ACM Siggraph or TOG

IEEE Transactions





S.Xiong, J.Zhang, J. Zheng, J.Cai, L.Liu (2014)

"Robust surface reconstruction via dictionary learning"

ACM Transactions on Graphics (SIGGRAPH Asia 2014), Vol.33, No.6, 2014.


Surface reconstruction from point cloud is of great practical importance in computer graphics. Existing methods often realize reconstruction via a few phases with respective goals, whose integration may not give an optimal solution. In this paper, to avoid the inherent limitations of multi-phase processing in the prior art, we propose a unified framework that treats geometry and connectivity construction as one joint optimization problem. The framework is based on dictionary learning in which the dictionary consists of the vertices of the reconstructed triangular mesh and the sparse coding matrix encodes the connectivity of the mesh. The dictionary learning is formulated as a constrained L_{2,q}-optimization (0 < q < 1), aiming to find the vertex position and triangulation that minimize an energy function composed of point-to-mesh metric and regularization. Our formulation takes many factors into account within the same framework, including distance metric, noise/outlier resilience, sharp feature preservation, no need to estimate normal, etc., thus providing a global and robust algorithm that is able to efficiently recover a piecewise smooth surface from dense data points with imperfections.


P.Song, C.W.Fu, P.Goswami, J. Zheng, N.Mitra, D.Cohen-Or (2013)

"Reciprocal frame structures made easy"

ACM Transactions on Graphics (SIGGRAPH 2013), Vol.32, No.4, 2013.


A reciprocal frame (RF) is a self-supported 3D structure made up of three or more sloping rods, which form a closed circuit. Large RF-structures built as complex grillages of one or a few similar circuits have an intrinsic beauty derived from their inherent self-similar and highly symmetric patterns. This paper presents an interactive computational tool for designing large RF-structures over a 3D guiding surface. Users can sketch designs with a wide variety of RF patterns and interactively modify the appearance and parameters with preview, and the tool ensures the design's structure connectivity and coherence. The work contains three key technical components: (1) we draw an analogy between RF-structures and plane tiling with regular polygons, and develop a computational scheme to generate coherent RF-tessellations from simple grammar rules; (2) we employ a conformal mapping to lift the 2D tessellation over a 3D guiding surface, allowing a real-time preview and efficient exploration of wide ranges of RF design parameters; and (3) we devise an optimization method to guarantee the collinearity of contact joints along each rod, while preserving the geometric properties of the RF-structure.


J. Zhang, J. Zheng, C. Wu, J. Cai (2012)

"Variational mesh decomposition"

ACM Transactions on Graphics (presented at SIGGRAPH 2012), Vol.31, No.3, 2012.


This paper presents a variational mesh decomposition algorithm that can efficiently partition a mesh into a prescribed number of segments. The algorithm extends the Mumford-Shah model to 3D meshes, which simultaneously handles segmentation and boundary smoothing. The efficiency is achieved by solving the Mumford-Shah model through a saddle-point problem that is solved by a fast primal-dual method. A pre-process step is also proposed to determine the number of segments that the mesh should be decomposed into. By incorporating this pre-processing step, the proposed algorithm can automatically segment a mesh into meaningful parts. Furthermore, user interaction is allowed by incorporating user's inputs into the variational model to reflect user's special intention.


T. Sederberg, D.Cardon, G.Finnigan, N.North, J. Zheng, T. Lyche (2004)

"T-spline simplification and local refinement"
ACM Transactions on Graphics
(SIGGRAPH 2004), Vol.23, No.3, 2004.


A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

T. Sederberg, J. Zheng, A. Bakenov, A. Nasri (2003)

"T-splines and T-NURCCs"
ACM Transactions on Graphics
(SIGGRAPH 2003), Vol.22, No.3, pp 477-484, 2003.


This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-splines permit T-junctions in their control grid, and thus support many valuable operations. T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull- Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. 



J. Zheng, T. Sederberg (2000)

"Estimating tessellation parameter intervals for rational curves and surfaces"

ACM Transactions on Graphics, Vol.19, No.1, pp 56-77, 2000.

This paper presents a method for determining a priori a constant parameter interval for tessellating a rational curve or surface such that the deviation of the curve or surface from its piecewise linear approximation is within a specified tolerance. The parameter interval is estimated based on information about second-order derivatives in the homogeneous coordinates, instead of using affine coordinates directly.

T. Sederberg, J. Zheng, D. Sewell, M. Sabin (1998)

"Non-uniform recursive subdivision surfaces"

ACM SIGGRAPH 1998 Proceedings, pp 387-394, 1998.


This paper develops rules for non-uniform Doo-Sabin and Catmull-Clark surfaces that generalize non-uniform tensor product Bspline surfaces to arbitrary topologies. This added flexibility allows, among other things, the natural introduction of features such as cusps, creases, and darts, while else where maintaining the same order of continuity as their uniform counterparts.


IEEE Transactions



Y.Ma, J. Zheng, J.Xie (2014) 

"Foldover-free mesh warping for constrained texture mapping" 

IEEE Transactions on Visualization and Computer Graphics, 2014.


Mapping texture onto 3D meshes with positional constraints is a popular technique that can effectively enhance the visual realism of geometric models. Such a process usually requires constructing a valid mesh embedding satisfying a set of positional constraints, which is known to be a challenging problem. This paper presents a novel algorithm for computing a foldover-free piecewise linear mapping with exact positional constraints. The algorithm begins with an unconstrained planar embedding, followed by iterative constrained mesh transformations. At the heart of the algorithm are radial basis function (RBF)-based warping and the longest edge bisection (LEB)-based refinement. A delicate integration of the RBF-based warping and the LEB-based refinement provides a provably foldover-free, smooth constrained mesh warping, which can handle a large number of constraints and output a visually pleasing mapping result without extra smoothing optimization. The experiments demonstrate the effectiveness of the proposed algorithm.


H.Zhou, J. Zheng, L.Wei (2014) 

"Representing images using curvilinear feature driven subdivision surfaces" 

IEEE Transactions on Image Processing, Vol.23, No.8, 2014.


This paper presents a subdivision-based vector graphics for image representation and creation. The graphics representation is a subdivision surface defined by a triangular mesh augmented with color attribute at vertices and feature attribute at edges. Special cubic B-splines are proposed to describe curvilinear features of an image. New subdivision rules are then designed accordingly, which are applied to the mesh and the color attribute to define the spatial distribution and piecewise-smoothly varying colors of the image. A sharpness factor is introduced to control the color transition across the curvilinear edges. In addition, an automatic algorithm is developed to convert a raster image into such a vector graphics representation. The algorithm first detects the curvilinear features of the image, then constructs a triangulation based on the curvilinear edges and feature attributes, and finally iteratively optimizes the vertex color attributes and updates the triangulation. Compared with existing vector-based image representations, the proposed representation and algorithm have the following advantages in addition to the common merits (such as editability and scalability): 1) they allow flexible mesh topology and handle images or objects with complicated boundaries or features effectively; 2) they are able to faithfully reconstruct curvilinear features, especially in modeling subtle shading effects around feature curves; and 3) they offer a simple way for the user to create images in a freehand style.


H.Zhu, J. Zheng, J.Cai, N.Thalmann (2013) 

"Object-level image segmentation using low level cues" 

IEEE Transactions on Image Processing, Vol.22, 2013.


