近年来，我们的社区中的兴趣重新提高了由表面网格，其体柔性内嵌或表面点云表示的3D对象的形状分析。部分地，通过增加RGBD摄像机的可用性以及计算机愿景，对自主驾驶，医学成像和机器人的应用来刺激这种兴趣。在这些设置中，频谱坐标由于能够以质量不变于等距变换而与定性不变的方式结合局部和全局形状属性，所示的形状表示的承诺。然而，令人惊讶的是，这种坐标迄今为止通常仅被认为是局部表面位置或衍生信息。在本文中，我们建议用内侧（物体宽度）信息配备光谱坐标，以便丰富它们。关键思想是通过邻接矩阵的权重耦合共享内侧球的曲面点。我们使用这个想法和计算它的算法开发一个光谱功能。物体宽度和内侧耦合的掺入具有直接的益处，如我们对象分类，对象分割和表面点对应的实验所示。

我们考虑如何通过鲁棒地计算其边界的缩回的在线过程直接提取最初未知的二维环境的路线图（也称为拓扑表示）。在本文中，我们首先在在线建设拓扑地图和执行控制法，以指导机器人到最近的未开发地区，首先介绍[1]。所提出的方法通过允许机器人在局部构造的地图上定位本身来操作，计算对环境（前沿）的未探究部分的路径，当机器人完全探索环境时计算稳健的终端条件，并实现环路闭合检测。所提出的算法导致机器人导航需求的平滑安全路径。所提出的方法是任何时间算法，其具有优点：它允许从激光扫描数据中获取激光扫描数据的主动创建拓扑映射。我们还提出了一种基于启发式的导航策略，其中机器人针对拓扑映射中的节点，该拓扑地图开放到空的空间。然后，我们通过呈现[1]中的工作，呈现一种利用特定光谱对应方法[2]的强度来扩展[1]的工作，以匹配从我们拓扑制作算法生成的映射环境。在这里，我们专注于实现一种可以使用AFF骨架来匹配映射环境的拓扑的系统。在两个给定地图和他们的AOF骷髅之间的拓扑匹配中，我们首先在两个不同环境的AFOF骨架上的点之间找到相应的通知。然后我们将环境的（2D）点对齐。我们还基于其提取的AOF骨架及其拓扑在两个给定的环境之间计算距离测量，作为对应点之间的匹配错误的总和。

在本文中，我们介绍了复杂的功能映射，它将功能映射框架扩展到表面上切线矢量字段之间的共形图。这些地图的一个关键属性是他们的方向意识。更具体地说，我们证明，与连锁两个歧管的功能空间的常规功能映射不同，我们的复杂功能图在面向的切片束之间建立了一个链路，从而允许切线矢量场的稳健和有效地传输。通过首先赋予和利用复杂的结构利用各个形状的切线束，所得到的操作变得自然导向，从而有利于横跨形状保持对应的取向和角度，而不依赖于描述符或额外的正则化。最后，也许更重要的是，我们演示了这些对象如何在功能映射框架内启动几个实际应用。我们表明功能映射及其复杂的对应物可以共同估算，以促进定向保存，规范的管道，前面遭受取向反转对称误差的误差。

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them.Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

基于简单的扩散层对空间通信非常有效的洞察力，我们对3D表面进行深度学习的新的通用方法。由此产生的网络是自动稳健的，以改变表面的分辨率和样品 - 一种对实际应用至关重要的基本属性。我们的网络可以在各种几何表示上离散化，例如三角网格或点云，甚至可以在一个表示上培训然后应用于另一个表示。我们优化扩散的空间支持，作为连续网络参数，从纯粹的本地到完全全球范围，从而消除手动选择邻域大小的负担。该方法中唯一的其他成分是在每个点处独立地施加的多层的Perceptron，以及用于支持方向滤波器的空间梯度特征。由此产生的网络简单，坚固，高效。这里，我们主要专注于三角网格表面，并且展示了各种任务的最先进的结果，包括表面分类，分割和非刚性对应。

Figure 1. Given input as either a 2D image or a 3D point cloud (a), we automatically generate a corresponding 3D mesh (b) and its atlas parameterization (c). We can use the recovered mesh and atlas to apply texture to the output shape (d) as well as 3D print the results (e).

