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Fast displacement probability profile approximation from HARDI using 4th-order tensors
Abstract
Cartesian tensor basis have been widely used to approximate spherical functions. In Medical Imaging, tensors of various orders have been used to model the diffusivity function in Diffusion-weighted MRI data sets. However, it is known that the peaks of the diffusivity do not correspond to orientations of the underlying fibers and hence the displacement probability profiles should be employed instead. In this paper, we present a novel representation of the probability profile by a 4<sup>th</sup> order tensor, which is a smooth spherical function that can approximate single-fibers as well as multiple-fiber structures. We also present a method for efficiently estimating the unknown tensor coefficients of the probability profile directly from a given high-angular resolution diffusion-weighted (HARDI) data set. The accuracy of our model is validated by experiments on synthetic and real HARDI datasets from a fixed rat spinal cord.
... The wave vector q is q = γδG/2π, with γ the nuclear gyromagnetic ratio and G the applied diffusion gradient vector. Various methods already exist to reconstruct the EAP [4,5,6,7,8,9,10,11,12]. Among the most commonly used methods, diffusion tensor imaging (DTI) [4] is limited by the Gaussian assumption of the free diffusion model, which excludes observed in vivo phenomena such as restriction, heterogeneity, anomalous diffusion, and finite boundary permeability. ...
... The results are promising but have not yet been applied on an in vivo brain. Other techniques suggest using multiple spherical shell acquisitions in order to reconstruct the features of the EAP, such as generalized high order tensors [6] based on cumulant expansions; or the composite and hindered restricted model of diffusion (CHARMED) [7]; or the diffusion kurtosis [8]; or the diffusion orientation transform (DOT) [9]; or hybrid diffusion properties of the EAP [11]; or a fourth order Cartesian tensor representation of the probability profile [12]. Unfortunately, for most of these methods, many DW measurements are needed. ...
... In order to relate the observed diffusion signal to the underlying tissue microstructure , we need to understand how the diffusion signal is influenced by the tissue geometry and its properties. Under the narrow pulse assumption, the relationship between the diffusion signal, E(q), in q-space and the EAP, P (R), in real space, is given by an inverse Fourier transform (IFT) [3] Various methods already exist to reconstruct the EAP [4,5,6,7,8,9,10,11,12]. Among the most commonly used methods, diffusion tensor imaging (DTI) [4] is limited by the Gaussian assumption of the free diffusion model, which excludes observed in vivo phenomena such as restriction, heterogeneity, anomalous diffusion, and finite boundary permeability. ...
Many recent single-shell high angular resolution diffusion imaging reconstruction techniques have been introduced to reconstruct orientation distribution functions (ODF) that only capture angular information contained in the diffusion process of water molecules. By also considering the radial part of the diffusion signal, the reconstruction of the ensemble average diffusion propagator (EAP) of water molecules can provide much richer information about complex tissue microstructure than the ODF. In this paper, we present diffusion propagator imaging (DPI), a novel technique to reconstruct the EAP from multiple shell acquisitions. The DPI solution is analytical and linear because it is based on a Laplace equation modeling of the diffusion signal. DPI is validated with ex vivo phantoms and also illustrated on an in vivo human brain dataset. DPI is shown to reconstruct EAP from only two b-value shells and approximately 100 diffusion measurements.
... In this continuous mixture, we use a mixture of von Mises-Fisher as the mixing density. Once the signal has been modeled, we use the method proposed in [7] to recover the displacement probability (one can also use any of the other published methods) from which orientations of fibers are extracted by fixing the radius magni- tude [36] (radial integration as proposed in [26] can also be used). In our mixture model we preserve the advantages that spherical deconvolution based methods have over multicompartmental methods as in our method, neither the number of components be pre-decided nor any non-linear optimization technique be used to obtain the mixture weights. ...
