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Welcome to visit us, the Group of Computer Aided Geometric Design and
Computer Graphics (CAGD&CG Group) at Department of Mathematics, Zhejiang University!
The core of computer aided geometric design and computer
graphics (CAGD&CG) is the fundamental problem of defining, representing,
and manipulating shape. CAGD&CG are vital tools in the task of modeling
and analyzing the physical world for scentists and engineers. Applications of
these technologies include the design and manufacture of car bodies, ship
hulls, airplane wings, and a wide variety of mechanical components and
assemblies. The scope of CAGD&CG is rather broad, including shape design
and analysis, solid modeling, mesh generation, automated design and
manufacture, computer animation, scientific visualization, image and signal
processing, and computer vision. Our CAGD&CG group explores the
fundamental mathematics associated with shape and applications of shape to
these areas.
Research
Areas People Publications Projects Teaching
Seminars Resources
Research Areas
- Mathematics of
curves and surfaces
- Shape
representations
- Digital geometry
processing
- Computer
animation
- Applications in
industry
- Image and signal processing
People
Recent Publications
List of All
Publications
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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. Proceedings
of Eurographics
Symposium on Geometry Processing 2008 (SGP 2008),
Copenhagen, July 2-4, 2008. Computer Graphics Forum,
2008, 27(5): 1495-1504. [PDF,
5.2M] [Talk,
9.8M] |
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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, 2007, 39(9),
772-782. [PDF,
1.9M]
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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, 2007, 18: 271-277.
(Proceedings of CASA 2007) [PDF,
1.5M] [Video,
1.5M] [Talk,
3.4M] |
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In this paper, 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 three applications including sketching
mesh editing, mesh cut and paste, and component based mesh morphing, for
geometry processing.
Zhongping Ji, Ligang
Liu, Zhonggui Chen, Guojin Wang. Easy Mesh Cutting. Proceedings of Eurographics,
Vienna, Austria, Sep., 2006.
Computer Graphics Forum, 2006, 25(3): 283-291. [PDF, 4.6M] [Video, 15M] [Talk, 3.7M]
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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. Proceedings
of 24th Computer Graphics International Conference, June 2006, Hangzhou China. Lecture Notes in
Computer Science, 2006, 4035: 160-171. [PDF, 3.9M] [Video, 6M]
[Talk, 1.5M]
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Oscar Kin-Chung
Au, Chiew-Lan Tai, Ligang Liu, and Hongbo Fu. Dual Laplacian editing for meshes.
IEEE Transactions on Visualization and Computer Graphics, 2006,
12(3): 386-395.
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.
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Zhongping Ji,
Ligang Liu, and Guojin Wang. Global
smoothing approach for arbitrary meshes with feature preservation. 9th
International CAD/Graphics 2005 conference, Hong Kong,
The Best Student
Paper Award.
This paper
presents a novel approach for surface smoothing with feature preservation
on arbitrary meshes. Laplacian operator is performed in a global way over
the mesh. The surface smoothing is formulated as a quadratic optimization
problem, which is easily solved a sparse linear system. The cost function
to be optimized penalizes deviations from the global Laplacian operator
while maintaining the overall shape of the original mesh. The features of
the original mesh can be preserved by adding feature constraints and
barycenter constraints in the system. Our approach is simple,
non-iterative, fast, and does not cause surface shrinkage and distortion.
Many experimental results are presented to show the applicability and flexibility
of the approach.
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Renjiang Zhang,
Ligang Liu, Weiyin Ma, and Guojin Wang. Construction of cubic
triangular patches with C1 continuity around a corner. 9th
International CAD/Graphics 2005 conference, Hong
Kong.
This paper
presents a novel approach for constructing a piecewise triangular cubic
polynomial surface with C1 continuity around a common corner vertex. A C1
continuity condition between two cubic triangular patches is first derived
using mixed directional derivatives. An approach for constructing a surface
with C1 continuity around a corner is then developed. Our approach is easy
and fast with the virtue of cubic reproduction, local shape
controllability, C2 continuous at the corner vertex. Some experimental
results are presented to show the applicability and flexibility of the
approach.
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Oscar Kin-Chung
Au, Chiew-Lan Tai, Hongbo Fu, and Ligang Liu. Mesh editing with curvature flow
Laplacian. Proceedings of Eurographics Symposium on Geometry
Processing, poster,
2005.
Recently,
differential information as local intrinsic feature descriptors has been
used for mesh editing. Given certain user input as constraints, a deformed
mesh is reconstructed by minimizing the changes in the differential
information. Since the differential information is encoded in the global
coordinate system, it must somehow be transformed to fit the orientation of
details in the deformed surface, otherwise distortion will appear. We
observe that visually desired deformed meshes should preserve both local
parameterization and geometry details. To find suitable representations for
these two types of information, we exploit certain properties of the
curvature flow Laplacian operator. Specifically, we consider the
coefficients of Laplacian operator as the parametrization information and
the magnitudes of the Laplacian coordinates as the geometry information.
