# Catalogue of Artificial Intelligence Techniques

View Maths as: Images | MathML

### Categories: Vision

Author(s): H.W. Hughes

Superquadrics are a set of parameterised volumetric shapes used for computer vision applications. They are useful because a wide range of solid shapes can be generated by use of only a few parameters. Surface patches can be defined by the surface of a superquadric volume. Additionally, more complex objects can be defined by combinations of simpler volumes using union, intersection and complement. Any superquadric may be expressed by the equation:

$\chi \left(\eta ,\omega \right)=\left(\begin{array}{c}a1{\mathrm{cos}}^{\epsilon 1}\eta {\mathrm{cos}}^{\epsilon 2}\omega \\ a2{\mathrm{cos}}^{\epsilon 1}{\mathrm{sin}}^{\epsilon 2}\omega \\ a3{\mathrm{sin}}^{\epsilon 1}\end{array}\right)$

where $-\pi /2\le \eta \le \pi /2$ and $-\pi \le \omega <\pi$, $a1$, $a2$ and $a3$ are the length, width and breadth respectively. $ϵ1$ and $ϵ2$ are the parameters which specify the shape of the superquadric. $ϵ1$ is the squareness parameter in the north-south direction. $ϵ2$ is the squareness parameter in the east-west direction. The advantage of this representation is that the normal vectors are given by:

$n\left(\eta ,\omega \right)=\left(\begin{array}{c}\frac{1}{a1}{\mathrm{cos}}^{2-\epsilon 1}\eta {\mathrm{cos}}^{2-\epsilon 2}\omega \\ \frac{1}{a2}{\mathrm{cos}}^{2-\epsilon 1}\eta {\mathrm{sin}}^{2-\epsilon 2}\omega \\ \frac{1}{a3}{\mathrm{sin}}^{2-\epsilon 1}\eta \end{array}\right)$

enabling the shape parameters to be recovered via surface normal information from image intensity, contour shape and the like. These mathematical solids are a subset of the more general family of Generalised Cylinders.

### References:

• Barr, A.H., Superquadrics and angle-preserving transformations IEEE Computer Graphics and Applications 1, 1--20.