Dr. Ligang LIU


State Key Lab of CAD&CG
Department of Mathematics
Zhejiang University



Address: Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Phone&Fax: (+86-571) 8795-3668
Web: http://www.math.zju.edu.cn/ligangliu

Research Interests    Publications    Patents    Professional Service    Teaching    Resources

Note: I had moved to the University of Science and Technology of China (USTC). Please visit my current website at USTC.  This page will NOT be updated any more.

Academic Background

Research Interests

Selected Publications

List of All Publications

  Please visit my new webpage at USTC for more papers since 2012.


The order of constructing elements of a drawing reflects how objects or things are conceptualized and schematized. While recovering the original order of compilation of a given drawing is challenging, if not ill-posed, we show that it is often possible to recover a plausible drawing order, especially when the drawing has clearly defined shape geometry, like in many line drawings. We present the key principles of drawing order from the field of drawing cognition, and map them to computational procedures in our framework. Our system produces plausible animations of sketching of input line drawings, with no or little user intervention. We test our algorithm on a range of input sketches, with varying degree of complexity and structure, and evaluate the quality of the inverse sketches via a user study. We also present applications to line drawing simplification and enhancement, gesture drawing synthesis, and drawing animation in the context of storytelling.

Hongbo Fu, Shizhe Zhou, Ligang Liu, Niloy J. Mitra. Animated Construction of Line Drawings. ACM Transactions on Graphics (Proc. SIGGRAPH ASIA), 30(6), Article No.133: 1-10, 2011. [Project page]

We introduce an algorithm for 3D object modeling where the user draws creative inspiration from an object captured in a single photograph. Our method leverages the rich source of photographs for creative 3D modeling. However, with only a photo as a guide, creating a 3D model from scratch is a daunting task. We support the modeling process by utilizing an available set of 3D candidate models. Specifically, the user creates a digital 3D model as a geometric variation from a 3D candidate. Our modeling technique consists of two major steps. The first step is a user-guided image-space object segmentation to reveal the structure of the photographed object. The core step is the second one, in which a 3D candidate is automatically deformed to fit the photographed target under the guidance of silhouette correspondence. The set of candidate models have been pre-analyzed to possess useful high-level structural information, which is heavily utilized in both steps to compensate for the ill-posedness of the modeling problems based only on content in a single image. Equally important, the structural information is preserved by the geometric variation so that the final product is coherent with its inherited structural information readily usable for subsequent model refinement or processing.

Kai Xu, Hanlin Zheng, Hao Zhang, Daniel Cohen-Or, Ligang Liu, Yueshan Xiong. Photo-Inspired Model-Driven 3D Object Modeling. ACM Transactions on Graphics (Proc. SIGGRAPH), 2011, 30(4), Article No.80: 1-10. [Project page]

We present a novel sketch-based tool, called iCutter (short for Intelligent Cutter), for cutting out semantic parts of 3D shapes. When a user performs a cutting task, he only needs to draw a freehand stroke to roughly specify where cuts should be made without much attention. Then iCutter intelligently returns the best cut that meets the user's intention and expectation. We develop a novel scheme for selecting the optimal isoline from a well-designed scalar field induced from the input stroke, which respects the part saliency as well as the input stroke. We demonstrate various examples to illustrate the flexibility and applicability of our iCutter tool.

Min Meng, Lubin Fan, Ligang Liu. iCutter: A Direct Cut Out Tool for 3D Shapes. Journal of Computer Animation and Virtual World,  22(4), 335-342, 2011. [Project page]

Given a 3-vertex-connected triangular planar graph and an embedding of its boundary vertices, can the interior vertices be embedded to form a valid triangulation? We describe an algorithm which decides this problem and produces such an embedding if it exists.

