This paper describes the design and delivery of a highly successful MOOC that uses MATLAB to teach introductory computer programming to a wide audience. The decisions behind the curriculum and assessment strategy are detailed, and the results are evaluated based on three sessions of the course that saw 80,000 active students, two million lecture views and 100,000 auto-graded programming assignment sets during the three sessions delivered in 2015.

Since its introduction in the early 1990s, the Iterative Closest Point (ICP) algorithm has become one of the most well-known methods for geometric alignment of 3D models. Given two roughly aligned shapes represented by two point sets, the algorithm iteratively establishes point correspondences given the current alignment of the data and computes a rigid transformation accordingly. From a statistical point of view, however, it implicitly assumes that the points are observed with isotropic Gaussian noise. In this paper, we show that this assumption may lead to errors and generalize the ICP such that it can account for anisotropic and inhomogenous localization errors. We 1) provide a formal description of the algorithm, 2) extend it to registration of partially overlapping surfaces, 3) prove its convergence, 4) derive the required covariance matrices for a set of selected applications, and 5) present means for optimizing the runtime. An evaluation on publicly available surface meshes as well as on a set of meshes extracted from medical imaging data shows a dramatic increase in accuracy compared to the original ICP, especially in the case of partial surface registration. As point-based surface registration is a central component in various applications, the potential impact of the proposed method is high.

A general approach to the first-order analysis of error in rigid point registration is presented that accommodates fiducial localization error (FLE) that may be inhomogeneous (varying from point to point) and anisotropic (varying with direction) and also accommodates arbitrary weighting that may also be inhomogeneous and anisotropic. Covariances are derived for target registration error (TRE) and for weighted fiducial registration error (FRE) in terms of covariances of FLE, culminating in a simple implementation that encompasses all combinations of weightings and anisotropy. Furthermore, it is shown that for ideal weighting, in which the weighting matrix for each fiducial equals the inverse of the square root of the cross covariance of its two-space FLE, fluctuations of FRE and TRE are mutually independent. These results are validated by comparison with previously published expressions and by simulation. Furthermore, simulations for randomly generated fiducial positions and FLEs are presented that show that correlation is negligible (correlation coefficient < 0.1) in the exact case for both ideal and uniform weighting (i.e., no weighting), the latter of which is employed in commercial surgical guidance systems. From these results we conclude that for these weighting schemes, while valid expressions exist relating the covariance of FRE to the covariance of TRE, there are no measures of the goodness of fit of the fiducials for a given registration that give to first order any information about the fluctuation of TRE from its expected value and none that give useful information in the exact case. Therefore, as estimators of registration accuracy, such measures should be approached with extreme caution both by the purveyors of guidance systems and by the practitioners who use them.

Presents the revised text for two paragraphs from the above titled paper (ibid., vol. 30, no. 3, pp. 679-693, Mar. 2011).

Gradient-echo (GE) echo planar imaging (EPI) is susceptible to both geometric distortions and signal loss. This paper presents a retrospective correction approach based on nonrigid image registration. A new physics-based intensity correction factor derived to compensate for intravoxel dephasing in GE EPI images is incorporated into a previously reported nonrigid registration algorithm. Intravoxel dephasing causes signal loss and thus intensity attenuation in the images. The new rephasing factor we introduce, which changes the intensity of a voxel in images during the registration, is used to improve the accuracy of the intensity-based nonrigid registration method and mitigate the intensity attenuation effect. Simulation-based experiments are first used to evaluate the method. A magnetic resonance (MR) simulator and a real field map are used to generate a realistic GE EPI image. The geometric distortion computed from the field map is used as the ground truth to which the estimated nonrigid deformation is compared. We then apply the algorithm to a set of real human brain images. The results show that, after registration, alignment between EPI and multi-shot, spin-echo images, which have relatively long acquisition times but negligible distortion, is improved and that signal loss caused by dephasing can be recovered.

In order to demonstrate the growth of the medical image registration field over the past decades, this paper presents the number of journal publications on this topic since 1988 until 2002. In a similar manner, trends in topics within the field of medical image registration are detected. Publications on computed tomography (CT) and magnetic resonance imaging (MRI) are rather constant through the years. Positron emission tomography (PET) and single photon emission computed tomography (SPECT), on the other hand, seem to loose ground to newly emerging functional imaging techniques, such as functional MRI (fMRI) whereas an increase in interest in registration of ultrasound (US) images was observed. Two topics in image registration that are currently considered hot are intraoperative and elastic registration. Although the interest in intraoperative registration strongly increased in the late 1990s, there seems to be a slight relative decrease in recent years. On the other hand, elastic registration has become a popular topic, reaching the highest numbers so far in 2002.

International audience

Guidance systems designed for neurosurgery, hip surgery, spine surgery and for approaches to other anatomy that is relatively rigid can use rigid-body transformations to accomplish image registration. These systems often rely on point-based registration to determine the transformation and many such systems use attached fiducial markers to establish accurate fiducial points for the registration, the points being established by some fiducial localization process. Accuracy is important to these systems, as is knowledge of the level of that accuracy. An advantage of marker-based systems, particularly those in which the markers are bone-implanted, is that registration error depends only on the fiducial localization and is, thus, to a large extent independent of the particular object being registered. Thus, it should be possible to predict the clinical accuracy of marker-based systems on the basis of experimental measurements made with phantoms or previous patients. For most registration tasks, the most important error measure is target registration error (TRE), which is the distance after registration between corresponding points not used in calculating the registration transform. In this paper, we derive an approximation to the distribution of TRE; this is an extension of previous work that gave the expected squared value of TRE. We show the distribution of the squared magnitude of TRE and that of the component of TRE in an arbitrary direction. Using numerical simulations, we show that our theoretical results are a close match to the simulated ones.

While laparoscopes are used for numerous minimally invasive (MI) procedures, MI liver resection and ablative surgery is infrequently performed. The paucity of cases is due to the restriction of the field of view by the laparoscope and the difficulty in determining tumor location and margins under video guidance. By merging MI surgery with interactive, image-guided surgery (IIGS), we hope to overcome localization difficulties present in laparoscopic liver procedures. One key component of any IIGS system is the development of accurate registration techniques to map image space to physical or patient space. This manuscript focuses on the accuracy and analysis of the direct linear transformation (DLT) method to register physical space with laparoscopic image space on both distorted and distortion-corrected video images. Experiments were conducted on a liver-sized plastic phantom affixed with 20 markers at various depths. After localizing the points in both physical and laparoscopic image space, registration accuracy was assessed for different combinations and numbers of control points (n) to determine the quantity necessary to develop a robust registration matrix. For n = 11, average target registration error (TRE) was 0.70 +/- 0.20 mm. We also studied the effects of distortion correction on registration accuracy. For the particular distortion correction method and laparoscope used in our experiments, there was no statistical significance between physical to image registration error for distorted and corrected images. In cases where a minimum number of control points (n = 6) are acquired, the DLT is often not stable and the mathematical process can lead to high TRE values. Mathematical filters developed through the analysis of the DLT were used to prospectively eliminate outlier cases where the TRE was high. For n = 6, prefilter average TRE was 17.4 +/- 153 mm for all trials; when the filters were applied, average TRE decreased to 1.64 +/- 1.10 mm for the remaining trials.

The primary objective of this study is to perform a blinded evaluation of two groups of retrospective image registration techniques, using as a gold standard a prospective marker-based registration method, and to compare the performance of one group with the other. These techniques have already been evaluated individually [27]. In this paper, however, we find that by grouping the techniques as volume based or surface based, we can make some interesting conclusions which were not visible in the earlier study. In order to ensure blindness, all retrospective registrations were performed by participants who had no knowledge of the gold-standard results until after their results had been submitted. Image volumes of three modalities: X-ray computed tomography (CT), magnetic resonance (MR), and positron emission tomography (PET) were obtained from patients undergoing neurosurgery at Vanderbilt University Medical Center on whom bone-implanted fiducial markers were mounted. These volumes had all traces of the markers removed and were provided via the Internet to project collaborators outside Vanderbilt, who then performed retrospective registrations on the volumes, calculating transformations from CT to MR and/or from PET to MR. These investigators communicated their transformations, again via the Internet, to Vanderbilt, where the accuracy of each registration was evaluated. In this evaluation, the accuracy is measured at multiple volumes of interest (VOI's). Our results indicate that the volume-based techniques in this study tended to give substantially more accurate and reliable results than the surface-based ones for the CT-to-MR registration tasks, and slightly more accurate results for the PET-to-MR tasks. Analysis of these results revealed that the rotational component of error was more pronounced for the surface-based group. It was also apparent that all of the registration techniques we examined have the potential to produce satisfactory results much of the time, but that visual inspection is necessary to guard against large errors.

