version 1.0 (1.15 MB) by
Viktor Witkovsky

Estimates the ellipse parameters and their uncertainties

EllipseFit4HC is an ellipse fitting algorithm based on first order Taylor expansion (linearization) of the originally nonlinear model.

EllipseFit4HC is suggested for uncertainty evaluation of the estimated phases and/or displacements, based on quadrature homodyne interferometer measurements (with the Heydemann Correction applied).

The Heydemann Correction is used to evaluate the phase in homodyne interferometer applications to correct the interferometer nonlinearities.

Here we assume that the measurement errors for x and y are independent (optionally correlated, with known correlation coefficient rho), with zero mean and common variance sigma^2.

The standard deviation of the measurement errors, sigma, is assumed to be small, such that the measurements are relatively close to the true, however unobservable ellipse curve, as is the case for typical interferometry measurements.

Moreover, due to numerical stability of the algorithm, it is reasonable to consider normalized measurements (x,y), i.e. such that the length of the main semiaxis of the fitted ellipse is close to 1.

Here we consider the following algebraic parametrization of the ellipse, (B,C,D,F,G), as it is typically used in the field of interferometry, see e.g. Wu, Su and Peng (1996):

x^2 + B*y^2 + C*x*y + D*x + F*y + G = 0,

and the geometric parametrization, (alpha_0,alpha_1,beta_0, beta_1, phi_0) of the form:

x(phi) = alpha_0 + alpha_1 * cos(phi)

y(phi) = beta_0 + beta_1 * sin(phi + phi_0).

where -pi/2 < phi0 < pi/2 is the phase offset, alpha_0, beta_0 denote the coordinates of the ellipse center (offsets), and alpha_1, beta_1 are the signal amplitudes.

The algorithm estimates the locally approximate BLUEs (Best Linear Unbiased Estimators) of the ellipse parameters, the BLUEs of the true signal values, say mu and nu (the values on the fitted ellipse), together with their covariance matrix, as suggested in Koening et al (2014).

This is based on iterative linearizations of the originally nonlinear model with nonlinear constraints on the model parameters. For more details see Kubacek (1988, p.152).

Based on that, ellipseFit4HC estimates also the geometric parameters (alpha_0, alpha_1, beta_0, beta_1, phi_0) and the N-dimensional vector of phases phi (and/or displacements) together with their standard uncertainties computed by using the delta method.

REFERENCES:

[1] Koening, R., Wimmer, G. and Witkovsky V.: Ellipse fitting by linearized nonlinear constrains to demodulate quadrature homodyne interferometer signals and to determine the statistical uncertainty of the interferometric phase. To appear in Measurement Science and Technology, 2014.

[2] Kubacek, L.: Foundations of Estimation Theory. Elsevier, 1988.

[3] Chien-Ming Wu, Ching-Shen Su and Gwo-Sheng Peng. Correction of nonlinearity in one-frequency optical interferometry. Measurement Science and Technology, 7 (1996), 520–524.

[4] Chernov N., Wijewickrema S.: Algorithms for projecting points onto conics. Journal of Computational and Applied Mathematics 251 (2013) 8–21.