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2005
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Geometric Construction of
Coordinates for Convex Polyhedra using Polar Duals
Proceedings of Eurographics Symposium on Geometry Processing, 2005, to appear
(Paper)
T. Ju, S. Schaefer, J. Warren, M.Desbrun
A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a
combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and
3D deformation. A related problem is to express a vector lying in a convex cone as a non-negative combination
of edge rays of this cone. This problem also arises in many applications such as planar graph embedding and
spherical parameterization. In this paper, we present a uned geometric construction for building these weighted
combinations using the notion of polar duals. We show that our method yields a simple geometric construction
for Wachspress's barycentric coordinates, as well as for constructing Colin de VerdiËre matrices from convex
polyhedraóa critical step in Lovasz's method with applications to parameterizations.
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Mean value coordinates for closed triangular meshes
Proceedings of ACM SIGGRAPH, 2005, to appear
(Paper)
T. Ju, S. Schaefer, J. Warren
Constructing a function that interpolates a set of values dned at
vertices of a mesh is a fundamental operation in computer graphics.
Such an interpolant has many uses in applications such as shading,
parameterization and deformation. For closed polygons, mean
value coordinates have been proven to be an excellent method for
constructing such an interpolant. In this paper, we generalize mean
value coordinates from closed 2D polygons to closed triangular
meshes. Given such a mesh P, we show that these coordinates
are continuous everywhere and smooth on the interior of P. The
coordinates are linear on the triangles of P and can reproduce linear
functions on the interior of P. To illustrate their usefulness, we
conclude by considering several interesting applications including
constructing volumetric textures and surface deformation.
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Building a 3D Atlas of the Mouse Brain
Ph.D Thesis
(Paper)
Tao Ju
Building and studying 3D representations of anatomical structures, such as the brain,
plays an important role in modern biology and medical science. While 3D imaging
methods such as MRI and CT have been widely applied, 2D imaging methods such
as optical microscopy typically generate images with much higher resolution. In this
thesis I describe how to construct a high-resolution 3D atlas of the mouse brain from
2D microscopic images. In particular, I focus on using computer graphics techniques,
such as image registration to correcting tissue distortions and polygonal modeling to
build surfaces representing the partitioning of anatomical regions. The methods are
being applied to construct a high-quality polygonal atlas of an adult mouse brain
from 350 histological tissue sections. While the resulting brain atlas will contribute
to a larger project of building a spatial database of gene expressions over the mouse
brain, the methods described in this thesis are well suited for the general purpose of
building polygonal atlases of arbitrary anatomical structures from tissue sections.
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2004
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Robust Repair of Polygonal Models
Proceedings of ACM SIGGRAPH, 2004
(Paper,
Slides,
Program)
Tao Ju
We present a robust method for repairing arbitrary polygon models.
The method is guaranteed to produce a closed surface that
partitions the space into disjoint internal and external volumes.
Given any model represented as a polygon soup, we construct an
inside/outside volume using an octree grid, and reconstruct the surface
by contouring. Our novel algorithm can efficiently process
large models containing millions of polygons and is capable of reproducing
sharp features in the original geometry.
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Automated Characterization of Gene Expression Patterns
with an Atlas of the Mouse Brain
Proceedings of EMBS, 2004 (Paper)
J. P. Carson, T. Ju, C. Thaller, J. Warren, M. Bello, I. Kakadiaris, W. Chiu, G. Eichele
A spatio-temporal map of gene activity in the
brain would be an important contribution to the understanding
of brain development, disease, and function. Such a resource is
now possible using high-throughput in situ hybridization, a
method for transcriptome-wide acquisition of cellular
resolution gene expression patterns in serial tissue sections.
However, querying an enormous quantity of image data
requires computational methods for describing and organizing
gene expression patterns in a consistent manner. In addressing
this, we have developed procedures for automated annotation
of gene expression patterns in the postnatal mouse brain.