Both sets of information are non-directional and non-linearly dependent on
the vertex positions. Thus, we propose a new editing framework which
iteratively updates both the vertex positions and the Laplacian coordinates
to reduce distortion in parametrization and geometry. Our method can
produce pleasing deformation results with simple user interaction not
possible with previous related work. In addition, since the magnitudes of
the Laplacian coordinates approximate the integrated mean curvatures, our
framework is useful for modifying mesh geometry via updating the curvature
field. We demonstrate this use in spherical parameterization and
non-shrinking smoothing.
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Wenhao Song,
Ligang Liu, and Xunnian Yang. Fast
and Robust Ridge-Ravine Detection on Triangular Meshes by Smooth Parametric
Functions. Proceedings of Pacific Graphics, poster, 2005.
This paper
presents a novel approach for detecting ridge-ravine lines on dense
triangular meshes. We first simplify the original mesh into to a domain
mesh by a local
fitting based approach which can preserve the local features. An arbitrary
smooth parametric surface is constructed defined on the domain mesh. Ridges
and ravines
are then accurately and efficiently computed and detected using the smooth
parametric function. Our approach treats shape noise by simplifying the
mesh into its domain mesh. Our method is fast, flexible, robust to noise,
and invariant to geometric transformations.We demonstrate the superiority
of our approach by experimental results.
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Wenhao Song
and Xunnian Yang. Free-form
deformation with weighted T-spline. The Visual Computer,21(3):
139-151, 2005.
A new method
of free-form deformation, w-TFFD, is proposed, for which an original shape
is deformed by using weighted T-spline volumes. We generalize T-splines
[24] to weighted T-spline volumes that also permit T-junctions. Weighted
T-spline volumes are a natural generalization of NURBS volumes but permit
more flexible control lattices. Thus, w-TFFD holds many virtues of
traditional FFDs and is more adaptive to objects with arbitrary topology or
complex shape. The lattices can be automatically generated and approximate
the shape of the object arbitrarily close by octree subdivision. Besides
constructing and deforming a multiresolution lattice, users can also sculpt
specific local details to their required shape by modifying weights. A set
of direct-acting tools that are similar to the techniques proposed by Noble
et al. [21] can be applied to w-TFFD.
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Xunnian Yang. Surface interpolation of meshes by
geometric subdivision. Computer-Aided Design, 37(5): 497-508,
2005.
also: Xunnian Yang. Surface interpolation of meshes with shape optimization.
Proceedings of Geometric Modeling and Processing, IEEE CS Press,
pp103-112, 2004.
Subdivision
surfaces are generated by repeated approximation or interpolation from
initial control meshes. In this paper two new nonlinear subdivision
schemes, face based subdivision scheme and normal based subdivision scheme,
are introduced for surface interpolation of triangular meshes. With a given
coarse mesh more and more details will be added to the surface when the
triangles have been split and refined. Because every intermediate mesh is a
piecewise linear approximation to the final surface, the first type of
subdivision scheme computes each new vertex as the solution to a least
square fitting problem of selected old vertices and their neighboring
triangles. Consequently, sharp features as well as smooth regions are
generated automatically. For the second type of subdivision, the
displacement for every new vertex is computed as a combination of normals
at old vertices. By computing the vertex normals adaptively, the limit
surface is $G^1$ smooth. The fairness of the interpolating surface can be
improved further by using the neighboring faces. Because the new vertices
by either of these two schemes depend on the local geometry, but not the
vertex valences, the interpolating surface inherits the shape of the
initial control mesh more fairly and naturally. Several examples are also
presented to show the efficiency of the new algorithms.
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Ligang Liu,
Bo Zhang, Baining Guo, and Heung-Yeung Shum. Polygonal shape blending with
topological evolutions. Journal of Computer Science and Technology,
2005, 20(1): 77-89.
We present a
general approach to morph between 2D shapes with different topologies.
All possible topological evolutions are classified into three types by
attaching three different topological cells. This formalism is resulted
from the Morse theory on the behavior of the 3D surface around a
non-degenerate critical point. Also we incorporate degenerate topological
evolutions into our framework which produce more attractive morphing
effects. The user controls the morph by specifying the types of
topological evolutions as well as the feature correspondences between the
source and target shapes. Some techniques are also provided to control
the vertex path during the morphing process. The amount of user input
required to produce a morph is directly proportional to the amount of
control the user wishes to impose on the process. The user may allow the
system to automatically generate the morph as well. Our approaches are
totally geometric based and are easy and fast enough in fully interactive
time. Many experimental results show the applicability and flexibility of
our approaches.
