Diffusion MRI (dMRI) is the primary imaging modality used to study brain microstructure in vivo. Reliable and computationally efficient parameter inference for common dMRI biophysical models is a challenging inverse problem, due to factors such as variable dimensionalities (reflecting the unknown number of distinct white matter fiber populations in a voxel), low signal-to-noise ratios, and non-linear forward models. These challenges have led many existing methods to use biologically implausible simplified models to stabilize estimation, for instance, assuming shared microstructure across all fiber populations within a voxel. In this work, we introduce a novel sequential method for multi-fiber parameter inference that decomposes the task into a series of manageable subproblems. These subproblems are solved using deep neural networks tailored to problem-specific structure and symmetry, and trained via simulation. The resulting inference procedure is largely amortized, enabling scalable parameter estimation and uncertainty quantification across all model parameters. Simulation studies and real imaging data analysis using the Human Connectome Project (HCP) demonstrate the advantages of our method over standard alternatives. In the case of the standard model of diffusion, our results show that under HCP-like acquisition schemes, estimates for extra-cellular parallel diffusivity are highly uncertain, while those for the intra-cellular volume fraction can be estimated with relatively high precision.

The Orientation Distribution Function (ODF) characterizes key brain microstructural properties and plays an important role in understanding brain structural connectivity. Recent works introduced Implicit Neural Representation (INR) based approaches to form a spatially aware continuous estimate of the ODF field and demonstrated promising results in key tasks of interest when compared to conventional discrete approaches. However, traditional INR methods face difficulties when scaling to large-scale images, such as modern ultra-high-resolution MRI scans, posing challenges in learning fine structures as well as inefficiencies in training and inference speed. In this work, we propose HashEnc, a grid-hash-encoding-based estimation of the ODF field and demonstrate its effectiveness in retaining structural and textural features. We show that HashEnc achieves a 10 % enhancement in image quality while requiring 3 $$\times $$ less computational resources than current methods. Our code can be found at https://github.com/MunzerDw/NODF-HashEnc .

Parcellation of anatomically segregated cortical and subcortical brain regions is required in diffusion MRI (dMRI) analysis for region-specific quantification and better anatomical specificity of tractography. Most current dMRI parcellation approaches compute the parcellation from anatomical MRI (T1- or T2-weighted) data, using tools such as FreeSurfer or CAT12, and then register it to the diffusion space. However, the registration is challenging due to image distortions and low resolution of dMRI data, often resulting in mislabeling in the derived brain parcellation. Furthermore, these approaches are not applicable when anatomical MRI data is unavailable. As an alternative we developed the Deep Diffusion Parcellation (DDParcel), a deep learning method for fast and accurate parcellation of brain anatomical regions directly from dMRI data. The input to DDParcel are dMRI parameter maps and the output are labels for 101 anatomical regions corresponding to the FreeSurfer Desikan-Killiany (DK) parcellation. A multi-level fusion network leverages complementary information in the different input maps, at three network levels: input, intermediate layer, and output. DDParcel learns the registration of diffusion features to anatomical MRI from the high-quality Human Connectome Project data. Then, to predict brain parcellation for a new subject, the DDParcel network no longer requires anatomical MRI data but only the dMRI data. Comparing DDParcel's parcellation with T1w-based parcellation shows higher test-retest reproducibility and a higher regional homogeneity, while requiring much less computational time. Generalizability is demonstrated on a range of populations and dMRI acquisition protocols. Utility of DDParcel's parcellation is demonstrated on tractography analysis for fiber tract identification.