Yin Xu, Renjie Chen, Craig Gotsman, Ligang Liu. Embedding a Triangular Graph within a Given Boundary. Computer Aided Geometric Design, 28, 349-356, 2011. [Paper]

We present a novel progressive painting-based mesh cutout tool, called Paint Mesh Cutting, for interactive mesh segmentation. Different from the previous user interfaces, the user only needs to draw a single stroke on the foreground region and then obtains the desired cutting part at an interactive rate. Moreover, the user progressively paints the region of interest using a brush and have the instant feedback on cutting results as he/she drags the mouse. This is achieved by efficient graph-cut based optimizations based on the Gaussian mixture models (GMM) on the shape diameter function (SDF) metric of the shape. We demonstrate a number of various examples to illustrate the flexibility and applicability of our system and present a user study that supports the advantages of our user interface.

Lubin Fan, Ligang Liu, Kun Liu. Paint Mesh Cutting. Computer Graphics Forum (Proc. Eurographics), 30(2), 603-611, 2011. [Project page]

We introduce symmetry hierarchy of man-made objects, a high-level structural representation of a 3D model providing a symmetry-induced, hierarchical organization of the model's constituent parts. Given an input model segmented into primitive parts, we first build an initial graph which encodes inter-part symmetry and connectivity relations, as well as self-symmetries in individual parts. The symmetry hierarchy is constructed from the initial graph via recursive graph contraction consisting of two operations: grouping of parts by symmetries and assembly of connected sets of parts. The order of graph contraction is dictated by a set of precedence rules designed primarily to respect the law of symmetry in perceptual grouping and the principle of compactness of representation. The symmetry hierarchy of a man-made object naturally implies a hierarchical segmentation that is more meaningful than those produced by local geometric considerations. We demonstrate that such a hierarchical segmentation facilitates intra-class semantic labeling of shapes. We also develop applications of symmetry hierarchies for structural shape editing and detection of upright orientations.

Yanzhen Wang, Kai Xu, Jun Li, Hao Zhang, Ariel Shamir, Ligang Liu, Zhiquan Cheng, Yueshan Xiong. Symmetry Hierarchy of Man-Made Objects. Computer Graphics Forum (Proc.Eurographics), 30(2), 287-296, 2011. [Project page]

This paper presents an extensive comparative evaluation of five popular foreground/background sketch-based interactive mesh segmentation algorithms, addressing the quantitative assessment of the accuracy, efficiency, and stability of each algorithm. To facilitate the comparison, we have developed a complete framework with an intuitive and simple sketch-based interface to enable interactive mesh segmentation by marking strokes to specify the foreground and background with the mouse buttons, allowing us to quantify the algorithms in a unified manner. The evaluation has been performed via extensive user experiments in which each participant was assigned to segment models with the evaluated algorithms and the corresponding update of each segmentation was recorded as a new refinement when additional interactions were added. We then collected the segmentations from participants and evaluated them against the ground truth corpus constructed from the Princeton segmentation database. To investigate how well the interactive segmentations match the ground-truth, five metrics were used to measure the boundary and region accuracy of segmentations. By studying the experimental results, we have analyzed the performance of the evaluated algorithms and provided valuable insights into their characteristics.

Min Meng, Lubin Fan, Ligang Liu. A Comparative Evaluation of Foreground/Background Sketch-based Mesh Segmentation Algorithms. Computers &Graphics (Proc. Shape Modeling International), 35(3), 650-660, 2011. [Project page]

Sample point distributions possessing blue noise spectral characteristics play a central role in computer graphics, but are notoriously difficult to generate. We describe an algorithm to very efficiently generate these distributions. The core idea behind our method is to compute a Capacity-Constrained Delaunay Triangulation (CCDT), namely, given a simple polygon P in the plane, and the desired number of points n, compute a Delaunay triangulation of the interior of P with n Steiner points, whose triangles have areas which are as uniform as possible. This is computed iteratively by alternating update of the point geometry and triangulation connectivity. The vertex set of the CCDT is shown to have good blue noise characteristics, comparable in quality to those of state-of-the-art methods, achieved at a fraction of the runtime. Our CCDT method may be applied also to an arbitrary density function to produce non-uniform point distributions. These may be used to halftone grayscale images.