Point cloud is an important type of geometric data structure. Due to its irregular format, most researchers transform such data to regular 3D voxel grids or collections of images. This, however, renders data unnecessarily voluminous and causes issues. In this paper, we design a novel type of neural network that directly consumes point clouds, which well respects the permutation invariance of points in the input. Our network, named PointNet, provides a unified architecture for applications ranging from object classification, part segmentation, to scene semantic parsing. Though simple, PointNet is highly efficient and effective. Empirically, it shows strong performance on par or even better than state of the art. Theoretically, we provide analysis towards understanding of what the network has learnt and why the network is robust with respect to input perturbation and corruption.

我们提出了一种新的方法，可以在点云对之间进行无监督的形状对应学习。我们首次尝试适应经典的局部线性嵌入算法（LLE）（最初是为非线性维度降低）的形状对应关系的。关键思想是通过首先获得低维点云的高维邻域保护嵌入，然后使用局部线性转换对源和目标嵌入对齐，从而找到形状之间的密集对应。我们证明，使用新的LLE启发的点云重建目标学习嵌入会产生准确的形状对应关系。更具体地说，该方法包括一个端到端的可学习框架，该框架是提取高维邻域保护的嵌入，估算嵌入空间中的局部线性变换，以及通过基于差异测量的构建构建的概率密度函数的对准形状，并重建形状。目标形状。我们的方法强制将形状的嵌入在对应中，以放置在相同的通用/规范嵌入空间中，最终有助于正规化学习过程，并导致形状嵌入之间的简单最近的邻居接近以找到可靠的对应关系。全面的实验表明，新方法对涵盖人类和非人类形状的标准形状信号基准数据集进行了明显的改进。

Point cloud learning has lately attracted increasing attention due to its wide applications in many areas, such as computer vision, autonomous driving, and robotics. As a dominating technique in AI, deep learning has been successfully used to solve various 2D vision problems. However, deep learning on point clouds is still in its infancy due to the unique challenges faced by the processing of point clouds with deep neural networks. Recently, deep learning on point clouds has become even thriving, with numerous methods being proposed to address different problems in this area. To stimulate future research, this paper presents a comprehensive review of recent progress in deep learning methods for point clouds. It covers three major tasks, including 3D shape classification, 3D object detection and tracking, and 3D point cloud segmentation. It also presents comparative results on several publicly available datasets, together with insightful observations and inspiring future research directions.

本文介绍了一组数字方法，用于在不变（弹性）二阶Sobolev指标的设置中对3D表面进行Riemannian形状分析。更具体地说，我们解决了代表为3D网格的参数化或未参数浸入式表面之间的测量学和地球距离的计算。在此基础上，我们为表面集的统计形状分析开发了工具，包括用于估算Karcher均值并在形状群体上执行切线PCA的方法，以及计算沿表面路径的平行传输。我们提出的方法从根本上依赖于通过使用Varifold Fidelity术语来为地球匹配问题提供轻松的变异配方，这使我们能够在计算未参数化表面之间的地理位置时强制执行重新训练的独立性，同时还可以使我们能够与多用途算法相比，使我们能够将表面与vare表面进行比较。采样或网状结构。重要的是，我们演示了如何扩展放松的变分框架以解决部分观察到的数据。在合成和真实的各种示例中，说明了我们的数值管道的不同好处。

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 Euclideanstructured 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-, graphand 3D shape analysis and show that it consistently outperforms previous approaches.

