The majority of current research on sensor network localization focuses on wireless sensor networks deployed on two-dimensional (2D) plane or in three-dimensional (3D) space, very few on the 3D terrain surface. However, many real-world applications require large-scale sensor networks deployed on the surface of a complex 3D terrain. Compared with planar and 3D network localization, terrain surface network localization generates unique and fundamental hardness. We explore 3D surface network localization with terrain models. A digital terrain model (DTM), available to the public with a variable resolution of up to one meter, is a 3D representation of a terrain's surface. It is commonly built using remote sensing technology or from land surveying and can be easily converted to a triangular mesh. Given a sensor network deployed on the surface of a 3D terrain with one-hop distance information available, we can extract a triangular mesh from the connectivity graph of the network. The constraint that the sensors must be on the known 3D terrain's surface ensures that the triangular meshes of the network and the terrain's surface overlap and approximate the same geometric shape. The basic idea of the localization algorithms is to map the two triangular meshes extracted from the connectivity graph of a sensor network and the DTM of its deployed terrain surface to the plane. The two meshes mapped to the plane can be easily aligned if the location information of anchor nodes is available. We introduce a fully distributed algorithm to construct a well-aligned mapping between the two triangular meshes in the plane based on anchor nodes information. However, accidents may happen on anchor nodes. We then introduce an anchor-free algorithm to extract feature points with geometric properties intrinsic to surface distances and independent of the embedding of the two meshes in 3D. The matched feature points induce transformations to align the two meshes in the plane. With the aligned triangular meshes of a network and its deployed terrain surface, each sensor node of the network can easily locate reference grid points from the DTM of the terrain to calculate its own geographic location. We carry out extensive simulations under various scenarios to evaluate the overall performance of the proposed algorithms with different factors such as the one-hop distance measurement error, the resolution of a DTM, and the performance of the algorithm in the situation of connectivity only.

This paper considers a fundamental problem of designing routing scheme resilient to node or link failures for wireless sensor networks deployed on a surface of a complex-connected three-dimensional (3D) setting. Instead of heuristically detouring around the failed path, we borrow homotopy, an important topological concept, to effectively create and evaluate the diversity of alternative paths. We propose a tessellation-free and GPS-free method to compute paths with different homotopy types on surface networks. A source node greedily forwards a packet to its destination based on the computed nodes’ virtual planar coordinates. When the current path fails, the source node can flexibly choose another greedy path from a different homotopy type to deliver the packet. The proposed algorithms are distributed and scalable to both the size and genus number of a surface network. We evaluate the performance of the proposed routing scheme under three different failure models. Simulation results show that our method achieves the best performance under geographically correlated failure models compared with other resilient routing schemes. We also compare our routing scheme with existing state-of-the-art ones specifically designed for surface networks when a network is failure free. Our method achieves the lowest stretch factor.

Social networking has been one of the fastest growing information technologies as evidenced by the popularity of online social network (OSN) sites. These highly active OSNs generate an enormous volume of data as well as work load every day. A cost-effective solution is horizontal scaling where an OSN is partitioned and deployed on a set of low-cost servers. The goal of the paper is to achieve an optimal partitioning by minimizing the overall cost (sum of the inter-server write traffic cost and moving cost) while maintaining a load balance across servers. Given the NP-hardness of the problem, we introduce a deep learning based model for incremental online learning and dynamic link prediction. We then propose a Dynamic Link Prediction based online algorithm named FLOAT that incorporates the predicted future link information into the online user assignment. Relying upon future and current link based node relocation/swap gain estimations (Adjusted Server Change Benefit (ASCB) and Adjusted Server Exchange Benefit (ASXB), FLOAT strategically assigns user nodes across servers. The simulation results confirm that a projected benefit based on the knowledge of future links help reduce the overall cost significantly compared with existing algorithms, at the same time, maintaining a low inter-server write traffic cost.

