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OrientMesh-Toolkit: Extracting and Visualizing Oriented Bounding Boxes from 3D Triangular Meshes

Basic Information

Project Overview: Extracting and Visualizing Oriented Bounding Boxes (OBB) from Triangular Meshes using C++ and Python

Triangular meshes1 stand as a cornerstone in computer graphics, essential in the portrayal and animation of 3D objects. A critical aspect of processing these meshes involves extracting Oriented Bounding Boxes (OBB)2. OrientMesh Toolkit presents an optimal solution to this task, fusing efficient algorithms with an interactive visual interface. This combination not only simplifies OBB extraction but also enhances user comprehension through vivid visual representations.

Problem to Solve

Triangular meshes, particularly when it comes to intricate surface detailing, can be both complex and computationally intensive. Operations on these meshes often demand significant resources. Having a precise oriented bounding box aids in simplifying many computational geometry problems related to surface meshes, like collision detection, mesh simplification, and many more. However, generating an optimal OBB for surface meshes is a non-trivial task. OrientMesh aims to offer a reliable, optimized, and easily-integrable solution for this challenge.

Prospective Users

OrientMesh Toolkit targets game developers, computer science students, researchers, and professionals working with 3D models, especially in areas requiring accurate bounding boxes like game development, simulation, and augmented reality.

System Architecture

OrientMesh Toolkit utilizes C++ for its core computational tasks, ensuring performance optimization. Moreover, Python is employed for its bindings, making it seamlessly usable for a wider audience and also aiding in GUI development and design.

Components:

  • Triangular Mesh Data Structures (C++): This component serves as the foundation, hosting the mesh structures essential for subsequent operations.
  • Algorithm for Optimal OBB Extraction (C++): A dedicated component that undertakes the crucial task of extracting the Oriented Bounding Boxes from the meshes.
  • API (C++ and Python): Interface that facilitates seamless communication between different modules and allows extensibility for future needs.
  • Python Bindings Using pybind11: This bridges the gap between C++ computations and Python's versatility, opening up the toolkit for a wider user base.
  • Interactive GUI (Python PyQt): A visual platform enabling users to visualize the mesh and extracted OBB interactively.

Mesh data structure:

class Mesh {
public:
    // Default constructor
    Mesh(); 
    // Copy constructor                             
    Mesh(const Mesh& mesh);                   

    // Read mesh from file
    bool read(const std::string& fileName);  
    // Write mesh to file
    bool write(const std::string& fileName);  

    // Mesh components
    std::vector<HalfEdge> halfEdges;          // Collection of HalfEdges
    std::vector<Vertex> vertices;             // Vertex list
    std::vector<Eigen::Vector3d> uvs;         // UV coordinates for texturing
    std::vector<Eigen::Vector3d> normals;     // Vertex normals
    std::vector<Edge> edges;                  // Edge list
    std::vector<Face> faces;                  // Face list
    std::vector<HalfEdgeIter> boundaries;     // Boundary HalfEdges for open meshes
};

BoundingBox data structure:

class BoundingBox {
public:
    // Default constructor
    BoundingBox();

    // Constructor initializing with specified min and max components
    BoundingBox(const Eigen::Vector3d& min0, const Eigen::Vector3d& max0);

    // Constructor initializing with a single point
    BoundingBox(const Eigen::Vector3d& p);

    // Destructor
    virtual ~BoundingBox(){};

    // Expand bounding box to include a point
    void expandToInclude(const Eigen::Vector3d& p);

    // Expand bounding box to include another bounding box
    void expandToInclude(const BoundingBox& b);

    // Return the maximum dimension of the bounding box
    int maxDimension() const;

    // Check if this bounding box contains another bounding box and calculate the distance
    bool contains(const BoundingBox& boundingBox, double& dist) const;

    // Compute an axis-aligned bounding box from a set of vertices
    void computeAxisAlignedBox(std::vector<Vertex>& vertices);

    // Compute an oriented bounding box using principal component analysis from a set of vertices
    void computeOrientedBox(std::vector<Vertex>& vertices);

    // Member variables
    Eigen::Vector3d min;
    Eigen::Vector3d max;
    Eigen::Vector3d extent;
    std::vector<Eigen::Vector3d> orientedPoints;
    std::string type;
};

System Flow Description:

In the OrientMesh workflow, everything begins with the user providing 3D object data in the form of a .obj file, which encapsulates information about vertices, edges, and faces. Initially, this data is ingested and stored by the Mesh class into specialized C++ data structures. Throughout this process, we leverage the Eigen library3 for advanced linear algebra operations, ensuring both robustness and efficiency.