In a previous study we demonstrated that automatic retrospective registration algorithms can frequently register magnetic resonance (MR) and computed tomography (CT) images of the brain with an accuracy of better than 2 mm, but in that same study we found that such algorithms sometimes fail, leading to errors of 6 mm or more. Before these algorithms can be used routinely in the clinic, methods must be provided for distinguishing between registration solutions that are clinically satisfactory and those that are not. One approach is to rely on a human observer to inspect the registration results and reject images that have been registered with insufficient accuracy. In this paper, we present a methodology for evaluating the efficacy of the visual assessment of registration accuracy. Since the clinical requirements for level of registration accuracy are likely to be application dependent, we have evaluated the accuracy of the observer's estimate relative to six thresholds: 1-6 mm. The performance of the observers was evaluated relative to the registration solution obtained using external fiducial markers that are screwed into the patient's skull and that are visible in both MR and CT images. This fiducial marker system provides the gold standard for our study. Its accuracy is shown to be approximately 0.5 mm. Two experienced, blinded observers viewed five pairs of clinical MR and CT brain images, each of which had each been misregistered with respect to the gold standard solution. Fourteen misregistrations were assessed for each image pair with misregistration errors distributed between 0 and 10 mm with approximate uniformity. For each misregistered image pair each observer estimated the registration error (in millimeters) at each of five locations distributed around the head using each of three assessment methods. These estimated errors were compared with the errors as measured by the gold standard to determine agreement relative to each of the six thresholds, where agreement means that the two errors lie on the same side of the threshold. The effect of error in the gold standard itself is taken into account in the analysis of the assessment methods. The results were analyzed by means of the Kappa statistic, the agreement rate, and the area of receiver-operating-characteristic (ROC) curves. No assessment performed well at 1 mm, but all methods performed well at 2 mm and higher. For these five thresholds, two methods agreed with the standard at least 80% of the time and exhibited mean ROC areas greater than 0.84. One of these same methods exhibited Kappa statistics that indicated good agreement relative to chance (Kappa > 0.6) between the pooled observers and the standard for these same five thresholds. Further analysis demonstrates that the results depend strongly on the choice of the distribution of misregistration errors presented to the observers.

Guidance systems designed for neurosurgery, hip surgery, and spine surgery, and for approaches to other anatomy that is relatively rigid can use rigid-body transformations to accomplish image registration. These systems often rely on point-based registration to determine the transformation, and many such systems use attached fiducial markers to establish accurate fiducial points for the registration, the points being established by some fiducial localization process. Accuracy is important to these systems, as is knowledge of the level of that accuracy. An advantage of marker-based systems, particularly those in which the markers are bone-implanted, is that registration error depends only on the fiducial localization error (FLE) and is thus to a large extent independent of the particular object being registered. Thus, it should be possible to predict the clinical accuracy of marker-based systems on the basis of experimental measurements made with phantoms or previous patients. This paper presents two new expressions for estimating registration accuracy of such systems and points out a danger in using a traditional measure of registration accuracy. The new expressions represent fundamental theoretical results with regard to the relationship between localization error and registration error in rigid-body, point-based registration. Rigid-body, point-based registration is achieved by finding the rigid transformation that minimizes "fiducial registration error" (FRE), which is the root mean square distance between homologous fiducials after registration. Closed form solutions have been known since 1966. The expected value (FRE2) depends on the number N of fiducials and expected squared value of FLE, (FLE-2, but in 1979 it was shown that (FRE2) is approximately independent of the fiducial configuration C. The importance of this surprising result seems not yet to have been appreciated by the registration community: Poor registrations caused by poor fiducial configurations may appear to be good due to a small FRE value. A more critical and direct measure of registration error is the "target registration error" (TRE), which is the distance between homologous points other than the centroids of fiducials. Efforts to characterize its behavior have been made since 1989. Published numerical simulations have shown that (TRE2) is roughly proportional to (FLE2)/N and, unlike (FRE2), does depend in some way on C. Thus, FRE, which is often used as feedback to the surgeon using a point-based guidance system, is in fact an unreliable indicator of registration-accuracy. In this work we derive approximate expressions for (TRE2), and for the expected squared alignment error of an individual fiducial. We validate both approximations through numerical simulations. The former expression can be used to provide reliable feedback to the surgeon during surgery and to guide the placement of markers before surgery, or at least to warn the surgeon of potentially dangerous fiducial placements; the latter expression leads to a surprising conclusion: Expected registration accuracy (TRE) is worst near the fiducials that are most closely aligned! This revelation should be of particular concern to surgeons who may at present be relying on fiducial alignment as an indicator of the accuracy of their point-based guidance systems.