Wireless Rechargeable Sensor Network (WRSN), consisting of sensors with rechargeable batteries and mobile chargers, has become a promising solution to the energy limitation problem in Wireless Sensor Networks (WSNs). Charger scheduling optimization focuses on optimizing the trajectory of mobile chargers to prolong the life of a WRSN system. Charger scheduling optimization problems are in general NP-hard. Previous solutions using traditional algorithms often require problem-specific design and a trade-off between the performance and computing time. An insight into these optimization problems is that a domain-specific charger scheduling strategy could be learned automatically when the objective function of an optimization problem is considered as a reward. We model charger scheduling optimization problems using a weighted graph and consider the objective function as a cumulative reward of charging sensors along a charging path in one cycle. We then build a deep reinforcement learning based framework to solve a diverse range of charger scheduling optimization problems. The biggest advantage of the framework is that an optimal charger scheduling strategy can be learned from previous experiences, i.e., different graphs with various sizes. A framework also simplifies the complexity of algorithm design for individual charger scheduling optimization problem. We compare the performance of algorithms based on the proposed framework with traditional ones on a set of selected charger scheduling optimization problems. They outperform all existing algorithms.

With the remarkable proliferation of intelligent mobile devices and fast growing broadband wireless technology, social networking is undergoing explosive growth in recent years as more and more users access social networks via mobile platforms. It is often expensive or even impossible to deploy a large online social network (OSN) on a single server. A cost-effective approach is horizontal scaling, where the OSN is partitioned and deployed on a set of low-cost servers. In this research, we study the problem of minimum-cost balanced partitioning of OSNs. Our goal is to achieve the best partitioning by minimizing the total inter-server traffic cost and at the same time balancing the load among servers. Given its NP-hardness, we propose new techniques and explore efficient heuristics to address the problem, especially for extremely large OSNs with an enormous volume of social nodes, social connections, and social data. Our key contributions include a localized approach with O(δ2) complexity to explicitly calculate the projected gain in inter-server traffic cost (named Server Change Benefit (SCB)). Built upon this technique, we devise two algorithms that offer online and offline solutions to achieving minimum-cost balanced partitioning of OSNs. The online algorithm is fast and highly efficient to process newly arrival individual nodes. The offline algorithm uses the current online result as a starting point. It further reduces inter-server traffic cost by applying relocation and swapping. It employs a merging process to group the nodes according to the social structure and swap the groups with similar size to further reduce the total inter-server traffic cost. We implement both algorithms and evaluate them based on a variety of real-world OSN datasets from Facebook, Arxiv, Gnutella, Amazon, and Twitter. The simulations demonstrate that the proposed algorithms can significantly reduce the execution time by an average of three folds and at the same time yield supreme performance (i.e., inter-server traffic cost) in comparison with existing solutions.

The recent advances in sensors, actuators, robots, and mobile wireless communication technologies have acceleratedinterest in autonomous networked robots (ANRs), where theindividual robots coordinate among themselves to complete atask, e.g., to explore or monitor a Field of Interest (FoI). Byteamwork, which is especially important in complex tasks, ANRsystem expresses much more capacity than traditional staticsensor networks. Existing work focuses on improving the coverageperformance of a group of ANRs within a single FoI. In thisresearch, we consider a group of ANRs that are instructed toexplore a number of FoIs. After they complete a task at currentFoI, they move to the next one, which may be far away fromcurrent one and the shape can also vary dramatically. Ourresearch focuses on how to efficiently enable such transition. TheANRs must be able to redeploy themselves to desired positionsin the new FoI based on distributed algorithms. Besides, to avoidunexpected event breaks network's integrity, the ANRs shouldpreserve their local connectivities as much as they can andorganize themselves as a whole network without any isolatednodes during the transition. Furthermore, considering energyconsumption, such relocation algorithm should work at the costof reasonable total moving distance. We study this problemand call it optimal marching of autonomous networked robots. The proposed algorithms guarantee global connectivity, andpreserve local connectivities as much as possible at negligiblecost of moving distance. Additionally, ANRs can automaticallyadjust their deployment density in the new FoI based on therequirements of various tasks or regions.