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Wujun Che, Xunnian
Yang, and Guozhao Wang. Skeleton-driven
2D distance field metamorphosis using intrinsic shape parameters. Graphical
Models, 66(2):102-126, 2004.
In this
article a novel algorithm is presented for 2-D shape interpolation using
the intrinsic shape parameters of a piecewise linear curve. The skeletons
of two given shapes are computed and the smooth transformation of distance
fields is driven by metamorphosis from the skeleton of the source object to
that of the target one. We introduce feature graphs, linear forms of
skeletons, to guide the construction of intermediate skeleton. If the
topologies of the source object and the target one are different, their
feature graphs will be automatically extended with equivalent topologies.
Then we apply the technique of intrinsic shape parameters to the smooth
transition of the extended feature graphs, which will guide the
metamorphosis of the skeletons. Not only can the new approach be capable of
morphing between objects with different topological genus, but also the
topologies and the shapes of the intermediate objects can be controlled
efficiently.
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Ligang Liu, Guopu
Wang, Bo Zhang, Baining Guo and Heung-Yeung Shum, Perceptually based approach for
planar morphing, Proceedings of Pacific Graphics'2004, IEEE
Computer Society, Seoul, Korea, Oct., 2004, pp. 111-120.
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.
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Xunnian Yang. Curve fitting and fairing using conic
splines. Computer-Aided Design, 36(5): 461-472, 2004.
We present an efficient
geometric algorithm for conic spline curve fitting and fairing through
conic arc scaling. Given a set of planar points, we first construct a
tangent continuous conic spline by interpolating the points with a
quadratic Bezier spline curve or fitting the data with a smooth arc spline.
The arc spline can be represented as a piecewise quadratic rational Bezier
spline curve. For parts of the G1 conic spline without an inflection, we
can obtain a curvature continuous conic spline by adjusting the tangent
direction at the joint point and scaling the weights for every two adjacent
rational Bezier curves. The unwanted curvature extrema within conic
segments or at some joint points can be removed efficiently by scaling the
weights of the conic segments or moving the joint points along the normal
direction of the curve at the point. In the end, a fair conic spline curve
is obtained that is G2 continuous at convex or concave parts and G1
continuous at inflection points. The main advantages of the method lies in
two aspects, one advantage is that we can construct a curvature continuous
conic spline by a local algorithm, the other one is that the curvature plot
of the conic spline can be controlled efficiently. The method can be used
in the field where fair shape is desired by interpolating or approximating
a given point set. Numerical examples from simulated and real data are
presented to show the efficiency of the new method.
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Guozhao Wang,
Qinyu Chen, and Minhua Zhou. NUAT
B-Spline curves. Computer Aided Geometric Design, 21(2):
193-205, 2004.
This paper presents
a new kind of splines, called non-uniform algebraic-trigonometric B-splines
(NUAT B-splines), generated over the space spanned by {1,
t, . . ., t\(k-3), cos t, sin t} in which k is an arbitrary
integer larger than or equal to 3.We show that the NUAT B-splines share
most properties of the usual polynomial B-splines. The subdivision formulae
of this new kind of curves are given. The generation of tensor product
surfaces by these new splines is straightforward.
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Guojin Wang, Kai
Tang, and Chiew-Lan Tai. Parametric
representations of the surface pencil with a same spatial geodesic. Computer-Aided
Design, 36(5): 447-459, 2004.
In this paper, we
study the problem of constructing a family of surfaces from a given spatial
geodesic curve. We derive a parametric representation for a surface pencil
whose members share the same geodesic curve as an isoparametric curve. By
utilizing the Frenet trihedron frame along the given geodesic, we express
the surface pencil as a linear combination of the components of this local
coordinate frame, and derive the necessary and sufficient conditions for
the coefficients to satisfy both the geodesic and the isoparametric
requirements. We illustrate and verify the method by finding exact surface
pencil formulations for some simple surfaces, such as surfaces of
revolution and ruled surfaces. Finally, we demonstrate the use of this
method in a garment design application.
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Hongwei Lin,
Chiew-Lan Tai, Guojin Wang. A mesh
reconstruction algorithm driven by intrinsic property of point cloud. Computer-Aided
Design, 36(1): 1-9, 2004.
This paper
presents an algorithm for reconstructing a triangle mesh surface from a
given point cloud. Starting with a seed triangle, the algorithm grows a
partially reconstructed triangle mesh by selecting a new point based on an
intrinsic property of the point cloud, namely, the sampling uniformity
degree. The reconstructed mesh is essentially an approximate minimum-weight
triangulation to the point cloud constrained to be on a two-dimensional
manifold. Thus, the reconstructed surface has only small topological
difference from the surface of the sampled object. Topological correct
reconstruction can be guaranteed by adding a post-processing step.
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