Dr. Ligang LIU

Associate Professor

Affiliations:
CAGD&CG Group
State Key Lab of CAD&CG
Department of Mathematics
College of Science
Zhejiang University

This paper presents a novel parameterization method for a nonclosed triangular mesh. For every flattened 1-ring neighbors, we choose a local coordinate frame, and the local geometry structure is represented as local parametric coordinates. Then the global optimal parametric coordinates are attained by aligning all the local parametric planes while preserving the local structure as much as possible. The boundary conditions are not necessary in our method, thus no high distortion appears around the boundary, and distortion is uniformly distributed over parametric domain. In addition, our method can operate directly on mesh surface which has holes without any preprocessing of surface partition. Furthermore, linear constraints are allowed in the parameterization in a least squares sense.

Zhonggui Chen, Ligang Liu, Zhenyue Zhang, Guojin Wang. Surface Parameterization via Aligning Optimal Local Flattening. Proceedings of ACM Symposium on Solid and Physical Modeling (SPM 2007), Beijing, China, pp.291-296, 2007. [PDF, 0.5M] [Talk, 2.7M][Poster, 0.3M]

This paper presents a novel approach for constructing a piecewise triangular cubic polynomial surface with C1 continuity around a common corner vertex. A C1 continuity condition between two cubic triangular patches is first derived using mixed directional derivatives. An approach for constructing a surface with C1 continuity around a corner is then developed. Our approach is easy and fast with the virtue of cubic reproduction, local shape controllability, C2 continuous at the corner vertex. Some experimental results are presented to show the applicability and flexibility of the approach.

Renjiang Zhang, Ligang Liu, Weiyin Ma, and Guojin Wang. Construction of cubic triangular patches with C1 continuity around a corner. Proceedings of The 9th International Conference on Computer Aided Design and Computer Graphics, 2005, HongKong, IEEE Computer Society, pp. 73-78. [PDF, 0.4M] [Talk slides, 0.5M]

Recently, differential information as local intrinsic feature descriptors has been used for mesh editing. Given certain user input as constraints, a deformed mesh is reconstructed by minimizing the changes in the differential information. Since the differential information is encoded in the global coordinate system, it must somehow be transformed to fit the orientation of details in the deformed surface, otherwise distortion will appear. We observe that visually desired deformed meshes should preserve both local parameterization and geometry details. To find suitable representations for these two types of information, we exploit certain properties of the curvature flow Laplacian operator. Specifically, we consider the coefficients of Laplacian operator as the parametrization information and the magnitudes of the Laplacian coordinates as the geometry information. Both sets of information are non-directional and non-linearly dependent on the vertex positions. Thus, we propose a new editing framework which iteratively updates both the vertex positions and the Laplacian coordinates to reduce distortion in parametrization and geometry. Our method can produce pleasing deformation results with simple user interaction not possible with previous related work. In addition, since the magnitudes of the Laplacian coordinates approximate the integrated mean curvatures, our framework is useful for modifying mesh geometry via updating the curvature field. We demonstrate this use in spherical parameterization and non-shrinking smoothing.

Oscar Kin-Chung Au, Chiew-Lan Tai,  Hongbo Fu, and Ligang Liu. Mesh editing with curvature flow Laplacian. Proceedings of Eurographics Symposium on Geometry Processing, 2005. [PDF, 2.2M] [Poster, 20M]

We present a general approach to morph between 2D shapes with different topologies. All possible topological evolutions are classified into three types by attaching three different topological cells. This formalism is resulted from the Morse theory on the behavior of the 3D surface around a non-degenerate critical point. Also we incorporate degenerate topological evolutions into our framework which produce more attractive morphing effects. The user controls the morph by specifying the types of topological evolutions as well as the feature correspondences between the source and target shapes. Some techniques are also provided to control the vertex path during the morphing process. The amount of user input required to produce a morph is directly proportional to the amount of control the user wishes to impose on the process. The user may allow the system to automatically generate the morph as well. Our approaches are totally geometric based and are easy and fast enough in fully interactive time. Many experimental results show the applicability and flexibility of our approaches.

Ligang Liu, Bo Zhang, Baining Guo, and Heung-Yeung Shum. Polygonal shape blending with topological evolutions. Journal of Computer Science and Technology, 2005, 20(1): 77-89.  [PDF, 0.8M]

We present a novel approach for establishing vertex correspondences between two planar shapes. Correspondences are established between the perceptual feature points extracted from both source and target shapes. A similarity metric between two feature points is defined using the intrinsic properties of their local neighborhoods. The optimal correspondence is found by an efficient dynamic programming technique. Our approach treats shape noise by allowing discarding small feature points, which introduces skips in the traversal of the dynamic programming graph. Our method is fast, feature preserving, and invariant to geometric transformations. We demonstrate the superiority of our approach over other approaches by experimental results.