Distributed in-network data-centric processing aims to reduce energy consumed for communication and establish a self-contained data storage, retrieval, aggregation, and query sensor system that focuses more on the data itself rather than the identities of the individual network nodes. Double-ruling-based schemes support efficient in-network data-centric information storage and retrieval, especially for aggregated data, since all data with different types generated in a network can be conveniently retrieved along any single retrieval curve. Previous double-ruling-based research focuses on two-dimensional (2-D) wireless sensor networks where a 2-D planar setting is assumed. With increasing interests in deploying wireless sensors in three-dimensional (3-D) space for various applications, it is urgent yet fundamentally challenging to design double-ruling-based approach in general 3-D sensor networks because double-ruling-based schemes in general have much harder geometric constraints than other distributed in-network data-centric processing schemes. In this research, we propose a geographic location-free double-ruling-based approach for general 3-D sensor networks with possibly complicated topology and geometric shapes. Without the knowledge of the geographic location and the distance bound, a query simply travels along a simple curve with the guaranteed success to retrieve aggregated data through time and space with one or different types across the network. Extensive simulations and comparisons show the proposed scheme with low cost and a balanced traffic load.

This research focuses on distributed and localized algorithms for precise boundary detection in 3-D wireless networks. Our objectives are twofold. First, we aim to identify the nodes on the boundaries of a 3-D network, which serve as a key attribute that characterizes the network, especially in such geographic exploration tasks as terrain and underwater reconnaissance. Second, we construct locally planarized 2-manifold surfaces for inner and outer boundaries in order to enable available graph theory tools to be applied on 3-D surfaces, such as embedding, localization, partition, and greedy routing among many others. To achieve the first objective, we propose a Unit Ball Fitting (UBF) algorithm that discovers a majority of boundary nodes, followed by a refinement algorithm, named Isolated Fragment Filtering (IFF), to remove isolated nodes that are misinterpreted as boundary nodes. Based on the identified boundary nodes, we develop an algorithm that constructs a locally planarized triangular mesh surface for each 3-D boundary. Our proposed scheme is localized, requiring information within 1-hop neighborhood only. We further extend the schemes for online boundary detection in mobile sensor networks aiming to achieve low overhead. Our simulation and experimental results demonstrate that the proposed algorithms can effectively identify boundary nodes and surfaces, even under high measurement errors.

The majority of current research on sensor network localization focuses on wireless sensor networks deployed on two dimensional (2D) plane or in three dimensional (3D) space, very few on 3D surface. However, many real world applications require large-scale sensor networks deployed on the surface of a complex 3D terrain. Compared with planar and 3D network localizations, surface network localization generates unique and fundamental hardness. In this research, we explore 3D surface network localization with terrain model. A digital terrain model (DTM), available to public with a variable resolution up to one meter, is a 3D representation of a terrain's surface. It is commonly built using remote sensing technology or from land surveying and can be easily converted to a triangular mesh. Given a sensor network deployed on the surface of a 3D terrain with one-hop distance information available, we can extract a triangular mesh from the connectivity graph of the network. The constraint that the sensors must be on the known 3D terrain's surface ensures that the triangular meshes of the network and the DTM of the terrain's surface approximate the same geometric shape and overlap. We propose a fully distributed algorithm to construct a well-aligned mapping between the two triangular meshes. Based on this mapping, each sensor node of the network can easily locate reference grid points from the DTM to calculate its own geographic location. We carry out extensive simulations under various scenarios to evaluate the overall performance of the proposed localization algorithm. We also discuss the possibility of 3D surface network localization with mere connectivity and the results are promising.

We propose a distributed and deterministic routing algorithm with constant storage, communication and computation overhead, dubbed trace-routing, for strong-connected 3D wireless sensor networks. Its basic idea is to construct a virtual cutting plane that intersects boundary surface to yield a trace, along which a routing path with guaranteed delivery can be established. We prove the correctness of trace-routing under both continuous and discrete settings. We implement the trace-routing algorithm on Crossbow sensors and carry out extensive simulations to evaluate its routing efficiency.