... Yet another technique seeks an alternative representation called the fiber orientation distribution (from the Q-Ball images) from which the optimal fiber orientations can be derived ([13], [4]). In this work we used the method proposed in [7], though any of the above mentioned methods would also work. Once the water displacement probability has been estimated , orientations of the underlying distinct fiber bundles can be recovered using the spherical function p(r) which is extracted from the volume of P (r 0 r) by either fixing r 0 ([26]) or by integrating over r 0 ([36]) and then finding the maxima of p(r). ...
... In this continuous mixture, we use a mixture of von Mises-Fisher as the mixing density. Once the signal has been modeled, we use the method proposed in [7] to recover the displacement probability (one can also use any of the other published methods) from which orientations of fibers are extracted by fixing the radius magnitude [36] (radial integration as proposed in [26] can also be used). In our mixture model we preserve the advantages that spherical deconvolution based methods have over multicompartmental methods as in our method, neither the number of components be pre-decided nor any non-linear optimization technique be used to obtain the mixture weights. ...
In this paper we propose a method for reconstructing the Diffusion Weighted Magnetic Resonance (DW-MR) signal at each lattice point using a novel continuous mixture of von Mises-Fisher distribution functions. Unlike most existing methods, neither does this model assume a fixed functional form for the MR signal attenuation (e.g. 2nd or 4th order tensor) nor does it arbitrarily fix important mixture parameters like the number of components. We show that this continuous mixture has a closed form expression and leads to a linear system which can be easily solved. Through extensive experimentation with synthetic data we show that this technique outperforms various other state-of-the-art techniques in resolving fiber crossings. Finally, we demonstrate the effectiveness of this method using real DW-MRI data from rat brain and optic chiasm.
... There are only few reconstruction techniques which take advantage of the Cartesian Fourier Transform, e.g. Generalized DTI 2 (GDTI2) (section-3.4.2), the approach proposed in [115] and the approach we have proposed in chapter-5. Diffusion Kurtosis Imaging (DKI) (section-3.4.4) is also closely related, since it models the diffusion signal in the multivariate polynomial basis. ...
... We have begun addressing this problem and preliminary results were presented in [116]. The methods in [115] and in chapter-5 do not consider complete functional bases for modelling the signal, which restricts the types of signal functions that can be estimated in these incomplete functional bases. ...
... Inferring the micro-structure of the tissues requires to reconstruct the diusion pdf from raw diusion signals S(q) which, in absence of measurement noise, read A(q) = |A (q)| (theoretical diusion signals Adjusting the raw diusion signal to a Cartesian lattice by means of bilinear interpolation yields Hybrid Diusion Imaging (HYDI) [12]. Other model-free methods based on MSI sampling provide estimates of the EAP by expanding the raw diusion signals by means of (i) 4-th order tensors [13,14], (ii) the spherical polar Fourier basis up to a given order [15,16] or (iii) the solid harmonics basis up to a given order giving its name to Diusion Propagator Imaging (DPI) [17]. An MSI-based method for the reconstruction of the full diusion pdf (including symmetric and asymmetric dislacements) is also proposed in [18], using the Gram-Charlier expansion of the raw diusion signal up to a given order. ...
... This expression can be simplied by introducing: α = α(t; µ, κ, R) = Re z + |z| 2 , and , β = β(t; µ, κ, R) = Im z 2 (Re z + |z|) , (13) provided that these quantities are well-dened (Re z + |z| > 0), which leads to: ...
