Analyzing the geometric and semantic properties of 3D point clouds through the deep networks is still challenging due to the irregularity and sparsity of samplings of their geometric structures. This paper presents a new method to define and compute convolution directly on 3D point clouds by the proposed annular convolution. This new convolution operator can better capture the local neighborhood geometry of each point by specifying the (regular and dilated) ring-shaped structures and directions in the computation. It can adapt to the geometric variability and scalability at the signal processing level. We apply it to the developed hierarchical neural networks for object classification, part segmentation, and semantic segmentation in large-scale scenes. The extensive experiments and comparisons demonstrate that our approach outperforms the state-of-the-art methods on a variety of standard benchmark datasets (e.g., ModelNet10, ModelNet40, ShapeNet-part, S3DIS, and ScanNet).
Besides point cloud based methods, several approaches have been proposed to develop convolutional networks on 3D meshes for shape analysis. Geodesic CNN @cite_6 is an extension of the Euclidean CNNs to non-Euclidean domains and is based on a local geodesic system of polar coordinates to extract local patches. Anisotropic CNN @cite_37 is another generalization of Euclidean CNNs to non-Euclidean domains, where classical convolutions are replaced by projections over a set of oriented anisotropic diffusion kernels. Mixture Model Networks (MoNet) @cite_17 generalizes deep learning methods to non-Euclidean domains (graphs and manifolds) by combining previous methods, e.g., classical Euclidean CNN, Geodesic CNN, and Anisotropic CNN. MoNet proposes a new type of kernel in parametric construction. Directionally Convolutional Networks (DCN) @cite_44 applies convolution operation on the triangular mesh of 3D shapes to address part segmentation problem by combining local and global features. Lastly, Surface Networks @cite_11 propose upgrades to Graph Neural Networks to leverage extrinsic differential geometry properties of 3D surfaces for increasing their modeling power.
@cite_37 @cite_17 @cite_6 @cite_44 @cite_11
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Convolutional neural networks have achieved extraordinary results in many computer vision and pattern recognition applications; however, their adoption in the computer graphics and geometry processing communities is limited due to the non-Euclidean structure of their data. In this paper, we propose Anisotropic Convolutional Neural Network (ACNN), a generalization of classical CNNs to non-Euclidean domains, where classical convolutions are replaced by projections over a set of oriented anisotropic diffusion kernels. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes, a fundamental problem in geometry processing, arising in a wide variety of applications. We tested ACNNs performance in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks. 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. 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. Deep learning has achieved a remarkable performance breakthrough in several fields, most notably in speech recognition, natural language processing, and computer vision. In particular, convolutional neural network (CNN) architectures currently produce state-of-the-art performance on a variety of image analysis tasks such as object detection and recognition. Most of deep learning research has so far focused on dealing with 1D, 2D, or 3D Euclidean-structured data such as acoustic signals, images, or videos. Recently, there has been an increasing interest in geometric deep learning, attempting to generalize deep learning methods to non-Euclidean structured data such as graphs and manifolds, with a variety of applications from the domains of network analysis, computational social science, or computer graphics. In this paper, we propose a unified framework allowing to generalize CNN architectures to non-Euclidean domains (graphs and manifolds) and learn local, stationary, and compositional task-specific features. We show that various non-Euclidean CNN methods previously proposed in the literature can be considered as particular instances of our framework. We test the proposed method on standard tasks from the realms of image-, graph-and 3D shape analysis and show that it consistently outperforms previous approaches. Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data ( patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks. In this paper, we propose a generalization of convolutional neural networks (CNN) to non-Euclidean domains for the analysis of deformable shapes. Our construction is based on localized frequency analysis (a generalization of the windowed Fourier transform to manifolds) that is used to extract the local behavior of some dense intrinsic descriptor, roughly acting as an analogy to patches in images. The resulting local frequency representations are then passed through a bank of filters whose coefficient are determined by a learning procedure minimizing a task-specific cost. Our approach generalizes several previous methods such as HKS, WKS, spectral CNN, and GPS embeddings. Experimental