This paper focuses on greedy routing in wireless networks deployed on 2-D and 3-D surfaces. It introduces a distributed embedding scheme based on the conformal map theory. The proposed scheme identifies the convex hull of each boundary and employs Yamabe flow to compute flat metric under convex hull boundary condition to establish virtual coordinates. Such virtual coordinates are then used for greedy routing. Since the proposed embedding algorithm maps the outer boundary to a convex shape and an interior concave void to a circle-like convex polygon, it effectively eliminates local minimum and attains guaranteed delivery. At the same time, it introduces a small distortion only and consequently achieves a low stretch factor. Our simulations show that its stretch factor is lower than any existing greedy embedding algorithms. Moreover, the proposed scheme is merely based on local connectivity and consumes a small constant storage, thus scaling to arbitrarily large networks.

The medial axis of a shape provides a compact abstraction of its global topology and a proximity of its geometry. The construction of medial axis in two-dimensional (2D) sensor networks has been discussed in the literature, in support of several applications including routing and navigation. In this work, we first reveal the challenges of constructing medial axis in a three-dimensional (3D) sensor network. With more complicated geometric features and complex topology shapes, previous methods proposed for 2D settings cannot be extended easily to 3D networks. Then we propose a distributed algorithm with linear time complexity and communication cost to build a well-structured medial axis of a 3D sensor network without knowing its global shape or global position information. Furthermore we apply the computed medial axis for safe navigation and distributed information storage and retrieval in 3D sensor networks. Simulations are carried out to demonstrate the efficiency of the proposed medial axis-based applications in various 3D sensor networks.

We address the problem of in-network information processing, storage, and retrieval in three-dimensional (3D) sensor networks in this research. We propose a geographic location free double-ruling-based scheme for large-scale 3D sensor networks. The proposed approach does not require a 3D sensor network with a regular cube shape or uniform node distribution. Without the knowledge of the geographic location and the distance bound, a data query simply travels along a simple curve with the guaranteed success to retrieve aggregated data through time and space with one or different types across the network. Simulations and comparisons show the proposed approach with low cost and a balanced traffic load.

Location awareness is imperative for a variety of sensing applications and network operations. Although a diversity of GPS-less and GPS-free solutions have been developed recently for autonomous localization in wireless sensor networks, they primarily target at 2D planar or 3D volumetric settings. There exists unique and fundamental hardness to extend them to 3D surface. The contributions of this work are twofold. First, it proposes a theoretically-proven algorithm for the 3D surface localization problem. Seeing the challenges to localize general 3D surface networks and the solvability of the localization problem on single-value (SV) surface, this work proposes the {\em cut-and-sew} algorithm that takes a divide-and-conquer approach by partitioning a general 3D surface network into SV patches, which are localized individually and then merged into a unified coordinates system. The algorithm is optimized by discovering the minimum SV partition, an optimal partition that creates a minimum set of SV patches. Second, it develops practically-viable solutions for real-world sensor network settings where the inputs are often noisy. The proposed algorithm is implemented and evaluated via simulations and experiments in an indoor testbed. The results demonstrate that the proposed cut-and-sew algorithm achieves perfect 100% localization rate and the desired robustness against measurement errors.

A wireless sensor network can be represented by a graph. While the network graph is extremely useful, it often exhibits undesired irregularity. Therefore, special treatment of the graph is required by a variety of network algorithms and protocols. In particular, many geometry-oriented algorithms depend on a type of subgraph called Delaunay triangulation. However, when location information is unavailable, it is nontrivial to achieve Delaunay triangulation by using connectivity information only. The only connectivity-based algorithm available for Delaunay triangulation is built upon the property that the dual graph for a Voronoi diagram is a Delaunay triangulation. This approach, however, often fails in practical wireless sensor networks because the boundaries of Voronoi cells can be arbitrarily short in discrete sensor network settings. In a sensor network with connectivity information only, it is fundamentally unattainable to correctly judge neighboring cells when a Voronoi cell boundary is less than one hop. Consequently, the Voronoi diagram-based Delaunay triangulation fails. The proposed algorithm employs a distributed approach to perform centroidal Voronoi tessellation, and constructs its dual graph to yield Delaunay triangulation. It exhibits several distinctive properties. First, it eliminates the problem due to short cell boundaries and thus effectively avoids crossing edges. Second, the proposed algorithm is proven to converge and succeed in constructing a Delaunay triangulation, if the CVT cell size is greater than a constant threshold. Third, the established Delaunay triangulation consists of close-to-equilateral triangles, benefiting a range of applications such as geometric routing, localization, coverage, segmentation, and data storage and processing. Extensive simulations are carried out under various 2D network models to evaluate the effectiveness and efficiency of the proposed CVT-based triangulation algorithm.

