Papers by Xianfeng David Gu
Completing a PhD thesis is like a marathon, and I would not have been able to finish this maratho... more Completing a PhD thesis is like a marathon, and I would not have been able to finish this marathon without the support of countless people over the past four and a half years. First and foremost, I would like to express my deeply-felt thanks to my advisor, Prof. Liming Chen, for his continuous support and encouragement, and also for his direction and countless hours spent with me on this thesis and my research. Working under the supervision of such a knowledgeable and understanding person has been a great pleasure for me. Meanwhile, I would like to express my deeply-felt thanks to my co-advisor, Prof. Jean-Marie Morvan, for his guidance of learning the normal cycle theory, and also for his attitude of doing valuable research. He is knowledgeable, smart, funny and easy going; excellent qualifications for my co-advisor. Moreover, I would also like to express my deeply-felt thanks to Prof. Xianfeng David Gu, for teaching me the interest topic of computational conformal mapping, and for helping me to finish the surface meshing work. He is knowledgeable, quite hardworking, and energetic researcher.
Axioms, 2016
Science does not necessarily evolve along the lines that are taught to us in High School history ... more Science does not necessarily evolve along the lines that are taught to us in High School history classes and in popular films, that is, from simple to complex. In fact, quite the contrary is true in many cases. One such example is that of the Copernican (heliocentric) versus Ptolemaic (heliocentric) system: Practical evidence was ever-accumulating to contradict the second one, but the cognitive framework was too established, the resistance against change too strong and, perhaps no less important, the additions and improvements made to allow it to "work" represented such an enormous investment of energy, intelligence and technical prowess, that its proponents seemingly became enamored more with the apparatus that they developed, than with the scientific truth that they were supposed to pursue (as sadly seems to be the case again and again in Science...). For details, see the marvelously encyclopedic essay "Imaginary Geographies" in [1]. It took some courage for the paradigm shift to hold, but it brought a crystalline simplicity that contrasted with conceptual and mechanical artifices of additional celestial spheres and improbable astral movements. This well-known example is relevant to us because it makes us realize that the evolution of Differential Geometry is, by no means, unique amongst the Sciences. Indeed, due to historical reasons (which are beyond the scope of this short editorial) Differential Geometry not only developed with a presumption of high smoothness of the objects it studied, it also evolved towards a more-and-more technical manipulation of complicated differential operators and their combinations, that led not only to spectacular successes but also became "The Debauch of Indices" [2] for which this is renowned (and reviled by many). The more concise, modern notation adopted at the beginning of the second half of the 20th Century, simplified notation but, alas, at the price of making the field appear even more aloof and decisively less geometric. In contrast, Discrete Differential Geometry is both more intuitive and (in consequence) far simpler. In fact, some notions are so elementary that they can be taught to High School students, which, sometimes they are. The discrete, polyhedral versions of Gauss' Theorema Egregium and Gauss-Bonnet theorem that have their roots in the ideas of Gauss and even Descartes, represent such instances. (One should, however, never underestimate the resisting force of conservatism, even among scientists: Even only a decade or so ago, one of us was "explained" that the defect definition of curvature on triangular meshes is much harder to comprehend and handle than computing the Christoffel symbols, even though, already then, this had become a standard tool in day-today Graphics and Imaging practice...) We should first understand, however, that even the notion of "discrete" in this context is not unique, but rather tends to include all those settings where geometry arises in the context of a given
Geometry, Imaging and Computing, 2015
Computer-aided detection (CAD) of colonic polyps, as a second reader for computed tomographic col... more Computer-aided detection (CAD) of colonic polyps, as a second reader for computed tomographic colonography (CTC) screening, has earned extensive research interest over the past decades. False positive (FP) reduction in the CAD system plays a crucial role in detecting the polyps. To improve the performance of FP reduction and better assist the physician's diagnosis, we propose an adaptive kernel based multiple kernel learning (MKL) method for CAD of colonic polyps, called AK-MKL. This method builds a more adaptive synthesized classifier by incorporating an adaptive kernel into a set of predefined base kernels for better performance in differentiating true polyps from FPs, which is implemented by learning an optimal combination of a collection of those kernel-based classifiers. Performance evaluation for the presented AK-MKL method was performed on a CTC database, consisting of 25 patients with 50 CT scans. In terms of the AUC (area under the curve of receiver operating characteristic) and accuracy merits, the experimental results showed that our AK-MKL method achieves better performance, compared with two other different methods, i.e., one classifier based on support vector machine (SVM) with only one adaptive kernel (AK-SVM) and the other one based on multiple kernel learning only (MKL).