Yin Xu, Ligang Liu, Craig Gotsman, Steven J. Gortler. Capacity-Constrained Delaunay Triangulation for Point Distributions. Computers &Graphics (Proc. Shape Modeling International), 35(3), 510-516, 2011. [Project page]

We present a new method for enhancing shape features of a mesh surface by moving mesh vertices from low-curvature regions to high-curvature regions or feature regions. The movement of the vertices, also called vertex flow, is driven by minimizing an objective function defined to take into account several important considerations in mesh improvement. First, a new edge-based energy term is used to measure the uniformity of the approximation error of a target shape by a mesh surface. Clearly, given a fixed number of triangle faces of a mesh surface approximating an underlying target surfaces, the approximation is made more uniform by placing more faces are used in higher-curvature regions and fewer faces in lower-curvature regions. Therefore, the minimization of this edge-based energy term provides a strong force to move mesh vertices towards high-curvature regions. Second, to maintain faithful shape approximation during vertex flow, a distance-error term is included to penalize the displacement of mesh vertices along normal directions of the underlying surface, and a novel local quadratic model is employed to efficiently minimize this term. Third, a fairing term is used to ensure the smoothness of the resulting mesh surface. Our method enhances significantly shape features even at a low sampling rate and is useful to several feature-aware geometry processing operations, such as simplification and perceptual line drawing.

Zhongping Ji, Ligang Liu, Bin Wang, Wenping Wang. Feature Enhancement by Vertex Flow for 3D Shapes. Computer-Aided Design and Applications, 8(5): 649-664, 2011. [Paper]

Detailed and interleaving structures, common on many man-made objects, are often created using cane, coils, metal wires, rods, etc. The delicate structures, although manufactured using simple procedures, are challenging to scan and reconstruct. We observe that such structures are inherently 1D in nature, and hence are naturally represented using an arrangement of generating curves. We refer to the resultant surfaces as arterial surfaces. In this paper we propose a novel approach for scanning, analyzing, and manipulating such arterial surfaces. The core of the algorithm is a novel deformable model, called arterial snake, that simultaneously captures the topology and geometry of the arterial objects. The recovered snakes produce a natural decomposition of the raw scans, with the decomposed parts often capturing meaningful object sections. We demonstrate the robustness of our algorithm on a variety of relief and arterial objects corrupted with noise, outliers, and with large parts missing. We present a range of applications including reconstruction, topology repairing, and manipulation of the arterial surface by directly controlling the underlying curve network and the associated profile section sections, which are difficult to perform without such the recovered shape structure.

Guo Li, Ligang Liu, Hanlin Zheng, Niloy Mitra. Analysis, Reconstruction and Manipulation using Arterial Snakes. ACM Transactions on Graphics (Proc. ACM SIGGRAPH ASIA), 29(5), Article No.152: 1-10, 2010. [Project page]


We present an easy-to-use image retouching technique for realistic reshaping of human bodies in a single image. A model-based approach is taken by integrating a 3D whole-body morphable model into the reshaping process to achieve globally consistent editing effects. A novel body-aware image warping approach is introduced to reliably transfer the reshaping effects from the model to the image, even under moderate fitting errors. Thanks to the parametric nature of the model, our technique parameterizes the degree of reshaping by a small set of semantic attributes, such as weight and height. It allows easy creation of desired reshaping effects by changing the full-body attributes, while producing visually pleasing results even for loosely-dressed humans in casual photographs with a variety of poses and shapes.

