of such meshes or triangulations should be "well-distributed," or satisfy certain "sampling conditions." This work clarifies and extends some algorithms for finding such well-distributed vertices, by showing that they can be regarded as finding e-nets or Delone sets in appropriate metric spaces. In some cases where such Delone properties were already understood, such as for meshes to approximate smooth manifolds that bound convex bodies, the upper and lower bound results are extended to more general manifolds; in particular, under some general conditions, the minimum Hausdorff distance for a mesh with n simplices to a d-manifold M is Θ((∫M√|κ(x)| n)2 d) as n ⋺ ∞, where κ(x) is the Gaussian curvature at point x ∈ M. We also relate these constructions to Dudley's approximation scheme for convex bodies, which can be interpreted as involving an e-net in a metric space whose distance function depends on surface normals.