Superquadric in Computer Vision Research

Superquadrics have been employed in computer vision and robotics problems related to object recognition. A high level description of [Ferr93] and [Terz91] serve as examples.

Using superquadrics for object recognition

[Ferr93] describes the use of superquadrics, in particular superellipsoids, in fitting geometric shapes to laser rangefinder data obtained by a robot. The laser rangefinder retrieves an estimate of surface points of objects in a scene. A 3-step computational model is employed which converts these point estimates into surface estimates, to parts, and finally to objects. We restrict ourselves to a discussion of the last modeling stage because it is the stage which involves superquadrics.

Once the rangefinder data has been transformed into surface regions, it is desired to find a volumetric element to model the 3-dimensional characteristics of each region. This problem is underdetermined because the rangefinder data usually cannot completely describe the 3-dimensional quality of an object. That is, sections of an object are often occluded and therefore an estimate of this unknown portion of the object must be made.

In order to fit a superquadric to a surface region, 11 parameters must be determined: three extent parameters (, , ), two shape parameters ( and ), three translation parameters, and three rotation parameters. Ferrie et al. employ a two step modeling approach:

  1. Determine an initial fit for the surface region using an ellipsoid model (recall that an ellipsoid is a superellipsoid with = = 1). The critical issue in this stage is to determine a reasonable set of rotation and translation parameters for the relative position of the ellipsoid
  2. Minimize the error trying to fit the superellipsoid to the given surface patch. In this step, the extent and shape parameters are adjusted according to an error minimization algorithm.
Sample reconstructions of a 3D toy doll model can be found in [Ferr93].

Scene reconstruction and recognition using deformable superquadrics

In [Terz91], Terzopoulos and Metaxas extend the basic parameterized form of the superquadric to include local control parameters. They call this new class of dynamic objects deformable superquadrics.

The motivation for this extension draws from work in shape reconstruction and object recognition. These two problems represent fundamental levels in a conceptual hierarchy of how a human interprets a visual scene. Shape reconstruction is a low level process where sensor data is interpreted to regenerate objects in a scene making as few assumptions as possible about the objects. Because the interpretation attempts to be as unassuming as possible, a geometric model which has a large amount of modeling freedom is desired. Spline models have often filled this role.

Object recognition is a higher level process whose goal is to absract from the detailed data in order to characterize objects in a scene. Generally, objects in the scene are matched with precalculated objects which are stored in a database. A parameterized model is desired so that the database can be kept at a reasonable size. A flexible parameterized class of objects such as superquadrics are often employed.

[Terz91] presents a new geometric model which combines the freedom of splines with the generality of superquadrics. In particular, it is desired to reconstruct objects as in [Ferr93] but to use the new local deformation capabilities of deformable superquadrics to obtain a closer fit to the sensor data. Terzopoulos and Metaxas summarize the enhanced flexibility of deformable superquadrics as follows:

"The local degrees of freedom of deformable superquadrics allow the reconstruction of fine scale structure and the natural irregularities of real world data, whereas the global degrees of freedom capture the salient features of shape that are innate to natural parts and appropriate for matching against object prototypes."

The mathematical details of their model can be found in [Terz91].