This paper considers the problem of automatically segmenting an image into a small number of regions that correspond to objects conveying semantics or high-level structure. While such object-level segmentation usually requires additional high-level knowledge or learning process, we explore what low level cues can produce for this purpose. Our idea is to construct a feature vector for each pixel, which elaborately integrates spectral attributes, color Gaussian Mixture Models and geodesic distance, such that it encodes global color and spatial cues as well as global structure information. Then we formulate the Potts variational model in terms of the feature vectors to provide a variational image segmentation algorithm that is performed in the feature space. We also propose a heuristic approach to automatically select the number of segments. The use of feature attributes enables the Potts model to produce regions that are coherent in color and position, comply with global structures corresponding to objects or parts of objects and meanwhile maintain a smooth and accurate boundary.


Y.Cai, N.Chia, D.Thalmann, N.Kee, J. Zheng, N.Thalmann (2013) 

"Design and development of a virtual dolphinarium for children with autism" 

IEEE Transactions on Neural System and Rehabilitation Engineering, Vol.21, No.2, pp 208-217.


The recent proliferation of virtual reality technology applications in the autism therapy to promote learning and positive behavior among such children has produced optimistic results in developing a variety of skills and abilities in them. Dolphin-assisted therapy has also become a topic of public and research interest for autism intervention and treatment. This paper presents an innovative design and development of a virtual dolphinarium for potential autism intervention. Instead of emulating the swimming with dolphins, our virtual dolphin interaction program allows children with autism to act as dolphin trainers at the poolside and to learn (non-verbal) communication through hand gestures with the virtual dolphins. Immersive visualisation and gesture-based interaction are implemented to engage children with autism within an immersive room equipped with a curved screen spanning a 320 degree and a high-end 5-panel projection system. The paper also reports a pilot study to establish trial protocol of autism screening to explore the participants' readiness for the virtual dolphin interaction.


T.Nguyen, J.Cai, J. Zhang, J. Zheng (2012) 

"Robust interactive image segmentation using convex active contour" 

IEEE Transactions on Image Processing, Vol.21. No.8, pp 3734-3743.


The state-of-the-art interactive image segmentation algorithms are sensitive to the user inputs and often unable to produce an accurate boundary with a small amount of user interaction. They frequently rely on laborious user editing to refine the segmentation boundary. In this paper, we propose a robust and accurate interactive method based on the recently developed continuous-domain convex active contour model. The proposed method exhibits many desirable properties of an effective interactive image segmentation algorithm, including robustness to user inputs and different initializations, the ability to produce a smooth and accurate boundary contour, and the ability to handle topology changes. Experimental results on a benchmark data set show that the proposed tool is highly effective and outperforms the state-of-the-art interactive image segmentation algorithms.



J. Zhang, J. Zheng, J.Cai (2011) 

"Interactive mesh cutting using constrained random walks" 

IEEE Transactions on Visualization and Computer Graphics, Vol.17, No.3, pp 357-367.


This paper considers the problem of interactively finding the cutting contour to extract components from an existing mesh. First, we propose a constrained random walks algorithm that can add constraints to the random walks procedure and thus allows for a variety of intuitive user inputs. Second, we design an optimization process that uses the shortest graph path to derive a nice cut contour. Then a new mesh cutting algorithm is developed based on the constrained random walks plus the optimization process. Within the same computational framework, the new algorithm provides a novel user interface  for interactive mesh cutting that supports three typical user inputs and also their combinations: 1) foreground/background seed inputs: the user draws strokes specifying seeds for “foreground” (i.e., the part to be cut out) and “background” (i.e., the rest); 2) soft constraint inputs: the user draws strokes on the mesh indicating the region which the cuts should be made nearby; and 3) hard constraint inputs: the marks which the cutting contour must pass. The algorithm uses feature sensitive metrics that are based on surface geometric properties and cognitive theory. The integration of the constrained random walks algorithm, the optimization process, the feature sensitive metrics and the varieties of user inputs makes the algorithm intuitive, flexible, and effective as well. The experimental examples show that the proposed cutting method is fast, reliable, and capable of producing good results reflecting user intention and geometric attributes.


W.Yang, J.Cai, J. Zheng, J. Luo (2010) 

"User-friendly interactive image segmentation through unified combinatorial user inputs" 

IEEE Transactions on Image Processing, Vol.19, No.9, pp 2470-2479.


One weakness in the existing interactive image segmentation algorithms is the lack of more intelligent ways to understand the intention of user inputs. In this paper, we advocate the use of multiple intuitive user inputs to better reflect a user's intention. In particular, we propose a constrained random walks algorithm that facilitates the use of three types of user inputs: (1) foreground and background seed input, (2) soft constraint input, and (3) hard constraint input, as well as their combinations. The foreground and background seed input allows a user to draw strokes to specify foreground and background seeds. The soft constraint input allows a user to draw strokes to indicate the region that the boundary should pass through. The hard constraint input allows a user to specify the pixels that the boundary must align with. Our proposed method supports all three types of user inputs in one coherent computational framework consisting of a constrained random walks and a local editing algorithm, which allows more precise contour refinement. Experimental results on two benchmark data sets show that the proposed framework is highly effective and can quickly and accurately segment a wide variety of natural images with ease.


W. Guan, J.Cai, J.Zhang, J. Zheng  (2010) 

"Progressive coding and illumination and view dependent transmission of 3D meshes using R-D optimization" 

IEEE Transactions on Circuits and Systems for Video Technology, Vol.20, No.4, pp 575-586.


For transmitting complex 3D models over bandwidth-limited networks, efficient mesh coding and transmission are indispensable. The state-of-the-art 3D mesh transmission system employs wavelet-based progressive mesh coder, which converts an irregular mesh into a semi-regular mesh and directly applies the zerotree-like image coders to compress the wavelet vectors, and view-dependent transmission, which saves the transmission bandwidth through only delivering the visible portions of a mesh model. We propose methods to improve both progressive mesh coding and transmission based on thorough rate-distortion (RD) analysis. In particular, by noticing that the dependency among the wavelet coefficients generated in remeshing is not being considered in the existing approaches, we propose to introduce a preprocessing step to scale up the wavelets so that the inherent dependency of wavelets can be truly understood by the zerotree-like image compression algorithms. The weights used in the scaling process are carefully designed through thoroughly analyzing the distortions of wavelets at different refinement levels. For the transmission part, we propose to incorporate the illumination effects into the existing view-depend progressive mesh transmission system to further improve the performance. We develop a novel distortion model that considers both illumination distortion and geometry distortion. Based on our proposed distortion model, given the viewing and lighting parameters, we are able to optimally allocate bits among different segments in real time. Simulation results show significant improvements in both progressive compression and transmission.



W.Yang, J. Zheng, J.Cai, S.Rahardja, C.Chen (2009) 

"Natural and seamless image composition with color control" 

IEEE Transactions on Image Processing, Vol.18. No.11, pp 2584-2592.


While the state-of-the-art image composition algorithms subtly handle the object boundary to achieve seamless image copy-and-paste, it is observed that they are unable to preserve the color fidelity of the source object, often require quite an amount of user interactions, and often fail to achieve realism when there exists salient discrepancy between the background textures in the source and destination images. These observations motivate our research towards color controlled natural and seamless image composition with least user interactions. In particular, based on the Poisson image editing framework, we first propose a variational model that considers both the gradient constraint and the color fidelity. The proposed model allows users to control the coloring effect caused by gradient domain fusion. Second, to have less user interactions, we propose a distance-enhanced random walks algorithm, through which we avoid the necessity of accurate image segmentation while still able to highlight the foreground object. Third, we propose a multiresolution framework to perform image compositions at different subbands so as to separate the texture and color components to simultaneously achieve smooth texture transition and desired color control. The experimental results demonstrate that our proposed framework achieves better and more realistic results for images with salient background color or texture differences, while providing comparable results as the state-of-the-art algorithms for images without the need of preserving the object color fidelity and without significant background texture discrepancy.