最近有一项激烈的活动在嵌入非常高维和非线性数据结构的嵌入中，其中大部分在数据科学和机器学习文献中。我们分四部分调查这项活动。在第一部分中，我们涵盖了非线性方法，例如主曲线，多维缩放，局部线性方法，ISOMAP，基于图形的方法和扩散映射，基于内核的方法和随机投影。第二部分与拓扑嵌入方法有关，特别是将拓扑特性映射到持久图和映射器算法中。具有巨大增长的另一种类型的数据集是非常高维网络数据。第三部分中考虑的任务是如何将此类数据嵌入中等维度的向量空间中，以使数据适合传统技术，例如群集和分类技术。可以说，这是算法机器学习方法与统计建模（所谓的随机块建模）之间的对比度。在论文中，我们讨论了两种方法的利弊。调查的最后一部分涉及嵌入$ \ mathbb {r}^ 2 $，即可视化中。提出了三种方法：基于第一部分，第二和第三部分中的方法，$ t $ -sne，UMAP和大节。在两个模拟数据集上进行了说明和比较。一个由嘈杂的ranunculoid曲线组成的三胞胎，另一个由随机块模型和两种类型的节点产生的复杂性的网络组成。

许多天然形状的大部分特征特征集中在太空中的几个地区。例如，人类和动物具有独特的头形，而椅子和飞机等无机物体则由具有特定几何特征的良好定位功能部件制成。通常，这些特征是密切相关的 - 四足动物中面部特征的修改应引起身体结构的变化。但是，在形状建模应用中，这些类型的编辑是最难的编辑。他们需要高精度，但也需要全球对整个形状的认识。即使在深度学习时代，获得满足此类要求的可操作表征也是一个开放的问题，构成了重大限制。在这项工作中，我们通过将数据驱动的模型定义为线性操作员（网状拉普拉斯的变体）来解决此问题，该模型的光谱捕获了手头形状的全局和局部几何特性。对这些光谱的修改被转化为相应表面的语义有效变形。通过明确将全局与本地表面特征分离，我们的管道允许执行本地编辑，同时保持全局风格的连贯性。我们凭经验证明了我们的基于学习的模型如何推广以塑造在培训时间看不到的表示，并且我们系统地分析了本地运营商在各种形状类别上的不同选择。

3D点云的卷积经过广泛研究，但在几何深度学习中却远非完美。卷积的传统智慧在3D点之间表现出特征对应关系，这是对差的独特特征学习的内在限制。在本文中，我们提出了自适应图卷积（AGCONV），以供点云分析的广泛应用。 AGCONV根据其动态学习的功能生成自适应核。与使用固定/各向同性核的解决方案相比，AGCONV提高了点云卷积的灵活性，有效，精确地捕获了不同语义部位的点之间的不同关系。与流行的注意力体重方案不同，AGCONV实现了卷积操作内部的适应性，而不是简单地将不同的权重分配给相邻点。广泛的评估清楚地表明，我们的方法优于各种基准数据集中的点云分类和分割的最新方法。同时，AGCONV可以灵活地采用更多的点云分析方法来提高其性能。为了验证其灵活性和有效性，我们探索了基于AGCONV的完成，DeNoing，Upsmpling，注册和圆圈提取的范式，它们与竞争对手相当甚至优越。我们的代码可在https://github.com/hrzhou2/adaptconv-master上找到。

近年来，由于其表达力和灵活性，神经隐式表示在3D重建中获得了普及。然而，神经隐式表示的隐式性质导致缓慢的推理时间并且需要仔细初始化。在本文中，我们重新审视经典且无处不在的点云表示，并使用泊松表面重建（PSR）的可分辨率配方引入可分化的点对网格层，其允许给予定向的GPU加速的指示灯的快速解决方案点云。可微分的PSR层允许我们通过隐式指示器字段有效地和分散地桥接与3D网格的显式3D点表示，从而实现诸如倒角距离的表面重建度量的端到端优化。因此，点和网格之间的这种二元性允许我们以面向点云表示形状，这是显式，轻量级和富有表现力的。与神经内隐式表示相比，我们的形状 - 点（SAP）模型更具可解释，轻量级，并通过一个级别加速推理时间。与其他显式表示相比，如点，补丁和网格，SA​​P产生拓扑无关的水密歧管表面。我们展示了SAP对无知点云和基于学习的重建的表面重建任务的有效性。