Sensor deployment is a fundamental issue in a wireless sensor network, which often dictates the overall network performance. Previous studies on sensor deployment mainly focused on sensor networks on 2D plane or in 3D volume. In this paper, we tackle the problem of optimal sensor deployment on 3D surfaces, aiming to achieve the highest overall sensing quality. In general, the reading of a sensor node exhibits unreliability, which often depends on the distance between the sensor and the target to be sensed, as observed in a wide range of applications. Therefore, with a given set of sensors, a sensor network offers different accuracy in data acquisition when the sensors are deployed in different ways in the Field of Interest (FoI). We formulate this optimal surface deployment problem in terms of sensing quality by introducing a general function to measure the unreliability of monitored data in the entire sensor network. We present its optimal solution and propose a series of algorithms for practical implementation. Extensive simulations are conducted on various 3D mountain surface models to demonstrate the effectiveness of the proposed algorithms.

This work aims to address the problem of localization in 3D surface wireless sensor networks. First, it reveals the unique hardness in localization on 3D surface in comparison with the well-studied localization problems in 2D and 3D space, and offers useful insight into the necessary conditions to achieve desired localizability. Second, it formulates the localization problem under a practical setting with estimated link distances (between nearby nodes) and nodal height measurements, and introduces a layered approach to promote the localizability of such 3D surface sensor networks. Crossbow sensor-based experiments and large-scale simulations are carried out to evaluate the performance of the proposed localization algorithm. The numeric results show that it can effectively improve localizable rate and achieve low location errors and computational overhead, with the desired tolerability to measurement errors and high scalability to large-size wireless sensor networks.

In this work we develop a distributed boundary detection algorithm, dubbed Coconut, for 3D wireless sensor networks. It first constructs a tetrahedral structure to delineate the approximate geometry of the 3D sensor network, producing a set of “sealed” triangular boundary surfaces for separating non-boundary nodes and boundary node candidates. The former are hollowed out immediately while the latter are further refined to yield the final boundary nodes and fine-grained boundary surfaces. The proposed Coconut algorithm offers several salient features. First, it requires connectivity information only, with no need for localization or distance measurement. Second, it does not rely on particular communication models, but only assumes a constant maximum transmission range, which is generally known in practical wireless sensor networks. Third, it is robust to sensor distribution, effectively identifying boundaries in both uniformly and non-uniformly distributed sensor networks. We prove the correctness of the algorithm and quantitatively demonstrate its effectiveness via simulations under various network models.

Compared with its 2D counterpart, the scalability problem is greatly exacerbated in a 3D wireless sensor network. In this paper, we propose a scalable routing algorithm, dubbed Bubble Routing. It preprocesses global knowledge via a distributed algorithm, such that a node only needs to store a small constant information to make correct and efficient local routing decisions and achieve guaranteed delivery at the same time. More specifically, the proposed bubble routing algorithm first decompose a 3D network into a set of hollow spherical cells (HSCs). A continuous and one-to-one mapping is applied and a virtual tree structure is established inside each HSC to enable greedy routing. On the other hand, routing across HSCs is guided by a small routing table whose size is bounded by the number of interior holes. Our simulation results show that bubble routing can achieve guaranteed data delivery, low stretch factor, and well balanced traffic load.

This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and non-uniform nodal distribution. In contrast to current state-of-art connectivity-based localization methods, the proposed algorithm is fully distributed, where each node only needs the information of its neighbors, without cumbersome partitioning and merging process. The algorithm is highly scalable, with limited error propagation and linear computation and communication cost with respect to the size of the network. Moreover, the algorithm is theoretically guaranteed and numerically stable. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing superior performance of the proposed algorithm.

