arXiv subjects

Zuoqiang Shi

Publications and source records attributed to Zuoqiang Shi.

At least 37 records · Page 2Linked to original sources

arXiv2025

Kernel Variational Inference Flow for Nonlinear Filtering Problem

We present a novel particle flow for sampling called kernel variational inference flow (KVIF). KVIF do not require the explicit formula of the target distribution which is usually unknown in filtering problem. Therefore, it can be applied to construct filters with higher accuracy in the update stage. Such an improvement has theoretical assurance. Some numerical experiments for comparison with other classical filters are also demonstrated.

math.OC↗

arXiv2025

Fast and Memory-efficient Non-line-of-sight Imaging with Quasi-Fresnel Transform

Non-line-of-sight (NLOS) imaging seeks to reconstruct hidden objects by analyzing reflections from intermediary surfaces. Existing methods typically model both the measurement data and the hidden scene in three dimensions, overlooking the inherently two-dimensional nature of most hidden objects. This oversight leads to high computational costs and substantial memory consumption, limiting practical applications and making real-time, high-resolution NLOS imaging on lightweight devices challenging. In this paper, we introduce a novel approach that represents the hidden scene using two-dimensional functions and employs a Quasi-Fresnel transform to establish a direct inversion formula between the measurement data and the hidden scene. This transformation leverages the two-dimensional characteristics of the problem to significantly reduce computational complexity and memory requirements. Our algorithm efficiently performs fast transformations between these two-dimensional aggregated data, enabling rapid reconstruction of hidden objects with minimal memory usage. Compared to existing methods, our approach reduces runtime and memory demands by several orders of magnitude while maintaining imaging quality. The substantial reduction in memory usage not only enhances computational efficiency but also enables NLOS imaging on lightweight devices such as mobile and embedded systems. We anticipate that this method will facilitate real-time, high-resolution NLOS imaging and broaden its applicability across a wider range of platforms.

cs.CV↗

arXiv2025

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 measuring 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 $\partial Ω$, and it has been previously used for point normal orientation and surface reconstruction. In this work, we introduce a novel approach to evaluate and optimize the quality of point clouds based on the winding clearness. We observe that point clouds with less noise generally exhibit better winding clearness. Accordingly, we propose an objective function that quantifies the error in winding clearness, solely utilizing the coordinates of the point clouds. Moreover, we demonstrate that the winding clearness error is differentiable and can serve as a loss function in point cloud processing. We present this observation from two aspects: 1) We update the coordinates of the points by back-propagating the loss function for individual point clouds, resulting in an overall improvement without involving a neural network. 2) We incorporate winding clearness as a geometric constraint in the diffusion-based 3D generative model and update the network parameters to generate point clouds with less noise. Experimental results demonstrate the effectiveness of optimizing the winding clearness in enhancing the point cloud quality. Notably, 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.

cs.GR↗

arXiv2025

An FDM-sFEM scheme on time-space manifolds and its superconvergence analysis

We study superconvergent discretization of the Laplace-Beltrami operator on time-space product manifolds with Neumann temporal boundary values, which arise in the context of dynamic optimal transport on general surfaces. We propose a coupled scheme that combines finite difference methods in time with surface finite element methods in space. By establishing a new summation by parts formula and proving the supercloseness of the semi-discrete solution, we derive superconvergence results for the recovered gradient via post-processing techniques. In addition, our geometric error analysis is implemented within a novel framework based on the approximation of the Riemannian metric. Several numerical examples are provided to validate and illustrate the theoretical results.

math.NA↗

arXiv2025

OpenBreastUS: Benchmarking Neural Operators for Wave Imaging Using Breast Ultrasound Computed Tomography

Accurate and efficient simulation of wave equations is crucial in computational wave imaging applications, such as ultrasound computed tomography (USCT), which reconstructs tissue material properties from observed scattered waves. Traditional numerical solvers for wave equations are computationally intensive and often unstable, limiting their practical applications for quasi-real-time image reconstruction. Neural operators offer an innovative approach by accelerating PDE solving using neural networks; however, their effectiveness in realistic imaging is limited because existing datasets oversimplify real-world complexity. In this paper, we present OpenBreastUS, a large-scale wave equation dataset designed to bridge the gap between theoretical equations and practical imaging applications. OpenBreastUS includes 8,000 anatomically realistic human breast phantoms and over 16 million frequency-domain wave simulations using real USCT configurations. It enables a comprehensive benchmarking of popular neural operators for both forward simulation and inverse imaging tasks, allowing analysis of their performance, scalability, and generalization capabilities. By offering a realistic and extensive dataset, OpenBreastUS not only serves as a platform for developing innovative neural PDE solvers but also facilitates their deployment in real-world medical imaging problems. For the first time, we demonstrate efficient in vivo imaging of the human breast using neural operator solvers.