W. Guan, J. Cai, J. Zheng, C.W.Chen (2008) 

"Segmentation-based view-dependent 3D graphics model transmission" 

IEEE Transactions on Multimedia, Vol.10. No.5, pp 724-734.


For wireless network based graphics applications, a key challenge is how to efficiently transmit complex 3-D models over bandwidth-limited wireless channels. Most existing 3-D mesh transmission systems do not consider such a view-dependent delivery issue, and thus transmit unnecessary portions of 3-D mesh models, which leads to the waste in precious wireless network bandwidth. In this paper, we propose a novel view-dependent 3-D model transmission scheme, where a 3-D model is partitioned into a number of segments, each segment is then independently coded using the MPEG-4 3DMC coding algorithm, and finally only the visible segments are selected and delivered to the client. Moreover, we also propose analytical models to find the optimal number of segments so as to minimize the average transmission size. Simulation results show that such a view-based 3-D model transmission is able to substantially save the transmission bandwidth and therefore has a significant impact on wireless graphics applications.


J. Zheng, Y.Y. Cai (2006) 

"Interpolation over arbitrary topology meshes using a two-phase subdivision scheme" 

IEEE Transactions on Visualization and Computer Graphics, Vol.12. No.3, pp 301-310.


This paper presents a two-phase process, based on a topological modification of the control mesh and a subsequent Catmull-Clark subdivision, to construct a smooth surface that interpolates some or all of the vertices of a mesh with arbitrary topology. It is also possible to constrain the surface to have specified tangent planes at an arbitrary subset of the vertices to be interpolated. The method has following features: (1) it is guaranteed to always work and the computation is numerically stable; (2) there is no need to solve a system of linear equations and the whole computation complexity is O(K) where K is the number of the vertices; and (3) each vertex can be associated with a scalar shape handle for local shape control. These features make interpolation using Catmull-Clark surfaces simple and thus make the new method itself suitable for interactive freeform shape design.





T.Sederberg, J.Zheng (2015)

"Birational quadrilateral maps" 

Computer Aided Geometric Design, Vol. 32, No.1, pp 1-4.


A generic planar quadrilateral defines a 2:1 bilinear map. We show that by assigning an appropriate weight to one vertex of any planar quadrilateral, we can create a map whose inverse is rational linear.



X. Yang, J. Zheng (2013) 

"Curvature tensor computation by piecewise surface interpolation"

Computer-Aided Design, Vol.45, No.12, pp 1639-1650.


Estimating principal curvatures and principal directions of a subjacent, unknown, smooth surface represented by a triangular mesh is an important step in computer aided design and graphics applications. This paper presents a new method for curvature tensor estimation on a triangular mesh by replacing flat triangles with triangular parametric patches and computing Taubin integral---a 3x3 symmetric matrix in integral formulation---based on the parametric patches. Principal curvatures and principal directions are then computed from Taubin integral. The method has two novel technical ingredients: (1) a closed form expression of Taubin integral is derived; and (2) an explicit scheme is developed for constructing a cubic triangular Bezier patch from a triangle with three corner normals, which interpolates the three corner positions and normals with its boundary curves having local cylindrical precision at their two endpoints. All the computations in the new method are direct in closed form, and except for the approximation of the subjacent surface by piecewise interpolating patches, all other calculations are exact. These features make the new method accurate, efficient and robust.



Y. Wang, J. Zheng (2013) 

"Curvature-guided adaptive T-spline surface fitting"

Computer-Aided Design, Vol.45, No.8-9, pp 1095-1107.


The problem of fitting spline surfaces to triangular mesh models is of importance in computer-aided design. Many fitting algorithms have been developed. This paper proposes several novel plug-and-play components or strategies: the use of -splines for fitting, a curvature-guided strategy, faithful re-parameterization and initial spline knot re-placement, which can be used to enhance fitting algorithms. An adaptive -spline fitting algorithm integrating these components and strategies is also presented. The fitting algorithm can generate spline surfaces that well respect the geometrical features of input mesh models and have a more compact representation..



J. Pan, J. Zheng, G.Zhao (2013) 

"Blind watermarking of NURBS curves and surfaces"

Computer-Aided Design, Vol.45, No.2, pp 144-153.


This paper presents two watermarking methods for NURBS curves and surfaces. Both methods are blind, shape-preserving and data amount-preserving. These features are often required in watermarking of CAD models. The first method is based on the replacement of exterior knots of NURBS. Watermarks are embedded into the cross-ratio of four knots in the knot vector(s) and the method is robust to affine transformation, Möbius reparameterization, and interior knot insertion and removal operations. The second method is based on reparameterization using Möbius transformation. Watermarks are embedded into the ratio of two knot intervals selected from the knot vector(s) and the method is robust to affine transformation and linear reparameterization. The capacity of the first method is m−1 for a degree m NURBS curve and m+n−2 for a degree m×n NURBS surface, and the capacity of the second method is 1 for a curve and 2 for a surface. Experiments have been conducted to demonstrate the shape-preserving, data amount-preserving properties and robustness.



X. Yang, J. Zheng (2012) 

"Approximate T-spline surface skinning"

Computer-Aided Design, Vol.44, No.12, pp 1269-1276.


This paper considers the problem of constructing a smooth surface to fit rows of data points. A special class of T-spline surfaces is examined, which is characterized to have a global knot vector in one parameter direction and individual knot vectors from row to row in the other parameter direction. These T-spline surfaces are suitable for lofted surface interpolation or approximation. A skinning algorithm using these T-spline surfaces is proposed, which does not require the knot compatibility of sectional curves. The algorithm consists of three main steps: generating sectional curves by interpolating data points of each row by a B-spline curve; computing the control curves of a skinning surface that interpolates the sectional curves; and approximating each control curve by a B-spline curve with fewer knots, which results in a T-spline surface. Compared with conventional B-spline surface skinning, the proposed T-spline surface skinning has two advantages. First, the sectional curves and the control curves of a T-spline surface can be constructed independently. Second, the generated T-spline skinning surface usually has much fewer control points than a lofted B-spline surface that fits the data points with the same error bound. Experimental examples have demonstrated the effectiveness of the proposed algorithm.



X. Li, J. Zheng (2012) 

"An alternative method for constructing interpolatory subdivision from approximating subdivision"

Computer Aided Geometric Design, Vol.29, pp 474-484.


This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbelt's interpolatory scheme, and Loop subdivision and the butterfly algorithm.


X. Li, J. Zheng, T. Sederberg, T.Hughes, M.Scott (2012) 

"On linear independence of T-spline blending functions"

Computer Aided Geometric Design, Vol.29, No.1, pp 63-76.


This paper shows that, for any given T-spline, the linear independence of its blending functions can be determined by computing the nullity of the T-spline-to-NURBS transform matrix. The paper analyzes the class of T-splines for which no perpendicular T-node extensions intersect, and shows that the blending functions for any such T-spline are linearly independent.


P. Chiang, J. Zheng, K. Mak, N.Thalmann, Y.Cai (2012) 

"Progressive surface reconstruction for heart mapping procedure"

Computer-Aided Design, Vol.44, No.4, pp 289-299.