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Landmark-driven, Atlas-based Segmentation of Mouse
Brain Tissue Images Containing Gene Expression Data
Proceedings of MICCAI, 2004
(Paper)
Ioannis A.
Kakadiaris, Musodiq Bello, Shiva Arunachalam, Wei Kang, Tao Ju,
Joe Warren, James Carson, Wah Chiu, Christina Thaller, and Gregor Eichele
Associating specific gene activity with specific functional
locations in the brain anatomy results in a greater understanding of the role of the
gene's products.
To perform such an association for a large amount of data, reliable
automated methods that characterize the distribution of gene expression in
relation to a standard anatomical model are required.
In this paper, we present an
anatomical landmark detection method that has been incorporated into an atlasbased
segmentation. The addition of this technique significantly increases the accuracy
of automated atlas-deformation. The resulting large-scale annotation will
help scientists interpret gene expression patterns more rapidly and accurately.
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Hodograph Turtles
Proceedings of IASTED CGIM, 2004 (Paper, Slides)
Tao Ju and Ron Goldman.
In contrast to the classical turtle studied in turtle graphics,
here we study examples of a hodograph turtle and its extensions. We show that
some shapes are easier to generate using a hodograph turtle
instead of the classical turtle. More importantly, we can
extract information about the program of the classical turtle
from the geometry generated by a hodograph turtle. We
shall see that especially for complicated turtle paths such as
fractals, the path of a hodograph turtle is often much simpler
and easier to understand than the path of the classical
turtle. This simplicity reects the simplicity of the underlying
turtle program which is not always evident from the
actual fractal path traversed by the classical turtle.
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Turtle Geometry in Computer Graphics and Computer Aided Design
Computer-Aided Design, 36(14): 1471-1482, 2004 (Paper)
Ron Goldman, Scott Schaefer, and Tao Ju
LOGO is a programming language incorporating turtle graphics, originally devised for
teaching computing to young children in elementary and middle schools. Here we advocate the use
of LOGO to help introduce some of the basic concepts of computer graphics and computer aided
design to undergraduate and graduate students in colleges and universities. We shall show how to
motivate affine coordinates and affine transformations, fractal curves and iterated function systems,
relaxation methods and subdivision schemes from elementary notions in turtle geometry and turtle
programming.
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Recursive Turtle Programs and Iterated Affine Transformations
Computer and Graphics, 28(6): 991-1004, 2004.
(Paper)
Tao Ju,
Scott Schaefer and Ron Goldman.
Recursive turtle programs (RTP) and iterated affine transformations (IAT) are two popular methods
for generating fractals. We show that these two models are equivalent in their expressive power.
Conversion algorithms in both directions are presented explicitly from the structure of the RTP
and the affine transformations in the IAT.
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2003
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A geometry database for gene expression data
Proceedings of Symposium on Geomtry Processing, 2003.
(Paper, Slides)
Tao Ju,
Joe Warren, Gregor Eichele, Christina Thaller, Wah Chiu and James Carson
As the logical next step after sequencing the mouse genome,
biologists have developed laboratory methods for determining where
each of the 20K genes in the mouse genome is generating proteins.
Applying these methods to the mouse brain, biologists are
currently generating large numbers of 2D cross-sectional images
that record the expression pattern for each gene in the mouse
genome. In this paper, we describe the structure of a geometric
database for the mouse brain that allows biologists to organize
and search this gene expression data. The central component of
this database is a standard atlas that explicitly partitions the
mouse brain into key anatomical subregions. This atlas is
represented as a Catmull-Clark subdivision mesh with anatomical
subregions separated by a network of B-spline crease curves.
Due to this
partitioning of the subdivision mesh, user queries comparing
expression data between various genes can be restricted to
anatomical subregions without difficulty while the
multi-resolution structure of the subdivision mesh allows these
queries to be processed efficiently. The database and searching tools are available
at www.geneatlas.org.