Diffusion magnetic resonance imaging (dMRI) is the reference \emph{in vivo} modality to study the connectivity of the brain white matter. Images obtained through dMRI are indeed related to the probability density function (pdf) of displacement of water molecules subject to restricted diffusion in the brain white matter. The knowledge of this diffusion pdf is therefore of primary importance. Several methods have been devised to provide an estimate of it from noisy dMRI signal intensities. They include popular diffusion tensor imaging (DTI) as well as higher-order methods. These approaches suffer from important drawbacks. Standard DTI cannot directly cope with multiple fiber orientations. Higher-order approaches can alleviate these limitations but at the cost of increased acquisition time. In this research report, we propose, in the same vein as DTI, a new parametric model of the diffusion pdf with a reasonably low number of parameters, the estimation of which does not require acquisitions longer than those used in clinics for DTI. This model also accounts for multiple fiber orientations. It is based on the assumption that, in a voxel, diffusing water molecules are divided into compartments. Each compartment is representative of a specific fiber orientation (which defines two opposite directions). In a given compartment, we further assume that water molecules that diffuse along each direction are in equal proportions. We then focus on modeling the pdf of the displacements of water molecules that diffuse only along one of the two directions. Under this model, we derive an analytical relation between the dMRI signal intensities and the parameters of the diffusion pdf. We exploit it to estimate these parameters from noisy signal intensities. We carry out a cone-of-uncertainty analysis to evaluate the accuracy of the estimation of the fiber orientations and we evaluate the angular resolution of our method. Finally, we show promising results on real data and propose a visualization of the diffusion parameters which is very informative to the neurologist.
... The modified model lead to an analytical series expansion of the EAP in Hermite polynomials. In [94], the authors proposed to use tensors to describe a single R 0shell of the EAP, P(R 0 r ||r|| ). They used Hermite polynomials to describe the dMRI signal, since under certain constraints the Fourier transform of Hermite polynomials are homogeneous forms or tensors. ...
... They used Hermite polynomials to describe the dMRI signal, since under certain constraints the Fourier transform of Hermite polynomials are homogeneous forms or tensors. Note that [93] and [94] used the same dual Fourier bases but in the opposite spaces to analytically resolve the Fourier transform. ...
Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathemat-ical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it re-views a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.
... In the case of the mixture of von Mises-Fisher distributions, the Fourier integral cannot be computed analytically either, and no approximation formula has been reported to date. An efficient way to estimate the displacement probability from this model is to use the general method presented in [41] for approximating the probability from a set of DW-MR acquired images. In our particular case of the von Mises-Fisher model, the recovered signal attenuation can be computed from the approximated mixture model by evaluating Eq. ...
... The set of vectors can be constructed by tessellating the icosahedron on the unit hemi-sphere. After having computed the S i = S(g i )/S 0 the set of S i can be considered as a new high angular resolution DW-MR data set and used by the algorithm in [41] for estimating the displacement probability as a 4 th-order Cartesian tensor. ...
Concepts from Information Theory have been used quite widely in Image Processing, Computer Vision and Medical Image Analysis for several decades now. Most widely used concepts are that of KL-divergence, minimum description length (MDL), etc. These concepts have been popularly employed for image registration, segmentation, classification etc. In this chapter we review several methods, mostly developed by our group at the Center for Vision, Graphics & Medical Imaging in the University of Florida, that glean concepts from Information Theory and apply them to achieve analysis of Diffusion-Weighted Magnetic Resonance (DW-MRI) data.
This relatively new MRI modality allows one to non-invasively infer axonal connectivity patterns in the central nervous system. The focus of this chapter is to review automated image analysis techniques that allow us to automatically segment the region of interest in the DWMRI image wherein one might want to track the axonal pathways and also methods to reconstruct complex local tissue geometries containing axonal fiber crossings. Implementation results illustrating the algorithm application to real DW-MRI data sets are depicted to demonstrate the effectiveness of the methods reviewed.
... Finally the approximation of the Fourier transform P (r) k,k of the modelled diffusion signal E(q) k,k can be computed for any order k1 = k2 = k. At this point it is interesting to note the method in [9]. In this paper the authors express the spherical profile of the EAP as a 4th order tensor function evaluated on a sphere. ...
... It is claimed that this basis is the inverse Fourier transform of the 4th order tensor function. This is true only because the function is constrained to a sphere (in [9] Eq.2). Our solution, at first sight, might resemble this approach but differs fundamentally from their work by the facts that the EAP we compute is not restricted to a sphere and is defined on R 3 . ...
