On Seeing Spaghetti: A Novel Self-Adjusting Seven Parameter Hough Space for Analyzing Flexible Extruded Objects

Abstract

We present a model for flexible extruded objects, such as wires, tubes, or grommets, and demonstrate a novel, self-adjusting seven-dimensional Hough transform that derives and analyzes their three-space curved axes from position and surface normal information. The method is purely local and is very cheap to compute. The model considers such objects as piecewise toroidal, and decomposes the seven parameters of a torus into three nested subspaces, the structure of which counteract the errors implicit in the analysis of objects of great size and/or small curvature. We believe it is the first example of a parameter space structure designed to cluster ill-conditioned hypotheses together so that they can be easily detected and ignored. This work complements existing shape-from-contour approaches for analyzing tori: it uses no edge information, and it does not require the solution of high-degree non-linear equations by iterative techniques. Most of the results, including the conditions for the existence of more that one solution (phantom "anti-tori"), have been verified using a symbolic mathematical analysis system. We present, in the environment of the IBM Con VEx system, robust results on both synthetic CAD-CAM range data (the hasp of a lock), and actual range data (a knotted piece of coaxial cable), and discuss several system tuning issues.