Point clouds are characterized by irregularity and unstructuredness, which pose challenges in efficient data exploitation and discriminative feature extraction. In this paper, we present an unsupervised deep neural architecture called Flattening-Net to represent irregular 3D point clouds of arbitrary geometry and topology as a completely regular 2D point geometry image (PGI) structure, in which coordinates of spatial points are captured in colors of image pixels. \mr{Intuitively, Flattening-Net implicitly approximates a locally smooth 3D-to-2D surface flattening process while effectively preserving neighborhood consistency.} \mr{As a generic representation modality, PGI inherently encodes the intrinsic property of the underlying manifold structure and facilitates surface-style point feature aggregation.} To demonstrate its potential, we construct a unified learning framework directly operating on PGIs to achieve \mr{diverse types of high-level and low-level} downstream applications driven by specific task networks, including classification, segmentation, reconstruction, and upsampling. Extensive experiments demonstrate that our methods perform favorably against the current state-of-the-art competitors. We will make the code and data publicly available at https://github.com/keeganhk/Flattening-Net.

Spectral geometric methods have brought revolutionary changes to the field of geometry processing. Of particular interest is the study of the Laplacian spectrum as a compact, isometry and permutation-invariant representation of a shape. Some recent works show how the intrinsic geometry of a full shape can be recovered from its spectrum, but there are approaches that consider the more challenging problem of recovering the geometry from the spectral information of partial shapes. In this paper, we propose a possible way to fill this gap. We introduce a learning-based method to estimate the Laplacian spectrum of the union of partial non-rigid 3D shapes, without actually computing the 3D geometry of the union or any correspondence between those partial shapes. We do so by operating purely in the spectral domain and by defining the union operation between short sequences of eigenvalues. We show that the approximated union spectrum can be used as-is to reconstruct the complete geometry [MRC*19], perform region localization on a template [RTO*19] and retrieve shapes from a database, generalizing ShapeDNA [RWP06] to work with partialities. Working with eigenvalues allows us to deal with unknown correspondence, different sampling, and different discretizations (point clouds and meshes alike), making this operation especially robust and general. Our approach is data-driven and can generalize to isometric and non-isometric deformations of the surface, as long as these stay within the same semantic class (e.g., human bodies or horses), as well as to partiality artifacts not seen at training time.

动态MRI可以捕获具有高对比度的软组织器官中的时间解剖变化，但是获得的序列通常遭受有限的体积覆盖，这使得器官形状轨迹的高分辨率重建在时间研究中的主要挑战。由于腹部器官形状的变异性跨越时间和受试者，本研究的目的是朝向3D致密速度测量来完全覆盖整个表面并提取有意义的特征，其特征在于观察到的器官变形并实现临床作用或决定。我们在深呼吸运动期间提出了一种用于表征膀胱表面动力学的管道。对于紧凑的形状表示，首先使用重建的时间体积来使用LDDMM框架建立专用的动态4D网状序列。然后，我们从诸如网格伸长和失真的机械参数执行器官动力学的统计表征。由于我们将器官引用作为非平面，因此我们还使用平均曲率变化为度量来量化表面演变。然而，曲率的数值计算强烈地取决于表面参数化。为了应对这一依赖性，我们采用了一种用于表面变形分析的新方法。独立于参数化并最小化测地曲线的长度，通过最小化Dirichlet能量，它使表面曲线平滑地朝向球体。 eulerian PDE方法用于从曲线缩短流中导出形状描述符。使用Laplace Beltrami操作员特征函数来计算各个运动模式之间的接口，用于球形映射。用于提取用于局部控制的模拟形状轨迹的表征相关曲线的应用演示了所提出的形状描述符的稳定性。

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing, along with a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas. We then summarize recent advances in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning.