By Sung Joon Ahn
Due to the continued growth of sensor know-how, the supply of 3-D c- eras is already foreseeable. those cameras are able to producing a wide set of dimension issues inside a truly short while. there's a number of three-D digital camera - plications within the ?elds of robotics, fast product improvement and electronic factories. so that it will not just visualize the purpose cloud but in addition to acknowledge 3-D item m- els from the purpose cloud after which extra method them in CAD platforms, ef?cient and solid algorithms for 3D info processing are required. For the au- matic segmentation and popularity of such geometric primitives as airplane, sphere, cylinder, cone and torus in a three-D aspect cloud, ef?cient software program has lately been constructed on the Fraunhofer IPA through Sung Joon Ahn. This publication describes intimately the whole set of ‘best-?t’ algorithms for common curves and surfaces in house that are hired within the Fraunhofer software.
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Additional info for Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space
15). 1 Problem Definition The ultimate goal of the orthogonal distance fitting of a model feature to a set of given points in space is the determination of the model parameters which minimize the square sum of the minimum distances between the given points and the model feature. e. not only the square sum but also every single distance between the given points and the model feature should be minimized. 2 Generalized Orthogonal Distance Fitting 23 where X : coordinate column vector of the coordinate column vector of the on the model feature, d : distance column vector, given points corresponding points with weighting matrix or error covariance matrix (positive definite) P : nonsingular symmetric matrix .
Fortunately, the parameter update is generally moderate in the second half-phase of the outer iteration (after the first 5–10 outer iteration cycles with most fitting tasks) and the minimum distance point in XYZ frame changes its position incrementally between two consecutive outer iteration cycles. 12) must be mentioned. 12) is used (Fig. 5). g. with every 5–10th outer iteration cycle) from the given point during the second half-phase of the outer iteration The strategy for an efficient and inexpensive way of finding the minimum distance point is summarized below: 42 3.
E. algebraic fitting and geometric fitting. These are differentiated by their respective definition of the error measure to be minimized , , , . Although the 10 1. Introduction terminology originated with the fitting problems of implicit curves/surfaces, it reflects the research efforts made in past decades to develop fitting algorithms for general curves/surfaces. A highly representative class of curves/surfaces is the algebraic curve/surface described by an implicit polynomial equation with the algebraic parameters b (polynomial coefficients) for algebraic plane curves, and, for algebraic surfaces.