Understanding the effect of bias in fiducial localization error on point-based rigid-body registration

A new 2D-3D registration gold-standard dataset for the hip joint based on uncertainty modeling

Purpose

Estimation of the accuracy of 2D‐3D registration is paramount for a correct evaluation of its outcome in both research and clinical studies. Publicly available datasets with standardized evaluation methodology are necessary for validation and comparison of 2D‐3D registration techniques. Given the large use of 2D‐3D registration in biomechanics, we introduced the first gold standard validation dataset for computed tomography (CT)‐to‐x‐ray registration of the hip joint, based on fluoroscopic images with large rotation angles. As the ground truth computed with fiducial markers is affected by localization errors in the image datasets, we proposed a new methodology based on uncertainty propagation to estimate the accuracy of a gold standard dataset.

Methods

The gold standard dataset included a 3D CT scan of a female hip phantom and 19 2D fluoroscopic images acquired at different views and voltages. The ground truth transformations were estimated based on the corresponding pairs of extracted 2D and 3D fiducial locations. These were assumed to be corrupted by Gaussian noise, without any restrictions of isotropy. We devised the multiple projective points criterion (MPPC) that jointly optimizes the transformations and the noisy 3D fiducial locations for all views. The accuracy of the transformations obtained with the MPPC was assessed in both synthetic and real experiments using different formulations of the target registration error (TRE), including a novel formulation of the TRE (uTRE) derived from the uncertainty analysis of the MPPC.

Results

The proposed MPPC method was statistically more accurate compared to the validation methods for 2D‐3D registration that did not optimize the 3D fiducial positions or wrongly assumed the isotropy of the noise. The reported results were comparable to previous published works of gold standard datasets. However, a formulation of the TRE commonly found in these gold standard datasets was found to significantly miscalculate the true TRE computed in synthetic experiments with known ground truths. In contrast, the uncertainty‐based uTRE was statistically closer to the true TRE.

Conclusions

We proposed a new gold standard dataset for the validation of CT‐to‐X‐ray registration of the hip joint. The gold standard transformations were derived from a novel method modeling the uncertainty in extracted 2D and 3D fiducials. Results showed that considering possible noise anisotropy and including corrupted 3D fiducials in the optimization resulted in improved accuracy of the gold standard. A new uncertainty‐based formulation of the TRE also appeared as a good alternative to the unknown true TRE that has been replaced in previous works by an alternative TRE not fully reflecting the gold standard accuracy.

General first-order target registration error model considering a coordinate reference frame in an image-guided surgical system

Point-based rigid registration (PBRR) techniques are widely used in many aspects of image-guided surgery (IGS). Accurately estimating target registration error (TRE) statistics is of essential value for medical applications such as optically surgical tool-tip tracking and image registration. For example, knowing the TRE distribution statistics of surgical tool tip can help the surgeon make right decisions during surgery. In the meantime, the pose of a surgical tool is usually reported relative to a second rigid body whose local frame is called coordinate reference frame (CRF). In an n-ocular tracking system, fiducial localization error (FLE) should be considered inhomogeneous, that means FLE is different between fiducials, and anisotropic that indicates FLE is different in all directions. In this paper, we extend the TRE estimation algorithm relative to a CRF from homogeneous and anisotropic to heterogeneous FLE cases. Arbitrary weightings can be assumed in solving the registration problems in the proposed TRE estimation algorithm. Monte Carlo simulation results demonstrate the proposed algorithm's effectiveness for both homogeneous and inhomogeneous FLE distributions. The results are further compared with those using the other two algorithms. When FLE distribution is anisotropic and homogeneous, the proposed TRE estimation algorithm's performance is comparable with that of the first one. When FLE distribution is heterogeneous, proposed TRE estimation algorithm outperforms the other two classical algorithms in all test cases when ideal weighting scheme is adopted in solving two registrations. Possible clinical applications include the online estimation of surgical tool-tip tracking error with respect to a CRF in IGS. Graphical Abstract This paper provides the target registration error model considering a coordinate reference frame in surgical navigation.

Comparative Study of Two Pose Measuring Systems Used to Reduce Robot Localization Error

The performance of marker-based, six degrees of freedom (6DOF) pose measuring systems is investigated. For instruments in this class, the pose is derived from locations of a few three-dimensional (3D) points. For such configurations to be used, the rigid-body condition—which requires that the distance between any two points must be fixed, regardless of orientation and position of the configuration—must be satisfied. This report introduces metrics that gauge the deviation from the rigid-body condition. The use of these metrics is demonstrated on the problem of reducing robot localization error in assembly applications. Experiments with two different systems used to reduce the localization error of the same industrial robot yielded two conflicting outcomes. The data acquired with one system led to substantial reduction in both position and orientation error of the robot, while the data acquired with a second system led to comparable reduction in the position error only. The difference is attributed to differences between metrics used to characterize the two systems.