Tractography from high-dimensional diffusion magnetic resonance imaging (dMRI) data allows brain's structural connectivity analysis. Recent dMRI studies aim to compare connectivity patterns across subject groups and disease populations to understand subtle abnormalities in the brain's white matter connectivity and distributions of biologically sensitive dMRI derived metrics. Existing software products focus solely on the anatomy, are not intuitive or restrict the comparison of multiple subjects. In this paper, we present the design and implementation of FiberStars, a visual analysis tool for tractography data that allows the interactive visualization of brain fiber clusters combining existing 3D anatomy with compact 2D visualizations. With FiberStars, researchers can analyze and compare multiple subjects in large collections of brain fibers using different views. To evaluate the usability of our software, we performed a quantitative user study. We asked domain experts and non-experts to find patterns in a tractography dataset with either FiberStars or an existing dMRI exploration tool. Our results show that participants using FiberStars can navigate extensive collections of tractography faster and more accurately. All our research, software, and results are available openly.

Joint relaxation-diffusion measurements can provide new insight about the tissue microstructural properties. Most recent methods have focused on inverting the Laplace transform to recover the joint distribution of relaxation-diffusion. However, as is well-known, this problem is notoriously ill-posed and numerically unstable. In this work, we address this issue by directly computing the joint moments of transverse relaxation rate and diffusivity, which can be robustly estimated. To zoom into different parts of the joint distribution, we further enhance our method by applying multiplicative filters to the joint probability density function of relaxation and diffusion and compute the corresponding moments. We propose an approach to use these moments to compute several novel scalar indices to characterize specific properties of the underlying tissue microstructure. Furthermore, for the first time, we propose an algorithm to estimate diffusion signals that are independent of echo time based on the moments of the marginal probability density function of diffusion. We demonstrate its utility in extracting tissue information not contaminated with multiple intra-voxel relaxation rates. We compare the performance of four types of filters that zoom into tissue components with different relaxation and diffusion properties and demonstrate it on an in-vivo human dataset. Experimental results show that these filters are able to characterize heterogeneous tissue microstructure. Moreover, the filtered diffusion signals are also able to distinguish fiber bundles with similar orientations but different relaxation rates. The proposed method thus allows to characterize the neural microstructure information in a robust and unique manner not possible using existing techniques.

Coherence and causality measures are often used to analyze the influence of one region on another during analysis of functional brain networks. The analysis methods usually involve a regression problem, where the signal of interest is decomposed into a mixture of regressor and a residual signal. In this paper, we revisit this basic problem and present solutions that provide the minimal-entropy residuals for different types of regression filters, such as causal, instantaneously causal, and noncausal filters. Using optimal prediction theory, we derive several novel frequency-domain expressions for partial coherence, causality, and conditional causality analysis. In particular, our solution provides a more accurate estimation of the frequency-domain causality compared with the classical Geweke causality measure. Using synthetic examples and in vivo resting-state functional magnetic resonance imaging data from the human connectome project, we show that the proposed solution is more accurate at revealing frequency-domain linear dependence among high-dimensional signals.

The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK)-which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

The registration of a pair of point sets as well as the estimation of their pointwise correspondences is a challenging and important task in computer vision. In this paper, we present a method to estimate the diffeomorphic deformation, together with the pointwise correspondences, between a pair of point sets. Many of the registration problems are iteratively solved by estimating the correspondence, locally optimizing certain cost functionals over the rigid or similarity or affine transformation group, then estimating the correspondence again, and so on. This type of approach, however, is well-known to be susceptible to suboptimal local solutions. In this paper, we first adopt the perspective of treating the registration as a posterior estimation optimization problem and solve it accordingly via a particle-filtering framework. Second, within such a framework, the diffeomorphic registration is performed to correct the nonlinear deformation of the points. In doing so, we provide a solution less susceptible to local minima. We provide the experimental results, which include challenging medical data sets where the two point sets differ by 180 (°) rotation as well as local deformations, to highlight the algorithm's capability of robustly finding the more globally optimal solution for the registration task.