The composite imaging of measured cardiac properties on a reconstructed endocardial surface allows for the diagnosis of cardiac arrhythmia and myocardial infarct. This paper presents a new method for the progressive reconstruction of an endocardial surface during a heart mapping procedure. A generic mesh is first aligned with a set of anchor points to obtain a first approximation of the surface. Subsequent deformations are constrained by the preservation of local surface characteristics and the fidelity of new contract points. The mesh is refined by local subdivision and its geometrical shape is further improved by edge swapping. Compared to prior art, the new method can reconstruct a realistic surface from a set of sparse and random data. It can advantageously provide a smooth reconstruction at initial acquisition and ensure a geometrical consistency between consecutive reconstructions. The accurate reconstruction of a heart chamber provides important visual cues for an interventionalist to decide on the next mapping site, thus constructively influencing the final diagnosis.


W. Chen, J. Zheng, Y.Cai (2010) 

"Kernel modeling for molecular surfaces using a uniform solution" 

Computer-Aided Design, Vol 42, No.4, pp 267-278.


This paper proposes to use rational Bézier surfaces as a uniform approach to model all three types of molecular surfaces: van der Waals surface, solvent accessible surface and solvent excluded surface. The solution consists of three steps: topology modeling, boundary modeling and surface modeling. First, using weighted α-shape, topology modeling creates two networks to describe the neighboring relationship of the molecular atoms. Second, boundary modeling derives all boundary arcs from the networks. Third, surface modeling constructs all three types of molecular surfaces patch-by-patch based on the networks and the boundary arcs. For SES, the singularity is specially treated to avoid self-intersections. Instead of approximation, the proposed solution can produce precise shapes of molecular surfaces. Since rational Bézier representation is much simpler than trimmed NURBS, computational load can be significantly saved when dealing with molecular surface modeling. This research shows that the rational Bézier representations, more specifically, bi-cubic or 2×4 rational Bézier surfaces, are sufficient for kernel modeling of molecular surfaces and related applications.


J. Zheng (2005)

"Minimizing the maximal ratio of weights of a rational Bezier curve"

Computer Aided Geometric Design, Vol. 22, No. 3, pp 275-280


This paper presents a solution to the problem of reparameterizing a rational curve by a Moebius transformation such that the maximal ratio of weights in the reparameterized representation is minimized. The problem is reduced to solving a linear programming problem, which can be solved directly and simply. The result can be used to reparameterize rational curves so as to yield tight bounds on derivatives.


W.Li, S.Xu, J.Zheng, G.Zhao  (2004)

"Target curvature driven fairing algorithm for planar cubic B-spline curves" 

Computer Aided Geometric Design, Vol. 21., No.5., pp.499-513.


This paper proposes to use target curvature plots to identify bad points or bad curve segments of a given B-spline curve. Then the control points of the curve are modified by a local constrained optimization, which involves the shape fairness  and the coherence to the original design. The target curvature plots are prescribed by designers according to their design intention.


X.Song, T.Sederberg, J.Zheng, R.Farouki, J.Hass  (2004)

"Linear perturbation methods for topologically consistent representations of free-form surface intersection" 

Computer Aided Geometric Design, Vol. 21, No.3.


By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in R^3 of parameter-domain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. 

T. Sederberg, J. Zheng, Song X. (2004) 

"A conjecture on tangent intersections of surface patches"

Computer Aided Geometric Design, Vol.21, No.1


This paper provides some evidences to show that if two surface patches intersect with G1 continuity along an entire curve, the probability is one that the curve is rational. This idea has significance for surface intersection algorithms.

J. Zheng, T. Sederberg (2003)

"Gaussian and mean curvatures of rational Bezier patches"

Computer Aided Geometric Design, Vol. 20, No. 6, pp 297-301


This paper derives formulae for Gaussian and mean curvatures for rational Bézier surface patches. The formulae are expressed in terms of simple geometric quantities (lengths and areas) obtained from the control mesh. These formulae provide more geometric intuition and are easier to compute than the generic formulae from differential geometry. Both the tensor product and triangular patch cases are addressed.


T. Sederberg, J. Zheng, Song X. (2003) 

"Knot intervals and multi-degree splines" 

Computer Aided Geometric Design, Vol.20, No.7, pp 455-468.

This paper studies the merits of using knot interval notation for B-spline curves. Using knot interval notation, the paper introduces MD-splines, which are B-spline-like curves that are comprised of polynomial segments of various degrees (MD stands for  "multi-degree"). The paper focuses on MD-splines of degree 1, 2, and 3, as well as degree 1 and n

Chen F., J. Zheng, T. Sederberg (2001) 

"The mu-basis of a rational ruled surface"

Computer Aided Geometric Design, Vol.18, No.1, pp 61-72.


This paper presents a simple algorithm for computing the mu-basis for a rational ruled surface. The mu-basis consists of two polynomials p(x,y,z,s) and q(x,y,z,s) that are linear in x,y,z and degree mu and m-mu in s respectively, where m is the degree of the implicit equation. The implicit equation of the surface is then obtained by merely taking the resultant of p and q with respect to s. This implicitization algorithm is faster and/or more robust than previous methods.

J. Zheng, Wang G.Z., Liang Y. (1995) 

"GCn continuity conditions for adjacent rational parametric surfaces"

Computer Aided Geometric Design, Vol.12, No.2, pp 111-129.


This paper derives the constraints on the homogeneous surface belonging to a certain rational surface that are both necessary and sufficient to ensure that the rational surface is nth-order geometric continuous. This gives up the strong restriction that requires the homogeneous surface to be as smooth as the rational surface. Further the conditions for the rectangular rational Bézier patches are developed, and some simple and practical sufficient conditions are presented which might give a valid means for the construction of GCn connecting surfaces.

J. Zheng, Wang G.Z., Liang Y. (1992) 

"Curvature continuity between adjacent rational Bezier patches"

Computer Aided Geometric Design, Vol.9, No.5, pp 321-335.


This paper discusses the curvature continuity between two adjacent rational Bézier surfaces. The necessary and sufficient conditions are derived, and further, a series of simple sufficient conditions are developed. With them one can both check the geometric continuity between two surfaces and construct a rational surface possessing curvature continuity with a given rational patch along a certain boundary. 





W.Zhang, J. Zheng, N.Magnenat Thalmann (2015) 

"Real-time subspace integration for example-based elastic material"

Computer Graphics Forum (EUROGRAPHICS 2015), Vol.34, No.2.


Example-based material allows simulating complex material behaviors in an art-directed way. This paper presents a method for fast subspace integration for example-based elastic material, which is suitable for real-time simulation in computer graphics. At the core of the method is the formulation of a new potential using example-based Green strain tensors. By using this potential, the deformation can be attracted towards the example-based deformation feature space, the example weights can be explicitly obtained and the internal force can be decomposed into the conventional one and an additional one induced by the examples. The real-time subspace integration is then developed with subspace integration costs independent of geometric complexity, and both the reduced conventional internal force and additional one being cubic polynomials in reduced coordinates. Experiments demonstrate that our method can achieve real-time simulation while providing comparable quality with the prior art.



C. Chen, J.Cai, J. Zheng, T. Cham, G. Shi (2015) 

"Kinect depth recovery using a color-guided, region-adaptive, and depth-selective framework" 

ACM Transactions on Intelligent Systems and Technology, Vol.6, No.2, 2015.