results show that the proposed approach allows learning class-specific shape descriptors significantly outperforming recent state-of-the-art methods on standard benchmarks. Convolutional neural networks (CNNs) are inherently subject to invariable filters that can only aggregate local inputs with the same topological structures. It causes that CNNs are allowed to manage data with Euclidean or grid-like structures (e.g., images), not ones with non-Euclidean or graph structures (e.g., traffic networks). To broaden the reach of CNNs, we develop structure-aware convolution to eliminate the invariance, yielding a unified mechanism of dealing with both Euclidean and non-Euclidean structured data. Technically, filters in the structure-aware convolution are generalized to univariate functions, which are capable of aggregating local inputs with diverse topological structures. Since infinite parameters are required to determine a univariate function, we parameterize these filters with numbered learnable parameters in the context of the function approximation theory. By replacing the classical convolution in CNNs with the structure-aware convolution, Structure-Aware Convolutional Neural Networks (SACNNs) are readily established. Extensive experiments on eleven datasets strongly evidence that SACNNs outperform current models on various machine learning tasks, including image classification and clustering, text categorization, skeleton-based action recognition, molecular activity detection, and taxi flow prediction. Code will be available. Person re-identification is challenging due to the large variations of pose, illumination, occlusion and camera view. Owing to these variations, the pedestrian data is distributed as highly-curved manifolds in the feature space, despite the current convolutional neural networks (CNN)’s capability of feature extraction. However, the distribution is unknown, so it is difficult to use the geodesic distance when comparing two samples. In practice, the current deep embedding methods use the Euclidean distance for the training and test. On the other hand, the manifold learning methods suggest to use the Euclidean distance in the local range, combining with the graphical relationship between samples, for approximating the geodesic distance. From this point of view, selecting suitable positive (i.e. intra-class) training samples within a local range is critical for training the CNN embedding, especially when the data has large intra-class variations. In this paper, we propose a novel moderate positive sample mining method to train robust CNN for person re-identification, dealing with the problem of large variation. In addition, we improve the learning by a metric weight constraint, so that the learned metric has a better generalization ability. Experiments show that these two strategies are effective in learning robust deep metrics for person re-identification, and accordingly our deep model significantly outperforms the state-of-the-art methods on several benchmarks of person re-identification. Therefore, the study presented in this paper may be useful in inspiring new designs of deep models for person re-identification. Deep neural networks have enjoyed remarkable success for various vision tasks, however it remains challenging to apply CNNs to domains lacking a regular underlying structures such as 3D point clouds. Towards this we propose a novel convolutional architecture, termed SpiderCNN, to efficiently extract geometric features from point clouds. SpiderCNN is comprised of units called SpiderConv, which extend convolutional operations from regular grids to irregular point sets that can be embedded in ( R ^n ), by parametrizing a family of convolutional filters. We design the filter as a product of a simple step function that captures local geodesic information and a Taylor polynomial that ensures the expressiveness. SpiderCNN inherits the multi-scale hierarchical architecture from classical CNNs, which allows it to extract semantic deep features. Experiments on ModelNet40 demonstrate that SpiderCNN achieves state-of-the-art accuracy (92.4 ) on standard benchmarks, and shows competitive performance on segmentation task. Surfaces serve as a natural parametrization to 3D shapes. Learning surfaces using convolutional neural networks (CNNs) is a challenging task. Current paradigms to tackle this challenge are to either adapt the convolutional filters to operate on surfaces, learn spectral descriptors defined by the Laplace-Beltrami operator, or to drop surfaces altogether in lieu of voxelized inputs. Here we adopt an approach of converting the 3D shape into a ‘geometry image’ so that standard CNNs can directly be used to learn 3D shapes. We qualitatively and quantitatively validate that creating geometry images using authalic parametrization on a spherical domain is suitable for robust learning of 3D shape surfaces. This spherically parameterized shape is then projected and cut to convert the original 3D shape into a flat and regular geometry image. We propose a way to implicitly learn the topology and structure of 3D shapes using geometry images encoded with suitable features. We show the efficacy of our approach to learn 3D shape surfaces for classification