Shizhe Zhou, Hongbo Fu, Ligang Liu, Daniel Cohen-Or, Xiaoguang Han. Parametric Reshaping of Human Bodies in Images. Transactions on Graphics (Proc. ACM SIGGRAPH), 29(4), Article No.126: 1-10, 2010. [Project page]

We present a novel approach to localization of sensors in a network given a subset of noisy intersensor distances. The algorithm is based on "stitching" together local structures by solving an optimization problem requiring the structures to fit together in an "As-Rigid-As-Possible" manner, hence the name ARAP. The local structures consist of reference "patches" and reference triangles, both obtained from inter-sensor distances. We elaborate on the relationship between the ARAP algorithm and other state-of-the-art algorithms, and provide experimental results demonstrating that ARAP is significantly less sensitive to sparse connectivity and measurement noise. We also show how ARAP may be distributed.

Lei Zhang, Ligang Liu, Craig Gotsman and Steven J. Gortler. An As-Rigid-As-Possible Approach to Sensor Network Localization. ACM Transactions on Sensor Networks, 6(4), Article 35: 1-21, 2010. [Project page]

This paper presents a novel modeling system, called B-Mesh, for generating base meshes of 3D articulated shapes. The user only needs to draw a one-dimensional skeleton and to specify key balls at the skeletal nodes. The system then automatically generates a quad dominant initial mesh. Further subdivision and evolution are performed to refine the initial mesh and generate a quad mesh which has good edge flow along the skeleton directions. The user can also modify and manipulate the shape by editing the skeleton and the key balls and can easily compose new shapes by cutting and pasting existing models in our system. The mesh models generated in our system greatly benefit the sculpting operators for sculpting modeling and skeleton-based animation.

Zhongping Ji, Ligang Liu, Yigang Wang. B-Mesh: A Modeling System for Base Meshes of 3D Articulated Shapes. Computer Graphics Forum (Proceedings of Pacific Graphics), 29(7): 2169-2178, 2010. [Project page]

Humans have always sought to achieve aesthetics in art. In this paper, we present a novel approach for retargeting images to different aspect ratios while improving the composition aesthetics of the results. A simpler computational aesthetic energy is proposed and used to drive the salient objects and prominent lines to move towards their corresponding optimal positions. A mesh-based warping scheme is presented to transform the images while protecting the visual appearance of salient objects. The objective function is quadratic and thus it can be quickly minimized by solving a sparse linear system. The retargeting results are generated in realtime while the user changes the aspect ratios of the target images. A variety of experiments have shown the applicability and effectiveness of our algorithm.

Ligang Liu, Yong Jin, Qingbiao Wu. Realtime aesthetic image retargeting. Proc. Eurographics Workshop on Computational Aesthetic in Graphics, Visualization, and Imaging, pp.1-8, 2010. [Paper] [Demo] [Talk]

Aesthetic images evoke an emotional response that transcends mere visual appreciation. In this work we develop a novel computational means for evaluating the composition aesthetics of a given image based on measuring several well-grounded composition guidelines. A compound operator of crop-and-retarget is employed to change the relative position of salient regions in the image and thus to modify the composition aesthetics of the image. We propose an optimization method for automatically producing a maximally-aesthetic version of the input image. We validate the performance of the method and show its effectiveness in variety of experiments.

We demonstrate that aesthetics can be evaluated computationally with high enough accuracy to be useful. This opens a new avenue for various applications to be enhanced by the ability to automatically assign aesthetic scores. For example, optimally aesthetic views of 3D models can be automatically generated, the layout of documents can be arranged to maximize aesthetic criteria, and appealing logos can be generated given a set of user requirements.

Ligang Liu, Renjie Chen, Lior Wolf, Daniel Cohen-Or. Optimizing Photo Composition. Computer Graphics Forum (Proceedings of Eurographics), 29(2): 469-478, 2010. [Project page]

We present a novel approach for interactive content-aware image resizing. The resizing is performed on warping a triangular mesh over the image, which captures the image saliency information as well as the underlying image features. The warped triangular mesh and the horizontal and vertical scales of all triangles are simultaneously obtained by a quadratic optimization which can be achieved by solving a sparse linear system. Our approach can preserve the shapes of curved features in the resized images. The resizing operation can be performed in an interactive rate which makes the proposed approach practically useful for realtime image resizing. To guarantee foldover free resizing result, we modify the optimization to a standard quadratic programming. A number of experimental results have shown that our approach has obtained pleasing results and outperforms the previous approaches.