Considering the existing depth recovery approaches have different limitations when applying to Kinect depth data, in this paper, we propose to integrate their effective features including adaptive support region selection, reliable depth selection and color guidance together under an optimization framework for Kinect depth recovery. In particular, we formulate our depth recovery as an energy minimization problem, which solves the depth hole-filling and denoising simultaneously. The energy function consists of a fidelity term and a regularization term, which are designed according to the Kinect characteristics. Our framework inherits and improves the idea of guided filtering by incorporating structure information and prior knowledge of Kinect noise model. Through analysing the solution to the optimization framework, we also derive a local filtering version which provides an efficient and effective way of improving the existing filtering techniques.



Y.Zhang, J. Zheng, N.Magnenat Thalmann (2014) 

"Example-guided anthropometric human body modeling" 

The Visual Computer, 2014.


This paper presents an example-guided, anthropometry-based modeling method for creating 3D human body models from users’ input of partial anthropometric measurements with a given example dataset. Rather than directly  forming a mapping between the partial measurements and the  body model, we first estimate a set of chosen 30 measurements from the input based on the example-oriented measurement analysis. We then create an initial 3D model using the  example-oriented radial basis function model that maps the set of 30 measurements to the body shape space and is established based on the given examples. We finally refine the 3D model by constrained optimization to create the target body model. Our method has several advantages: (1) the created model is guaranteed to match the input measurements and reflects the shape characteristics of examples; (2) the input requirement is modest, which makes it useful in practice; and (3) the information of both the measurements and examples is fully utilized. We  demonstrate the effectiveness, accuracy, flexibility and extensibility of the method by various experimental evaluations and a Kinect-based body customization application.



Q.Duan, J.Cai, J. Zheng (2014) 

"Compressive environment matting" 

The Visual Computer, 2014.


The existing high-quality environment matting methods usually require the capturing of a few thousand sample images and spend a few hours in data acquisition. In this paper, a novel environment matting algorithm is proposed to capture and extract the environment matte data effectively and efficiently. First, the recently developed compressive sensing theory is incorporated to reformulate the environment matting problem and simplify the data acquisition process. Next, taking into account the special properties of light refraction and reflection effects of transparent objects, two advanced priors, group clustering and Gaussian priors, as well as other basic constraints are introduced during the matte data recovery process to combat with the limited image samples, suppress the effects of the measurement noise resulted from data acquisition, and faithfully recover the sparse environment matte data. Compared with most of the existing environment matting methods, our algorithm significantly simplifies and accelerates the environment matting extraction process while still achieving high-accurate composition results.



D.Xu, Q.Duan, J. Zheng, J.Zhang, J. Cai, T.Cham (2014) 

"Recovering surface details under general unknown illumination using shading and coarse multi-view stereo" 

IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2014.


Reconstructing the shape of a 3D object from multi-view images under unknown, general illumination is a fundamental problem in computer vision and high quality reconstruction is usually challenging especially when high detail is needed. This paper presents a total variation (TV) based approach for recovering surface details using shading and multi-view stereo (MVS). Behind the approach are our two important observations: (1) the illumination over the surface of an object tends to be piecewise smooth and (2) the recovery of surface orientation is not sufficient for reconstructing geometry, which were previously overlooked. Thus we introduce TV to regularize the lighting and use visual hull to constrain partial vertices. The reconstruction is formulated as a constrained TV minimization problem that treats the shape and lighting as unknowns simultaneously. An augmented Lagrangian method is proposed to quickly solve the TV-minimization problem. As a result, our approach is robust, stable and is able to efficiently recover high quality of surface details even starting with a coarse MVS. These advantages are demonstrated by the experiments with synthetic and real world examples.



Y.Li, W.Chen, Y.Cai, A.Nasri, J. Zheng (2015) 

"Surface skinning using periodic T-spline in semi-NURBS form" 

Journal of Computational and Applied Mathematics, Vol.273. pp.116-131, Jan 2015.


NURBS skinning is a powerful and effective process in Computer Aided Geometric Design. It constructs a surface by interpolating a set of cross sectional NURBS curves. These curves however may not be compatible, i.e., they have different knot vectors. This incompatibility is conventionally solved by knot refinement bringing all curves to share the same knot vector, which leads to an explosion in the number of control points defining the skinned surface. Another disadvantage of NURBS skinning is the difficulty of local modification: adjusting one cross section may result in a global change of the surface. In this paper, we present a method for surface skinning using periodic -splines, which is able to handle closed cross sections, support local modifications and control smoothness along the cross sectional curves. We provide explicit formulae for constructing such -spline skinned surfaces, which avoid solving a large system of equations. Experimental results and theoretical analysis confirm that our approach is better than NURBS skinning as it generates surfaces with fewer control points.


P.Song, C.W.Fu, P.Goswami, J. Zheng, N.Mitra, D.Cohen-Or (2014)

"An interactive computational design tool for large reciprocal frame structures"

Nexus Network Journal, Vol.16, No.1, pp.109-118, 2014.


This paper presents the detail of our interactive tool for designing reciprocal frame (RF) structures. In general, our tool addresses the RF design problem with three major steps: (1) it supports the design of RF-tessellation by connecting RF patterns and plane tiling; (2) it delivers interactive preview and exploration of RF designs in 3D space through conformal mapping; and (3) it performs a novel optimization method to arrange the rods in the RF-structures, so that we can ensure rod collinear contacts in the structures. This paper supplements our previous work with implementation details, user interface design and operations, as well as a preliminary study and various new results we devised from the tool.



X.Yang, J. Zheng, D.Wang (2014) 

"A computational approach to joint line detection on triangular meshes" 

Engineering with Computers, Vol.30, No.4, pp.583-597.


This paper presents formulae for evaluating differential quantities at vertices of triangular meshes that may approximate potential piecewise smooth surfaces with discontinuous normals or discontinuous curvatures at the joint lines. We also define the C1 and C2 discontinuity measures for surface meshes using changing rates of one-sided curvatures or changing rates of curvatures across mesh edges. The curvatures are computed discretely as of local interpolating surfaces that lie within a tolerance to the mesh. Together with proper estimation of local shape parameters, the obtained discontinuity measures own properties like sensitivity to salient joint lines and being scale invariant. A simple algorithm is finally developed for detection of C1 or C2 discontinuity joint lines on triangular meshes with even highly non-uniform triangulations. Several examples are provided to demonstrate the effectiveness of the proposed method.



T.Nguyen, J.Cai, J. Zheng, J.Li (2013) 

"Interactive object segmentation from multi-view images" 

Journal of Visual Communication and Image Representation, Vol.24, No.4, pp 477-485.


This paper presents a non-trivial solution to the following problem: given a set of calibrated or un-calibrated multi-view images, by interactively cutting 3 to 4 images, can the foreground object of the rest images be quickly cutout automatically and accurately? Our idea is to integrate 3D segmentation with 2D segmentation so as to combine their advantages. Our proposed system iteratively performs 2D and 3D segmentation, where the 3D segmentation results are used to initialize 2D segmentation and ensure the silhouette consistency among different views and the 2D segmentation results are used to provide more accurate cues for the 3D segmentation. The experimental results show that the proposed system is able to generate highly accurate segmentation results, even for some challenging real-world multi-view image sequences, with a small amount of user input.



H.Zhou, J. Zheng, L.Wei (2013) 

"Texture aware image segmentation using graph cuts and active contours" 

Pattern Recognition, Vol.46. No.6, pp 1719-1733.


