Implicit representation of shapes as level sets of multilayer perceptrons has recently flourished in different shape analysis, compression, and reconstruction tasks. In this paper, we introduce an implicit neural representation-based framework for solving the inverse obstacle scattering problem in a mesh-free fashion. We express the obstacle shape as the zero-level set of a signed distance function which is implicitly determined by network parameters. To solve the direct scattering problem, we implement the implicit boundary integral method. It uses projections of the grid points in the tubular neighborhood onto the boundary to compute the PDE solution directly in the level-set framework. The proposed implicit representation conveniently handles the shape perturbation in the optimization process. To update the shape, we use PyTorch's automatic differentiation to backpropagate the loss function w.r.t. the network parameters, allowing us to avoid complex and error-prone manual derivation of the shape derivative. Additionally, we propose a deep generative model of implicit neural shape representations that can fit into the framework. The deep generative model effectively regularizes the inverse obstacle scattering problem, making it more tractable and robust, while yielding high-quality reconstruction results even in noise-corrupted setups.

机器学习的最近进步已经创造了利用一类基于坐标的神经网络来解决视觉计算问题的兴趣，该基于坐标的神经网络在空间和时间跨空间和时间的场景或对象的物理属性。我们称之为神经领域的这些方法已经看到在3D形状和图像的合成中成功应用，人体的动画，3D重建和姿势估计。然而，由于在短时间内的快速进展，许多论文存在，但尚未出现全面的审查和制定问题。在本报告中，我们通过提供上下文，数学接地和对神经领域的文学进行广泛综述来解决这一限制。本报告涉及两种维度的研究。在第一部分中，我们通过识别神经字段方法的公共组件，包括不同的表示，架构，前向映射和泛化方法来专注于神经字段的技术。在第二部分中，我们专注于神经领域的应用在视觉计算中的不同问题，超越（例如，机器人，音频）。我们的评论显示了历史上和当前化身的视觉计算中已覆盖的主题的广度，展示了神经字段方法所带来的提高的质量，灵活性和能力。最后，我们展示了一个伴随着贡献本综述的生活版本，可以由社区不断更新。

Inverse medium scattering solvers generally reconstruct a single solution without an associated measure of uncertainty. This is true both for the classical iterative solvers and for the emerging deep learning methods. But ill-posedness and noise can make this single estimate inaccurate or misleading. While deep networks such as conditional normalizing flows can be used to sample posteriors in inverse problems, they often yield low-quality samples and uncertainty estimates. In this paper, we propose U-Flow, a Bayesian U-Net based on conditional normalizing flows, which generates high-quality posterior samples and estimates physically-meaningful uncertainty. We show that the proposed model significantly outperforms the recent normalizing flows in terms of posterior sample quality while having comparable performance with the U-Net in point estimation.

Figure 1: DeepSDF represents signed distance functions (SDFs) of shapes via latent code-conditioned feed-forward decoder networks. Above images are raycast renderings of DeepSDF interpolating between two shapes in the learned shape latent space. Best viewed digitally.

Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or SIRENs, are ideally suited for representing complex natural signals and their derivatives. We analyze SIREN activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how SIRENs can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine SIRENs with hypernetworks to learn priors over the space of SIREN functions. Please see the project website for a video overview of the proposed method and all applications.