Yong Jin, Ligang Liu, Qingbiao Wu. Nonhomogeneous Scaling Optimization for Realtime Image Resizing. The Visual Computer (Proceedings of CGI), 26(6-8): 769-778, 2010. [Project page]

We present a new method for content-aware image resizing based on a framework of global optimization. We show that the basic resizing problem can be formulated as a convex quadratic program. Furthermore, we demonstrate how the basic framework may be extended to prevent foldovers of the underlying mesh; encourage the magnification of salient regions; and preserve straight line structures. We show results demonstrating the effectiveness of the proposed method by comparing with four leading competitor methods.

Renjie Chen, Daniel Freedman, Zachi Karni, Craig Gotsman, Ligang Liu. Content-Aware Image Resizing by Quadratic Programming. Proc. CVPR Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment (NORDIA), 1-8, 2010. (The Best Paper Award) [Paper]

We present a new approach to simultaneously denoise and parameterize unorganized point cloud data. This is achieved by minimizing an appropriate energy function defined on the point cloud and its parameterization. An iterative algorithm to minimize the energy is described. The key ingredient of our approach is an "as-rigid-as-possible" meshless parameterization to map a point cloud with disk topology to the plane without building the connectivity of the point cloud. Then 2D triangulation method can be applied to the planar parameterization to provide triangle connectivity for the 2D points, which can be transferred back to the 3D point cloud to form a triangle mesh surface. We also show how to generalize the approach to meshes with closed topology of any genus. Experimental results have shown that our approach can effectively denoise the point cloud and our meshless parameterization can preserve local distances in the point cloud, resulting in a more regular 3D triangle mesh, compared to other methods.

Lei Zhang, Ligang Liu, Craig Gotsman, Hua Huang. Mesh reconstruction by meshless denoising and parameterization, Computers&Graphics (Proc. SMI), 34(3): 198-208, 2010. [Paper] [Talk]

The Delaunay triangulation of a planar point set is a fundamental construct in computational geometry. A simple algorithm to generate it is based on edge flips in convex quads. We characterize the effect of a single edge flip in a triangulation on the geometric Laplacian of the triangulation, which leads to a simpler and shorter proof of a theorem of Rippa that the Dirichlet energy of any piecewise-linear scalar function on a triangulation obtains its minimum on the Delaunay triangulation. Using Rippa’s theorem, we provide a spectral characterization of the Delaunay triangulation, namely that the spectrum of the geometric Laplacian is minimized on this triangulation. This spectral theorem then leads to a simpler proof of a theorem of Musin that the harmonic index also obtains its minimum on the Delaunay triangulation.

Renjie Chen, Yin Xu, Craig Gotsman, Ligang Liu. A Spectral Characterization of the Delaunay Triangulation. Computer Aided Geometric Design, 27(4): 295-300, 2010. [Paper]

Manifold learning have attracted considerable attention over the last decade. The most frequently used functional is the l2-norm of the gradient of the function. In this paper, we consider the linear manifold learning problem by minimizing this functional with appropriate constraint. We provide theoretical analysis on both the functional and the constraint, which shows the affine hulls of the manifold and the connected components are essential to linear manifold learning problem. Based on the theoretical analysis, we introduce a novel linear manifold learning algorithm called approximately harmonic projection (AHP). Unlike canonical linear methods such as principal component analysis, our method is sensitive to the connected components. This makes our method especially applicable to data clustering. We conduct several experimental results on three real data sets to demonstrate the effectiveness of our proposed method.

