A NOVEL INTRINSIC UNSCENTED KALMAN FILTER FOR TRACTOGRAPHY FROM HARDI*

The unscented Kalman filter (UKF) was recently introduced in literature for simultaneous multi-tensor estimation and tractography. This UKF however was not intrinsic to the space of diffusion tensors. Lack of this key property leads to inaccuracies in the multi-tensor estimation as well as in tractography. In this paper, we propose an novel intrinsic unscented Kalman filter (IUKF) in the space of symmetric positive definite matrices, which can be used for simultaneous recursive estimation of multi-tensors and tractography from diffusion weighted MR data. In addition to being more accurate, IUKF retains all the advantages of UKF for instance, multi-tensor estimation is only performed in the places where it is needed for tractography, which would be much more efficient than the two stage process involved in methods that do tracking post diffusion tensor estimation. The accuracy and effectiveness of the proposed method is demonstrated via real data experiments.

Kernel Fisher discriminant for shape-based classification in epilepsy.

In this paper, we present the application of kernel Fisher discriminant in the statistical analysis of shape deformations that indicate the hemispheric location of an epileptic focus. The scans of two classes of patients with epilepsy, those with a right and those with a left anterior medial temporal lobe focus (RATL and LATL), as validated by clinical consensus and subsequent surgery, were compared to a set of age and sex matched healthy volunteers using both volume and shape based features. Shape-based features are derived from the displacement field characterizing the non-rigid deformation between the left and right hippocampi of a control or a patient as the case may be. Using the shape-based features, the results show a significant improvement in distinguishing between the controls and the rest (RATL and LATL) vis-a-vis volume-based features. Using a novel feature, namely, the normalized histogram of the 3D displacement field, we also achieved significant improvement over the volume-based feature in classifying the patients as belonging to either of the two classes LATL or RATL, respectively. It should be noted that automated identification of hemispherical foci of epilepsy has not been previously reported.

DT-MRI denoising and neuronal fiber tracking.

Diffusion tensor imaging can provide the fundamental information required for viewing structural connectivity. However, robust and accurate acquisition and processing algorithms are needed to accurately map the nerve connectivity. In this paper, we present a novel algorithm for extracting and visualizing the fiber tracts in the CNS, specifically in the brain. The automatic fiber tract mapping problem will be solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing of the diffusion-weighted data (prior to tensor calculation) is achieved via a weighted TV-norm minimization, which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Neuronal fibers are then traced by calculating the integral curves of this vector field. Results are expressed using three modes of visualization: (1) Line integral convolution produces an oriented texture which shows fiber pathways in a planar slice of the data. (2) A streamtube map is generated to present a 3D view of fiber tracts. Additional information, such as degree of anisotropy, can be encoded in the tube radius, or by using color. (3) A particle system form of visualization is also presented. This mode of display allows for interactive exploration of fiber connectivity with no additional preprocessing.

Image registration via level-set motion: applications to atlas-based segmentation.

Image registration is an often encountered problem in various fields including medical imaging, computer vision and image processing. Numerous algorithms for registering image data have been reported in these areas. In this paper, we present a novel curve evolution approach expressed in a level-set framework to achieve image intensity morphing and a simple non-linear PDE for the corresponding coordinate registration. The key features of the intensity morphing model are that (a) it is very fast and (b) existence and uniqueness of the solution for the evolution model are established in a Sobolev space as opposed to using viscosity methods. The salient features of the coordinate registration model are its simplicity and computational efficiency. The intensity morph is easily achieved via evolving level-sets of one image into the level-sets of the other. To explicitly estimate the coordinate transformation between the images, we derive a non-linear PDE-based motion model which can be solved very efficiently. We demonstrate the performance of our algorithm on a variety of images including synthetic and real data. As an application of the PDE-based motion model, atlas based segmentation of hippocampal shape from several MR brain scans is depicted. In each of these experiments, automated hippocampal shape recovery results are validated via manual "expert" segmentations.

A new & robust information theoretic measure and its application to image alignment.

In this paper we develop a novel measure of information in a random variable based on its cumulative distribution that we dub cumulative residual entropy (CRE). This measure parallels the well known Shannon entropy but has the following advantages: (1) it is more general than the Shannon Entropy as its definition is valid in the discrete and continuous domains, (2) it possess more general mathematical properties and (3) it can be easily computed from sample data and these computations asymptotically converge to the true values. Based on CRE, we define the cross-CRE (CCRE) between two random variables, and apply it to solve the image alignment problem for parameterized (3D rigid and affine) transformations. The key strengths of the CCRE over using the mutual information (based on Shannon's entropy) are that the former has significantly larger tolerance to noise and a much larger convergence range over the field of parameterized transformations. We demonstrate these strengths via experiments on synthesized and real image data.