We propose a novel probabilistic framework to merge information from diffusion weighted imaging tractography and resting-state functional magnetic resonance imaging correlations to identify connectivity patterns in the brain. In particular, we model the interaction between latent anatomical and functional connectivity and present an intuitive extension to population studies. We employ the EM algorithm to estimate the model parameters by maximizing the data likelihood. The method simultaneously infers the templates of latent connectivity for each population and the differences in connectivity between the groups. We demonstrate our method on a schizophrenia study. Our model identifies significant increases in functional connectivity between the parietal/posterior cingulate region and the frontal lobe and reduced functional connectivity between the parietal/posterior cingulate region and the temporal lobe in schizophrenia. We further establish that our model learns predictive differences between the control and clinical populations, and that combining the two modalities yields better results than considering each one in isolation.

Despite the relative recency of its inception, the theory of compressive sampling (aka compressed sensing) (CS) has already revolutionized multiple areas of applied sciences, a particularly important instance of which is medical imaging. Specifically, the theory has provided a different perspective on the important problem of optimal sampling in magnetic resonance imaging (MRI), with an ever-increasing body of works reporting stable and accurate reconstruction of MRI scans from the number of spectral measurements which would have been deemed unacceptably small as recently as five years ago. In this paper, the theory of CS is employed to palliate the problem of long acquisition times, which is known to be a major impediment to the clinical application of high angular resolution diffusion imaging (HARDI). Specifically, we demonstrate that a substantial reduction in data acquisition times is possible through minimization of the number of diffusion encoding gradients required for reliable reconstruction of HARDI scans. The success of such a minimization is primarily due to the availability of spherical ridgelet transformation, which excels in sparsifying HARDI signals. What makes the resulting reconstruction procedure even more accurate is a combination of the sparsity constraints in the diffusion domain with additional constraints imposed on the estimated diffusion field in the spatial domain. Accordingly, the present paper describes an original way to combine the diffusion- and spatial-domain constraints to achieve a maximal reduction in the number of diffusion measurements, while sacrificing little in terms of reconstruction accuracy. Finally, details are provided on an efficient numerical scheme which can be used to solve the aforementioned reconstruction problem by means of standard and readily available estimation tools. The paper is concluded with experimental results which support the practical value of the proposed reconstruction methodology.

Visualization and analysis of the micro-architecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, the conventional diffusion tensor imaging (DTI) used for estimating the local orientations of neural fibers is incapable of performing reliably in the situations when a voxel of interest accommodates multiple fiber tracts. In this case, a much more accurate analysis is possible using the high angular resolution diffusion imaging (HARDI) that represents local diffusion by its apparent coefficients measured as a discrete function of spatial orientations. In this note, a novel approach to enhancing and modeling the HARDI signals using multiresolution bases of spherical ridgelets is presented. In addition to its desirable properties of being adaptive, sparsifying, and efficiently computable, the proposed modeling leads to analytical computation of the orientation distribution functions associated with the measured diffusion, thereby providing a fast and robust analytical solution for q-ball imaging.

We propose algorithms for tracking the boundary contour of a deforming object from an image sequence, when the nonaffine (local) deformation over consecutive frames is large and there is overlapping clutter, occlusions, low contrast, or outlier imagery. When the object is arbitrarily deforming, each, or at least most, contour points can move independently. Contour deformation then forms an infinite (in practice, very large), dimensional space. Direct application of particle filters (PF) for large dimensional problems is impractically expensive. However, in most real problems, at any given time, most of the contour deformation occurs in a small number of dimensions ("effective basis space") while the residual deformation in the rest of the state space ("residual space") is small. This property enables us to apply the particle filtering with mode tracking (PF-MT) idea that was proposed for such large dimensional problems in recent work. Since most contour deformation is low spatial frequency, we propose to use the space of deformation at a subsampled set of locations as the effective basis space. The resulting algorithm is called deform PF-MT. It requires significant modifications compared to the original PF-MT because the space of contours is a non-Euclidean infinite dimensional space.

