The Use of the l1 and l∞ Norms in Fitting Parametric Curves and Surfaces to Data

Abstract

Given a family of curves or surfaces in R^s, an important problem is that of finding a member of the family which gives a “best” fit to m given data points. There are many application areas, for example metrology, computer graphics, pattern recognition, and the most commonly used criterion is the least squares norm. However, there may be wild points in the data, and a more robust estimator such as the l₁ norm may be more appropriate. On the other hand, the object of modelling the data may be to assess the quality of a manufactured part, so that accept/reject decisions may be required, and this suggests the use of the Chebyshev norm.

We consider here the use of the l₁ and l_∞ norms in the context of fitting to data curves and surfaces defined parametrically. There are different ways to formulate the problems, and we review here formulations, theory and methods which generalize in a natural way those available for least squares. As well as considering methods which apply in general, some attention is given to a fundamental fitting problem, that of lines in three dimensions. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)