Binbin Lin, Xiaofei He, Yuan Zhou, Ligang Liu, Ke Lu. Approximately harmonic projection: Theoretical analysis and an algorithm. Pattern Recognition, 43: 3307-3313, 2010.[Paper]

In this paper, we propose an image driven shape deformation approach for stylizing a 3D mesh using styles learned from existing 2D illustrations. Our approach models a 2D illustration as a planar mesh and represents the shape styles with four components: the object contour, the context curves, user-specified features and local shape details. After the correspondence between the input model and the 2D illustration is established, shape stylization is formulated as a style-constrained differential mesh editing problem. A distinguishing feature of our approach is that it allows users to directly transfer styles from hand-drawn 2D illustrations with individual perception and cognition, which are difficult to be identified and created with 3D modeling and editing approaches. We present a sequence of challenging examples including unrealistic and exaggerated paintings to illustrate the effectiveness of our approach.

Guanghua Tan, Wei Chen, Ligang Liu. Image Driven Shape Deformation using Styles. Journal of Zhejiang University (SCIENCE C), 11(1): 27-35, 2010. [Paper]

In order to synthesize distortionless texture on surfaces, we develop a direct and accurate approach for local resampling in vector fields, and then use the approach to synthesize textures on 2-D manifold surfaces directly from a texture exemplar. Regular-grid patches produced by the local resampling are used as building blocks for texture synthesis. Then texture optimization and patch-based sampling are generalized to synthesize texture directly in vector fields. The first scheme can create texture of higher quality; however the second scheme is faster and simpler and works well for a wide range of textures. Users can control the vector field on the mesh to generate textures with local variations including the orientation and scale. Many experimental results are presented to demonstrate the ease of use and reliable results provided by our system.

Renjie Chen, Ligang Liu, Guangchang Dong. Local resampling for patch-based texture synthesis in vector fields. International Journal of Computer Applications in Technology, 38(1/2/3): 124-133, July, 2010. [Paper]

The essence of a 3D shape can often be well captured by its salient feature curves. In this paper, we explore the use of salient curves in synthesizing intuitive, shape-revealing textures on surfaces. Our texture synthesis is guided by two principles: matching the direction of the texture patterns to those of the salient curves, and aligning the prominent feature lines in the texture to the salient curves exactly. We have observed that textures synthesized by these principles not only fit naturally to the surface geometry, but also visually reveal, even reinforce, the shape’s essential characteristics. We call these feature-aligned shape texturing. Our technique is fully automatic, and introduces two novel technical components in vector field-guided texture synthesis: an algorithm that orients the salient curves on a surface for constrained vector field generation, and a feature-to-feature texture optimization.

Kai Xu, Daniel Cohen-Or, Tao Ju, Ligang Liu, Hao Zhang, Shizhe Zhou, Yueshan Xiong. Feature-Aligned Shape Texturing. ACM Transactions on Graphics (Proc. ACM SIGGRAPH ASIA), 28(5), Article No. 108: 1-7, 2009. [Project page]

While many 3D objects exhibit various forms of global symmetries, prominent intrinsic symmetries which exist only on parts of an object are also well recognized. Such partial symmetries are often seen as more natural than a global one, even when the symmetric parts are under complex pose. We introduce an algorithm to extract partial intrinsic reflectional symmetries (PIRS) of a 3D shape. Given a closed 2-manifold mesh, we develop a voting scheme to obtain an intrinsic reflectional symmetry axis (IRSA) transform, which is a scalar field over the mesh that accentuates prominent IRSAs of the shape. We then extract a set of explicit IRSA curves on the shape based on a refined measure of local reflectional symmetry support along a curve. The iterative refinement procedure combines IRSA-induced region growing and region-constrained symmetry support refinement to improve accuracy and address potential issues arising from rotational symmetries in the shape. We show how the extracted IRSA curves can be incorporated into a conventional mesh segmentation scheme so that the implied symmetry cues can be utilized to obtain more meaningful results. We also demonstrate the use of IRSA curves for symmetry-driven part repair.

