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.
For example, GCNN @cite_16 and ACNN @cite_8 construct a local geodesic polar coordinate system on a manifold, formulating the convolution as @math where @math is a local interpolation function with interpolation domain an isotropic disc for GCNN and an anisotropic ellipse for ACNN. A local geodesic polar coordinate system on a manifold can also be transformed to a 2-dimensional planar coordinate system on its tangent plane. Such transformation is the mapping @math which is defined by the inverse exponential map: @math with @math being a point in the local geodesic polar coordinate system at @math with coordinates @math . With this, we can easily interpret ACNN within the framework of ). Indeed, ACNN essentially chooses @math as the directions of the principal curvature at point @math . For GCNN, on the other hand, it avoids choosing a specific vector field on the manifold by taking max-pooling among all possible directions of @math at each point. Such definition of convolution, however, ignores the correspondence of the convolution kernels at different locations.
@cite_16 @cite_8
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Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks (GCNN), a generalization of the convolutional networks (CNN) paradigm to non-Euclidean manifolds. Our construction is based on a local geodesic system of polar coordinates to extract "patches", which are then passed through a cascade of filters and linear and non-linear operators. The coefficients of the filters and linear combination weights are optimization variables that are learned to minimize a task-specific cost function. We use GCNN to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence. Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks (GCNN), a generalization of the convolutional neural networks (CNN) paradigm to non-Euclidean manifolds. Our construction is based on a local geodesic system of polar coordinates to extract "patches", which are then passed through a cascade of filters and linear and non-linear operators. The coefficients of the filters and linear combination weights are optimization variables that are learned to minimize a task-specific cost function. We use ShapeNet to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence. Abstract In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N , the proposed algorithm's complexity is O ( N 2 log N ) , just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT's, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT's and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid. Learned local descriptors based on Convolutional Neural Networks (CNNs) have achieved significant improvements on patch-based benchmarks, whereas not having demonstrated strong generalization ability on recent benchmarks of image-based 3D reconstruction. In this paper, we mitigate this limitation by proposing a novel local descriptor learning approach that integrates geometry constraints from multi-view reconstructions, which benefits the learning process in terms of data generation, data sampling and loss computation. We refer to the proposed descriptor as GeoDesc, and demonstrate its superior performance on various large-scale benchmarks, and in particular show its great success on challenging reconstruction tasks. Moreover, we provide guidelines towards practical integration of learned descriptors in Structure-from-Motion (SfM) pipelines, showing the good trade-off that GeoDesc delivers to 3D reconstruction tasks between accuracy and efficiency.