I've obtained my PhD in Aug 2010 at the Department of Computer Science and Engineering of The Ohio State University under the supervision of Prof. Tamal K. Dey. I was then a postdoc with Prof. Scott Schaefer at Texas A&M University until Jan 2011.
I used POVRAY to render those images, here is another ray-traced image which I made in one of my computer graphics courses.
Positive Gordon-Wixom Coordinates. Josiah Manson, Kuiyu Li, Scott Schaefer. Computer Aided Design 2011, 43(11): 1422-1426 (2011) [Paper]
Abstract: We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in 2D. Our method extends weighted Gordon-Wixom interpolation to non-convex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with smooth boundaries. We achieve these properties by using the distance to lines tangent to the boundary curve to define a weight function that is positive and smooth. We derive closed-form expressions for arbitrary polygons in 2D and compare the basis functions of our coordinates with several other types of barycentric coordinates.
Persistent Heat Signature for Pose-oblivious Matching of Incomplete Models. Tamal K. Dey, Kuiyu Li, Chuanjiang Luo, Pawas Ranjan, Issam Safa, Yusu Wang. Eurographics Symposium on Geometry Processing 2010 (SGP 2010) [Paper]
Abstract: Although understanding of shape features in the context of shape matching and retrieval has made considerable progress in recent years, the case for partial and incomplete models in presence of pose variations still begs a robust and efficient solution. A signature that encodes features at multi-scales in a pose invariant manner is more appropriate for this case. The Heat Kernel Signature function from spectral theory exhibits this multi-scale property. We show how this concept can be merged with the persistent homology to design a novel efficient pose-oblivious matching algorithm for all models, be they partial, incomplete, or complete. We make the algorithm scalable so that it can handle large data sets. Several test results show the robustness of our approach.
Isotopic Reconstruction of Surfaces with Boundaries. Tamal K. Dey, Kuiyu Li, Edgar A. Ramos, Rephael Wenger. Eurographics Symposium on Geometry Processing 2009 (Computer Graphics Forum 28(5): 1371-1382) [Paper]
Abstract: We present an algorithm for the reconstruction of a surface with boundaries (including a non-orientable one) in three dimensions from a sufficiently dense sample. It is guaranteed that the output is isotopic to the unknown sampled surface. No previously known algorithm guarantees isotopic or homeomorphic reconstruction of surfaces with boundaries. Our algorihtm is surprisingly simple. It `peels' slivers greedily from an alpha-complex of a sample of the surface. No other post-processing is necessary. We provide several experimental results from an implementation of our basic algorithm and also a modified version of it.
Cut Locus and Topology from Surface Point Data. Tamal K. Dey, Kuiyu Li. Symposium on Computational Geometry 2009 (SoCG 2009) [Paper]
Abstract: A cut locus of a point p in a compact Riemannian manifold M is defined as the set of points where minimizing geodesics issued from p stop being minimizing. It is known that a cut locus contains most of the topological information of M. Our goal is to utilize this property of cut loci to decipher the topology of M from a point sample. Recently it has been shown that Rips complexes can be built from a point sample P of M systematically to compute the Betti numbers, the rank of the homology groups of M. Rips complexes can be computed easily and therefore are favored over others such as restricted Delaunay, alpha, Cech, and witness complex. However, the sizes of the Rips complexes tend to be large. Since the dimension of a cut locus is lower than that of the manifold M, a subsample of P approximating the cut locus is usually much smaller in size and hence admits a relatively smaller Rips complex.
In this paper we explore the above approach for point data sampled from surfaces embedded in any high dimensional Euclidean space. We present an algorithm that computes a subsample P0 of a sample P of a 2-manifold where P0 approximates a cut locus. Empirical results show that the first Betti number of M can be computed from the Rips complexes built on these subsamples. The sizes of these Rips complexes are much smaller than the one built on the original sample of M.
Persistence-based Handle and Tunnel Loops Computation Revisited for Speed Up. Tamal K. Dey, Kuiyu Li. Shape Modeling International 2009 (Computers & Graphics 33(3): 351-358) [Paper]
Abstract: Loops in surfaces associated with topological features such as handles and tunnels are important entities in many applications including surface parameterization, feature identification, and topological simplification. Recently, a persistent homology based algorithm has been proposed to compute them. The algorithm has several advantages including its simplicity, combinatorial nature and independence from computing other extra structures. In this paper, we propose changes to this loop computation algorithm based on some novel observations. These changes reduce the computation time of the algorithm dramatically. In particular, our experimental results show that the suggested changes achieve considerable speed up for large data sets without sacrificing loop qualities.
Abstract: Many applications such as topology repair, model editing, surface parameterization, and feature recognition benefit from computing loops on surfaces that wrap around their 'handles' and 'tunnels'. Computing such loops while optimizing their geometric lengths is difficult. On the other hand, computing such loops without considering geometry is easy but may not be very useful. In this paper we strike a balance by computing topologically correct loops that are also geometrically relevant. Our algorithm is a novel application of the concepts from topological persistence introduced recently in computational topology. The usability of the computed loops is demonstrated with some examples in feature identification and topology simplification.
On Computing Handle and Tunnel Loops. Tamal K. Dey, Kuiyu Li, Jian Sun. IEEE Proc. Internat. Conf. Cyberworlds 2007 (Computer-Aided Design) [Paper]
Abstract: Many applications seek to identify features like handles and tunnels in a shape bordered by a surface embedded in three dimensions. To this end we define handle and tunnel loops on surfaces which can help identifying these features. We show that a closed surface of genus g always has g handle and g tunnel loops induced by the embedding. For a class of shapes that retract to graphs, we characterize these loops by a linking condition with these graphs. These characterizations lead to algorithms for detection and generation of these loops. We provide an implementation with applications to feature detection and topology simplification to show the effectiveness of the method.