Superellipsoid
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter
, and whose vertical sections through the center are superellipses with the squareness parameter
[2][3] In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics.
[4][5] Superellipsoids have an rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates.
[6] It becomes an important geometric primitive widely used in computer vision,[6][5][7] robotics,[4] and physical simulation.
[8] The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape.
[9] This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning.
[4] A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic: Piet Hein's supereggs are also special cases of superellipsoids.
The basic superellipsoid is defined by the implicit function The parameters
are positive real numbers that control the squareness of the shape.
Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent
, which is Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent
, stretched horizontally by a factor w that depends on the sectioning plane.
In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent
The basic shape above extends from −1 to +1 along each coordinate axis.
The general superellipsoid is obtained by scaling the basic shape along each axis by factors
The implicit function is [2] Similarly, the surface of the superellipsoid is defined by the equation
[2] The superellipsoid has a parametric representation in terms of surface parameters
[3] In computer vision and robotic applications, a superellipsoid with a general pose in the 3D Euclidean space is usually of more interest.
relative to the world frame, the implicit function of a general posed superellipsoid surface defined the world frame is[6]
The volume encompassed by the superelllipsoid surface can be expressed in terms of the beta functions
Recoverying the superellipsoid (or superquadrics) representation from raw data (e.g., point cloud, mesh, images, and voxels) is an important task in computer vision,[11][7][6][5] robotics,[4] and physical simulation.
[11] The goal is to find out the optimal set of superellipsoid parameters
SE(3) is the pose of the superellipsoid frame with respect to the world coordinate.
[12] The first one is constructed directly based on the implicit function[11]
The minimization of the objective function provides a recovered superellipsoid as close as possible to all the input points
The other objective function tries to minimized the radial distance between the points and the superellipsoid.
A probabilistic method called EMS is designed to deal with noise and outliers.
[6] In this method, the superellipsoid recovery is reformulated as a maximum likelihood estimation problem, and an optimization method is proposed to avoid local minima utilizing geometric similarities of the superellipsoids.
The method is further extended by modeling with nonparametric bayesian techniques to recovery multiple superellipsoids simultaneously.
