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.
The newly proposed PTC @cite_15 defines convolution directly on the manifold, while uses tangent planes to transport kernels by a properly chosen parallel transport. PTC can be equivalently cast into the form of ) using the inverse exponential map, and implementation of the proposed parallel transported is realized through choosing specific vector fields @math guided by a Eikonal equation for transforming vectors along geodesic curves on manifolds.
@cite_15
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Deep generative models learn a mapping from a low-dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. Abstract Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation. The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with Log-Euclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low efficiency especially when the dimensionality of SPD matrix is high. To overcome this limitation, we propose a novel metric learning approach to work directly on logarithms of SPD matrices. Specifically, our method aims to learn a tangent map that can directly transform the matrix logarithms from the original tangent space to a new tangent space of more discriminability. Under the tangent map framework, the novel metric learning can then be formulated as an optimization problem of seeking a Mahalanobis-like matrix, which can take the advantage of traditional metric learning techniques. Extensive evaluations on several image set classification tasks demonstrate the effectiveness of our proposed metric learning method. A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair @math of points of M there exists a trajectory of S from m to @math . Here a trajectory of S is a curve which is an integral curve of some @math or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class @math , @math , or @math , there exists a completely controllable set S consisting of two vector fields of class @math .