在本文中，我们考虑使用Palentir在两个和三个维度中对分段常数对象的恢复和重建，这是相对于当前最新ART的显着增强的参数级别集（PALS）模型。本文的主要贡献是一种新的PALS公式，它仅需要一个单个级别的函数来恢复具有具有多个未知对比度的分段常数对象的场景。我们的模型比当前的多对抗性，多对象问题提供了明显的优势，所有这些问题都需要多个级别集并明确估计对比度大小。给定对比度上的上限和下限，我们的方法能够以任何对比度分布恢复对象，并消除需要知道给定场景中的对比度或其值的需求。我们提供了一个迭代过程，以找到这些空间变化的对比度限制。相对于使用径向基函数（RBF）的大多数PAL方法，我们的模型利用了非异型基函数，从而扩展了给定复杂性的PAL模型可以近似的形状类别。最后，Palentir改善了作为参数识别过程一部分所需的Jacobian矩阵的条件，因此通过控制PALS扩展系数的幅度来加速优化方法，固定基本函数的中心，以及参数映射到图像映射的唯一性，由新参数化提供。我们使用X射线计算机断层扫描，弥漫性光学断层扫描（DOT），Denoising，DeonConvolution问题的2D和3D变体证明了新方法的性能。应用于实验性稀疏CT数据和具有不同类型噪声的模拟数据，以进一步验证所提出的方法。

We consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the $L^2$ distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment. We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research.

我们引入了一个神经隐式框架，该框架利用神经网络的可区分特性和点采样表面的离散几何形状，以将它们作为神经隐含函数的级别集近似。为了训练神经隐式函数，我们提出了近似签名距离函数的损失功能，并允许具有高阶导数的术语，例如曲率的主要方向之间的对齐方式，以了解更多几何细节。在训练过程中，我们考虑了基于点采样表面的曲率的不均匀采样策略，以优先考虑点更多的几何细节。与以前的方法相比，这种抽样意味着在保持几何准确性的同时更快地学习。我们还介绍了神经表面（例如正常矢量和曲率）的分析差异几何公式。

Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets.In this paper we offer a new paradigm for computing high fidelity implicit neural representations directly from raw data (i.e., point clouds, with or without normal information). We observe that a rather simple loss function, encouraging the neural network to vanish on the input point cloud and to have a unit norm gradient, possesses an implicit geometric regularization property that favors smooth and natural zero level set surfaces, avoiding bad zero-loss solutions.We provide a theoretical analysis of this property for the linear case, and show that, in practice, our method leads to state of the art implicit neural representations with higher level-of-details and fidelity compared to previous methods.

将3D坐标映射到签名距离函数（SDF）或占用值的神经网络具有启用对象形状的高保真隐式表示。本文开发了一种新的形状模型，允许通过优化连续符号定向距离功能（SDDF）来合成新颖距离视图。与Deep SDF模型类似，我们的SDDF配方可以代表整个类别的形状并从部分输入数据中跨越形状填写或插入。与SDF不同，该SDF在任何方向上测量到最近表面的距离，SDDF测量给定方向的距离。这允许训练没有3D形状监控的SDDF模型，仅使用距离测量，从深度相机或激光雷达传感器易获得。我们的模型还通过直接在任意位置和观察方向上直接预测距离，去除像表面提取或渲染的后处理步骤。与深色视角综合技术不同，例如培训高容量黑盒型号的神经辐射字段，我们的模型通过构造SDDF值沿着观察方向线性降低的性质。这种结构约束不仅导致维度降低，而且还提供了关于SDDF预测的准确性的分析信心，无论到物体表面的距离如何。

物理信息的神经网络（PINN）是神经网络（NNS），它们作为神经网络本身的组成部分编码模型方程，例如部分微分方程（PDE）。如今，PINN是用于求解PDE，分数方程，积分分化方程和随机PDE的。这种新颖的方法已成为一个多任务学习框架，在该框架中，NN必须在减少PDE残差的同时拟合观察到的数据。本文对PINNS的文献进行了全面的综述：虽然该研究的主要目标是表征这些网络及其相关的优势和缺点。该综述还试图将出版物纳入更广泛的基于搭配的物理知识的神经网络，这些神经网络构成了香草·皮恩（Vanilla Pinn）以及许多其他变体，例如物理受限的神经网络（PCNN），各种HP-VPINN，变量HP-VPINN，VPINN，VPINN，变体。和保守的Pinn（CPINN）。该研究表明，大多数研究都集中在通过不同的激活功能，梯度优化技术，神经网络结构和损耗功能结构来定制PINN。尽管使用PINN的应用范围广泛，但通过证明其在某些情况下比有限元方法（FEM）等经典数值技术更可行的能力，但仍有可能的进步，最著名的是尚未解决的理论问题。

我们提出了一种方法，可以在神经SDF渲染器中相对于几何场景参数自动计算正确的梯度。最近基于物理的可区分渲染技术用于网格采样来处理不连续性，尤其是在对象轮廓上，但是SDF没有简单的参数形式，可用于采样。取而代之的是，我们的方法建立在区域采样技术的基础上，并为SDFS开发了连续的翘曲功能，以解决这些不连续性。我们的方法利用了在SDF中编码的表面的距离，并在球形示踪剂点上使用正交来计算此翘曲功能。我们进一步表明，这可以通过对要点进行次采样来使神经SDF的方法进行。我们可区分的渲染器可用于优化从多视图图像中的神经形状，并对最近基于SDF的反向渲染方法产生可比较的3D重建，而无需2D分割掩码来指导几何形状优化，而无需对几何形状进行体积近似。

我们对通过歧管（例如球形，Tori和其他隐式表面）描述的复杂几何形状的学习生成模型感兴趣。现有（欧几里德）生成模型的当前延伸仅限于特定几何形状，并且通常遭受高计算成本。我们介绍了Moser Flow（MF），是连续标准化流量（CNF）系列内的一类新的生成型号。 MF还通过解决方案产生CNF，然而，与其他CNF方法不同，其模型（学习）密度被参数化，因为源（先前）密度减去神经网络（NN）的发散。分歧是局部线性差分操作员，易于近似和计算歧管。因此，与其他CNFS不同，MF不需要在训练期间通过颂歌求解器调用或反向。此外，将模型密度明确表示为NN的发散而不是作为颂歌的解决方案有助于学习高保真密度。从理论上讲，我们证明了MF在合适的假设下构成了通用密度近似器。经验上，我们首次证明了流动模型的使用从一般曲面采样，并在挑战地球和气候的挑战性几何形状和现实世界基准中实现了密度估计，样本质量和培训复杂性的显着改善科学。

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

远期操作员的计算成本和选择适当的先前分布的计算成本挑战了贝叶斯对高维逆问题的推断。摊销的变异推理解决了这些挑战，在这些挑战中，训练神经网络以近似于现有模型和数据对的后验分布。如果以前看不见的数据和正态分布的潜在样品作为输入，则预处理的深神经网络（在我们的情况下是有条件的正常化流量）几乎没有成本的后验样品。然而，这种方法的准确性取决于高保真训练数据的可用性，由于地球的异质结构，由于地球物理逆问题很少存在。此外，准确的摊销变异推断需要从训练数据分布中汲取观察到的数据。因此，我们建议通过基于物理学的校正对有条件的归一化流量分布来提高摊销变异推断的弹性。为了实现这一目标，我们不是标准的高斯潜在分布，我们通过具有未知平均值和对角线协方差的高斯分布来对潜在分布进行参数化。然后，通过最小化校正后分布和真实后验分布之间的kullback-leibler差异来估算这些未知数量。尽管通用和适用于其他反问题，但通过地震成像示例，我们表明我们的校正步骤可提高摊销变异推理的鲁棒性，以相对于源实验数量的变化，噪声方差以及先前分布的变化。这种方法提供了伪像有限的地震图像，并评估其不确定性，其成本大致与五个反度迁移相同。

Neural 3D implicit representations learn priors that are useful for diverse applications, such as single- or multiple-view 3D reconstruction. A major downside of existing approaches while rendering an image is that they require evaluating the network multiple times per camera ray so that the high computational time forms a bottleneck for downstream applications. We address this problem by introducing a novel neural scene representation that we call the directional distance function (DDF). To this end, we learn a signed distance function (SDF) along with our DDF model to represent a class of shapes. Specifically, our DDF is defined on the unit sphere and predicts the distance to the surface along any given direction. Therefore, our DDF allows rendering images with just a single network evaluation per camera ray. Based on our DDF, we present a novel fast algorithm (FIRe) to reconstruct 3D shapes given a posed depth map. We evaluate our proposed method on 3D reconstruction from single-view depth images, where we empirically show that our algorithm reconstructs 3D shapes more accurately and it is more than 15 times faster (per iteration) than competing methods.

高维时空动力学通常可以在低维子空间中编码。用于建模，表征，设计和控制此类大规模系统的工程应用通常依赖于降低尺寸，以实时计算解决方案。降低维度的常见范例包括线性方法，例如奇异值分解（SVD）和非线性方法，例如卷积自动编码器（CAE）的变体。但是，这些编码技术缺乏有效地表示与时空数据相关的复杂性的能力，后者通常需要可变的几何形状，非均匀的网格分辨率，自适应网格化和/或参数依赖性。为了解决这些实用的工程挑战，我们提出了一个称为神经隐式流（NIF）的一般框架，该框架可以实现大型，参数，时空数据的网格不稳定，低级别表示。 NIF由两个修改的多层感知器（MLP）组成：（i）shapenet，它分离并代表空间复杂性，以及（ii）参数，该参数解释了任何其他输入复杂性，包括参数依赖关系，时间和传感器测量值。我们演示了NIF用于参数替代建模的实用性，从而实现了复杂时空动力学的可解释表示和压缩，有效的多空间质量任务以及改善了稀疏重建的通用性能。

综合照片 - 现实图像和视频是计算机图形的核心，并且是几十年的研究焦点。传统上，使用渲染算法（如光栅化或射线跟踪）生成场景的合成图像，其将几何形状和材料属性的表示为输入。统称，这些输入定义了实际场景和呈现的内容，并且被称为场景表示（其中场景由一个或多个对象组成）。示例场景表示是具有附带纹理的三角形网格（例如，由艺术家创建），点云（例如，来自深度传感器），体积网格（例如，来自CT扫描）或隐式曲面函数（例如，截短的符号距离）字段）。使用可分辨率渲染损耗的观察结果的这种场景表示的重建被称为逆图形或反向渲染。神经渲染密切相关，并将思想与经典计算机图形和机器学习中的思想相结合，以创建用于合成来自真实观察图像的图像的算法。神经渲染是朝向合成照片现实图像和视频内容的目标的跨越。近年来，我们通过数百个出版物显示了这一领域的巨大进展，这些出版物显示了将被动组件注入渲染管道的不同方式。这种最先进的神经渲染进步的报告侧重于将经典渲染原则与学习的3D场景表示结合的方法，通常现在被称为神经场景表示。这些方法的一个关键优势在于它们是通过设计的3D-一致，使诸如新颖的视点合成捕获场景的应用。除了处理静态场景的方法外，我们还涵盖了用于建模非刚性变形对象的神经场景表示...

Implicit fields have been very effective to represent and learn 3D shapes accurately. Signed distance fields and occupancy fields are the preferred representations, both with well-studied properties, despite their restriction to closed surfaces. Several other variations and training principles have been proposed with the goal to represent all classes of shapes. In this paper, we develop a novel and yet fundamental representation by considering the unit vector field defined on 3D space: at each point in $\mathbb{R}^3$ the vector points to the closest point on the surface. We theoretically demonstrate that this vector field can be easily transformed to surface density by applying the vector field divergence. Unlike other standard representations, it directly encodes an important physical property of the surface, which is the surface normal. We further show the advantages of our vector field representation, specifically in learning general (open, closed, or multi-layered) surfaces as well as piecewise planar surfaces. We compare our method on several datasets including ShapeNet where the proposed new neural implicit field shows superior accuracy in representing any type of shape, outperforming other standard methods. The code will be released at https://github.com/edomel/ImplicitVF