Automated fiducial marker detection and fiducial point localization in CT images for lung biopsy image-guided surgery systems

In the lung biopsy image-guided surgery systems, the fiducial markers are used for point-based registration of the patient space to the CT image space. Fiducial marker detection and fiducial point localization in CT images have great influence on the accuracy of registration and guidance. This study proposes a fiducial marker detection approach based on the features of marker image slice sequences and a fiducial point localization approach according to marker projection images, without depending on the priori-knowledge of the marker default parameters provided by the manufacturers. The accuracy of our method was validated based on a CT image dataset of 24 patients. The experimental results showed that all 144 markers of 24 patients were correctly detected, and the fiducial points were localized with the average error of 0.35 mm. In addition, the localization accuracy of the proposed method was improved by an average of 12.5% compared with the accuracy of the previous method using the marker default parameters provided by the manufacturers. Thus, the study demonstrated that the proposed detection and localization methods are accurate and robust, which is quite encouraging to meet the requirement of future clinical applications in the image guided lung biopsy and surgery systems.

Estimating Maximum Target Registration Error Under Uniform Restriction of Fiducial Localization Error in Image Guided System

In this paper, we investigate the estimation of the maximum target registration error (TRE) magnitude of the target location while using point-based rigid registration in the image guided system. Under the uniform restriction of fiducial localization error (FLE) magnitude, we explicitly formulate the estimation as an optimization problem. Through analyzing the approximated problem which assumes the rigidity of the fiducial set holds with the perturbation of FLE, we present a strict lower bound for the maximum TRE magnitude. The simulations show that the lower bound is close to the actual maximum TRE magnitude for the target locations lying far away from the fiducial points. Unlike the expected TRE magnitude in which all fiducial points contribute, the lower bound is only related to the fiducial points serving as the vertices of the convex hull of the fiducial set. Our analysis provides a new perspective of investigating the problem of TRE estimation and is helpful for the surgeons to learn about the worst situation during using the image guided system.

Experimental validation of predicted application accuracies for computer-assisted (CAS) intraoperative navigation with paired-point registration

PURPOSE

The target registration error (TRE) is a crucial parameter to estimate the potential usefulness of computer-assisted navigation intraoperatively. Both image-to-patient registration on base of rigid-body registration and TRE prediction methods are available for spatially isotropic and anisotropic data. This study presents a thorough validation of data obtained in an experimental operating room setting with CT images.

METHODS

Optical tracking was used to register a plastic skull, an anatomic specimen, and a volunteer to their respective CT images. Plastic skull and anatomic specimen had implanted bone fiducials for registration; the volunteer was registered with anatomic landmarks. Fiducial localization error, fiducial registration error, and total target error (TTE) were measured; the TTE was compared to isotropic and anisotropic error prediction models. Numerical simulations of the experiment were done additionally.

RESULTS

The user localization error and the TTE were measured and calculated using predictions, both leading to results as expected for anatomic landmarks and screws used as fiducials. TRE/TTE is submillimetric for the plastic skull and the anatomic specimen. In the experimental data a medium correlation was found between TRE and target localization error (TLE). Most of the predictions of the application accuracy (TRE) fall in the 68% confidence interval of the measured TTE. For the numerically simulated data, a prediction of TTE was not possible; TRE and TTE show a negligible correlation.

CONCLUSION

Experimental application accuracy of computer-assisted navigation could be predicted satisfactorily with adequate models in an experimental setup with paired-point registration of CT images to a patient. The experimental findings suggest that it is possible to run navigation and prediction of navigation application accuracy basically defined by the spatial resolution/precision of the 3D tracker used.

Selection of Fiducial Locations and Performance Metrics for Point-Based Rigid-Body Registration

A method is described to select the location and number of fiducials used in point-based, rigid-body registration of two coordinate frames. Two indices are introduced which are used to search for the optimum configuration of fiducials. They can be used to quickly evaluate a large number of configurations because no actual registration is involved in their calculation. Furthermore, configurations yielding small values of the indices correlate well with configurations which result in optimum registrations. Three registration performance metrics are discussed, and it is shown that optimization of different metrics leads to different selection of fiducial configurations. If an optimized configuration is selected as a starting configuration of N fiducials, the addition of extra fiducials does not significantly improve the registration in most cases. This work is based on 3D data acquired with three different instruments, each having different noise and bias characteristics.