Ligang Liu, Guopu Wang, Bo Zhang, Baining Guo and Heung-Yeung Shum, Perceptually based approach for planar morphing, Proceedings of Pacific Graphics'2004, IEEE Computer Society, Seoul, Korea, Oct., 2004, pp. 111-120. [PDF, 0.6M] [Talk slides, 0.5M] [Video, 6.7M]

There exist many computer graphics techniques which could achieve high quality tree generation. However, only a few works focus on realistic modeling of tree bark. Difficulties lie in the complex appearance of the bark surfaces which are largely determined by their mesostructures. Unlike traditional physical based methods, in this paper, we present an appearance-based method to model tree bark surfaces from a single image. We address three main issues here: feature specification, height field assignment and texture correction. For feature specification, we use texton channel analysis to specify a variant of common bark features, including ironbark, vertical and horizontal fractures, tessellation, furrowed cork, and lenticels. For height field assignment, we develop an intuitive and easy-to-use user interface (UI). Here similarity-based texture editing is used for assigning height fields within a texton channel mask. For texture correction, we use the modeled height fields to eliminate the underlying lighting effects in a captured texture. Our modeling system is image-based: it takes as input a bark image and produces as output a textured height field representing a bark sample. We demonstrate that our method is an effective and easy-to-use technique to interactively model a variety of photo realistic bark surfaces.

Xi Wang, Lifeng Wang, Ligang Liu, Shimin Hu and Baining Guo. Interactive modeling of tree bark. Proceedings of Pacific Graphics'2003, IEEE Computer Society, Alberta, Canada, Oct, 2003, pp. 83-90. [PDF, 2.9M] [Talk slides, 4.6M] [Video, 51.3M]

The bidirectional texture function (BTF) is a 6D function that can describe textures arising from both spatially-variant surface reflectance and surface mesostructures. In this paper, we present an algorithm for synthesizing the BTF on an arbitrary surface from a sample BTF. A main challenge in surface BTF synthesis is the requirement of a consistent mesostructure on the surface, and to achieve that we must handle the large amount of data in a BTF sample. Our algorithm performs BTF synthesis based on surface textons, which extract essential information from the sample BTF to facilitate the synthesis. We also describe a general search strategy, called the k-coherent search, for fast BTF synthesis using surface textons. A BTF synthesized using our algorithm not only looks similar to the BTF sample in all viewing/lighthing conditions but also exhibits a consistent mesostructure when viewing and lighting directions change. Moreover, the synthesized BTF fits the target surface naturally and seamlessly. We demonstrate the effectiveness of our algorithm with sample BTFs from various sources, including those measured from real-world textures.

Xin Tong, Jingdan Zhang, Ligang Liu, Xi Wang, Baining Guo, and Heung-Yeung Shum. Synthesis of bi-directional texture functions on arbitrary surfaces. Proceedings of SIGGRAPH'2002, ACM Transaction on Graphics, 2002, 21(3): 665-672. [PDF, 2.8M] [Talk slides, 9.3M] [Video, 26.0M]

The matrix forms for curves and surfaces were largely promoted in CAD/CAM. In this paper we have presented two matrix representation formulations for arbitrary degree NURBS curves and surfaces explicitly other than recursively. The two approaches are derived from the computation of divided difference and the Marsden identity respectively. The explicit coefficient matrix of B-spline with equally spaced knot and Bézier curves and surfaces are obtained by these formulae. The coefficient formulae and the coefficient matrix formulae developed in this paper express non-uniform B-spline functions of arbitrary degree in explicit polynomial and matrix forms. They are useful for the evaluation and conversion of NURBS curves and surfaces in CAD/CAM systems. They will promote the application of product data-exchange standards in industry.

Ligang Liu and Guojin Wang. Explicit representation for NURBS curves and surfaces. Computer Aided Geometric Design, 2002, 19(6): 409-419. [PDF, 0.1M]

The objective of this paper is to provide an efficient and reliable algorithm for representing and evaluating the boundary of the interval Bézier curve. The boundary is represented by a sequence of Bézier curve segments with same degree and line segments in the order they are encountered when marching counter-clockwise along its boundary. The boundary can also be represented as a single B-spline curve having the same degree with the interval Bézier curve. Our algorithm may be easily extended to 3D interval Bézier curves and interval NURBS curves with minor modifications. Some examples illustrate our algorithm.

Hongwei Lin, Ligang Liu, and Guojin Wang. Boundary evaluation for interval Bézier curves. Computer-Aided Design, 2002, 34(9), 637-646. [PDF, 0.3M]