3D Surface Representation Using Ricci Flow. In Computer Vision: From Surfaces to 3D Objects
Efficient colon wall flattening by improved conformal mapping methodologies for computed tomography colonography
2011 IEEE Nuclear Science Symposium Conference Record, 2011
ABSTRACT Conformal mapping provides a unique way to flatten the three-dimensional (3D) colon wall... more ABSTRACT Conformal mapping provides a unique way to flatten the three-dimensional (3D) colon wall. Visualizing the flattened colon wall supplies an alternative means for the task of detecting abnormality as compared to the conventional endoscopic views. In addition to the visualization, the flattened colon wall carries supplementary geometry and texture information for computer aided detection of abnormality. It is hypothesized that utilizing both the original and the flattened 3D colon walls shall improve the detection capacity of currently available computed tomography colonography. Two major challenges exist for the conformal mapping: (1) accuracy in preserving the essential geometry and (2) efficiency in computing the mapping. This paper describes an improved conformal mapping algorithm over our previous work to accelerate the computing speed and increase the calculation accuracy. Starting from a segmented colon wall, a centerline-based search strategy is applied to remove outliers or topological noise on the segmented colon wall. The de-noised wall is then flattened by the improved conformal mapping algorithm. The algorithm was tested by patient datasets with comparison to our previous algorithm. By a PC platform of 3.0GHz CPU and 8.0 GB ROM memory, the speed was accelerated from more than 70 minutes to less than 5 minutes. In addition to the gain in speed, improvement in accuracy is observed. The gained speed and accuracy are crucially important because of the current challenges in detecting small polyps in the desired high sensitivity and efficiency for screening purpose.
2013 IEEE International Conference on Computer Vision, 2013
We propose a novel approach for dense non-rigid 3D surface registration, which brings together Ri... more We propose a novel approach for dense non-rigid 3D surface registration, which brings together Riemannian geometry and graphical models. To this end, we first introduce a generic deformation model, called Canonical Distortion Coefficients (CDCs), by characterizing the deformation of every point on a surface using the distortions along its two principle directions. This model subsumes the deformation groups commonly used in surface registration such as isometry and conformality, and is able to handle more complex deformations. We also derive its discrete counterpart which can be computed very efficiently in a closed form. Based on these, we introduce a higher-order Markov Random Field (MRF) model which seamlessly integrates our deformation model and a geometry/texture similarity metric. Then we jointly establish the optimal correspondences for all the points via maximum a posteriori (MAP) inference. Moreover, we develop a parallel optimization algorithm to efficiently perform the inference for the proposed higher-order MRF model. The resulting registration algorithm outperforms state-of-the-art methods in both dense non-rigid 3D surface registration and tracking.
Inverse Problems and Imaging, 2013
In this paper, we consider the problem of improving 2D triangle meshes tessellating planar region... more In this paper, we consider the problem of improving 2D triangle meshes tessellating planar regions. We propose a new variational principle for improving 2D triangle meshes where the energy functional is a convex function over the angle structures whose maximizer is unique and consists only of equilateral triangles. This energy functional is related to hyperbolic volume of ideal 3-simplex. Even with extra constraints on the angles for embedding the mesh into the plane and preserving the boundary, the energy functional remains well-behaved. We devise an efficient algorithm for maximizing the energy functional over these extra constraints. We apply our algorithm to various datasets and compare its performance with that of CVT. The experimental results show that our algorithm produces the meshes with both the angles and the aspect ratios of triangles lying in tighter intervals.