Kai Xu, Hao Zhang, Andrea Tagliasacchi, Ligang Liu, Guo Li, Min Meng, Yueshan Xiong. Partial Intrinsic Reflectional Symmetry of 3D Shapes. ACM Transactions on Graphics (Proc. ACM SIGGRAPH ASIA), 28(5), Article No. 138: 1-10, 2009.[Project page]

This paper presents a new approach, called cubic clipping, for computing all the roots of a given polynomial within an interval. In every iterative computation step, two cubic polynomials are generated to enclose the graph of the polynomial within the interval of interest. A sequence of intervals is then obtained by intersecting the sequence of strips with the abcissa axis. The sequence of these intervals converges to the corresponding root with the convergence rate 4 for the single roots, 2 for the double roots and super-linear 4/3 for the triple roots. Numerical examples show that cubic clipping has many expected advantages over Bezier clipping and quadratic clipping. We also extend our approach by enclosing the graph of the polynomial using two lower degree k<n polynomials by degree reduction. The sequence of intervals converges to the corresponding root of multiplicity s with convergence rate (k+1)/s.

Ligang Liu, Lei Zhang, Binbin Lin, Guojin Wang. Fast Approach for Computing Roots of Polynomials using Cubic Clipping. Computer Aided Geometric Design, 26(5): 547-559, 2009. [Paper]

We present a novel approach to parameterize a mesh with disk topology to the plane in a shape-preserving manner. Our key contribution is a local/global algorithm, which combines a local mapping of each 3D triangle to the plane, using transformations taken from a restricted set, with a global "stitch" operation of all triangles, involving a sparse linear system. The local transformations can be taken from a variety of families, e.g. similarities or rotations, generating different types of parameterizations. In the first case, the parameterization tries to force each 2D triangle to be an as-similar-as-possible version of its 3D counterpart. This is shown to yield results identical to those of the LSCM algorithm. In the second case, the parameterization tries to force each 2D triangle to be an as-rigid-as-possible version of its 3D counterpart. This approach preserves shape as much as possible. It is simple, effective, and fast, due to pre-factoring of the linear system involved in the global phase. Experimental results show that our approach provides almost isometric parameterizations and obtains more shape-preserving results than other state-of-the-art approaches.

We present also a more general "hybrid" parameterization model which provides a continuous spectrum of possibilities, controlled by a single parameter. The two cases described above lie at the two ends of the spectrum. We generalize our local/global algorithm to compute these parameterizations. The local phase may also be accelerated by parallelizing the independent computations per triangle.

Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, Steven J. Gortler. A Local/Global Approach to Mesh Parameterization. Computer Graphics Forum (Proc. Eurographics Symposium on Geometry Processing (SGP)), 27(5): 1495-1504, 2008. [Project page]

This paper presents a global optimization operator for arbitrary meshes. The global optimization operator is composed of two main terms, one part is the global Laplacian operator of the mesh which keeps the fairness and another is the constraint condition which reserves the fidelity to the mesh. The global optimization operator is formulized as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Our global mesh optimization approach can be effectively used in at least three applications: smoothing the noisy mesh, improving the simplified mesh, and geometric modeling with subdivision-connectivity. Many experimental results are presented to show the applicability and flexibility of the approach.

Ligang Liu, Chiew-Lan Tai, Zhongping Ji, Guojin Wang. Non-Iterative Approach for Global Mesh Optimization. Computer-Aided Design, 39(9): 772-782, 2007. [Project page]

Recently, animations with deforming objects have been frequently used in various computer graphics applications. Morphing of objects is one of the techniques which realize shape transformation between two or more existing objects. In this paper, we present a novel morphing approach for 3D triangular meshes with the same topology. The basic idea of our method is to interpolate the mean curvature flow of the input meshes as the curvature flow Laplacian operator encodes the intrinsic local information of the mesh. The in-between meshes are recovered from the interpolated mean curvature flow in the dual mesh domain due to the simplicity of the neighborhood structure of dual mesh vertices. Our approach can generate visual pleasing and physical plausible morphing sequences and avoid the shrinkage and kinks appeared in the linear interpolation method. Experimental results are presented to show the applicability and flexibility of our approach.

