arXiv subjects

Xuecheng Tai

Publications and source records attributed to Xuecheng Tai.

5 recordsLinked to original sources

arXiv2026

Computing Smooth Geodesics under Two-Sided Curvature Bounds with Applications to Robotics and Image Analysis

Curvature of planar curves serves as a key regularization term for computing second-order minimal paths, due to its tight relevance to desirable geometric properties such as smoothness, rigidity, and elasticity. In this paper, we tackle a more challenging problem in computational physics and geometry problem: tracking minimal paths whose curvature is constrained by arbitrary upper and lower bounds. For that purpose, we propose a new curvature-bounded geodesic model, developed under the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) framework. It provides strong geometric control over minimal paths by enforcing curvature range constraints, whose paths are smooth and of bounded curvature limitation. We also present a discretization scheme for the Hamiltonian and the HJB PDE incorporating curvature bounds, allowing efficient solver for estimating numerical solutions to the model. Finally, we illustrate the capability of the proposed curvature-bounded geodesic model in applications of robot path planning and curvilinear structures tracking from images. Numerical experiments demonstrate that the proposed curvature-bounded geodesic model serves as a powerful and robust tool for finding satisfactory paths.

cs.RO↗

arXiv2023

Preconditioned Algorithm for Difference of Convex Functions with applications to Graph Ginzburg-Landau Model

In this work, we propose and study a preconditioned framework with a graphic Ginzburg-Landau functional for image segmentation and data clustering by parallel computing. Solving nonlocal models is usually challenging due to the huge computation burden. For the nonconvex and nonlocal variational functional, we propose several damped Jacobi and generalized Richardson preconditioners for the large-scale linear systems within a difference of convex functions algorithms framework. They are efficient for parallel computing with GPU and can leverage the computational cost. Our framework also provides flexible step sizes with a global convergence guarantee. Numerical experiments show the proposed algorithms are very competitive compared to the singular value decomposition based spectral method.

math.NA↗

arXiv2022

Geodesic Models with Convexity Shape Prior

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.

cs.CV↗

arXiv2020

Efficient ADMM and Splitting Methods for Continuous Min-cut and Max-flow Problems

The Potts model has many applications. It is equivalent to some min-cut and max-flow models. Primal-dual algorithms have been used to solve these problems. Due to the special structure of the models, convergence proof is still a difficult problem. In this work, we developed two novel, preconditioned, and over-relaxed alternating direction methods of multipliers (ADMM) with convergence guarantee for these models. Using the proposed preconditioners or block preconditioners, we get accelerations with the over-relaxation variants of preconditioned ADMM. The preconditioned and over-relaxed Douglas-Rachford splitting methods are also considered for the Potts model. Our framework can handle both the two-labeling or multi-labeling problems with appropriate block preconditioners based on Eckstein-Bertsekas and Fortin-Glowinski splitting techniques.

math.OC↗

arXiv2019

Data-driven Method for 3D Axis-symmetric Object Reconstruction from Single Cone-beam Projection Data

In this paper we consider 3D axis-symmetric (AS) object reconstruction from single cone-beam x-ray projection data. Traditional x-ray CT fails to capture fleeting state of material due to the long time for data acquisition at all angles. Therefore, AS object is devised to investigate the instant deformation of material under pulse change of environment because single projection data is enough to reconstruct its inner structure. Previous reconstruction methods are layer by layer, and ignore the longitudinal tilt of x-ray paths. We propose a regularization method using adaptive tight frame to reconstruct the 3D AS object structure simultaneously. Alternating direction method is adopted to solve the proposed model. More importantly, a numerical algorithm is developed to compute imaging matrix. Experiments on simulation data verify the effectiveness of our method

math.NA↗