Triangulation serves as the basis for many geometry-based algorithms in wireless sensor networks. In this paper we propose a distributed algorithm that produces a triangulation for an arbitrary sensor network, with no constraints on communication model or granularity of the triangulation. We prove its correctness in 2D, and further extend it to sensor networks deployed on 3D open and closed surfaces. Our simulation results show that the proposed algorithms can tolerate distance measurement errors, and thus work well under practical sensor network settings and effectively promote the performance a range of applications that depend on triangulations.

With both computational complexity and storage space bounded by a small constant, greedy routing is recognized as an appealing approach to support scalable routing in wireless sensor networks. However, significant challenges have been encountered in extending greedy routing from 2D to 3D space. In this research we develop decentralized solutions to achieve greedy routing in 3D sensor networks. Our proposed approach is based on a unit tetrahedron cell (UTC) mesh structure. We propose a distributed algorithm to realize volumetric harmonic mapping of the UTC mesh under spherical boundary condition. It is a one-to-one map that yields virtual coordinates for each node in the network. Since a boundary has been mapped to a sphere, node-based greedy routing is always successful thereon. At the same time, we exploit the UTC mesh to develop a face-based greedy routing algorithm, and prove its success at internal nodes. To deliver a data packet to its destination, face-based and node-based greedy routing algorithms are employed alternately at internal and boundary UTCs, respectively. As far as we know, this is the first work that realizes truly deterministic greedy routing with constant-bounded storage and computation in 3D wireless sensor networks.

Segmentation decomposes a network with complex and irregular shape into a set of subnetworks, each under a simple boundary condition without bottlenecks. It has a wide spectrum of applications in routing, coverage, localization, backbone construction and maintenance, and in-network data centric storage and retrieval. To our best knowledge, this is the first work that tackles the segmentation problem in 3D wireless sensor networks. We propose a fully distributed 3D segmentation scheme with mere network connectivity information. Each node on boundary computes its injectivity radius, which reflects the narrowness of the corresponding boundary area and thus is employed to locate the undesired bottlenecks. A cluster of connected boundary nodes with similar smallest injectivity radii form a bottleneck segment. A recursive process is applied to identify a set of such bottlenecks, which together divide the network boundary into segments. An internal non-boundary node simply joins the nearest segment, thus completing the segmentation of the entire 3D sensor network. Our simulations show that the proposed algorithm works efficiently under various sensor network models with different boundary conditions and noise levels, always yielding appropriate segmentation results. We further demonstrate that segmentation can effectively promote the performance of a range of applications in 3D wireless sensor networks.

This research focuses on distributed and localized algorithms for precise boundary detection in 3D wireless networks. Our objectives are in two folds. First, we aim to identify the nodes on the boundaries of a 3D network, which serve as a key attribute that characterizes the network, especially in such geographic exploration tasks as terrain and underwater reconnaissance. Second, we construct locally planarized 2-manifold surfaces for inner and outer boundaries, in order to enable available graph theory tools to be applied on 3D surfaces, such as embedding, localization, partition, and greedy routing among many others. To achieve the first objective, we propose a Unit Ball Fitting (UBF) algorithm that discovers a set of potential boundary nodes, followed by a refinement algorithm, named Isolated Fragment Filtering (IFF), which removes isolated nodes that are misinterpreted as boundary nodes by UBF. Based on the identified boundary nodes, we develop an algorithm that constructs a locally planarized triangular mesh surface for each 3D boundary. Our proposed scheme is localized, requiring information within one-hop neighborhood only. Our simulation results demonstrate that the proposed algorithms can effectively identify boundary nodes and surfaces, even under high measurement errors. As far as we know, this is the first work for discovering boundary nodes and constructing boundary surfaces in 3D wireless networks.

The centroidal Voronoi tessellation (CVT) has found versatile applications in geometric modeling, computer graphics, and visualization. In this paper, we extend the concept of the CVT from Euclidean space to hyperbolic space. A novel hyperbolic CVT energy is defined, and the relationship between minimizing this energy and the hyperbolic CVT is proved. We also show by our experimental results that the hyperbolic CVT has the similar property as its Euclidean counterpart where the sites are uniformly distributed according to given density values. Two algorithms -- Lloyd's algorithm and the L-BFGS algorithm -- are adopted to compute the hyperbolic CVT, and the convergence of Lloyd's algorithm is proved. As an example of the application, we utilize the hyperbolic CVT to compute uniform partitions and high-quality remeshing results for high-genus (genus>1) surfaces.

Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user-defined field topologies. The user has full control of the number, positions, and indexes of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility such that it can design N-RoSy fields with single singularity and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.

Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the potential value of the method for geometric database indexing, shape comparison and classification.

Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the efficiency and efficacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison.

Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.

Polycube T-spline has been formulated elegantly that can unify T-splines and manifold splines to define a new class of shape representations for surfaces of arbitrary topology by using polycube map as its parametric domain. In essense, The data fitting quality using polycube T-splines hinges upon the construction of underlying polycube maps. Yet, existing methods for polycube map construction exhibit some disadvantages. For example, existing approaches for polycube map construction either require projection of points from a 3D surface to its polycube approximation, which is therefore very difficult to handle the cases when two shapes differ significantly; or compute the map by conformally deforming the surfaces and polycubes to the common canonical domain and then construct the map using function composition, which is challenging to control the location of singularities and makes it hard for the data-fitting and hole-filling processes later on.

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in computer graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art inter-surface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is sorely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards numerically computing globally optimal maps. Consequently, our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching.

Summary form only given. In this paper we generalize the shortest path algorithm to the shortest cycles in each homotopy class on a surface with arbitrary topology, utilizing the universal covering space (UCS) in algebraic topology. In order to store and handle the UCS, we propose a two-level data structure which is efficient for storage and easy to process. We also pointed several practical applications for our shortest cycle algorithms and the UCS data structure.

This paper develops a novel computational technique to define and construct powerful manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite their earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g -- 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promising to promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

Surface classification is one of the most fundamental problems in geometric modeling. Surfaces can be classified according to their conformal structures. In general, each topological equivalent class has infinite conformally equivalent classes.

Three-dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition, and classification. However, shape matching with noise, occlusion, and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps including harmonic maps, conformal maps, and least-squares conformal maps with regards to 3D shape matching. As a result, we propose a novel and computationally efficient shape matching framework by using least-squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise. Therefore, highly accurate and efficient 3D shape matching algorithms can be achieved by using the above three parametric maps. Finally, the robustness of least-squares conformal maps is evaluated and analyzed comprehensively in 3D shape matching with occlusion, noise, and resolution variation. In order to further demonstrate the performance of our proposed method, we also conduct a series of experiments on two computer vision applications, i.e., 3D face recognition and 3D nonrigid surface alignment and stitching.

3D surface matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition and classification. However, surface matching with noise, occlusion and clutter is a challenging problem. In this paper, we analyze a family of conformal geometric maps including harmonic maps, conformal maps and least squares conformal maps with regards to 3D surface matching. As a result, we propose a novel and computationally efficient surface matching framework that uses least squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D surface-matching problem to a 2D image-matching problem, by comparing the resulting 2D conformal geometric maps, which are stable, insensitive to resolution changes and robust to occlusion and noise. Therefore, highly accurate and efficient 3D surface matching algorithms can be achieved by using conformal geometric maps. Finally, the performance of conformal geometric maps is evaluated and analyzed comprehensively in 3D surface matching with occlusion, noise and resolution variation. We also provide a series of experiments on real 3D face data that achieve high recognition rates.

We present an efficient colon flattening algorithm using a conformal structure, which is angle-preserving and minimizes the global distortion. Moreover, our algorithm is general as it can handle high genus surfaces. First, the colon wall is segmented and extracted from the CT data set of the abdomen. The topology noise (i.e., minute handle) is located and removed automatically. The holomorphic 1-form, a pair of orthogonal vector fields, is then computed on the 3D colon surface mesh using the conjugate gradient method. The colon surface is cut along a vertical trajectory traced using the holomorphic 1-form. Consequently, the 3D colon surface is conformally mapped to a 2D rectangle. The flattened 2D mesh is then rendered using a direct volume rendering method accelerated with the GPU. Our algorithm is tested with a number of CT data sets of real pathological cases, and gives consistent results. We demonstrate that the shape of the polyps is well preserved on the flattened colon images, which provides an efficient way to enhance the navigation of a virtual colonoscopy system.

Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces. Algorithms designed for planar domains based on a specific geometry can be systematically generalized to surface domains via the corresponding geometric structure. For example, polar form splines with planar domains are based on affine invariants. Polar form splines can be generalized to manifold splines on the surfaces which admit affine structures and are equipped with affine geometries.Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. All surfaces admit real projective structures and are equipped with real projective geometry. Because of their general existence, both hyperbolic structures and real projective structures have the potential to replace the role of affine structures in defining manifold splines.This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The method is based on a novel geometric tool - discrete variational Ricci flow. Any metric surface admits a special uniformization metric, which is conformal to its original metric and induces constant curvature. Ricci flow is an efficient method to calculate the uniformization metric, which determines the hyperbolic structure and real projective structure.The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure.More importantly, this work introduces the framework of general geometric structures, which enable different geometries to be defined on manifolds and lay down the theoretical foundation for many important applications in geometric modeling.

Topological concepts and techniques have been broadly applied in computer graphics and geometric modeling. However, the homotopy type of a mapping between two surfaces has not been addressed before. In this paper, we present a novel solution to the problem of computing continuous maps with different homotopy types between two arbitrary triangle meshes with the same topology. Inspired by the rich theory of topology as well as the existing body of work on surface mapping, our newly-developed mapping techniques are both fundamental and unique, offering many attractive advantages. First, our method allows the user to change the homotopy type or global structure of the mapping with minimal intervention. Moreover, to locally affect shape correspondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuristics to ensure validity. In addition to acting as a useful tool for computer graphics applications, our method can be used as a rigorous and practical mechanism for the visualization of abstract topological concepts such as homotopy type of surface mappings, homology basis, fundamental domain, and universal covering space. At the core of our algorithm is a procedure for computing the canonical homology basis and using it as a common cut graph for any surface with the same topology. We demonstrate our results by applying our algorithm to shape morphing in this paper.

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.

Bayesian belief networks (BBN) are a widely studied graphical model for representing uncertainty and probabilistic interdependence among variables. One of the factors that restricts the model's wide acceptance in practical applications is that the general inference with BBN is NP-hard. This is also true for the maximum a posteriori probability (MAP) problem, which is to find the most probable joint value assignment to all uninstantiated variables, given instantiation of some variables in a BBN. To circumvent the difficulty caused by MAP's computational complexity, we suggest in this paper a neural network approximation approach. With this approach, a BBN is treated as a neural network without any change or transformation of the network structure, and the node activation functions are derived based on an energy function defined over a given BBN. Three methods are developed. They are the hill-climbing style discrete method, the simulated annealing method, and the continuous method based on the mean field theory. All three methods are for BBN of general structures, with the restriction that nodes of BBN are binary variables. In addition, rules for applying these methods to noisy-or networks are also developed, which may lead to more efficient computation in some cases. These methods' convergence is analyzed, and their validity tested through a series of computer experiments with two BBN of moderate size and complexity. Although additional theoretical and empirical work is needed, the analysis and experiments suggest that this approach may lead to effective and accurate approximation for MAP problems.

The maximum a posteriori probability (MAP) problem is to find the most probable instantiation of all uninstantiated variables, given an instantiation of a set of variables in a Bayesian belief network (BBN). MAP is known to be NP-hard. To circumvent the high computational complexity, we propose a neural network approach based on the mean field theory to approximate the MAP problem. In this approach, a given BBN is treated as a neural network with an energy function defined in such a way that the MAP solution corresponds to the global minimum energy state. The mean field equation is then derived. We also propose a method called resettling to further improve the solution accuracy. A series of computer experiment shows that this approach may lead to effective and accurate solutions to MAP problems.

We suggest a neural network approach to probabilistic inference in Bayesian belief networks (BBN). This is demonstrated by solving maximum a posteriori probability (MAP) problems, which are known to be NP-hard. In this approach, a belief network is treated as a neural network without any structural changes, and the node activation functions are derived based on the probabilistic calculus of the BBN. Three models are proposed and their convergence analyzed. Computer experiments with two non-trivial example BBN show that this approach may lead to effective approximation methods for MAP.