After that, the parsed data from the Mesh class is passed onto the BoundingBox class. Here, utilizing methodologies such as Principal Component Analysis (PCA)4, the OBB is computed from the provided mesh data more efficiently. The BoundingBox assesses the eigen vectors of the object's distribution to determine a 3D box that minimally encapsulates the object.

Upon completion of the OBB extraction, all data is amalgamated through the BoundingBox class. Harnessing the power of pybind11 and PyQt, marrying the efficiency of C++ with the flexibility of Python, this interface acts as a bridge between C++ and Python, facilitating users to conveniently access and utilize OBB data and its associated operations.

Algorithm

3D Mesh Oriented Bounding Box (OBB) using PCA

  1. Extract vertex coordinates from the mesh:
$$\mathbf{p}_i, \quad i = 1, 2, \ldots, N$$
  1. Compute the mean of the vertex coordinates:
$$\mathbf{m} = \frac{1}{N} \sum_{i=1}^{N} \mathbf{p}_i$$
  1. Center the vertex coordinates:
$$\mathbf{q}_i = \mathbf{p}_i - \mathbf{m}$$
  1. Compute the covariance matrix:
$$\mathbf{C} = \frac{1}{N} \sum_{i=1}^{N} \mathbf{q}_i \mathbf{q}_i^T$$
  1. Decompose the covariance matrix to get the principal directions:
$$\mathbf{C} \mathbf{v}_j = \lambda_j \mathbf{v}_j$$

where $\mathbf{v}_j$ are the eigenvectors (principal directions) and $lambda_j$ are the eigenvalues of $\mathbf{C}$.

  1. Project vertices onto the principal directions to get the dimensions of the OBB:
$$D_j = \max_{i} (\mathbf{p}_i \cdot \mathbf{v}_j) - \min_{i} (\mathbf{p}_i \cdot \mathbf{v}_j), \quad j = 1, 2, 3$$

where $D_1, D_2, D_3$ represent the dimensions ($L, W, H$ ) respectively.

Build the Environment

  • To install the on Ubuntu/Debian like linux operating systems
sudo apt-get update
sudo apt-get install -y build-essential make libeigen3-dev freeglut3-dev python3 python3-pip python3-pytest python3-numpy
  • Install the Python dependencies
pip3 install -r python/requirements.txt
  • Build the project
make

How to use

  • Test the project
make test
  • UI Demo the project
make demo

References

  1. Oriented Bounding Boxes in VEX
  2. cpm-glm-aabb
  3. Trimesh
  4. PCA-OBB (GitHub Repository)
  5. SIMD-OBB (GitHub Repository)
  6. Chia-Tche Chang, Bastien Gorissen, and Samuel Melchior. 2011. Fast oriented bounding box optimization on the rotation group SO(3,ℝ). ACM Trans. Graph. 30, 5, Article 122 (October 2011), 16 pages. https://doi.org/10.1145/2019627.2019641
  7. Vitsas, N., Evangelou, I., Papaioannou, G. and Gkaravelis, A. (2023), Parallel Transformation of Bounding Volume Hierarchies into Oriented Bounding Box Trees. Computer Graphics Forum, 42: 245-254. https://doi.org/10.1111/cgf.14758

Footnotes

  1. Triangle mesh ↩

  2. Oriented Bounding Boxes on Wikipedia ↩

  3. Eigen OBB (Library Source Code) ↩

  4. Principal Component Analysis (PCA) ↩