This paper presents a method designed to register preoperative computed tomography (CT) images to vertebral surface points acquired intraoperatively from ultrasound (US) images or via a tracked probe. It also presents a comparison of the registration accuracy achievable with surface points acquired from the entire posterior surface of the vertebra to the accuracy achievable with points acquired only from the spinous process and central laminar regions. Using a marker-based method as a reference, this work shows that submillimetric registration accuracy can be obtained even when a small portion of the posterior vertebral surface is used for registration. It also shows that when selected surface patches are used, CT slice thickness is not a critical parameter in the registration process. Furthermore, the paper includes qualitative results of registering vertebral surface points in US images to multiple CT slices. The method has been tested with US points and physical points on a plastic spine phantom and with simulated data on a patient CT scan.

Most previously reported registration techniques that align three-dimensional image volumes by matching geometrical features such as points or surfaces use a single type of feature. We recently reported a hybrid registration technique that uses a weighted combination of multiple geometrical feature shapes. In this study we use the weighted geometrical feature (WGF) algorithm to register computed tomography (CT) images of the head to physical space using the skin surface only, the bone surface only, and various weighted combinations of these surfaces and one fiducial point (centroid of a bone-implanted marker). We use data acquired from 12 patients that underwent temporal lobe craniotomies for the resection of cerebral lesions. We evaluate and compare the accuracy of the registrations obtained using these various approaches by using as a reference gold standard the registration obtained using three bone-implanted markers. The results demonstrate that a combination of geometrical features can improve the accuracy of CT-to-physical space registration. Point-based registration requires a minimum of three noncolinear points. The position of a bone-implanted marker can be determined much more accurately than that of a skin-affixed marker or an anatomic landmark. A major disadvantage of using bone-implanted markers is that an invasive procedure is required to implant each marker. By combining surface information, the WGF algorithm allows registration to be performed using only one or two such markers. One important finding is that the use of a single very accurate point (a bone-implanted marker) allows very accurate surface-based registration to be achieved using very few surface points. Finally, the WGF algorithm, which not only allows the combination of multiple types of geometrical information but also handles point-based and surface-based registration as degenerate cases, could form the foundation of a "flexible" surgical navigation system that allows the surgeon to use what he considers the method most appropriate for an individual clinical situation.

In this paper, we describe an extrinsic-point-based, interactive image-guided neurosurgical system designed at Vanderbilt University, Nashville, TN, as part of a collaborative effort among the Departments of Neurological Surgery, Computer Science, and Biomedical Engineering. Multimodal image-to-image (II) and image-to-physical (IP) registration is accomplished using implantable markers. Physical space tracking is accomplished with optical triangulation. We investigate the theoretical accuracy of point-based registration using numerical simulations, the experimental accuracy of our system using data obtained with a phantom, and the clinical accuracy of our system using data acquired in a prospective clinical trial by six neurosurgeons at four medical centers from 158 patients undergoing craniotomies to resect cerebral lesions. We can determine the position of our markers with an error of approximately 0.4 mm in X-ray computed tomography (CT) and magnetic resonance (MR) images and 0.3 mm in physical space. The theoretical registration error using four such markers distributed around the head in a configuration that is clinically practical is approximately 0.5-0.6 mm. The mean CT-physical registration error for the phantom experiments is 0.5 mm and for the clinical data obtained with rigid head fixation during scanning is 0.7 mm. The mean CT-MR registration error for the clinical data obtained without rigid head fixation during scanning is 1.4 mm, which is the highest mean error that we observed. These theoretical and experimental findings indicate that this system is an accurate navigational aid that can provide real-time feedback to the surgeon about anatomical structures encountered in the surgical field.