and retrieval tasks on non-rigid and rigid shape datasets. We propose a system for surface completion and inpainting of 3D shapes using denoising autoencoders with convolutional layers, learnt on local patches. Our method uses height map based local patches parameterized using 3D mesh quadrangulation of the low resolution input shape. This provides us sufficient amount of local 3D patch dataset to learn deep generative Convolutional Neural Networks (CNNs) for the task of repairing moderate sized holes. We design generative networks specifically suited for the 3D encoding following ideas from the recent progress in 2D inpainting, and show our results to be better than the previous methods of surface inpainting that use linear dictionary. We validate our method on both synthetic shapes and real world scans. While deep convolutional neural networks (CNN) have been successfully applied for 2D image analysis, it is still challenging to apply them to 3D anisotropic volumes, especially when the within-slice resolution is much higher than the between-slice resolution and when the amount of 3D volumes is relatively small. On one hand, direct learning of CNN with 3D convolution kernels suffers from the lack of data and likely ends up with poor generalization; insufficient GPU memory limits the model size or representational power. On the other hand, applying 2D CNN with generalizable features to 2D slices ignores between-slice information. Coupling 2D network with LSTM to further handle the between-slice information is not optimal due to the difficulty in LSTM learning. To overcome the above challenges, we propose a 3D Anisotropic Hybrid Network (AH-Net) that transfers convolutional features learned from 2D images to 3D anisotropic volumes. Such a transfer inherits the desired strong generalization capability for within-slice information while naturally exploiting between-slice information for more effective modelling. The focal loss is further utilized for more effective end-to-end learning. We experiment with the proposed 3D AH-Net on two different medical image analysis tasks, namely lesion detection from a Digital Breast Tomosynthesis volume, and liver and liver tumor segmentation from a Computed Tomography volume and obtain the state-of-the-art results. This paper presents Point Convolutional Neural Networks (PCNN): a novel framework for applying convolutional neural networks to point clouds. The framework consists of two operators: extension and restriction, mapping point cloud functions to volumetric functions and vise-versa. A point cloud convolution is defined by pull-back of the Euclidean volumetric convolution via an extension-restriction mechanism. The point cloud convolution is computationally efficient, invariant to the order of points in the point cloud, robust to different samplings and varying densities, and translation invariant, that is the same convolution kernel is used at all points. PCNN generalizes image CNNs and allows readily adapting their architectures to the point cloud setting. Evaluation of PCNN on three central point cloud learning benchmarks convincingly outperform competing point cloud learning methods, and the vast majority of methods working with more informative shape representations such as surfaces and or normals. Graph convolutional neural networks (Graph-CNNs) extend traditional CNNs to handle data that is supported on a graph. Major challenges when working with data on graphs are that the support set (the vertices of the graph) do not typically have a natural ordering, and in general, the topology of the graph is not regular (i.e., vertices do not all have the same number of neighbors). Thus, Graph-CNNs have huge potential to deal with 3D point cloud data which has been obtained from sampling a manifold. In this paper we develop a Graph-CNN for classifying 3D point cloud data, called PointGCN1. The architecture combines localized graph convolutions with two types of graph downsampling operations (also known as pooling). By the effective exploration of the point cloud local structure using the Graph-CNN, the proposed architecture achieves competitive performance on the 3D object classification benchmark ModelNet, and our architecture is more stable than competing schemes. Convolutional neural networks (CNNs) have massively impacted visual recognition in 2D images, and are now ubiquitous in state-of-the-art approaches. CNNs do not easily extend, however, to data that are not represented by regular grids, such as 3D shape meshes or other graph-structured data, to which traditional local convolution operators do not directly apply. To address this problem, we propose a novel graph-convolution operator to establish correspondences between filter weights and graph neighborhoods with arbitrary connectivity. The key novelty of our approach is that these correspondences are dynamically computed from features learned by the network, rather than relying on predefined static coordinates over the graph as in previous work. We obtain excellent experimental results that significantly improve over previous state-of-the-art shape correspondence results. This shows that our approach can learn effective shape representations from raw input coordinates, without