On the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity

In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition

Alternately denoising and reconstructing unoriented point sets

We propose a new strategy to bridge point cloud denoising and surface reconstruction by alternately updating the denoised point clouds and the reconstructed surfaces. In Poisson surface reconstruction, the implicit function is generated by a set of smooth basis functions centered at the octnodes. When the octree depth is properly selected, the reconstructed surface is a good smooth approximation of the noisy point set. Our method projects the noisy points onto the surface and alternately reconstructs and projects the point set. We use the iterative Poisson surface reconstruction (iPSR) to support unoriented surface reconstruction. Our method iteratively performs iPSR and acts as an outer loop of iPSR. Considering that the octree depth significantly affects the reconstruction results, we propose an adaptive depth selection strategy to ensure an appropriate depth choice. To manage the oversmoothing phenomenon near the sharp features, we propose a $\lambda$-projection method, which means to project the noisy points onto the surface with an individual control coefficient $\lambda_{i}$ for each point. The coefficients are determined through a Voronoi-based feature detection method. Experimental results show that our method achieves high performance in point cloud denoising and unoriented surface reconstruction within different noise scales, and exhibits well-rounded performance in various types of inputs. The source code is available at~\url{https://github.com/Submanifold/AlterUpdate}.Comment: Accepted by Computers & Graphics from CAD/Graphics 202

An axiomatized PDE model of deep neural networks

Inspired by the relation between deep neural network (DNN) and partial differential equations (PDEs), we study the general form of the PDE models of deep neural networks. To achieve this goal, we formulate DNN as an evolution operator from a simple base model. Based on several reasonable assumptions, we prove that the evolution operator is actually determined by convection-diffusion equation. This convection-diffusion equation model gives mathematical explanation for several effective networks. Moreover, we show that the convection-diffusion model improves the robustness and reduces the Rademacher complexity. Based on the convection-diffusion equation, we design a new training method for ResNets. Experiments validate the performance of the proposed method

Diffusion Mechanism in Residual Neural Network: Theory and Applications

Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data points and is a critical component for achieving high classification accuracy. Many existing deep learning approaches directly impose the fusion loss when training neural networks. In this work, inspired by the convection-diffusion ordinary differential equations (ODEs), we propose a novel diffusion residual network (Diff-ResNet), internally introduces diffusion into the architectures of neural networks. Under the structured data assumption, it is proved that the proposed diffusion block can increase the distance-diameter ratio that improves the separability of inter-class points and reduces the distance among local intra-class points. Moreover, this property can be easily adopted by the residual networks for constructing the separable hyperplanes. Extensive experiments of synthetic binary classification, semi-supervised graph node classification and few-shot image classification in various datasets validate the effectiveness of the proposed method

Point Normal Orientation and Surface Reconstruction by Incorporating Isovalue Constraints to Poisson Equation

Oriented normals are common pre-requisites for many geometric algorithms based on point clouds, such as Poisson surface reconstruction. However, it is not trivial to obtain a consistent orientation. In this work, we bridge orientation and reconstruction in implicit space and propose a novel approach to orient point clouds by incorporating isovalue constraints to the Poisson equation. Feeding a well-oriented point cloud into a reconstruction approach, the indicator function values of the sample points should be close to the isovalue. Based on this observation and the Poisson equation, we propose an optimization formulation that combines isovalue constraints with local consistency requirements for normals. We optimize normals and implicit functions simultaneously and solve for a globally consistent orientation. Owing to the sparsity of the linear system, an average laptop can be used to run our method within reasonable time. Experiments show that our method can achieve high performance in non-uniform and noisy data and manage varying sampling densities, artifacts, multiple connected components, and nested surfaces

Winding Clearness for Differentiable Point Cloud Optimization

We propose to explore the properties of raw point clouds through the \emph{winding clearness}, a concept we first introduce for assessing the clarity of the interior/exterior relationships represented by the winding number field of the point cloud. In geometric modeling, the winding number is a powerful tool for distinguishing the interior and exterior of a given surface $\partial \Omega$, and it has been previously used for point normal orientation and surface reconstruction. In this work, we introduce a novel approach to assess and optimize the quality of point clouds based on the winding clearness. We observe that point clouds with reduced noise tend to exhibit improved winding clearness. Accordingly, we propose an objective function that quantifies the error in winding clearness, solely utilizing the positions of the point clouds. Moreover, we demonstrate that the winding clearness error is differentiable and can serve as a loss function in optimization-based and learning-based point cloud processing. In the optimization-based method, the loss function is directly back-propagated to update the point positions, resulting in an overall improvement of the point cloud. In the learning-based method, we incorporate the winding clearness as a geometric constraint in the diffusion-based 3D generative model. Experimental results demonstrate the effectiveness of optimizing the winding clearness in enhancing the quality of the point clouds. Our method exhibits superior performance in handling noisy point clouds with thin structures, highlighting the benefits of the global perspective enabled by the winding number

Few-shot Non-line-of-sight Imaging with Signal-surface Collaborative Regularization

The non-line-of-sight imaging technique aims to reconstruct targets from multiply reflected light. For most existing methods, dense points on the relay surface are raster scanned to obtain high-quality reconstructions, which requires a long acquisition time. In this work, we propose a signal-surface collaborative regularization (SSCR) framework that provides noise-robust reconstructions with a minimal number of measurements. Using Bayesian inference, we design joint regularizations of the estimated signal, the 3D voxel-based representation of the objects, and the 2D surface-based description of the targets. To our best knowledge, this is the first work that combines regularizations in mixed dimensions for hidden targets. Experiments on synthetic and experimental datasets illustrated the efficiency and robustness of the proposed method under both confocal and non-confocal settings. We report the reconstruction of the hidden targets with complex geometric structures with only $5 \times 5$ confocal measurements from public datasets, indicating an acceleration of the conventional measurement process by a factor of 10000. Besides, the proposed method enjoys low time and memory complexities with sparse measurements. Our approach has great potential in real-time non-line-of-sight imaging applications such as rescue operations and autonomous driving.Comment: main article: 10 pages, 7 figures supplement: 11 pages, 24 figure