Detection of the set of centers of maximal disks is a key step toward generation of accurate skeletons on the basis of distance maps. Algorithms using approximate distance metrics are abundant and their theory has been well established. However, the resulting skeletons may be inaccurate and sensitive to rotation. In this paper, we study methods for detecting maximal disks from distance maps based on the exact Euclidean metric. We first show that no previous algorithm identifies the exact set of discrete maximal disks under Euclidean distance metric. We then propose new algorithms and show that they produce the exact set of maximal disks. The effectiveness of our algorithms is demonstrated with numerous examples.

The skeleton is an important representation for shape analysis. A common approach for generating discrete skeletons takes three steps: 1) computing the distance map, 2) detecting maximal disks from the distance map, and 3) linking the centers of maximal disks (CMDs) into a connected skeleton. Algorithms using approximate distance metrics are abundant and their theory has been well established. However, the resulting skeletons may be inaccurate and sensitive to rotation. In this paper, we study methods for generating skeletons based on the exact Euclidean metric. We first show that no previous algorithms identify the exact set of discrete maximal disks under the Euclidean metric. We then propose new algorithms and show that they are correct. To link CMDs into connected skeletons, we examine two prevalent approaches: connected thinning and steepest ascent. We point out that the connected thinning approach does not work properly for Euclidean distance maps. Only the steepest ascent algorithm produces skeletons that are truly medially placed. The resulting skeletons have all the desirable properties: they have the same simple connectivity as the figure, they are well-centered, they are insensitive to rotation, and they allow exact reconstruction. The effectiveness of our algorithms is demonstrated with numerous examples.

Analysis of brain images often requires accurate localization of cortical convolutions. Although magnetic resonance (MR) brain images offer sufficient resolution for identifying convolutions in theory, the nature of tomographic imaging prevents clear definition of convolutions in individual slices. Existing methods for solving this problem rely on heuristic adaptation of brain atlases created from a small number of individuals. These methods do not usually provide high accuracy because of large biological variations among individuals. The authors propose to localize convolutions by linking realistic visualizations of the cortical surface with the original image volume. They have developed a system so that a user can quickly localize key convolutions in several visualizations of an entire brain surface. Because of the links between the visualizations and the original volume, these convolutions are simultaneously localized in the original image slices. In the process of the authors' development, they have implemented a fast and easy method for visualizing cortical surfaces in MR images, thereby making their scheme usable in practical applications.

The authors present a weighted geometrical feature (WGF) registration algorithm. Its efficacy is demonstrated by combining points and a surface. The technique is an extension of Besl and McKay's (1992) iterative closest point (ICP) algorithm. The authors use the WGF algorithm to register X-ray computed tomography (CT) and T2-weighted magnetic resonance (MR) volume head images acquired from eleven patients that underwent craniotomies in a neurosurgical clinical trial. Each patient had five external markers attached to transcutaneous posts screwed into the outer table of the skull. The authors define registration error as the distance between positions of corresponding markers that are not used for registration. The CT and MR images are registered using fiducial paints (marker positions) only, a surface only, and various weighted combinations of points and a surface. The CT surface is derived from contours corresponding to the inner surface of the skull. The MR surface is derived from contours corresponding to the cerebrospinal fluid (CSF)-dura interface. Registration using points and a surface is found to be significantly more accurate then registration using only points or a surface.

It is shown that it is possible to provide an effective, patient-specific correction to geometrical distortion in MRI (magnetic resonance imaging) for use in stereotaxy. The authors examine two correction techniques, each based on the acquisition of an additional image of the patient utilizing a reversed readout gradient. Information from the additional image is combined with that of the standard image to provide geometrically corrected coordinates for targets identified by a neurosurgeon. They have applied the technique retrospectively to images of a patient who has undergone stereotaxic surgery, and they show that large geometrical errors are corrected.>

A technique for producing geometrically accurate magnetic resonance images (MRIs) with undistorted intensity in the face of high levels of static field inhomogeneity arising from either source is presented. The technique requires the acquisition of two images of the same object with altered gradients. On the basis of a knowledge of these gradients it employs an automatic postprocessing step that exploits some invariant characteristics of the distortions to produce a rectified image from the two acquired images. No phantom imaging is involved and no operator interaction is required. The technique is theoretically justified and compared to other techniques, and experimental results that show that the technique works are presented. The improved accuracy in geometry and intensity may improve reliability of stereotactic surgery, may enhance the feasibility of both clinical and industrial imaging via external fields, and may increase the resolution of microscopic imaging.

No abstract available.