We describe a technique that uses tractography to drive the local fiber model estimation. Existing techniques use independent estimation at each voxel so there is no running knowledge of confidence in the estimated model fit. We formulate fiber tracking as recursive estimation: at each step of tracing the fiber, the current estimate is guided by those previous. To do this we perform tractography within a filter framework and use a discrete mixture of Gaussian tensors to model the signal. Starting from a seed point, each fiber is traced to its termination using an unscented Kalman filter to simultaneously fit the local model to the signal and propagate in the most consistent direction. Despite the presence of noise and uncertainty, this provides a causal estimate of the local structure at each point along the fiber. Using two- and three-fiber models we demonstrate in synthetic experiments that this approach significantly improves the angular resolution at crossings and branchings. In vivo experiments confirm the ability to trace through regions known to contain such crossing and branching while providing inherent path regularization.

Segmentation involves separating an object from the background in a given image. The use of image information alone often leads to poor segmentation results due to the presence of noise, clutter or occlusion. The introduction of shape priors in the geometric active contour (GAC) framework has proved to be an effective way to ameliorate some of these problems. In this work, we propose a novel segmentation method combining image information with prior shape knowledge, using level-sets. Following the work of Leventon et al., we propose to revisit the use of PCA to introduce prior knowledge about shapes in a more robust manner. We utilize kernel PCA (KPCA) and show that this method outperforms linear PCA by allowing only those shapes that are close enough to the training data. In our segmentation framework, shape knowledge and image information are encoded into two energy functionals entirely described in terms of shapes. This consistent description permits to fully take advantage of the Kernel PCA methodology and leads to promising segmentation results. In particular, our shape-driven segmentation technique allows for the simultaneous encoding of multiple types of shapes, and offers a convincing level of robustness with respect to noise, occlusions, or smearing.

Presented at British Machine Vision Conference 2007, University of Warwick, UK, September 10-13, 2007.

This paper proposes a novel method to apply the standard graph cut technique to segmenting multimodal tensor valued images. The Riemannian nature of the tensor space is explicitly taken into account by first mapping the data to a Euclidean space where non-parametric kernel density estimates of the regional distributions may be calculated from user initialized regions. These distributions are then used as regional priors in calculating graph edge weights. Hence this approach utilizes the true variation of the tensor data by respecting its Riemannian structure in calculating distances when forming probability distributions. Further, the non-parametric model generalizes to arbitrary tensor distribution unlike the Gaussian assumption made in previous works. Casting the segmentation problem in a graph cut framework yields a segmentation robust with respect to initialization on the data tested.

In this note, we propose a method to perform segmentation on the tensor manifold, that is, the space of positive definite matrices of given dimension. In this work, we explicitly use the Riemannian structure of the tensor space in designing our algorithm. This structure has already been utilized in several approaches based on active contour models which separate the mean and/or variance inside and outside the evolving contour. We generalize these methods by proposing a new technique for performing segmentation by separating the entire probability distributions of the regions inside and outside the contour using the Bhattacharyya metric. In particular, this allows for segmenting objects with multimodal probability distributions (on the space of tensors). We demonstrate the effectiveness of our algorithm by segmenting various textured images using the structure tensor. A level set based scheme is proposed to implement the curve flow evolution equation.

The standard graph cut technique is a robust method for globally optimal image segmentations. However, because of its global nature, it is prone to capture outlying areas similar to the object of interest. This paper proposes a novel method to constrain the standard graph cut technique for tracking anywhere from one to several objects in regions of interest. For each object, we introduce a pixel penalty based upon distance from a region of interest and so segmentation is biased to remain in this area. Also, we employ a filter predicting the location of the object. The distance penalty is then centered at this location and adoptively scaled based on prediction confidence. This method is capable of tracking multiple interacting objects of different intensity profiles in both gray-scale and color imagery.