relying on shape descriptors. This article presents a novel approach for 3D mesh labeling by using deep Convolutional Neural Networks (CNNs). Many previous methods on 3D mesh labeling achieve impressive performances by using predefined geometric features. However, the generalization abilities of such low-level features, which are heuristically designed to process specific meshes, are often insufficient to handle all types of meshes. To address this problem, we propose to learn a robust mesh representation that can adapt to various 3D meshes by using CNNs. In our approach, CNNs are first trained in a supervised manner by using a large pool of classical geometric features. In the training process, these low-level features are nonlinearly combined and hierarchically compressed to generate a compact and effective representation for each triangle on the mesh. Based on the trained CNNs and the mesh representations, a label vector is initialized for each triangle to indicate its probabilities of belonging to various object parts. Eventually, a graph-based mesh-labeling algorithm is adopted to optimize the labels of triangles by considering the label consistencies. Experimental results on several public benchmarks show that the proposed approach is robust for various 3D meshes, and outperforms state-of-the-art approaches as well as classic learning algorithms in recognizing mesh labels. In this paper, we study the problem of semantic annotation on 3D models that are represented as shape graphs. A functional view is taken to represent localized information on graphs, so that annotations such as part segment or keypoint are nothing but 0-1 indicator vertex functions. Compared with images that are 2D grids, shape graphs are irregular and non-isomorphic data structures. To enable the prediction of vertex functions on them by convolutional neural networks, we resort to spectral CNN method that enables weight sharing by parametrizing kernels in the spectral domain spanned by graph Laplacian eigenbases. Under this setting, our network, named SyncSpecCNN, strives to overcome two key challenges: how to share coefficients and conduct multi-scale analysis in different parts of the graph for a single shape, and how to share information across related but different shapes that may be represented by very different graphs. Towards these goals, we introduce a spectral parametrization of dilated convolutional kernels and a spectral transformer network. Experimentally we tested SyncSpecCNN on various tasks, including 3D shape part segmentation and keypoint prediction. State-of-the-art performance has been achieved on all benchmark datasets. We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors. For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domain-independence. In this paper, we study the problem of semantic annotation on 3D models that are represented as shape graphs. A functional view is taken to represent localized information on graphs, so that annotations such as part segment or keypoint are nothing but 0-1 indicator vertex functions. Compared with images that are 2D grids, shape graphs are irregular and non-isomorphic data structures. To enable the prediction of vertex functions on them by convolutional neural networks, we resort to spectral CNN method that enables weight sharing by parameterizing kernels in the spectral domain spanned by graph laplacian eigenbases. Under this setting, our network, named SyncSpecCNN, strive to overcome two key challenges: how to share coefficients and conduct multi-scale analysis in different parts of the graph for a single shape, and how to share information across related but different shapes that may be represented by very different graphs. Towards these goals, we introduce a spectral parameterization of dilated convolutional kernels and a spectral transformer network. Experimentally we tested our SyncSpecCNN on various tasks, including 3D shape part segmentation and 3D keypoint prediction. State-of-the-art performance has been achieved on all benchmark datasets. We propose a generalization of convolutional neural networks (CNNs) to irregular domains, through the use of an inferred graph structure. In more details, we introduce a three-step methodology to create convolutional layers that are adapted to the signals to process: 1) From a training set of signals, infer a graph representing the topology on which they evolve; 2) Identify translation operators in the vertex domain; 3) Emulate a convolution operator by translating a localized kernel on the graph. Using these layers, a convolutional neural network is built, and is trained on the initial signals to perform a classification task. Contributions are twofold. First, we adapt a definition of translations on graphs to make them more robust to irregularities, and to take into account locality of the kernel. Second, we introduce a procedure to build CNNs from data. We apply our methodology on a scrambled version of the CIFAR-10 and Haxby datasets. Without using any knowledge on the signals, we significantly outperform existing methods. Moreover, our approach extends classical CNNs on images in the sense that such networks are a particular case of our approach when the inferred graph is a grid.