Invariant fitting of arbitrary single - extremum surfaces . Andrew

Besl and Jain's variable order surface fitting algorithm [1] is a useful method of constructing a noise-free reconstruction of 2jD range images with a small number of primitive regions. The use of bivariate polynomials as the approximation basis functions is linear, fast and easy to render robust. Seeding fits from regions classified by differential geometry is an important step towards a viewpoint invariant segmentation. However, in order to better approximate arbitrarily shaped surfaces, polynomials of high degree are needed. For a region-growing paradigm, the poor extrapolation power of high order polynomials slows convergence and generates "non-intuitive" segmentations when crossing curvature discontinuities. Such segmentations are difficult to match against traditional CAD-like models. Further, the instability of the segmentation makes invocation of the correct model from a large database extremely difficult. We show that these algorithms must of necessity trade representational richness for repeatability. In this paper we describe a new method of satisfying the requirement for high representational richness while retaining the ease of manipulation and recognition of single-extremum surface patches. By introducing a canonical reparameterised coordinate system, biquadratic patches can be made to approximate arbitrary single-extremum shapes in a viewpoint invariant manner. An iterative fitting algorithm is presented, which quickly converges to the appropriate description. Examples of the abilities of the new approach are supplied, and compared with alternative strategies.