The problem of segmenting a foreground object out from its complex background is of great interest in image processing and computer vision. Many interactive segmentation algorithms such as graph cut have been successfully developed. In this paper, we present four technical components to improve graph cut based algorithms, which are combining both color and texture information for graph cut, including structure tensors in the graph cut model, incorporating active contours into the segmentation process, and using a "softbrush" tool to impose soft constraints to refine problematic boundaries. The integration of these components provides an interactive segmentation method that overcomes the difficulties of previous segmentation algorithms in handling images containing textures or low contrast boundaries and producing a smooth and accurate segmentation boundary. Experiments on various images from the Brodatz, Berkeley and MSRC data sets are conducted and the experimental results demonstrate the high effectiveness of the proposed method to a wide range of images.



X.Wu, J. Zheng, C.Wu, Y.Cai (2013) 

"Variational structure-texture image decomposition on manifolds " 

Signal Processing 93(7): 1773-1784, 2013.


This paper considers the problem of decomposing an image defined on a manifold into a structural component and a textural component. We formulate such decomposition as a variational problem, in which the total variation energy is used for extracting the structural part and based on the properties of texture one of three norms, L2, L1 and G, is used in the fidelity term for the textural part. We develop efficient numerical methods to solve the proposed variational problems using augmented Lagrangian methods (ALM) when the manifold is represented by a triangular mesh. The contributions of the paper are two-fold: (1) We adapt the variational structure–texture image decomposition to manifolds, which takes the intrinsic property of manifolds into account. The non-quadratic fidelity terms with L1 and G norms are extended to 3D triangular meshes for the first time. (2) We show how to efficiently tackle the variational problems with non-linearity/non-differentiability terms by iteratively solving some sub-problems that either have closed form solutions or are to solve linear equations.



X.Li, J. Zheng (2013) 

"Interproximate curve subdivision" 

Journal of Computational and Applied Mathematics, Vol.244. pp.36-48, May 2013.


This paper presents a new curve subdivision algorithm called interproximate subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By the interproximate subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of C2 continuity. The interproximate subdivision also contains tension parameters assigned to vertices or edges for shape adjustment. Compared to the 4-point interpolatory subdivision scheme, the interproximate subdivision does not force the new inserted vertices to be interpolated and is thus expected to have improved behavior; and compared to the cubic B-spline refinement scheme, the interproximate subdivision is able to generate curves interpolating user-specified vertices. In addition, the paper also presents two extensions of the interproximate subdivision: one automatically adapts the tension parameters locally according to the geometry of the control polygon during the refinement to achieve convexity preservation and the other automatically relaxes the interpolating property of some vertices to achieve better shape behavior.



P. Chiang, Y.Cai, K. Mak, J. Zheng (2013) 

"A B-spline approach to phase unwrapping in tagged cardiac MRI for motion tracking"

Magnetic Resonance in Medicine 69: 1297-1309, 2013.


A novel B-Spline based approach to phase unwrapping in tagged magnetic resonance images is proposed for cardiac motion tracking. A bicubic B-spline is used to model the absolute phase. The phase unwrapping problem is formulated as a mixed integer optimization problem that minimizes the sum of the difference between the spatial gradients of absolute and wrapped phases, and the difference between the rewrapped and wrapped phases. In contrast to the existing techniques for motion tracking, the proposed approach can overcome the limitation of interframe half-tag displacement and increase the robustness of motion tracking. The paper further presents a hybrid harmonic phase imaging-B-spline method to take the advantage of the harmonic phase imaging method for small motion and the efficiency of the B-Spline approach for large motion. The proposed approach has been successively applied to a full set of cardiac MRI scans in both long and short axis slices with superior performance when compared with the harmonic phase imaging and quality guided path-following methods.



P. Chiang, J. Zheng, Y.Yu, K. Mak, C.Chui, Y.Cai (2013) 

"A VR simulator for intracardiac intervention"

IEEE Computer Graphics and Applications, Vol.33, No.1, pp 44-57, 2013.


The process of learning diagnosis and minimally invasive treatment of heart diseases takes substantial time due to the complex nature of the diseases and the high skill set required to manipulate surgical devices, especially in percutanous intra-cardiac cases. We are interested to develop a novel simulator to create a realistic and low-cost training environment for intra-cardiac intervention. This paper focuses on the concept, design and implementation of a Virtual Reality (VR) simulator for intra-cardiac intervention. Instead of using traditional Finite Element Method (FEM), a new geometrical-based method is proposed to model the interaction between an intra-cardiac catheter and the heart wall. A boundary-enhanced voxelization technique is developed to accelerate the process for detecting catheter-heart interactions. A tactile interface is designed incorporating a VR catheter unit to track the catheter movement within the virtual heart. A progressive approach is implemented to reconstruct the heart chamber for the application of heart mapping.


X. Yang, J. Zheng (2013) 

"Shape aware normal interpolation for curved surface shading from polyhedral approximation " 

The Visual Computer, Vol. 29 (3): 189-201, 2013.


Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. In this paper, we present two schemes based on the Gregory normal patch or the side-vertex interpolating normal for normal interpolation along with cubic Bezier triangles for rendering curved surfaces from rough triangular meshes. Compared to the previous quadratic normal interpolation method, the proposed two methods can generally produce more realistic shading results for rendering rough triangular meshes.



H.Zhou, J. Zheng, X.Yang (2012) 

"Euler arc splines for curve completion" 

Computers & Graphics, Vol.36. No.6, pp.642-650.


This paper introduces a special arc spline called an Euler arc spline as the basic form for visually pleasing completion curves. It is considered as an extension of an Euler curve in the sense that the points in the Euler curve are replaced by arcs. A simple way for specifying it, which is suitable for shape completion, is presented. It is shown that Euler arc splines have several properties desired by aesthetics of curves, in addition to computational simplicity and NURBS representation. An algorithm is proposed for curve completion using Euler arc splines. The development of the algorithm involves two optimization processes, which are converted into a single minimization problem in two variables solved by the Levenberg-Marquardt algorithm. Compared to previous methods, the proposed algorithm always guarantees the interpolation of two boundary conditions.


A. Nasri, K.Sinno, J. Zheng (2012) 

"Local T-spline surface skinning " 

The Visual Computer, Vol. 28., No.6-8, pp 787-797.


Skinning is a process of generating a surface from a set of given curves. In the interpolating approach, the incompatibility of the input NURBS curves are solved by knot insertion, which often leads to an explosion in the number of control points defining the skinned surface. In this paper, we present a solution to this problem using T-splines. Compared to existing approaches, a T-spline skinned surface interpolates a set of incompatible curves with a control mesh of fewer vertices. Moreover, our approach is a local construction.


Q.Duan, J. Zheng, J.Cai (2011) 

"Flexible and accurate transparent-object matting and compositing using refractive vector field" 

Computer Graphics Forum, Vol.30. No.6, pp 1812-1824.


In digital image editing, environment matting and compositing are fundamental and interesting operations that can capture and simulate the refraction and reflection effects of light from an environment. The state-of-the-art real-time environment matting and compositing method is short of flexibility, in the sense that it has to repeat the entire complex matte acquisition process if the distance between the object and the background is different from that in the acquisition stage, and also lacks accuracy, in the sense that it can only remove noises but not errors. In this paper, we introduce the concept of refractive vector and propose to use a refractive vector field as a new representation for environment matte. Such refractive vector field provides great flexibility for transparent-object environment matting and compositing. Particularly, with only one process of the matte acquisition and the refractive vector field extraction, we are able to composite the transparent object into an arbitrary background at any distance. Furthermore, we introduce a piecewise vector field fitting algorithm to simultaneously remove both noises and errors contained in the extracted matte data. Experimental results show that our method is less sensitive to artifacts and can generate perceptually good composition results for more general scenarios.


