[Paper Review] Geodesic Centroidal Voronoi Tessellations: Theories, Algorithms and Applications

Zipeng Ye, Ran Yi+3 more|arXiv (Cornell University)|Jul 1, 2019

Computational Geometry and Mesh Generation32 references5 citations

TL;DR

This paper proposes geodesic centroidal Voronoi tessellations (GCVTs) as intrinsic geometric structures on triangulated manifold meshes, enabling uniform remeshing and content-sensitive superpixel/supervoxel generation. It introduces efficient algorithms based on geodesic distance and Lloyd iteration, achieving O(nK) time complexity for image-based GCVT and RCVT construction with O(n) space, demonstrating strong applicability in computer graphics and vision.

ABSTRACT

Nowadays, big data of digital media (including images, videos and 3D graphical models) are frequently modeled as low-dimensional manifold meshes embedded in a high-dimensional feature space. In this paper, we summarized our recent work on geodesic centroidal Voronoi tessellations(GCVTs), which are intrinsic geometric structures on manifold meshes. We show that GCVT can find a widely range of interesting applications in computer vision and graphics, due to the efficiency of search, location and indexing inherent in these intrinsic geometric structures. Then we present the challenging issues of how to build the combinatorial structures of GCVTs and establish their time and space complexities, including both theoretical and algorithmic results.

Motivation & Objective

• To develop intrinsic geometric structures—GCVTs—on triangulated manifold meshes for use in computer graphics and vision.
• To address the challenge of constructing combinatorial GCVT structures with provable uniformity and efficiency on discrete, C⁰-continuous meshes.
• To establish theoretical and algorithmic foundations for time and space complexity of GCVT construction on manifold meshes.
• To enable practical applications such as uniform remeshing and content-sensitive superpixel/supervoxel generation using intrinsic geodesic metrics.
• To bridge the gap between computationally expensive geodesic methods and faster approximations via RCVT, ensuring accuracy and efficiency.

Proposed method

• Proposes GCVTs as Voronoi tessellations on manifolds where generators coincide with the geodesic mass centroids of their cells, ensuring uniformity.
• Employs the Lloyd method with geodesic distance to iteratively update generators and Voronoi regions, converging to GCVT via alternating Voronoi construction and centroid computation.
• Uses Dijkstra’s algorithm on mesh graphs to compute exact geodesic distances in O(n log n) time per iteration, with optimized label-correcting variants reducing it to O(n).
• Applies landmark multidimensional scaling (LMDS) to unfold geodesic Voronoi cells into R² for fast approximation of nominal mass centroids.
• Introduces restricted centroidal Voronoi tessellation (RCVT) as a fast approximation using Euclidean distance in the embedding space, avoiding costly geodesic computations.
• Analyzes time and space complexity, showing O(nK) time and O(n) space for both GCVT (via IMSLIC) and RCVT (via MSLIC) on images, with K iterations.

Experimental results

Research questions

• RQ1 How can centroidal Voronoi tessellations be generalized to intrinsic Riemannian manifolds represented as triangulated meshes?
• RQ2 What are the theoretical and algorithmic challenges in constructing combinatorial GCVT structures on discrete, non-smooth manifolds?
• RQ3 How can efficient and accurate geodesic distance computation be integrated into iterative GCVT construction for large-scale data?
• RQ4 To what extent can RCVT serve as a fast and accurate approximation to GCVT while preserving geometric uniformity?
• RQ5 What are the time and space complexities of GCVT and RCVT algorithms on manifold meshes, and how do they scale with input size?

Key findings

• GCVTs are proven to be uniform tessellations on manifold meshes, providing a stable and intrinsic geometric structure for data partitioning.
• The use of geodesic distance ensures that Voronoi cells remain connected, unlike standard Euclidean Voronoi tessellations on surfaces, which can produce disconnected cells.
• The IMSLIC algorithm constructs approximate GCVTs in O(nK) time and O(n) space per iteration using Dijkstra’s algorithm and LMDS-based centroid approximation.
• The MSLIC algorithm computes RCVTs in O(nK) time and O(n) space per iteration using Euclidean distance in the embedding space, significantly accelerating convergence.
• RCVT serves as an effective bridge between GCVT and CCVT, combining the geometric fidelity of GCVT with the computational efficiency of CCVT.
• Empirical results show that both GCVT and RCVT enable high-quality uniform remeshing and content-sensitive superpixel/supervoxel generation in images and 3D models.