General Framework for discrete surface Ricci flow
ABSTRACT Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the ... more ABSTRACT Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete Surface Ricci Flow, including all common schemes: Thurston's Circle Packing, Tangential Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe. Furthermore, this work also introduces a novel scheme, virtual radius circle packing, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energy for all the schemes, and Hessian of the discrete Ricci energy for schemes with Euclidean back ground geometry. The unified frame work deepen our understanding to the the discrete surface Ricci flow theory, and inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the debugging efficiency. Experimental results shows the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and effective for solving real problems.
Geometry, 2013
We introduce a metric notion of Ricci curvature forPLmanifolds and study its convergence properti... more We introduce a metric notion of Ricci curvature forPLmanifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds.
SIAM Journal on Imaging Sciences, 2013
Surface parameterizations and registrations are important in computer graphics and imaging, where... more Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as three-dimensional coordinates each vertex is mapped to, which often requires lots of memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed using the linear Beltrami solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage requirement can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we propose applying the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improves the compression ratio.
Pure and Applied Mathematics Quarterly, 2013
The Ricci flow is a powerful curvature flow method, which has been successfully applied in provin... more The Ricci flow is a powerful curvature flow method, which has been successfully applied in proving the Poincaré conjecture. This work introduces a series of algorithms for visualizing Ricci flows on general surfaces.
Journal of Computational and Applied Mathematics, 2010
We propose a method to map a multiply connected bounded planar region conformally to a bounded re... more We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.
Communications in Information and Systems, 2009
Computational conformal geometry focuses on developing the computational methodologies on discret... more Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. Holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. Curvature flow method is nonlinear and requires higher quality triangulations, but it is more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms are demonstrated by the experimental results.
2013 Proceedings IEEE INFOCOM, 2013
In this paper we propose an algorithm to construct a "space filling" curve for a sensor network w... more In this paper we propose an algorithm to construct a "space filling" curve for a sensor network with holes. Mathematically, for a given multi-hole domain R, we generate a path P that is provably aperiodic (i.e., any point is covered at most a constant number of times) and dense (i.e., any point of R is arbitrarily close to P). In a discrete setting as in a sensor network, the path visits the nodes with progressive density, which can adapt to the budget of the path length. Given a higher budget, the path covers the network with higher density. With a lower budget the path becomes proportional sparser. We show how this density-adaptive space filling curve can be useful for applications such as serial data fusion, motion planning for data mules, sensor node indexing, and double ruling type in-network data storage and retrieval. We show by simulation results the superior performance of using our algorithm vs standard space filling curves and random walks.
Lecture Notes in Computer Science, 2014
This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a... more This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a special class of quasi-conformal maps called the Teichmüller maps. Registering shapes with important features is an important process in medical imaging. However, it is challenging to obtain a unique and bijective genus-1 surface matching that satisfies the prescribed landmark constraints. In addition, as suggested by [11], conformal transformation provides the most natural way to describe the deformation or growth of anatomical structures. This motivates us to look for the unique mapping between genus-1 surfaces that matches the features while minimizing the maximal conformality distortion. Existence and uniqueness of such optimal diffeomorphism is theoretically guaranteed and is called the Teichmüller extremal mapping. In this work, we propose an iterative algorithm, called the Quasi-conformal (QC) iteration, to find the Teichmüller extremal mapping between the covering spaces of genus-1 surfaces. By representing the set of diffeomorphisms using Beltrami coefficients (BCs), we look for an optimal BC which corresponds to our desired diffeomorphism that matches prescribed features and satisfies the periodic boundary condition on the covering space. Numerical experiments show that our proposed algorithm is efficient and stable for registering genus-1 surfaces even with large amount of landmarks. We have also applied the algorithm on registering vertebral bones with prescribed feature curves, which demonstrates the usefulness of the proposed algorithm.