W.Chen, R.Yu, J. Zheng, Y.Cai, C.Au (2011) 

"Triangular Bezier sub-surfaces on a triangular Bezier surface" 

Journal of Computational and Applied Mathematics, Vol.235. No.17, pp.5001-5016.


This paper considers the problem of computing the Bezier representation for a triangular sub-patch on a triangular Bezier surface. The triangular sub-patch is defined as a composition of the triangular surface and a domain surface that is also a triangular Bezier patch. Based on de Casteljau recursions and shifting operators, previous methods express the control points of the triangular sub-patch as linear combinations of the construction points that are constructed from the control points of the triangular Bezier surface. The construction points contain too many redundancies. This paper derives a simple explicit formula that computes the composite triangular sub-patch in terms of the blossoming points that correspond to distinct construction points and then an efficient algorithm is presented to calculate the control points of the sub-patch.


P.Chiang, Y.Cai, K.Mak, E.Soe, C. Chui, J. Zheng (2011) 

"A geometric approach to the modeling of the catheter-heart interaction for VR simulation of intra-cardiac intervention" 

Computers & Graphics, Vol.35. No.5, pp.1013-1022.


Cardiac intervention is a minimally invasive diagnostic and therapeutic procedure used to treat cardiac diseases. The mapping of heart geometry with minimal visual assistance presents a technical challenge for interventional cardiologists attempting catheter navigation. This paper presents a geometric approach to modeling the catheter-heart interaction for VR simulations of catheter navigation within a heart chamber. Three types of modeling are used to model the interaction between the catheter and the heart wall: non-slip, pseudo-slip and slip modeling. A two-step shape memory process that minimizes the bending of and strain on the catheter is designed for catheter deformation for non-slip or pseudo-slip contact, and a progressive group linkage bending process that constrains the catheter curvature and position within the volume enclosure is designed for catheter deformation for slip contact. The proposed model is consistent with the observations made during the experiment. The model is able to deform the catheter in any free-state shape within the volume enclosure and is independent of local motion increment. Thus, it presents advantages in terms of complexity and real-time requirements.



J Zhang, J. Zheng, J. Cai (2010) 

"A diffusion approach to seeded image segmentation" 

IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2010, pp.2125-2132.


Seeded image segmentation is a popular type of supervised image segmentation in computer vision and image processing. Previous methods of seeded image segmentation treat the image as a weighted graph and minimize an energy function on the graph to produce a segmentation. In this paper, we propose to conduct the seeded image segmentation according to the result of a heat diffusion process in which the seeded pixels are considered to be the heat sources and the heat diffuses on the image starting from the sources. After the diffusion reaches a stable state, the image is segmented based on the pixel temperatures. To better control diffusion, we propose to incorporate the attributes of the image into the diffusion process, yielding an anisotropic diffusion method for image segmentation.


J. Zheng, Y. Wang (2010) 

"Periodic T-splines and tubular surface fitting" 

Curves and Surfaces 2010, pp.731-746.


This paper discusses a special type of T-spline surfaces called periodic T-splines that are closed in one parameter direction, and their application in tubular surface fitting. First, a global representation is proposed for representing periodic T-splines. This representation does not require repeating control points, which facilitates surface fitting process. Then, an algorithm for adaptively fitting periodic T-splines to a tubular triangular mesh that has the same topology as a cylinder is presented. The resulting periodic T-spline is obtained respecting the geometric distribution of the input mesh. The use of periodic T-splines for tubular surface fitting has at least two advantages: 1) adaptive fitting is easily achieved due to the local refinement of T-splines; 2) the algorithm avoids cutting the mesh to make it a disk topologically for conventional B-spline fitting due to the periodic representation and this overcomes the drawback of finding a good cutting path, which is usually difficult.


J.Zhang, C.Wu, J.Cai, J. Zheng, X.Tai (2010) 

"Mesh snapping: robust interactive mesh cutting using fast geodesic curvature flow" 

Computer Graphics Forum (Eurographics'2010), Vol.29, No.2, pp 517-526.


This paper considers the problem of interactively finding the cutting contour to extract components from a given mesh. A geodesic curvature flow based framework is proposed to solve this problem. Since in many cases the meaningful cutting contour on a 3D mesh is locally shortest in the sense of some weighted curve length, the geodesic curvature flow is an ideal tool for our problem. It evolves the cutting contour to the nearby local minimum. We should mention that the previous numerical scheme, discretized geodesic curvature flow (dGCF) is too slow and has not been applied to mesh segmentation. With a careful observation to dGCF, we devise here a fast computation scheme called fast geodesic curvature flow (FGCF), which only needs to solve a smaller and easier problem. The initial cutting contour is generated by a variant of random walks algorithm, which is very fast and gives reasonable cutting result with little user input. Experiment results on the benchmark mesh segmentation data set show that our proposed framework is robust to user input and capable of producing good results reflecting geometric features and human shape perception.


W. Chen, J. Zheng, Y. Cai (2010) 

"Monge mapping using hierarchical NURBS" 

The Visual Computer, Vol. 26, No.6-8, pp.779-789.


In this paper, Monge mapping technique is developed for detail and local shape modification of NURBS represented geometry in 3D environment. Based on multi-resolution and refinement schemes, hierarchical NURBS (H-NURBS) is investigated to design a mechanism for the purpose of carrying localized geometric information. Monge mapping on H-NURBS patch can be easily performed via simple cut&paster operation. Parametric control of the local shapes is developed to facilitate easier and better 3D local modeling.


M. Hu, J. Feng, J. Zheng (2010) 

"An additional branch free algebraic B-spline curve fitting method" 

The Visual Computer, Vol. 26, No.6-8, pp.801-811.


Algebraic curve fitting based on algebraic distance is simple, but it has a disadvantage of inclining to a trivial solution. Researchers therefore introduce some constraints into the objective function in order to avoid the trivial solution. However, this often causes additional branches. Fitting based on geometric distance can avoid additional branches, but it does not offer sufficient fitting precision. In this paper we present a new algebraic B-spline curve fitting method which combines both geometric distance and algebraic distance. The method first generates an initial curve by a distance field fitting that takes geometric distance as the objective function. Then local topology-preserving calibrations based on algebraic distance are performed so that each calibration doesn't produce any additional branches. In this way, we obtain an additional branch free fitting result whose precision is close to or even better than that produced by purely algebraic distance based methods.




C. Au, Y. Cai, J. Zheng, T. Woo (2010) 

"Orienting a protein model by crossing number to generate the characteristic views for identification" 

Computer Modeling in Engineering & Sciences, Vol. 68, No.3, pp.221-237.


A protein model (such as a ribbon model) can be created from the atomic coordinates in the protein data base files. These coordinates are obtained by X-ray crystallography or NMR spectroscopy with the protein arbitrarily oriented. As such, identifying or comparing a novel structure with a known item using protein model in the protein data base can be a timely process since a large number of transformations may be involved. The identification efficiency will be improved if the protein models are uniformly oriented. This paper presents an approach to orient a protein model to generate the characteristic views with minimum and maximum crossings respectively. The projection directions for these characteristic views are determined by a set of crossing maps (C-maps). Re-orientating the protein models in the protein data base to two characteristic views will facilitate the process of identification.


Y.Wang, J. Zheng (2010) 

"Tubular triangular mesh parameterization and applications" 

Journal of Visualization and Computer Animation, Vol 21, No.2, pp 91-102.


