Online Estimation of the Target Registration Error for n-Ocular Optical Tracking Systems

Abstract

Abstract. For current surgical navigation systems optical tracking is state of the art. The accuracy of these tracking systems is currently determined statically for the case of full visibility of all tracking targets. We propose a dynamic determination of the accuracy based on the visibility and geometry of the tracking setup. This real time estimation of accuracy has a multitude of applications. For multiple camera systems it allows reducing line of sight problems and guaranteeing a certain accuracy. The visualization of these accuracies allows surgeons to perform the procedures taking to the tracking accuracy into account. It also allows engineers to design tracking setups interactively guaranteeing a certain accuracy.Our model is an extension to the state of the art models of Fitzpatrick et al. [1] and Hoff et al. [2]. We model the error in the camera sensor plane. The error is propagated using the internal camera parameter, camera poses, tracking target poses, target geometry and marker visibility, in order to estimate the final accuracy of the tracked instrument.

A Theoretical Comparison of Different Target Registration Error Estimators

“…In this section, the maximum likelihood [8], spatial stiffness [4], transform of covariance [5], Wiles [7], and Fitzpatrick and West [2] algorithms are briefly explained. Then, it is shown that when FLE has an arbitrary Gaussian distribution, the first three algorithms converge to a unique solution; and when FLE has an identical and isotropic Gaussian distribution, all the algorithms converge to the same solution.…”

“…Sielhorst et al [5] employed the transform of covariance algorithm, introduced by Hoff and Vincent [6], to calculate the covariance matrix of the transformation parameters as follows:…”

“…In this paper, we mathematically show that, when FLE has a heterogenous Gaussian distribution, the spatial stiffness [4] and transform of covariance [5] algorithms both converge to the ML solution [8]. If FLE has an identical and anisotropic Gaussian distribution, the modified Wiles algorithm using weight LMS also converges to the ML solution.…”

“…Based on the spatial stiffness theory, they derived a solution which estimates the mean squared value of TRE in the presence of this noise distribution. Seilhorst et al [5], also in 2007, utilized the transformation of covariance method, proposed by Hoff and Vincent [6], to estimate the distribution of TRE when FLE has an inhomogeneous and anisotropic (heterogeneous) Gaussian distribution. Similarly, Wiles et al [7], in 2008, modified Fitzpatrick and West algorithm to derive another closed-form solution to estimate the mean squared value of TRE in the presence of identical and anisotropic Gaussian FLE.…”

Determining the point of minimum error for 6DOF pose uncertainty representation

“…In flexible configurations of multi-camera setups and under varying viewing conditions in dynamic scenes, however, a full error description becomes more difficult and cannot easily be interpreted anymore. Also, the real-time description of varying uncertainty in pose estimation has only recently been addressed in more detail ( [17], [2], [10]). Most approaches make use of the covariance matrix as a means of quantifying measurement errors and for the visualization of error ellipsoids.…”