cs.CV↗

arXiv2025

When surface evolution meets Fokker-Planck equation: a novel tangential velocity model for uniform parametrization

A common issue in simulating geometric evolution of surfaces is unexpected clustering of points that may cause numerical instability. We propose a novel artificial tangential velocity method for this matter. The artificial tangential velocity is generated from a surface density field governed by a Fokker-Planck equation to guide the point distribution. A target distribution matching algorithm is developed leveraging the surface Kullback-Leibler divergence of density functions. The numerical method is formulated within a fully meshless framework using the moving least squares approximation, thereby eliminating the need for mesh generation and allowing flexible treatment of unstructured point cloud data. Extensive numerical experiments are conducted to demonstrate the robustness, accuracy, and effectiveness of the proposed approach across a variety of surface evolution problems, including the mean curvature flow.

math.NA↗

arXiv2025

Interface Laplace Learning: Learnable Interface Term Helps Semi-Supervised Learning

We introduce a novel framework, called Interface Laplace learning, for graph-based semi-supervised learning. Motivated by the observation that an interface should exist between different classes where the function value is non-smooth, we introduce a Laplace learning model that incorporates an interface term. This model challenges the long-standing assumption that functions are smooth at all unlabeled points. In the proposed approach, we add an interface term to the Laplace learning model at the interface positions. We provide a practical algorithm to approximate the interface positions using k-hop neighborhood indices, and to learn the interface term from labeled data without artificial design. Our method is efficient and effective, and we present extensive experiments demonstrating that Interface Laplace learning achieves better performance than other recent semi-supervised learning approaches at extremely low label rates on the MNIST, FashionMNIST, and CIFAR-10 datasets.

cs.LG↗

arXiv2025

A seamless local-nonlocal coupling diffusion model with $H^1$ vanishing nonlocality convergence

Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in $H^1$ norm can be derived.

math.AP↗

arXiv2025

Wavelet-based Global Orientation and Surface Reconstruction for Point Clouds

Unoriented surface reconstruction is an important task in computer graphics and has extensive applications. Based on the compact support of wavelet and orthogonality properties, classic wavelet surface reconstruction achieves good and fast reconstruction. However, this method can only handle oriented points. Despite some improved attempts for unoriented points, such as iWSR, these methods perform poorly on sparse point clouds. To address these shortcomings, we propose a wavelet-based method to represent the mollified indicator function and complete both the orientation and surface reconstruction tasks. We use the modifying kernel function to smoothen out discontinuities on the surface, aligning with the continuity of the wavelet basis function. During the calculation of coefficient, we fully utilize the properties of the convolutional kernel function to shift the modifying computation onto wavelet basis to accelerate. In addition, we propose a novel method for constructing the divergence-free function field and using them to construct the additional homogeneous constraints to improve the effectiveness and stability. Extensive experiments demonstrate that our method achieves state-of-the-art performance in both orientation and reconstruction for sparse models. We align the matrix construction with the compact support property of wavelet basis functions to further accelerate our method, resulting in efficient performance on CPU. Our source codes will be released on GitHub.

cs.CG↗

arXiv2025

Asymptotically Compatible Error Bound of Finite Element Method for Nonlocal Diffusion Model with An Efficient Implementation

This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes, the error is bounded by $O(h^k + δ)$, where $h$ is the mesh size, $δ$ is the nonlocal horizon, and $k$ is the order of the FEM basis. Without shape regularity, the bound becomes $O(h^{k+1}/δ+ δ)$. In addition, we present an efficient implementation of the finite element method of nonlocal model. The direct implementation of the finite element method of nonlocal model requires computation of $2n$-dimensional integrals which are very expensive. For the nonlocal model with Gaussian kernel function, we can decouple the $2n$-dimensional integral to 2-dimensional integrals which reduce the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.

math.NA↗

arXiv2025

A Nonlocal Biharmonic Model with $Γ$-Convergence to Local Model and an Efficient Numerical Method

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with the clamped boundary condition ($u$ and $\frac{\partial u}{\partial n}$ are given on the boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, we propose an efficient finite element method to solve the nonlocal model and the numerical results verify the analytical properties of the nonlocal model and its solution.

