Steven Haker

Statistical validation of image segmentation quality based on a spatial overlap index.

RATIONALE AND OBJECTIVES To examine a statistical validation method based on the spatial overlap between two sets of segmentations of the same anatomy. MATERIALS AND METHODS The Dice similarity coefficient (DSC) was used as a statistical validation metric to evaluate the performance of both the reproducibility of manual segmentations and the spatial overlap accuracy of automated probabilistic fractional segmentation of MR images, illustrated on two clinical examples. Example 1: 10 consecutive cases of prostate brachytherapy patients underwent both preoperative 1.5T and intraoperative 0.5T MR imaging. For each case, 5 repeated manual segmentations of the prostate peripheral zone were performed separately on preoperative and on intraoperative images. Example 2: A semi-automated probabilistic fractional segmentation algorithm was applied to MR imaging of 9 cases with 3 types of brain tumors. DSC values were computed and logit-transformed values were compared in the mean with the analysis of variance (ANOVA). RESULTS Example 1: The mean DSCs of 0.883 (range, 0.876-0.893) with 1.5T preoperative MRI and 0.838 (range, 0.819-0.852) with 0.5T intraoperative MRI (P < .001) were within and at the margin of the range of good reproducibility, respectively. Example 2: Wide ranges of DSC were observed in brain tumor segmentations: Meningiomas (0.519-0.893), astrocytomas (0.487-0.972), and other mixed gliomas (0.490-0.899). CONCLUSION The DSC value is a simple and useful summary measure of spatial overlap, which can be applied to studies of reproducibility and accuracy in image segmentation. We observed generally satisfactory but variable validation results in two clinical applications. This metric may be adapted for similar validation tasks.

Optimal Mass Transport for Registration and Warping

Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge–Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the L2 Kantorovich–Wasserstein or “Earth Movers Distance” under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extend this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.

Differential and Numerically Invariant Signature Curves Applied to Object Recognition

We introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of É. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identical. The important examples of the Euclidean and equi-affine groups are discussed in detail. Secondly, we show how a new approach to the numerical approximation of differential invariants, based on suitable combination of joint invariants of the underlying group action, allows one to numerically compute differential invariant signatures in a fully group-invariant manner. Applications to a variety of fundamental issues in vision, including detection of symmetries, visual tracking, and reconstruction of occlusions, are discussed.

On the Laplace-Beltrami operator and brain surface flattening

In this paper, using certain conformal mappings from uniformization theory, the authors give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, the authors indicate how the procedure may be implemented using finite elements. Further, they show how the geometry of the brain surface may be studied using this approach.

New Approaches to Estimation of White Matter Connectivity in Diffusion Tensor MRI: Elliptic PDEs and Geodesics in a Tensor-Warped Space

We investigate new approaches to quantifying the white matter connectivity in the brain using Diffusion Tensor Magnetic Resonance Imaging data. Our first approach finds a steady-state concentration/ heat distribution using the three-dimensional tensor field as diffusion/ conductivity tensors. Our second approach casts the problem in a Riemannian framework, deriving from each tensor a local warping of space, and finding geodesic paths in the space. Both approaches use the information from the whole tensor, and can provide numerical measures of connectivity.

Robot-assisted needle placement in open MRI: System architecture, integration and validation

In prostate cancer treatment, there is a move toward targeted interventions for biopsy and therapy, which has precipitated the need for precise image-guided methods for needle placement. This paper describes an integrated system for planning and performing percutaneous procedures with robotic assistance under MRI guidance. A graphical planning interface allows the physician to specify the set of desired needle trajectories, based on anatomical structures and lesions observed in the patients registered pre-operative and pre-procedural MR images, immediately prior to the intervention in an open-bore MRI scanner. All image-space coordinates are automatically computed, and are used to position a needle guide by means of an MRI-compatible robotic manipulator, thus avoiding the limitations of the traditional fixed needle template. Automatic alignment of real-time intra-operative images aids visualization of the needle as it is manually inserted through the guide. Results from in-scanner phantom experiments are provided.

