Discrete Differential Geometry and Variational Methods

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Since 2005, this line of research has formed the mathematical foundation of my work in geometric computing. It develops discrete differential geometry and variational methods as a unifying framework for a broad spectrum of problems in imaging and graphics. By integrating geometric modeling, optimization, and partial differential equations, this research establishes principled formulations for both theoretical analysis and algorithmic design. These methods support high-level applications across 2D imaging and 3D graphics, architectural geometry, multimedia, computer-aided design, medical imaging, and wireless sensor networking. Although these domains differ in dimension and application context, they share a common computational structure: geometric quantities are discretized and formulated as variational or intrinsic problems, and solutions are obtained through structured optimization and differential geometry tools. This geometry-driven perspective provides strong theoretical foundations while enabling robust and scalable algorithms across diverse imaging and graphics applications.

Poisson Vector Graphics

Poisson Vector Graphics (PVG) is a unified vector graphics framework in which diffusion curves arise as a special case. By extending the classical diffusion-curve formulation, PVG introduces additional geometric primitives, including Poisson curves and Poisson regions, substantially enriching expressive power and modeling flexibility. As a geometry-aware and resolution-independent representation, PVG encodes visual appearance using a compact set of vector primitives reconstructed through a gradient-domain (Poisson) formulation. This enables faithful reproduction of smooth shading and fine details while supporting intuitive editing, stylization, and transfer. Together, these works establish PVG as a unified vectorization and editing pipeline that bridges 2D images, videos, and 3D surfaces with compact representation and high visual fidelity.

Architectural Geometry

These works investigate architectural geometry with an emphasis on designing freeform structures that are both expressive and structurally feasible. A central theme is the construction of self-supporting surfaces, an inherently over-constrained problem that is widely recognized as computationally challenging to solve in practice. In our TOG 2019 work, we introduced a fundamentally new computational framework by proving that 3D self-supporting surfaces can be characterized as the hyper-generatrix of 4D minimal hypersurfaces of revolution. This result establishes a completely different roadmap from conventional approaches, which typically reduce the problem via 3D-to-2D projection and compute approximate solutions restricted to height functions. In contrast, our formulation naturally accommodates non-height-function geometries and enables interactive exploration of the feasible solution space, provided the boundary conditions are valid. Building on this foundation, subsequent contributions address practical constructability constraints, including arch-beam layouts (TVCG 2025) and planar quadrilateral elements (CVMJ 2022). Beyond self-supporting surfaces, we further investigate discrete geometric structures that support modular construction (SIGGRAPH 2023) and tiling (SIGGRAPH 2010).

Voronoi Diagrams and Delaunay Triangulations/Meshes

This line of work builds a practical geometry engine around Voronoi–Delaunay structures, with an emphasis on intrinsic (geodesic) and weighted distances on meshes and in 3D. Starting from efficient and parallelizable constructions in the Euclidean setting (CAD 2013, CVMJ 2015), we move to surface-based Voronoi diagrams (CAD 2014, CGF 2015, SIGGRAPH Asia 2016, GMOD 2023). On the dual side, we exploit Voronoi–Delaunay duality to build intrinsic Delaunay triangulations (TOG 2017) and Delaunay meshes that improve numerical stability for downstream geometry processing, while also enabling controllable simplification (SIGGRAPH Asia 2015 & 2018). We also extend beyond classical Voronoi diagrams to weighted variants (e.g., Apollonius diagrams) and robust 3D algorithms, making these fundamental structures usable in real pipelines. Together, these works turn Voronoi/Delaunay theory into reliable, scalable, and application-ready tools for remeshing, parameterization, and geometric computation.

Computer-Generated Line Drawings

These works develop geometry-aware algorithms for extracting expressive, illustrator-style line drawings from 3D surfaces and volumes in real time. A unifying theme is to treat feature lines as outcomes of principled differential or variational signals defined on geometry, so that the resulting drawings are both visually meaningful (capturing salient shape cues) and computationally stable. Across the series, we introduce and refine families of line descriptors (e.g., photic-extremum-based and Laplacian-based lines), improve robustness using filtering ideas inspired by image processing (e.g., Difference-of-Gaussian), and design efficient GPU-friendly implementations suitable for interactive systems. Collectively, this line of research bridges non-photorealistic rendering and discrete differential geometry, turning low-level geometric operators into controllable, high-level visual communication tools.

