Invariant Fitting of Arbitrary Single-Extremum Surfaces

Besl and Jain's variable order surface tting algorithm 1] is a useful method of constructing a noise-free reconstruction of 2 1 2 D range images with a small number of primitive regions. The use of bivariate polynomi-als as the approximation basis functions is linear, fast and easy to render robust. Seeding ts from regions classiied by diierential 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 dis-continuities. Such segmentations are diicult to match against traditional CAD-like models. Further, the instability of the segmentation makes in-vocation of the correct model from a large database extremely diicult. 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, biquad-ratic patches can be made to approximate arbitrary single-extremum shapes in a viewpoint invariant manner. An iterative tting 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.