Showing papers by "Hongkai Zhao published in 2013"

Solving Partial Differential Equations on Point Clouds

Abstract: In this paper we present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parameterization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis using $K$ nearest neighbors. Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.

A local mesh method for solving PDEs on point clouds

Abstract: In this work, we introduce a numerical method to approximate differential operators and integrals on point clouds sampled from a two dimensional manifold embedded in $\mathbb{R}^n$. Global mesh structure is usually hard to construct in this case. While our method only relies on the local mesh structure at each data point, which is constructed through local triangulation in the tangent space obtained by local principal component analysis (PCA). Once the local mesh is available, we propose numerical schemes to approximate differential operators and define mass matrix and stiffness matrix on point clouds, which are utilized to solve partial differential equations (PDEs) and variational problems on point clouds. As numerical examples, we use the proposed local mesh method and variational formulation to solve the Laplace-Beltrami eigenproblem and solve the Eikonal equation for computing distance map and tracing geodesics on point clouds.

A Hybrid Reconstruction Method for Quantitative PAT

Abstract: The objective of quantitative photoacoustic tomography (qPAT) is to reconstruct the diffusion and absorption properties of a medium from data of absorbed energy distribution inside the medium. Mathematically, qPAT can be formulated as an inverse coefficient problem for the diffusion equation. Past research showed that if the boundary values of the coefficients are known, then the interior values of the coefficients can be uniquely and stably reconstructed with two well-chosen data sets. We propose a hybrid numerical reconstruction procedure for qPAT that uses both interior energy data and boundary current data. We show that these data allow the unique reconstruction of the boundary and interior values of the coefficients. The numerical implementation is based on reformulating the inverse coefficient problem as a nonlinear optimization problem. An explicit reconstruction scheme is utilized to eliminate the unknown coefficients inside the medium so that we need only minimize over the boundary values, which ...

Point Cloud Segmentation and Denoising via Constrained Nonlinear Least Squares Normal Estimates

Abstract: We first introduce a surface normal estimation procedure for point clouds capable of handling geometric singularities in the data, such as edges and corners. Our formulation is based on recasting the popular Principal Component Analysis (PCA) method as a constrained nonlinear least squares (NLSQ) problem. In contrast to traditional PCA, the new formulation assigns appropriate weights to neighboring points automatically during the optimization process in order to minimize the contributions of points located across singularities. We extend this strategy to point cloud denoising by combining normal estimation, point projection, and declustering into one NLSQ formulation. Finally, we propose a point cloud segmentation technique based on surface normal estimates and local point connectivity. In addition to producing consistently oriented surface normals, the process segments the point cloud into disconnected components that can each be segmented further into piecewise smooth components as needed.

Robust and Efficient Implicit Surface Reconstruction for Point Clouds Based on Convexified Image Segmentation

Abstract: We present an implicit surface reconstruction algorithm for point clouds. We view the implicit surface reconstruction as a three dimensional binary image segmentation problem that segments the entire space $$\mathbb R ^3$$ or the computational domain into an interior region and an exterior region while the boundary between these two regions fits the data points properly. The key points with using an image segmentation formulation are: (1) an edge indicator function that gives a sharp indicator of the surface location, and (2) an initial image function that provides a good initial guess of the interior and exterior regions. In this work we propose novel ways to build both functions directly from the point cloud data. We then adopt recent convexified image segmentation models and fast computational algorithms to achieve efficient and robust implicit surface reconstruction for point clouds. We test our methods on various data sets that are noisy, non-uniform, and with holes or with open boundaries. Moreover, comparisons are also made to current state of the art point cloud surface reconstruction techniques.

Exploring a charge-central strategy in the solution of Poisson's equation for biomolecular applications

Abstract: Continuum solvent treatments based on the Poisson–Boltzmann equation have been widely accepted for energetic analysis of biomolecular systems. In these approaches, the molecular solute is treated as a low dielectric region and the solvent is treated as a high dielectric continuum. The existence of a sharp dielectric jump at the solute–solvent interface poses a challenge to model the solvation energetics accurately with such a simple mathematical model. In this study, we explored and evaluated a strategy based on the “induced surface charge” to eliminate the dielectric jump within the finite-difference discretization scheme. In addition to the use of the induced surface charges in solving the equation, the second-order accurate immersed interface method is also incorporated to discretize the equation. The resultant linear system is solved with the GMRES algorithm to explicitly impose the flux conservation condition across the solvent–solute interface. The new strategy was evaluated on both analytical and realistic biomolecular systems. The numerical tests demonstrate the feasibility of utilizing induced surface charge in the finite-difference solution of the Poisson–Boltzmann equation. The analysis data further show that the strategy is consistent with theory and the classical finite-difference method on the tested systems. Limitations of the current implementations and further improvements are also analyzed and discussed to fully bring out its potential of achieving higher numerical accuracy.

Quantitative Fluorescence Photoacoustic Tomography

Abstract: Fluorescence photoacoustic tomography (fPAT) is a multimodality biomedical imaging technique that combines high-resolution ultrasound imaging with high-contrast fluorescence optical tomography. In the first step of fPAT, one utilizes the photoacoustic effect to recover the total absorbed energy map inside the media with ultrasound tomography. In the second step, called quantitative fPAT (QfPAT), one uses interior absorbed energy data to recover either the quantum efficiency or the concentration distribution or both of the fluorophores inside the media. The objective of this work is to derive the mathematical model for QfPAT and to study the corresponding inverse problems. We derive some uniqueness and stability results on these inverse problems and propose a few (often explicit) reconstruction algorithms. Numerical simulations based on synthetic data are presented to verify the theory and algorithms proposed.

A semi-implicit augmented IIM for Navier-Stokes equations with open and traction boundary conditions

Abstract: In this paper, a new Navier-Stokes solver based on a nite dierence approximation is proposed to solve incompressible o ws on irregular domains with open and traction boundary conditions, which can be applied to simulations of uid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For this type of problem, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier-Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM so that a fast Poisson solver can be used. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to uid structure interaction between an imcompressible uid and a compressible gas bubble is also presented.

Model Classes, Approximation, and Metrics for Dynamic Processing of Urban Terrain Data

Abstract: : Theory, algorithms, and software have been developed for the analysis and processing of point cloud sensor data for representation, analysis and visualization of complex urban terrain. These involve various parameterizations of terrain data based on implicit surface representations and adaptive multiscale methods that enable high resolution and enhance understanding of topology and geometric features. The wavelet and multi scale methods enable fast computation and allow for varying local resolution of the data depending on the local density of the point cloud. The implicit representations which are developed facilitate highly accurate approximation of signed distances to the sensed terrain surface. The level sets of the signed distance provide efficiently computed field of view from specified observation points. Collaboration among MURI focus groups has yielded hybrid methods incorporating the best features of both approaches. Simulation and field experiments have been conducted to test the MURI methodologies. These include problems of sensor assimilation for autonomous navigation of urban terrain, surveillance, secure route planning, line of sight, target acquisition and a host of related problems.