Estimating FLEimage distributions of manual fiducial localization in CT images

Purpose

The fiducial localization error distribution (FLE) and fiducial configuration govern the application accuracy of point-based registration and drive target registration error (TRE) prediction models. The error of physically localizing patient fiducials (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{patient}$$\end{document}FLEpatient) is negligible when a registration probe matches the implanted screws with mechanical precision. Reliable trackers provide an unbiased estimate of the positional error (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{tracker}$$\end{document}FLEtracker) with cheap repetitions. FLE further contains the localization error in the imaging data (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{image}$$\end{document}FLEimage), sampling of which in general is expensive and possibly biased. Finding the best techniques for estimating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{image}$$\end{document}FLEimage is crucial for the applicability of the TRE prediction methods.

Methods

We built a ground-truth (gt)-based unbiased estimator (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\hbox {FLE}}_\mathrm{gt}}$$\end{document}FLEgt^) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{image}$$\end{document}FLEimage from the samples collected in a virtual CT dataset in which the true locations of image fiducials are known by definition. Replacing true locations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{gt}$$\end{document}FLEgt by the sample mean creates a practical difference-to-mean (dtm)-based estimator (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\hbox {FLE}}_\mathrm{dtm}}$$\end{document}FLEdtm^) that is applicable on any dataset. To check the practical validity of the dtm estimator, ten persons manually localized nine fiducials ten times in the virtual CT and the resulting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{dtm}$$\end{document}FLEdtm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{gt}$$\end{document}FLEgt distributions were tested for statistical equality with a kernel-based two-sample test using the maximum mean discrepancy (MMD) (Gretton in J Mach Learn Res 13:723–773, 2012) statistics at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0.05$$\end{document}α=0.05.

Results

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{dtm}$$\end{document}FLEdtm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{gt}$$\end{document}FLEgt were found (for most of the cases) not to be statistically significantly different; conditioning them on persons and/or screws however yielded statistically significant differences much more often.

Conclusions

We conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\hbox {FLE}}_\mathrm{dtm}}$$\end{document}FLEdtm^ is the best candidate (within our model) for estimating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{image}$$\end{document}FLEimage in homogeneous TRE prediction models. The presented approach also allows ground-truth-based numerical validation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {FLE}}_\mathrm{image}$$\end{document}FLEimage estimators and (manual/automatic) image fiducial localization methods in phantoms with parameters similar to clinical datasets.

Cramér-Rao Lower Bound for Point Based Image Registration With Heteroscedastic Error Model for Application in Single Molecule Microscopy

The Cramér-Rao lower bound for the estimation of the affine transformation parameters in a multivariate heteroscedastic errors-in-variables model is derived. The model is suitable for feature-based image registration in which both sets of control points are localized with errors whose covariance matrices vary from point to point. With focus given to the registration of fluorescence microscopy images, the Cramér-Rao lower bound for the estimation of a feature's position (e.g., of a single molecule) in a registered image is also derived. In the particular case where all covariance matrices for the localization errors are scalar multiples of a common positive definite matrix (e.g., the identity matrix), as can be assumed in fluorescence microscopy, then simplified expressions for the Cramér-Rao lower bound are given. Under certain simplifying assumptions these expressions are shown to match asymptotic distributions for a previously presented set of estimators. Theoretical results are verified with simulations and experimental data.

New Protocol for Skin Landmark Registration in Image-Guided Neurosurgery: Technical Note

BACKGROUND

Newer versions of the commercial Medtronic StealthStation allow the use of only 8 landmark pairs for patient-to-image registration as opposed to 9 landmarks in older systems. The choice of which landmark pair to drop in these newer systems can have an effect on the quality of the patient-to-image registration.

OBJECTIVE

To investigate 4 landmark registration protocols based on 8 landmark pairs and compare the resulting registration accuracy with a 9-landmark protocol.

METHODS

Four different protocols were tested on both phantoms and patients. Two of the protocols involved using 4 ear landmarks and 4 facial landmarks and the other 2 involved using 3 ear landmarks and 5 facial landmarks. Both the fiducial registration error and target registration error were evaluated for each of the different protocols to determine any difference between them and the 9-landmark protocol.

RESULTS

No difference in fiducial registration error was found between any of the 8-landmark protocols and the 9-landmark protocol. A significant decrease (P < .05) in target registration error was found when using a protocol based on 4 ear landmarks and 4 facial landmarks compared with the other protocols based on 3 ear landmarks.

CONCLUSION

When using 8 landmarks to perform the patient-to-image registration, the protocol using 4 ear landmarks and 4 facial landmarks greatly outperformed the other 8-landmark protocols and 9-landmark protocol, resulting in the lowest target registration error.

An analysis of tracking error in image-guided neurosurgery

PURPOSE

This study quantifies some of the technical and physical factors that contribute to error in image-guided interventions. Errors associated with tracking, tool calibration and registration between a physical object and its corresponding image were investigated and compared with theoretical descriptions of these errors.

