1987 Reports

Recovery of superquadrics from depth information

Boult, Terrance E.; Gross, Ari D.

Superquadrics are a a class volumetric primitive which can model objects including rectangular solids with rounded corners, ellipsoids, octaheadrons, 8-pointed stars, hyperbolic sheets, and toroids with cross sections ranging from rectangles with rounded corners to elliptical regions. They can be stretched, bent, tapered and combined with boolean operations to model a wide range of objects. This paper discusses our progress at attempting to recover a subclass of superquadrics from 3D depth data. The first section of this paper presents a mathematical definition of superquadrics. Some of the rationale for using superquadrics for object recognition is then discussed. Briefly, superquadrics are flexible enough to represent a wide class of objects, but are simple enough to be recovered from 3d data. Additionally, the surface and its normal surface both have well defined inside-out functions which provide a useful tool for their recovery. The third section examines some of the difficulties to be encountered when modeling objects with superquadrics, or attempting to recover superquadrics from 3D data. These difficulties include the general problems of a non-orthogonal representation, difficulties of dealing with objects which are not exactly representable with CSG operations on the primitives, the need to recognize negative objects. Certain numerical instabilities and some problems caused by using the inside-out function as an approximation of the distance of a point from the superquadric. Our current system employs a nonlinear least square minimization technique on the inside-out function to recover the parameters. After discussing the details of the current system, the paper presents examples, using noisy synthetic data, where the system succe88fully uses multiple views to recover underlying superquadrics. Also presented are examples using range data, including the recovery of a negative superellipsiod. Some pros and cons of our approach as well as few conclusions, and a discussion of our planned future work appear in the final section. The main result is that least square minimization using the inside-out function allows both positive and negative instances of superellipsoids to be recovered from depth data. A second preliminary result is that a single view of a superquadric may not be sufficient for reconstruction without additional assumption.