Graph cut image segmentation with intensity information alone is prone to fail for objects with weak edges, in clutter, or under occlusion. Existing methods to incorporate shape are often too restrictive for highly varied shapes, use a single fixed shape which may be prone to misalignment, or are computationally intensive. In this note we show how highly variable nonlinear shape priors learned from training sets can be added to existing iterative graph cut methods for accurate and efficient segmentation of such objects. Using kernel principle component analysis, we demonstrate how a shape projection pre-image can induce an iteratively refined shape prior in a Bayesian manner. Examples of natural imagery show that both single-pass and iterative segmentation fail without such shape information.

Tracking deforming objects involves estimating the global motion of the object and its local deformations as a function of time. Tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape, but these are dependent on the chosen parametrization and cannot handle changes in curve topology. Geometric active contours provide a framework which is parametrization independent and allow for changes in topology. In the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects. To the best of our knowledge, this is the first attempt to implement an approximate particle filtering algorithm for tracking on a (theoretically) infinite dimensional state space.

This paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which active contours maximize the difference between a finite number of empirical moments of the "inside" and "outside" distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As an additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either over- or underestimation artifacts. Finally, a number of relevant segmentation results are demonstrated and some further research directions are discussed.

Tracking deforming objects involves estimating the global motion of the object and its local deformations as functions of time. Tracking algorithms using Kalman filters or particle filters (PFs) have been proposed for tracking such objects, but these have limitations due to the lack of dynamic shape information. In this paper, we propose a novel method based on employing a locally linear embedding in order to incorporate dynamic shape information into the particle filtering framework for tracking highly deformable objects in the presence of noise and clutter. The PF also models image statistics such as mean and variance of the given data which can be useful in obtaining proper separation of object and background.

Segmentation involves separating an object from the background. In this work, we propose a novel segmentation method combining image information with prior shape knowledge, within the level-set framework. Following the work of Leventon et al., we revisit the use of principal component analysis (PCA) to introduce prior knowledge about shapes in a more robust manner. To this end, we utilize Kernel PCA and show that this method of learning shapes outperforms linear PCA, by allowing only shapes that are close enough to the training data. In the proposed segmentation algorithm, shape knowledge and image information are encoded into two energy functionals entirely described in terms of shapes. This consistent description allows to fully take advantage of the Kernel PCA methodology and leads to promising segmentation results. In particular, our shape-driven segmentation technique allows for the simultaneous encoding of multiple types of shapes, and offers a convincing level of robustness with respect to noise, clutter, partial occlusions, or smearing.

We propose particle filtering algorithms for tracking on infinite (or large) dimensional state spaces. We consider the general case where state space may not be a vector space, we assume it to be a separable metric space (Polish space). In implementation, any such space is approximated by a finite but large dimensional vector, whose dimension may vary at every time. Monte Carlo sampling from a large dimensional system noise distribution is computationally expensive. Also, the number of particles required for accurate particle filtering increases with the number of independent dimensions of the system noise, making particle filtering even more expensive. But as long as the number of independent system noise dimensions is small, even if the total state space dimension is very large, a particle filtering algorithm can be implemented. In most large dim applications, it is fair to assume that "most of the state change" occurs in a small dimensional basis, which may be fixed or slowly time varying (approximated as piecewise constant). We use this assumption to propose efficient PF algorithms. These are analyzed and extended in N. Vaswani, (2006)

Geometric active contours are formulated in a manner which is parametrization independent. As such, they are amenable to representation as the zero level set of the graph of a higher dimensional function. This representation is able to deal with singularities and changes in topology of the contour. It has been used very successfully in static images for segmentation and registration problems where the contour (represented as an implicit curve) is evolved until it minimizes an image based energy functional. But tracking involves estimating the global motion of the object and its local deformations as a function of time. Some attempts have been made to use geometric active contours for tracking, but most of these minimize the energy at each frame and do not utilize the temporal coherency of the motion or the deformation. On the other hand, tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape. But these are dependent on the chosen parametrization and cannot handle changes in curve topology. In the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects.