l1 and l1 ODR Fitting of Geometric Elements (original) (raw)
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l1 and l∞ ODR Fitting of Geometric Elements
msl.uni-bonn.de
We consider the fitting of geometric elements, such as lines, planes, circles, cones, and cylinders, in such a way that the sum of distances or the maximal distance from the element to the data points is minimized. We refer to this kind of distance based fitting as orthogonal distance regression or ODR. We present a separation of variables algorithm for l1 and l∞ ODR fitting of geometric elements. The algorithm is iterative and allows the element to be given in either implicit form f (x, β) = 0 or in parametric form x = g(t, β), where β is the vector of shape parameters, x is a 2-or 3-vector, and s is a vector of location parameters. The algorithm may even be applied in cases, such as with ellipses, in which a closed form expression for the distance is either not available or is difficult to compute. For l1 and l∞ fitting, the norm of the gradient is not available as a stopping criterion, as it is not continuous. We present a stopping criterion that handles both the l1 and the l∞ case, and is based on a suitable characterization of the stationary points.
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In this paper, we present techniques for ellipsoid fitting which are based on minimizing the sum of the squares of the geometric distances between the data and the ellipsoid. The literature often uses "orthogonal fitting" in place of "geometric fitting" or "best-fit". For many different purposes, the best-fit ellipsoid fitting to a set of points is required. The problem offitting ellipsoid is encounteredfrequently intheimage processing, face recognition, computer games, geodesy etc. Today, increasing GPS and satellite measurementsprecisionwill allow usto determine amore realistic Earth ellipsoid. Several studies have shown that the Earth, other planets, natural satellites, asteroids and comets can be modeled as triaxial ellipsoids Burša and Šima (1980), Iz et all (2011). Determining the reference ellipsoid for the Earth is an important ellipsoid fitting application, because all geodetic calculations are performed on the reference ellipsoid. Algebraic fit...
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