A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from DWI.

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from diffusion weighted imaging (DWI). The constrained variational principle involves the minimization of a regularization term in an LP norm, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The original nonlinear form leads to a more accurate (when compared to the linearized form) estimated tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a possibly known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of cholesky factors and estimated. variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Both synthetic and real data experiments are shown to depict the performance of the tensor field estimation algorithm. Fiber tracts in a rat brain are then mapped using a particle system based visualization technique.

An accurate and efficient bayesian method for automatic segmentation of brain MRI.

Automatic three-dimensional (3-D) segmentation of the brain from magnetic resonance (MR) scans is a challenging problem that has received an enormous amount of attention lately. Of the techniques reported in the literature, very few are fully automatic. In this paper, we present an efficient and accurate, fully automatic 3-D segmentation procedure for brain MR scans. It has several salient features; namely, the following. 1) Instead of a single multiplicative bias field that affects all tissue intensities, separate parametric smooth models are used for the intensity of each class. 2) A brain atlas is used in conjunction with a robust registration procedure to find a nonrigid transformation that maps the standard brain to the specimen to be segmented. This transformation is then used to: segment the brain from nonbrain tissue; compute prior probabilities for each class at each voxel location and find an appropriate automatic initialization. 3) Finally, a novel algorithm is presented which is a variant of the expectation-maximization procedure, that incorporates a fast and accurate way to find optimal segmentations, given the intensity models along with the spatial coherence assumption. Experimental results with both synthetic and real data are included, as well as comparisons of the performance of our algorithm with that of other published methods.

Local frequency representations for robust multimodal image registration.

Automatic registration of multimodal images involves algorithmically estimating the coordinate transformation required to align the data sets. Most existing methods in the literature are unable to cope with registration of image pairs with large nonoverlapping field of view (FOV). We propose a robust algorithm, based on matching dominant local frequency image representations, which can cope with image pairs with large nonoverlapping FOV. The local frequency representation naturally allows for processing the data at different scales/resolutions, a very desirable property from a computational efficiency view point. Our algorithm involves minimizing-over all rigid/affine transformations--the integral of the squared error (ISE or L2 E) between a Gaussian model of the residual and its true density function. The residual here refers to the difference between the local frequency representations of the transformed (by an unknown transformation) source and target data. We present implementation results for image data sets, which are misaligned magnetic resonance (MR) brain scans obtained using different image acquisition protocols as well as misaligned MR-computed tomography scans. We experimently show that our L2E-based scheme yields better accuracy over the normalized mutual information.

An efficient motion estimator with application to medical image registration.

Image registration is a very important problem in computer vision and medical image processing. Numerous algorithms for registering single and multi-modal image data have been reported in these areas. Robustness as well as computational efficiency are prime factors of importance in image data registration. In this paper, a robust/reliable and efficient algorithm for estimating the transformation between two image data sets of a patient taken from the same modality over time is presented. Estimating the registration between two image data sets is formulated as a motion-estimation problem. We use a hierarchical optical flow motion model which allows for both global as well as local motion between the data sets. In this hierarchical motion model, we represent the flow field with a B-spline basis which implicitly incorporates smoothness constraints on the field. In computing the motion, we minimize the expectation of the squared differences energy function numerically via a modified Newton iteration scheme. The main idea in the modified Newton method is that we precompute the Hessian of the energy function at the optimum without explicitly knowing the optimum. This idea is used for both global and local motion estimation in the hierarchical motion model. We present examples of motion estimation on synthetic and real data (from a patient acquired during pre- and post-operative stages) and compare the performance of our algorithm with that of competing ones.

Fast numerical algorithms for fitting multiresolution hybrid shape models to brain MRI.

In this paper, we present new and fast numerical algorithms for shape recovery from brain MRI using multiresolution hybrid shape models. In this modeling framework, shapes are represented by a core rigid shape characterized by a superquadric function and a superimposed displacement function which is characterized by a membrane spline discretized using the finite-element method. Fitting the model to brain MRI data is cast as an energy minimization problem which is solved numerically. We present three new computational methods for model fitting to data. These methods involve novel mathematical derivations that lead to efficient numerical solutions of the model fitting problem. The first method involves using the nonlinear conjugate gradient technique with a diagonal Hessian preconditioner. The second method involves the nonlinear conjugate gradient in the outer loop for solving global parameters of the model and a preconditioned conjugate gradient scheme for solving the local parameters of the model. The third method involves the nonlinear conjugate gradient in the outer loop for solving the global parameters and a combination of the Schur complement formula and the alternating direction-implicit method for solving the local parameters of the model. We demonstrate the efficiency of our model fitting methods via experiments on several MR brain scans.