Triangular meshes are a popular geometric representation for 3D models used in computer graphics. Parameterization is a process that establishes a mapping between the surface of a model and a suitable domain. This paper considers the problem of parameterizing triangular meshes that have tubular shapes. Unlike an open mesh that is of plane topological type, a tubular mesh gives rise to some special issues in parameterization due to its mesh structure. This paper presents an edge-based parameterization method, in which the edges rather than the vertices of the mesh are treated as the target for parameterization. It first parameterizes the edges on the two boundaries of the tubular mesh, then parameterizes the internal edges based on the mean value coordinates, and finally computes the parameters of the mesh vertices. The method does not need cutting of the mesh. It improves conventional cutting-based algorithms, which cut the mesh to make it a disk topologically, and overcomes the problems of cutting paths that are the zigzag paths leading to suboptimal parameterizations and the difficulty in finding good cutting paths. Some applications such as surface fitting and texture mapping are also provided. 


J. Zheng (2009) 

"C1 NURBS representation of G1 composite rational Bezier curves" 

Computing, Vol.86. No2, pp 257-268.


This paper is concerned with the re-representation of a G1 composite rational Bezier curve. Although the rational Bezier curve segments that form the composite curve are G^1 continuous at their joint points, their homogeneous representations may not be even C0 continuous in the homogeneous space. In this paper, an algorithm is presented to convert the G1 composite rational Bezier curve into a NURBS curve whose nonrational homogeneous representation is C1 continuous in the homogeneous space. This re-representation process involves reparameterization using Moebius transformations, smoothing multiplication and parameter scaling transformations. While the previous methods may fail in some situations, the method proposed in this paper always works.


W. Chen, Y. Cai, J. Zheng (2008) 

"Constructing triangular meshes of minimal area" 

Computer-Aided Design and Applications, Vol.5. No.1-4, pp.508-518.


This paper is concerned with the problem of constructing an aesthetically pleasing triangular mesh with a given closed polygonal contour in three dimensional space as boundary. Triangular meshes of minimal area from all triangular meshes with the prescribed boundary are suggested as the candidates for this problem. An iterative algorithm of constructing such a triangular mesh from a given polygonal boundary is presented. Experimental examples show that the proposed algorithm is reliable and effective. Some related theoretical issues, possible extensions and applications are also discussed.


I. Chandrasekaran, Y. Cai, C.Cao, B. Lu, J. Zheng (2007) 

"Virtual reality prototyping of bio-molecules" 

Virtual and Physical Prototyping, Vol.2. No.1, pp.37-49.


Modelling, sharing and transmission of 3D graphics data of biomolecules are essential in many bio visualization tasks from collaborative research and education to molecular simulation and drug discovery. In the current paper, modelling and representing of bio-molecular structure for virtual and physical rapid prototyping is presented. Our aim is to devise a uniform solution for visualizing, browsing, interacting and prototyping of bio-molecules in various environments including internet, immersive virtual reality (VR), and rapid manufacturing. To do so, we use non uniform rational B-spline surfaces (NURBS) to represent protein secondary structure and surface structure. NURBS protein structures are then tessellated to form bio-molecular graphics models. Their triangular mesh representation is next extracted from their scene graph. A geometric optimization process is followed to make data compatible for their formatting in compact and consistent VR standard to support protein internet browsing, protein VR visualization, protein 3D rapid prototyping and crystal sub-surface laser engraving.


Y.Y.Cai, B.Lu, J. Zheng, L. Lin (2006) 

"Immersive protein gaming for bio edutainment" 

Simulation and Gaming, Vol.37. No.4, pp.466-275.


Learning through playing is one of the natural ways for knowledge and skill acquisition. Games have long been used as a tool for education, from concept building to problem solving. Through fun learning, students may further develop their curiosities and interest in their study. This paper addresses the issue of learning biomolecular structures by Virtual Reality (VR) gaming. A bio edutainment system is developed for protein structure learning. It consists of mainly three components: visualization, modeling and interaction, on top of various supporting technologies such as Graphics and Networking. Via virtual reach-in and hands-on, students can interact with amino acid sequences, protein a-helices, b-sheets and other protein structural information.


J. Zheng, Y.Y. Cai (2005) 

"Making Doo-Sabin surface interpolation always work over irregular meshes" 

The Visual Computer, Vol.21, No.4., pp 242-251


This paper presents a reliable method for constructing a control mesh whose Doo-Sabin subdivision surface interpolates the vertices of a given mesh with arbitrary topology. The method improves on existing techniques in two respects: (1) it is guaranteed to always work for meshes of arbitrary topological type; (2) there is no need to solve a system of linear equations to obtain the control points. Extensions to include normal vector interpolation and/or shape adjustment are also discussed.

J. Zheng, T. Sederberg, R. Johnson (2004)

"Least squares methods for solving differential equations using Bezier control points"

Applied Numerical Mathematics, Vol. 48, No.2


This paper investigates the use of the control points of the Bernstein- Bezier form for numerically solving differential equations. Two least squares type schemes based on degree raising and subdivision are proposed. The convergence of the methods applied to two-point boundary value problems is analyzed. 




J. Zheng, Wang G.Z. (2003) 

"Perturbing Bezier coefficients for best constrained degree reduction in L2-norm" 

Graphical Models, Vol.65, No.6., pp 351-368


This paper shows how the Bézier coefficients of a given degree n polynomial are perturbed, based on minimizing a weighted Euclidean norm, so that it can be reduced to a degree m (<n) polynomial with the constraint that continuity of a prescribed order is preserved at the two endpoints. Then the paper proves that the problem of finding a best L2-approximation over the interval [0,1] for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. The relevant weights are derived. 


J. Zheng, T. Sederberg, Chionh E.-W., D. Cox (2003)

"Implicitizing rational surfaces with base points using the method of moving surface"

Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics Series, Vol. 334, Ron Goldman and Rimvydas Krasauskas, eds., AMS, pp 151-168. ISBN 0-8218-3420-7.


The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. This is a unifying approach whereby tensor product surfaces, pure degree surfaces, and "corner-cut" surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k-1



T. Sederberg, J. Zheng (2002)

"Algebraic methods for computer aided geometric design"

Handbook of Computer Aided Geometric Design, G. Farin, J. Hoschek, M.-S. Kim, eds., Elsevier, North-Holland, pp.363-387. ISBN: 0-444-51104-0.


The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.



J. Zheng, T. Sederberg (2001)

 "A direct approach to computing the mu-basis of planar rational curves" 

Journal of Symbolic Computation, Vol.31, No.5, pp 619-629.


This paper presents an O(n2) algorithm, based on Gröbner basis techniques, to compute the small mu, Greek -basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n3). 


T. Sederberg, J. Zheng, K. Klimaszewski, T. Dokken (1999) 

"Approximate implicitization using monoid curves and surfaces"

Graphical Models and Image Processing, Vol.61, No.4, pp 177-198.

This paper presents an approach to finding an approximate implicit equation and an approximate inversion map of a rational parametric curve or surface. High accuracy of the approximation is achieved with a small number of low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and "phantom" branches normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors.

Wang G.Z., J. Zheng (1997) 

"Bounds on the moving control points of hybrid curves"

Graphical Models and Image Processing, Vol.59, No.1, pp 19-25.


This paper provides several methods to estimate the error bounds for the approximation to the moving control point of the hybrid curves. When the given rational Bézier curves satisfies the convergent conditions for moving control point of the hybrid curve, the polynomial Bézier curves that  approximate the rational Bézier curve can be obtained by replacing the moving control point with the special point.