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Morphing of rational b-spline curves and surfaces using mass distributions
Proceedings of Eurographics, short papers, 2003.
(Paper, Slides)
Tao
Ju and Ron Goldman.
A rational B-spline curve or surface is a
collection of points associated with a mass (weight) distribution.
These mass distributions can be used to exert local control over
the morph between two rational B-spline curves or surfaces. Here
we propose a technique for designing customized morphs by
attaching appropriate mass distributions to target B-spline curves
and surfaces. We also develop a user interface for this morphing
method that is easy to use and requires no knowledge of B-splines
on the part of the user. Examples to illustrate this morphing
method are presented, and applications to computer animation and
computer modelling are explored.
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Convex Contouring on Volumetric Data
The Visual Computer, 19: 513-525, 2003.
(Paper, Slides)
Tao Ju, Scott
Schaefer, Joe Warren.
In this paper we present a fast, table-driven isosurface
extraction technique on volumetric data. Unlike Marching Cubes or
other cell-based algorithms, the proposed polygonization generates
convex negative space inside individual cells, enabling fast
collision detection on the triangulated isosurface. In our
implementation, we are able to perform over 2 million point
classifications per second. The algorithm is driven by an
automatically constructed look-up table that stores compact
decision trees by sign configurations. The decision trees
determine triangulations dynamically by values at cell corners.
Using the same technique, we can perform fast, crack-free
multi-resolution contouring on nested grids of volumetric data.
The method can also be extended to extract isosurfaces on
arbitrary convex, space-filling polyhedra.
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2002
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Dual Contouring on Hermite Data
Proceedings of ACM SIGGRAPH, 2002.
(Paper)
Tao Ju, Frank
Losasso, Scott Schaefer and Joe Warren
This paper describes a new method for contouring a signed grid
whose edges are tagged by Hermite data (i.e; exact intersection
points and normals). This method avoids the need to explicitly
identify and process ”features” as required in previous Hermite contouring
methods. We extend this contouring method to the case of
multi-signed functions and demonstrate how to model textured contours
using multi-signed functions. Using a new, numerically stable
representation for quadratic error functions, we develop an octreebased
method for simplifying these contours and their textured regions.
We next extend our contouring method to these simplified
octrees. This new method imposes no constraints on the octree
(such as being a restricted octree) and requires no ”crack patching”.
We conclude with a simple test for preserving the topology of
both the contour and its textured regions during simplification.
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2001
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Modifying the shape of NURBS surfaces with geometric constraints
Computer Aided Design, 33(12): 903-912, 2001. (Paper)
Shi-Min Hu, You-Fu Li, Tao Ju and Xiang Zhu
NURBS surfaces are among the most commonly used parametric surfaces in CAGD and Computer Graphics. This
paper investigates shape modification of NURBS surfaces with geometric constraints, such as point, normal
vector, and curve constraints. Two new methods are presented by constrained optimization and energy
minimization. The former is based on minimizing changes in control net of surfaces, whereas the latter is
based on strain energy minimization. By these two methods, we change control points and weights of an
original surface, such that the modified surface satisfies the given constraints. Comparison results and
practical examples are also given.
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Approximate merging of a pair of Bezier curves
Computer Aided Design, 33(2): 125-136, 2001. (Paper)
Hu Shi-Min, Tong Ruo-Feng, Ju Tao and Sun Jia-Guang
This paper deals with the merging problem, i.e. to approximate two adjacent Bezier curves by a single Bezier curve.
A novel approach for
approximate merging is introduced in the paper by using the constrained optimization method. The basic idea of this
method is to find
conditions for the precise merging of Bezier curves first, and then compute the constrained optimization solution by
moving the control
points. "Discrete" coefficient norm in L2 sense and "squared difference integral" norm are used in our method.
Continuity at the endpoints of
curves are considered in the merging process, and approximate merging with points constraints are also discussed.
Further, it is shown that
the degree elevation of original Bezier curves will reduce the merging error.
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