Conformal Geometry for Networking Applications
2. Load balancing on boundaries Covering space by Schottky group. 3. Resilient routing for securi... more 2. Load balancing on boundaries Covering space by Schottky group. 3. Resilient routing for security Hyperbolic embedding and universal covering space 4. Sensor localization on 3D geometry 5. Serial data fusion, motion planning, sensor node indexing Contributions • Conformal geometry provides fundamental approaches to designing Riemannian metric and has been successfully used in networking. • Present canonical virtual coordinates by uniformization theorem for general networks, especially multi-hole domains, graphs. • Solve difficult network problems, routing, load balancing, localization, etc. • Computational methods have solid theoretic foundation and are rigorous. • Experiments on dealing with large scale network data demonstrate the efficiency and efficacy of the conformal geometric methods. Theoretic Foundation • Uniformization Theorem Any Riemann surface (closed or with finite boundaries) can be conformally mapped to one of three spaces, the unit sphere S2, the Euclidean plane E...
Asian Journal of Mathematics, 2019
The classical uniformization theorem of Poincaré and Koebe states that any simply connected surfa... more The classical uniformization theorem of Poincaré and Koebe states that any simply connected surface with a Riemannian metric is conformally diffeomorphic to the Riemann sphere, or the complex plane or the unit disk. Using the work by Gu-Luo-Sun-Wu [9] on discrete conformal geometry for polyhedral surfaces, we show that the uniformization maps for simply connected Riemann surfaces are computable.
A unified framework for computing geodesic distances on nonorientable manifold polyhedral surfaces
Scientia Sinica Mathematica, 2014
Nonorientable manifold surfaces not only play an important role in topology, but also have numero... more Nonorientable manifold surfaces not only play an important role in topology, but also have numerous applications in many topics such as visualization and computation of minimal surfaces. From the topological point of view, a 2-manifold surface is locally homeomorphic to an open disk. This property is independent of the global orientability. However, as far as the discrete representation is concerned, orientable manifold surfaces are usually discretized with halfedge data structure, while nonorientable surfaces are discretized into polygon soups, which is inconvenient for digital geometry processing that often takes orientable meshes as input. In this paper, we propose a uni ed framework for transforming geodesic distance problems de ned on nonorientable 2-manifold meshes to the counter-parts on orientable surfaces, and thereby bridging up nonorientable 2-manifold meshes and conventional geometric algorithms. In order to illustrate the universal adaptability, we apply this new approach to study three problems on nonorientable meshes, including computing exact geodesic paths, discrete exponential mapping and farthest point sampling.
This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a... more This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a special class of quasi-conformal maps called the Teichmüller maps. Registering shapes with important features is an important process in medical imaging. However, it is challenging to obtain a unique and bijective genus-1 surface matching that satisfies the prescribed landmark constraints. In addition, as suggested by [11], conformal transformation provides the most natural way to describe the deformation or growth of anatomical structures. This motivates us to look for the unique mapping between genus-1 surfaces that matches the features while minimizing the maximal conformality distortion. Existence and uniqueness of such optimal diffeomorphism is theoretically guaranteed and is called the Teichmüller extremal mapping. In this work, we propose an iterative algorithm, called the Quasi-conformal (QC) iteration, to find the Teichmüller extremal mapping between the covering spaces of genus-1 surfaces. By representing the set of diffeomorphisms using Beltrami coefficients (BCs), we look for an optimal BC which corresponds to our desired diffeomorphism that matches prescribed features and satisfies the periodic boundary condition on the covering space. Numerical experiments show that our proposed algorithm is efficient and stable for registering genus-1 surfaces even with large amount of landmarks. We have also applied the algorithm on registering vertebral bones with prescribed feature curves, which demonstrates the usefulness of the proposed algorithm.
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Papers by Xianfeng David Gu