Probabilistic Quadrics

Reference implementation of the Eurographics 2020 paper "Fast and Robust QEF Minimization using Probabilistic Quadrics".

• Publication page: https://graphics.rwth-aachen.de/probabilistic-quadrics
• Eurographics DL: https://diglib.eg.org/handle/10.1111/cgf13933
• Eurographics Talk: https://www.youtube.com/watch?v=uEzGgFdbFMA&t=3080s

@article {10.1111:cgf.13933,
    journal = {Computer Graphics Forum},
    title = {{Fast and Robust QEF Minimization using Probabilistic Quadrics}},
    author = {Trettner, Philip and Kobbelt, Leif},
    year = {2020},
    publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
    ISSN = {1467-8659},
    DOI = {10.1111/cgf.13933}
}

Dependencies

• a C++17 compiler

Usage

// our probabilistic quadrics
#include "probabilistic-quadrics.hh"

// some math library (see below for different options)
#include "minimal-math.hh"

// optional: typedef your quadric type
using quadric3 = pq::quadric<pq::minimal_math<float>>;
using dquadric3 = pq::quadric<pq::minimal_math<double>>;

// quadrics are value types with proper operator overloads
quadric3 q;
q = q + q;
q = q - q;
q = q * 3;
q = q / 2.5f;

// quadrics can be evaluated at positions
q(1, 2, 3);
q({1, 2, 3});
q(some_pos);

// quadrics can be created from coefficients
q = quadric3::from_coefficients(some_mat3, some_vec3, some_scalar);

// quadric minimizers can be computed (using matrix inversion internally)
pq::pos3 min_p = q.minimizer();

// some classical quadrics are predefined:
q = quadric3::point_quadric(some_pos);
q = quadric3::plane_quadric(some_pos, some_normal_vec);
q = quadric3::triangle_quadric(p0, p1, p2);

// our probabilistic plane quadrics in isotropic or general form:
float stddev_pos = ...;
float stddev_normal = ...;
pq:mat3 sigma_pos = ...;
pq:mat3 sigma_normal = ...;
q = quadric3::probabilistic_plane_quadric(mean_pos, mean_normal, stddev_pos, stddev_normal);
q = quadric3::probabilistic_plane_quadric(mean_pos, mean_normal, sigma_pos, sigma_normal);

// our probabilistic triangle quadrics in isotropic or general form:
float stddev_pos = ...;
pq:mat3 sigma_p0 = ...;
pq:mat3 sigma_p1 = ...;
pq:mat3 sigma_p2 = ...;
q = quadric3::probabilistic_triangle_quadric(p0, p1, p2, stddev_pos);
q = quadric3::probabilistic_triangle_quadric(p0, p1, p2, sigma_p0, sigma_p1, sigma_p2);

Math Class

Our code is written to be largely agnostic to the choice of the math library. The quadric type is templated on a trait class that abstracts the math code away. Different types for positions and vectors are supported but not required.

template <class ScalarT, class Pos3, class Vec3, class Mat3>
struct math;

The following math classes are tested:

• the built-in minimal-math.hh:

  #include "minimal-math.hh"
  pq::minimal_math<float>
  pq::minimal_math<double>

• Typed Geometry:

  #include <typed-geometry/tg.hh>
  pq::math<float, tg::pos3, tg::vec3, tg::mat3>
  pq::math<double, tg::dpos3, tg::dvec3, tg::dmat3>

• GLM:

  #include <glm/glm.hpp>
  pq::math<float, glm::vec3, glm::vec3, glm::mat3>
  pq::math<double, glm::dvec3, glm::dvec3, glm::dmat3>

• Eigen:

  #include <eigen3/Eigen/Core>
  pq::math<float, Eigen::Vector3f, Eigen::Vector3f, Eigen::Matrix3f>
  pq::math<double, Eigen::Vector3d, Eigen::Vector3d, Eigen::Matrix3d>

To make your custom math library work, it needs to provide the following operations:

• pos - pos -> vec
• pos + vec -> pos
• pos - vec -> pos
• vec + vec -> vec
• vec - vec -> vec
• vec * scalar -> vec
• vec / scalar -> vec
• pos * scalar -> pos
• pos / scalar -> pos
• mat * scalar -> mat
• mat * vec -> vec
• mat + mat -> mat
• mat - mat -> mat
• pos[int-literal] -> scalar&
• vec[int-literal] -> scalar&
• mat[col][row] -> scalar& OR mat(row, col) -> scalar&
• mat, pos and vec default constructor
• mat, pos, vec, and scalar behave like value types
• scalar must be constructable from integer literals where
• pos is a 3D position type
• vec is a 3D vector type
• pos and vec can be the same, e.g. glm::vec3
• mat is a 3x3 matrix type

License

This code is licensed under the MIT license.