MINIMIZING FLOWS FOR THE MONGE-KANTOROVICH PROBLEM ∗

In this work, we formulate a new minimizing flow for the optimal mass transport (Monge-Kantorovich) problem. We study certain properties of the flow, including weak solutions as well as short- and long-term existence. Optimal transport has found a number of applications, includ- ing econometrics, fluid dynamics, cosmology, image processing, automatic control, transportation, statistical physics, shape optimization, expert systems, and meteorology. 1. Introduction. In this paper, we derive a novel gradient descent flow for the computation of the optimal transport map (when it exists) in the Monge-Kantorovich framework. Besides being quite useful for the efficient computation of the transport map, we believe that the flow presented here is quite interesting from a theoretical point of view as well. In the present work, we undertake a study of some of its key

Multiscale 3-D Shape Representation and Segmentation Using Spherical Wavelets

This paper presents a novel multiscale shape representation and segmentation algorithm based on the spherical wavelet transform. This work is motivated by the need to compactly and accurately encode variations at multiple scales in the shape representation in order to drive the segmentation and shape analysis of deep brain structures, such as the caudate nucleus or the hippocampus. Our proposed shape representation can be optimized to compactly encode shape variations in a population at the needed scale and spatial locations, enabling the construction of more descriptive, nonglobal, nonuniform shape probability priors to be included in the segmentation and shape analysis framework. In particular, this representation addresses the shortcomings of techniques that learn a global shape prior at a single scale of analysis and cannot represent fine, local variations in a population of shapes in the presence of a limited dataset. Specifically, our technique defines a multiscale parametric model of surfaces belonging to the same population using a compact set of spherical wavelets targeted to that population. We further refine the shape representation by separating into groups wavelet coefficients that describe independent global and/or local biological variations in the population, using spectral graph partitioning. We then learn a prior probability distribution induced over each group to explicitly encode these variations at different scales and spatial locations. Based on this representation, we derive a parametric active surface evolution using the multiscale prior coefficients as parameters for our optimization procedure to naturally include the prior for segmentation. Additionally, the optimization method can be applied in a coarse-to-fine manner. We apply our algorithm to two different brain structures, the caudate nucleus and the hippocampus, of interest in the study of schizophrenia. We show: 1) a reconstruction task of a test set to validate the expressiveness of our multiscale prior and 2) a segmentation task. In the reconstruction task, our results show that for a given training set size, our algorithm significantly improves the approximation of shapes in a testing set over the Point Distribution Model, which tends to oversmooth data. In the segmentation task, our validation shows our algorithm is computationally efficient and outperforms the Active Shape Model algorithm, by capturing finer shape details

Capturing intraoperative deformations: research experience at Brigham and Women’s hospital

During neurosurgical procedures the objective of the neurosurgeon is to achieve the resection of as much diseased tissue as possible while achieving the preservation of healthy brain tissue. The restricted capacity of the conventional operating room to enable the surgeon to visualize critical healthy brain structures and tumor margin has lead, over the past decade, to the development of sophisticated intraoperative imaging techniques to enhance visualization. However, both rigid motion due to patient placement and nonrigid deformations occurring as a consequence of the surgical intervention disrupt the correspondence between preoperative data used to plan surgery and the intraoperative configuration of the patients brain. Similar challenges are faced in other interventional therapies, such as in cryoablation of the liver, or biopsy of the prostate. We have developed algorithms to model the motion of key anatomical structures and system implementations that enable us to estimate the deformation of the critical anatomy from sequences of volumetric images and to prepare updated fused visualizations of preoperative and intraoperative images at a rate compatible with surgical decision making. This paper reviews the experience at Brigham and Womens Hospital through the process of developing and applying novel algorithms for capturing intraoperative deformations in support of image guided therapy.

Automated atlas-based clustering of white matter fiber tracts from DTMRI

A new framework is presented for clustering fiber tracts into anatomically known bundles. This work is motivated by medical applications in which variation analysis of known bundles of fiber tracts in the human brain is desired. To include the anatomical knowledge in the clustering, we invoke an atlas of fiber tracts, labeled by the number of bundles of interest. In this work, we construct such an atlas and use it to cluster all fiber tracts in the white matter. To build the atlas, we start with a set of labeled ROIs specified by an expert and extract the fiber tracts initiating from each ROI. Affine registration is used to project the extracted fiber tracts of each subject to the atlas, whereas their B-spline representation is used to efficiently compare them to the fiber tracts in the atlas and assign cluster labels. Expert visual inspection of the result confirms that the proposed method is very promising and efficient in clustering of the known bundles of fiber tracts.