Jianwei Hu, Ligang Liu, Guozhao Wang. Dual Laplacian Morphing for Triangular Meshes. Computer Animation and Virtual Worlds (Proc. International Conference on Computer Animation and Social Agents (CASA)), 18: 271-277, 2007. [Project page]

We present Easy Mesh Cutting, an intuitive and easy-to-use mesh cutout tool. Users can cut meaningful components from meshes by simply drawing freehand sketches on the mesh. Our system provides instant visual feedback to obtain the cutting results based on an improved region growing algorithm using a feature sensitive metric. The cutting boundary can be automatically optimized or easily edited by users. Extensive experimentation shows that our approach produces good cutting results while requiring little skill or effort from the user and provides a good user experience. Based on the easy mesh cutting framework, we introduce two applications including sketch-based mesh editing and mesh merging for geometry processing.

Zhongping Ji, Ligang Liu, Zhonggui Chen, Guojin Wang. Easy Mesh Cutting. Computer Graphics Forum (Proc. Eurographics), 25(3): 283-291, 2006. [Project page]

Manifold parameterization considers the problem of parameterizing a given triangular mesh onto another mesh surface, which could be particularly plane or sphere surfaces. In this paper we propose a unified framework for manifold parameterization between arbitrary meshes with identical genus. Our approach does this task by directly mapping the connectivity of the source mesh onto the target mesh surface without any intermediate domain and partition of the meshes. The connectivity graph of source mesh is used to approximate the geometry of target mesh using least squares meshes. A subset of user specified vertices are constrained to have the geometry information of the target mesh. The geometry of the mesh vertices is reconstructed while approximating the known geometry of the subset by positioning each vertex approximately at the center of its immediate neighbors. This leads to a sparse linear system which can be effectively solved. Our approach is simple and fast with less user interactions. Many experimental results and applications are presented to show the applicability and flexibility of the approach.

Lei Zhang, Ligang Liu, Zhongping Ji, Guojin Wang. Manifold Parameterization. Lecture Notes in Computer Science (Proc. Computer Graphics International), 4035: 160-171, 2006. [Project page]

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 a global coordinate system, it must somehow be transformed to fit the orientations of details in the deformed surface, otherwise distortion will appear. We observe that visually pleasing deformed meshes should preserve both local parameterization and geometry details. We propose to encode these two types of information in the dual mesh domain due to the simplicity of the neighborhood structure of dual mesh vertices. Both sets of information are nondirectional and nonlinearly dependent on the vertex positions. Thus, we present a novel editing framework that iteratively updates both the primal vertex positions and the dual Laplacian coordinates to progressively reduce distortion in parametrization and geometry. Unlike previous related work, our method can produce visually pleasing deformations with simple user interaction, requiring only the handle positions, not local frames at the handles.

Oscar Kin-Chung Au, Chiew-Lan Tai, Ligang Liu, Hongbo Fu. Dual Laplacian Editing for Meshes. IEEE Transactions on Visualization and Computer Graphics, 12(3): 386-395, 2006. [Project page] [Paper]

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, Heung-Yeung Shum, Perceptually based Approach for Planar Morphing. Proc. Pacific Graphics, IEEE Computer Society, Seoul, Korea, Oct., pp. 111-120, 2004. [Paper] [Demo] [Talk]

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, Baining Guo. Interactive modeling of tree bark. Proc. Pacific Graphics, IEEE Computer Society, Alberta, Canada, Oct, pp. 83-90, 2003. [Paper] [Demo] [Talk]

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, Heung-Yeung Shum. Synthesis of Bi-directional Texture Functions on Arbitrary Surfaces. ACM Transactions on Graphics (Proc. ACM SIGGRAPH), 21(3): 665-672, 2002. [Paper] [Demo] [Talk]

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