Generalized Diffusion Tensor Imaging (GDTI) was developed to model complex Apparent Diffusivity Coefficient (ADC) using Higher Order Tensors (HOT) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile doesn't correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the Ensemble Average Propagator (EAP). Though interesting methods for estimating a positive ADC using 4th order diffusion tensors were developed, GDTI in general was overtaken by other approaches, e.g. the Orientation Distribution Function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper we present a novel closed-form approximation of the EAP using Hermite Polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate on 4th order diffusion tensors.
... There exists an identity in which d is a real ternary form d = d(g 1 , g 2 , g 3 ) of degree four which is positive semi-definite and the p i are quadratic forms with real coefficients. By using this theorem, Eq. (2) can be expressed as a sum of 3 squares of quadratic forms as (3) where, v is a properly chosen vector of monomials, (e.g. ...
... The displacement probability P(R) is given by the Fourier integral P(R) = ∫ E(q)exp(−2πiq·R)dq where q is the reciprocal space vector, E(q) is the signal value associated with vector q divided by the zero gradient signal and R is the displacement vector. In our experiments, we estimated the displacement probability profiles from the 4 th -order tensors using the method in [3]. ...
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing, a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. From this tensor approximation, one can compute useful scalar quantities (e.g. anisotropy, mean diffusivity) which have been clinically used for monitoring encephalopathy, sclerosis, ischemia and other brain disorders. It is now well known that this 2nd-order tensor approximation fails to capture complex local tissue structures, e.g. crossing fibers, and as a result, the scalar quantities derived from these tensors are grossly inaccurate at such locations. In this paper we employ a 4th order symmetric positive-definite (SPD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the SPD property. Several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates, which is a fundamental constraint since negative values of the diffusivity are not meaningful. In this paper we represent the 4th-order tensors as ternary quartics and then apply Hilbert's theorem on ternary quartics along with the Iwasawa parametrization to guarantee an SPD 4th-order tensor approximation from the DW-MRI data. The performance of this model is depicted on synthetic data as well as real DW-MRIs from a set of excised control and injured rat spinal cords, showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations.
... The water molecule displacement probability is given by the Fourier integral (1) where q is the reciprocal space vector, S(q) is the DW-MRI signal value associated with vector q, S 0 the zero gradient signal and r and r 0 is the direction and magnitude respectively of the displacement vector [17]. There are several existing methods for computing P(r 0 r) in which we either first reconstruct the signal S(q) and then evaluate Eq. 1 [4], or we directly estimate the displacement probability from given diffusion-weighted MR data [7,18]. Also, one may obtain an alternative representation called the fiber orientation distribution (from the Q-Ball images) from which one can find the optimal fiber orientations [13,19]. ...
... The data set was of size 16×16×16 and was generated by simulating the diffusion-weighted MR signal using the realistic simulation model in [16] (b-value=1250s/mm 2 , 81 gradient directions). After that, we estimated the displacement probability field (Fig. 1a) from the simulated signal by using the method in [18] (one can also use any other method). ...
In this paper we present a novel method for estimating a field of asymmetric spherical functions, dubbed tractosemas, given the intra-voxel displacement probability information. The peaks of tractosemas correspond to directions of distinct fibers, which can have either symmetric or asymmetric local fiber structure. This is in contrast to the existing methods that estimate fiber orientation distributions which are naturally symmetric and therefore cannot model asymmetries such as splaying fibers. We propose a method for extracting tractosemas from a given field of displacement probability iso-surfaces via a diffusion process. The diffusion is performed by minimizing a kernel convolution integral, which leads to an update formula expressed in the convenient form of a discrete kernel convolution. The kernel expresses the probability of diffusion between two neighboring spherical functions and we model it by the product of Gaussian and von Mises distributions. The model is validated via experiments on synthetic and real diffusion-weighted magnetic resonance (DW-MRI) datasets from a rat hippocampus and spinal cord.
