Comparative testing of ellipse-fitting algorithms: implications for analysis of strain and curvature T.J. Wynn a, * , S.A. Stewart b a TRACS International, Falcon House, Union Grove Lane, Aberdeen AB10 6XU, UK b BP Azerbaijan, c/o Chertsey Road, Sunbury on Thames, Middlesex TW16 7LN, UK Received 1 September 2004; received in revised form 26 June 2005; accepted 27 June 2005 Available online 11 August 2005 Abstract Several types of geological problem involve fitting an ellipse to sparse data in order to define a property such as strain or curvature. The sensitivity of ellipse-fitting algorithms to noise in input geological data is often poorly documented. Here we compare the performance of some well-known approaches to this problem in geology against each other and an algorithm developed for machine vision. The specific methods tested here are an analytical method, a Mohr diagram method, two least squares methods and a constrained ellipse approach. The algorithms were tested on artificial datasets of known elliptical and noise properties. These results allow the selection of an ellipse fitting method in a variety of geological applications and also allow an assessment of the absolute and relative accuracy of a chosen method for various combinations of sample numbers and noise levels. For the most accurate semi axis magnitude and orientation estimates with five or more input data, the ‘mean object’ least squares approach is recommended. However, the other least squares method also yields good results and is also suitable for three or four data points. Where curvature data is being assessed, the least squares method is preferred as it can handle negative principal curvature values. q 2005 Elsevier Ltd. All rights reserved. Keywords: Strain; Curvature; Noise; Ellipse-fitting; Least squares; Mohr-circle 1. Introduction Determination of principal vectors, or maximum and minimum magnitudes and directions from sparse or scattered data, is a feature of several types of problem in structural geology. Examples are determining two-dimen- sional strain in classic ‘stretched belemnite’-type problems (Lisle and Ragan, 1988 and references therein) and estimation of principal surface curvatures (Lisle and Robinson, 1995; Belfield, 2000). With these applications in mind, in this paper we investigate the sensitivity of prevailing algorithms to quality of input data. Strain (in two dimensions) and curvature can be represented by second order tensors (Ramsay and Huber, 1983). Tensors allow the description of the variation of a property independently of the co-ordinate system used to define the locations of the property. However, they are linear operators that send a vector to a vector so a description of the co-ordinate system is required for practical purposes. In two-dimensional Cartesian space, four components are generally required to fully define a tensor. In the examples mentioned here, only three components are needed since strain and surface curvature can be represented by symmetric second order tensors with mutually orthogonal principal axes (Ramsay and Huber, 1983; Lisle and Robinson, 1995). These properties of symmetrical second order tensors allow them to be represented by an ellipse, which are defined by the magnitudes of two mutually orthogonal principal axes and an orientation of the major axis. Strain or curvature data usually have to be measured in directions that are likely to be arbitrary with respect to the principal axes. Therefore, the principal axes have to be derived from these arbitrarily oriented measurements by fitting ellipses to the sampled data (Lisle and Ragan, 1988; Lisle and Robinson, 1995; Stewart and Podolski, 1998). Many methods are available for estimating principal axes from arbitrarily oriented measurements in geological Journal of Structural Geology 27 (2005) 1973–1985 www.elsevier.com/locate/jsg 0191-8141/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2005.06.010 * Corresponding author. Tel.: C44 1224 321213; fax: C44 1224 321214. E-mail address: tim.wynn@tracsint.com (T.J. Wynn).