METHODS

A precision milled linear testing apparatus was constructed to perform the measurements.

RESULTS

The tracking error was shown to increase in linear fashion with distance normal to the camera, and the tracking error ranged between 0.15 and 0.6 mm. The tool calibration error increased as a function of distance from the camera and the reference tool (0.2-0.8 mm). The fiducial registration error was shown to improve when more points were used up until a plateau value was reached which corresponded to the total fiducial localization error ([Formula: see text]0.8 mm). The target registration error distributions followed a [Formula: see text] distribution with the largest error and variation around fiducial points.

CONCLUSIONS

To minimize errors, tools should be calibrated as close as possible to the reference tool and camera, and tools should be used as close to the front edge of the camera throughout the intervention, with the camera pointed in the direction where accuracy is least needed during surgery.

Anatomical landmarks for point-matching registration in image-guided neurosurgery

BACKGROUND

Accurate patient to image registration is the core for successful image-guided neurosurgery. While skin adhesive markers (SMs) are widely used in point-matching registration, a proper implementation of anatomical landmarks (ALs) may overcome the inconvenience brought by the use of SMs.

METHODS

Using nine ALs, a set of three configurations of different combinations of them is proposed. These configurations are defined according to the required positioning of the patient's head during surgery and the resulting distribution of the expected target registration error (TRE). These configurations were first evaluated by simulation experiment using the data of 20 patients from two hospitals, and then testing the applicability of them in eight real clinical surgeries of neuronavigation.

RESULTS

The results of the simulation experiment showed that, by incorporating a fiducial registration error (FRE) of 3.5 mm measured in the clinical setting, the expected TRE in the whole skull was less than 2.5 mm, and the expected TRE in the whole brain was less than 1.75 mm when using all the nine ALs. A small TRE could also be achieved in the corresponding surgical field by using the other three configurations with less ALs. In the clinical experiment, the FLE ranges in the image and the patient space were 1.4-3.6 mm and 1.6-5.5 mm, respectively. The measured TRE and FRE were 3.1 ± 0.75 mm and 3.5 ± 0.17 mm, respectively.

CONCLUSIONS

The AL configurations proposed in this investigation provide sufficient registration accuracy and can help to avoid the disadvantages of SMs if used clinically.

Reduced risk trajectory planning in image-guided keyhole neurosurgery

PURPOSE

The authors present and evaluate a new preoperative planning method and computer software designed to reduce the risk of candidate trajectories for straight rigid tool insertion in image-guided keyhole neurosurgery.

METHODS

Trajectories are computed based on the surgeon-defined target and a candidate entry point area on the outer head surface on preoperative CT/MRI scans. A multiparameter risk card provides an estimate of the risk of each trajectory according to its proximity to critical brain structures. Candidate entry points in the outer head surface areas are then color-coded and displayed in 3D to facilitate selection of the most adequate point. The surgeon then defines and/or revised the insertion trajectory using an interactive 3D visualization of surrounding structures. A safety zone around the selected trajectory is also computed to visualize the expected worst-case deviation from the planned insertion trajectory based on tool placement errors in previous surgeries.

RESULTS

A retrospective comparative study for ten selected targets on MRI head scans for eight patients showed a significant reduction in insertion trajectory risk. Using the authors' method, trajectories longer than 30 mm were an average of 2.6 mm further from blood vessels compared to the conventional manual method. Average planning times were 8.4 and 5.9 min for the conventional technique and the authors' method, respectively. Neurosurgeons reported improved understanding of possible risks and spatial relations for the trajectory and patient anatomy.

CONCLUSIONS

The suggested method may result in safer trajectories, shorter preoperative planning time, and improved understanding of risks and possible complications in keyhole neurosurgery.

Fiducial optimization for minimal target registration error in image-guided neurosurgery

This paper presents new methods for the optimal selection of anatomical landmarks and optimal placement of fiducial markers in image-guided neurosurgery. These methods allow the surgeon to optimally plan fiducial marker locations on routine diagnostic images before preoperative imaging and to intraoperatively select the set of fiducial markers and anatomical landmarks that minimize the expected target registration error (TRE). The optimization relies on a novel empirical simulation-based TRE estimation method built on actual fiducial localization error (FLE) data. Our methods take the guesswork out of the registration process and can reduce localization error without additional imaging and hardware. Our clinical experiments on five patients who underwent brain surgery with a navigation system show that optimizing one marker location and the anatomical landmarks configuration reduced the TRE. The average TRE values using the usual fiducials setup and using the suggested method were 4.7 mm and 3.2 mm, respectively. We observed a maximum improvement of 4 mm. Reducing the target registration error has the potential to support safer and more accurate minimally invasive neurosurgical procedures.