IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 3, MARCH 2010 879 Target Tracking Errors for 5D and 6D Spatial Measurement Systems Andrew D. Wiles*, Member, IEEE, and Terry M. Peters, Fellow, IEEE Abstract—In recent years, magnetic tracking systems, whose fundamental unit of measurement is a 5D transformation (three translational and two rotational degrees-of-freedom), have be- come much more popular. Two 5D sensors can be combined to obtain a 6D transformation similar to the ones provided by the point-based registration in optical tracking. However, estimates of the tool tip uncertainty, which we have called the target tracking error (TTE) since no registration is explicitly performed, are not available in the same manner as their optical counterpart. If the systematic bias error can be corrected and estimates of the 5D or 6D fiducial localizer error (FLE) are provided in the form of zero mean normally distributed random variables in and , respectively, then the TTE can be modeled. In this paper, the required expressions that model the TTE as a function of the systematic bias, FLE and target location are derived and then val- idated using Monte Carlo simulations. We also show that the first order approximation is sufficient beyond the range of errors typi- cally observed during an image-guided surgery (IGS) procedure. Applications of the models are described for a minimally invasive intracardiac surgical guidance system and needle-based therapy systems. Together with the target registration error (TRE) statis- tical models for point-based registration, the models presented in this article provide the basic framework for estimating the total system measurement uncertainty for an IGS system. Future work includes developing TRE models for commonly used registration methods that do not already have them. Index Terms—Anisotropic fiducial localizer error, magnetic tracking, point-based registration, target tracking error (TTE), TTE model. I. INTRODUCTION T ARGET registration error (TRE) is the most appropriate measure of system accuracy developed for medical image registration and image-guided surgery (IGS). First introduced in the early 1990s [1], [2], the TRE is the error between the true target location and the target location inferred after regis- tration due to inherent measurement error of localizing the fidu- Manuscript received November 24, 2009; accepted December 14, 2009. Cur- rent version published March 03, 2010. This work was supported in part by the Heart and Stroke Foundation of Canada, in part by the Canadian Institutes of Health Research (CIHR), in part by the Natural Science and Engineering Research Council of Canada (NSERC), in part by the Ontario Research Fund (ORF), in part by the Canadian Foundation for Innovation (CFI), in part by the University of Western Ontario (UWO) Graduate Research Scholarships, and in part by Northern Digital Inc. (NDI). Asterisk indicates corresponding author. *A. D. Wiles was with the Department of Medical Biophysics, The University of Western Ontario, and the Robarts Research Institute, London, ON, N2V 1C5 Canada (e-mail: awiles@ndigital.com). T. M. Peters is with The University of Western Ontario and the Robarts Re- search Institute, London, ON N6A 5K8, Canada (e-mail: tpeters@imaging.ro- barts.ca). Digital Object Identifier 10.1109/TMI.2009.2039344 cial markers known as the fiducial localizer error (FLE). Targets consist of any anatomical object that is of interest to the radi- ologist for medical image registration or the surgeon in IGS. One of the most common registration methods is point-based registration whereby the distance between homologous points, e.g., fiducial markers, is minimized to obtain the best fit trans- formation. The remaining error between the measured fiducial markers and the transformed homologous points is the fiducial registration error (FRE). The FRE is often used as a metric to determine the quality of the registration, however as proven by Fitzpatrick et al. [3], [4] certain fiducial marker configurations could provide a low FRE but yield a large TRE and hence the FRE is “an unreliable indicator of registration accuracy.” There- fore, a statistical model that estimates the rms of the TRE as a function of fiducial geometry, target location and isotropic FLE was developed. The distribution of the TRE was later derived in [5]. Up to this point, the statistical model was being applied to medical image registration where the assumption of isotropic FLE was reasonable. However, West and Maurer [6] showed that the model could be used to evaluate the design of opti- cally tracked rigid bodies, since optical tracking systems used in IGS use point-based registration techniques to find the best-fit transformation of the rigid bodies attached to the surgical tools being tracked. Again, the TRE was identified to be a very im- portant metric since it identified the error at the surgical tool’s tip (target) as a function of the tool geometry and the 3D mea- surement error (FLE) of the optical markers (fiducials). This ob- servation also provided a framework for which the TRE of the optically tracked pointer tool could be used as the FLE for a point-based registration that maps the image coordinate frame to the patient coordinate frame in an IGS system. Using the TRE of the optically tracked tool as input in the patient reg- istration would allow for an estimation of the TRE for a given surgical target. Knowing this information allows the surgeon to make strategic decisions that would optimize the accuracy of the procedure. However, West and Maurer also pointed out the shortcoming of the original model in that the FLE is typically anisotropic for optical tracking systems, resulting in an incorrect estimate of the TRE. In practice, the error along the axis aligned with the viewing direction of the optical tracking system is typ- ically three to five times larger than the error in the orthogonal directions. This observation motivated the development of a new statistical model, whereby it is assumed the FLE is anisotropic [7] and the covariance and rms of the TRE is computed. In a related article [8] we demonstrated that the anisotropic TRE models from two individual rigid bodies can be combined to estimate the error associated with tracking a surgical tool rela- 0278-0062/$26.00 © 2010 IEEE