Parameterization, Meshing and Splines

Sketching Interfaces for Imaging and Graphics

Optimization for 3D Design

Dynamic 3D Data Compression

This line of research develops geometry-driven frameworks for efficient compression of dynamic 3D data, including human motion capture and time-varying facial expressions. Our methods can be broadly grouped into two categories. First, we introduced geometry video representations that convert time-varying 3D geometry into regular video-like signals, enabling the direct reuse of mature video coding techniques for highly efficient encoding of dynamic meshes and articulated motions. Second, beyond geometry videos, we developed optimization-based compression models for motion capture sequences using low-rank approximation, sparse representations, tensor decomposition, and learned decorrelation transforms. These methods exploit intrinsic geometric and temporal coherence to achieve compact, scalable, and low-latency representations. Together, this work demonstrates how geometric reformulation and principled optimization can significantly improve compression efficiency for complex 3D dynamic content.

Geometry-Aware Point Cloud Filtering and Feature Analysis

This line of research develops geometry-aware methods for robust filtering and feature analysis of 3D point clouds. Unlike grid-based image data, point clouds are unstructured and irregular, making classical signal processing techniques inadequate. We formulate filtering and feature detection problems using variational principles, intrinsic operators, and low-rank modeling, enabling accurate geometry preservation under noise and sampling irregularity. These methods provide strong robustness, theoretical grounding, and practical effectiveness for large-scale 3D data processing.

Geometric Approaches for Image Processing

We apply intrinsic geometry processing ideas to image analysis by modeling an image as a 2D manifold embedded in a higher-dimensional feature space (e.g., spatial coordinates + color), so that intrinsic distances capture both spatial proximity and appearance variation. Manifold SLIC (CVPR 2016) and Intrinsic Manifold SLIC (TPAMI 2018) extend the classic SLIC framework by clustering pixels using manifold-based intrinsic distances, producing superpixels that better align with object boundaries and fine-scale structures while remaining computationally efficient. Our subsequent work (ICCV 2019) further improves scalability and generality by enabling fast intrinsic-distance computation for 3D supervoxels, extending the framework to video over-segmentation. We also extend this geometric viewpoint beyond segmentation by constructing a field-aligned quadrilateral structure that follows dominant image directions, enabling compact and structure-aware image vectorization (CGF 2019).

Geometric Approaches for Wireless Sensor Networks

These works show that many problems in wireless sensor networks are fundamentally geometric, because sensors are embedded in physical space and collectively sample an underlying spatial domain. By modeling a network as a discrete surface or spatial graph and leveraging intrinsic constructions, such as harmonic maps, geometric parameterization, and Voronoi-based partitioning, we developed distributed and scalable algorithms for k-surface coverage and load balancing (ICDCS 2012, TWC 2015), boundary detection and parameterization in irregular 3D deployments (MobiHoc 2011, TON 2014), and topology-aware data management in networks with holes (SECON 2012). We further translated these geometric insights into practical indoor systems, including lightweight scene reconstruction for natural localization (IoT-J 2019) and multipath-enabled joint source localization and space scanning (SenSys 2014). Overall, geometry provides a unifying language that turns noisy, irregular sensing deployments into structured computations with strong robustness and scalability.

Geometric Approaches for Medical Imaging

These works demonstrate how intrinsic geometric representations, such as geodesic distances, conformal mappings, and discrete Ricci flow, provide principled and effective tools for analyzing complex 3D anatomical surfaces. We developed robust registration frameworks and statistically grounded morphometric analyses for vestibular systems and brainstem surfaces in adolescent idiopathic scoliosis (AIS). We also applied geodesic facial measurements derived from 3D stereophotogrammetry to quantify craniofacial variation and to stratify autism spectrum disorder (ASD) into clinically meaningful subgroups. Across these studies, geometry-driven modeling ensures consistent surface correspondences, remains stable under complex topology, and transforms rich surface geometry into interpretable biomarkers for quantitative medical morphometry.

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