math.AP↗

arXiv2025

A network based approach for unbalanced optimal transport on surfaces

In this paper, we present a neural network approach to address the dynamic unbalanced optimal transport problem on surfaces with point cloud representation. For surfaces with point cloud representation, traditional method is difficult to apply due to the difficulty of mesh generating. Neural network is easy to implement even for complicate geometry. Moreover, instead of solving the original dynamic formulation, we consider the Hamiltonian flow approach, i.e. Karush-Kuhn-Tucker system. Based on this approach, we can exploit mathematical structure of the optimal transport to construct the neural network and the loss function can be simplified. Extensive numerical experiments are conducted for surfaces with different geometry. We also test the method for point cloud with noise, which shows stability of this method. This method is also easy to generalize to diverse range of problems.

math-ph↗

arXiv2025

Solving Unbalanced Optimal Transport on Point Cloud by Tangent Radial Basis Function Method

In this paper, we solve unbalanced optimal transport (UOT) problem on surfaces represented by point clouds. Based on alternating direction method of multipliers algorithm, the original UOT problem can be solved by an iteration consists of three steps. The key ingredient is to solve a Poisson equation on point cloud which is solved by tangent radial basis function (TRBF) method. The proposed TRBF method requires only the point cloud and normal vectors to discretize the Poisson equation which simplify the computation significantly. Numerical experiments conducted on point clouds with varying geometry and topology demonstrate the effectiveness of the proposed method.

math-ph↗

arXiv2025

Neural Tangent Kernel of Neural Networks with Loss Informed by Differential Operators

Spectral bias is a significant phenomenon in neural network training and can be explained by neural tangent kernel (NTK) theory. In this work, we develop the NTK theory for deep neural networks with physics-informed loss, providing insights into the convergence of NTK during initialization and training, and revealing its explicit structure. We find that, in most cases, the differential operators in the loss function do not induce a faster eigenvalue decay rate and stronger spectral bias. Some experimental results are also presented to verify the theory.

cs.LG↗

arXiv2025

Reconstruction of dynamical systems from data without time labels

In this paper, we study the method to reconstruct dynamical systems from data without time labels. Data without time labels appear in many applications, such as molecular dynamics, single-cell RNA sequencing etc. Reconstruction of dynamical system from time sequence data has been studied extensively. However, these methods do not apply if time labels are unknown. Without time labels, sequence data becomes distribution data. Based on this observation, we propose to treat the data as samples from a probability distribution and try to reconstruct the underlying dynamical system by minimizing the distribution loss, sliced Wasserstein distance more specifically. Extensive experiment results demonstrate the effectiveness of the proposed method.

cs.LG↗

arXiv2025

Discontinuous Galerkin methods for the Laplace-Beltrami operator on point cloud

This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds. We build a novel geometric error analysis framework by estimating the error in the approximation of the Riemann metric tensor. The innovative framework serves as a fundamental tool for analyzing discontinuous Galerkin methods applied to the Laplace-Beltrami operator on possibly discontinuous geometry. We provide numerical examples on patchy surfaces reconstructed from point clouds to support our theoretical findings.

math.NA↗

arXiv2024

$Γ$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data

We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly impose Dirichlet boundary condition in nonlocal diffusion model. To achieve this, we formulate the Dirichlet boundary condition as a penalty term and use theory of $\varGamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\varGamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\varGamma$-convergence as long as the penalty term satisfies a few mild conditions. This work provides a theoretical foundation for approximate Dirichlet boundary condition in nonlocal diffusion model.

math.AP↗

arXiv2024

A Neural Network Framework for High-Dimensional Dynamic Unbalanced Optimal Transport

In this paper, we introduce a neural network-based method to address the high-dimensional dynamic unbalanced optimal transport (UOT) problem. Dynamic UOT focuses on the optimal transportation between two densities with unequal total mass, however, it introduces additional complexities compared to the traditional dynamic optimal transport (OT) problem. To efficiently solve the dynamic UOT problem in high-dimensional space, we first relax the original problem by using the generalized Kullback-Leibler (GKL) divergence to constrain the terminal density. Next, we adopt the Lagrangian discretization to address the unbalanced continuity equation and apply the Monte Carlo method to approximate the high-dimensional spatial integrals. Moreover, a carefully designed neural network is introduced for modeling the velocity field and source function. Numerous experiments demonstrate that the proposed framework performs excellently in high-dimensional cases. Additionally, this method can be easily extended to more general applications, such as crowd motion problem.

math.OC↗