A Buyer's Guide to Euclidean Elliptical Cylindrical and Conical Surface Fitting

The ability to construct CAD or other object models from edge and range data has a fundamental meaning in building a recognition and positioning system. While the problem of model fitting has been successfully addressed, the problem of efficient high accuracy and stability of the fitting is still an open problem. In the past researchers have used approximate distance functions rather than the real Euclidean distance because of computational efficiency. We now feel that machine speeds are sufficient to ask whether it is worth considering Euclidean fitting again. This paper address the problem of estimation of elliptical cylinder and cone surfaces to 3D data by a constrained Euclidean fitting. We study and compare the performance of various distance functions in terms of correctness, robustness and pose invariance, and present our results improving known fitting methods by closed form expressions of the real Euclidean distance. 1 Motivation Shape analysis of objects is a key problem in computer vision with several important applications in manufacturing, such as quality control and reverse engineering. However, the application of shape in computer vision has been limited to date by the difficulties in its computation. To build a recognition and positioning system based on implicit curves and surfaces it is imperative to solve the problem of how curves and surfaces can be fitted to the data extracted from single or multiple 3D images. The fitting process is necessary for automatically constructing object models and for building intermediate representations from observations during the recognition. Implicit polynomial surfaces are potentially among the most useful object or data representations for use in computer vision and image analysis. Their power appears by their ability to smooth noisy data, to interpolate through sparse or missing data, their compactness and their form being commonly used in numerous constructions. An implicit surface is the zero set of a smooth function of the variables: . Let be an implicit polynomial of degree given by