... Several imaging and analysis schemes, which use fewer measurements than traditional DSI, have recently been proposed in the literature (Wu and Alexander, 2007;Jensen et al., 2005;Assemlal et al., 2011;Merlet et al., 2012;Barmpoutis et al., 2008;Descoteaux et al., 2010;Zhang et al., 2012;Ye et al., 2011Ye et al., , 2012Hosseinbor et al., 2012). Each of these techniques captures a different aspect of the underlying tissue organization, which is missed by HARDI. ...
For accurate estimation of the ensemble average diffusion propagator (EAP), traditional multi-shell diffusion imaging (MSDI) approaches require acquisition of diffusion signals for a range of b-values. However, this makes the acquisition time too long for several types of patients, making it difficult to use in a clinical setting. In this work, we propose a new method for the reconstruction of diffusion signals in the entire q-space from highly undersampled sets of MSDI data, thus reducing the scan time significantly. In particular, to sparsely represent the diffusion signal over multiple q-shells, we propose a novel extension to the framework of spherical ridgelets by accurately modeling the monotonically decreasing radial component of the diffusion signal. Further, we enforce the reconstructed signal to have smooth spatial regularity in the brain, by minimizing the total variation (TV) norm. We combine these requirements into a novel cost function and derive an optimal solution using the Alternating Directions Method of Multipliers (ADMM) algorithm. We use a physical phantom data set with known fiber crossing angle of 45° to determine the optimal number of measurements (gradient directions and b-values) needed for accurate signal recovery. We compare our technique with a state-of-the-art sparse reconstruction method (i.e., the SHORE method of Cheng et al. (2010)) in terms of angular error in estimating the crossing angle, incorrect number of peaks detected, normalized mean squared error in signal recovery as well as error in estimating the return-to-origin probability (RTOP). Finally, we also demonstrate the behavior of the proposed technique on human in vivo data sets. Based on these experiments, we conclude that using the proposed algorithm, at least 60 measurements (spread over three b-value shells) are needed for proper recovery of MSDI data in the entire q-space.
... However, this technique requires many measurements, making it impractical to use in clinical settings. Consequently, other imaging and analysis schemes, which use fewer measurements have been proposed in [2][3][4][5][6]. These techniques acquire important information about the neural tissue, which is missed by HARDI methods, yet, only a few of these are used in clinical studies. ...
Estimation of the diffusion propagator from a sparse set of diffusion MRI (dMRI) measurements is a field of active research. Sparse reconstruction methods propose to reduce scan time and are particularly suitable for scanning un-coperative patients. Recent work on reconstructing the diffusion signal from very few measurements using compressed sensing based techniques has focussed on propagator (or signal) estimation at each voxel independently. However, the goal of many neuroscience studies is to use tractography to study the pathology in white matter fiber tracts. Thus, in this work, we propose a joint framework for robust estimation of the diffusion propagator from sparse measurements while simultaneously tracing the white matter tracts. We propose to use a novel multi-tensor model of diffusion which incorporates the biexponential radial decay of the signal. Our preliminary results on in-vivo data show that the proposed method produces consistent and reliable fiber tracts from very few gradient directions while simultaneously estimating the bi-exponential decay of the diffusion propagator.
... For example, the tensors that approximate the Bidirectional Reflectance Distribution Function (BRDF) [7] are antisymmetric, while the diffusion [10] and the structure tensors [46] are antipodally symmetric. Furthermore, certain applications demand that the estimated tensors be positive-definite since they model positive-valued physical quantities such as the diffusivity function or the displacement probability of water molecules [8]. In this paper, we are interested in the case of fully symmetric positive-definite tensors of various orders and hence for sake of simplicity, every reference to the term tensor will imply this particular case of tensors unless otherwise stated. ...
Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space [Formula: see text] of 2m(th)-order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on [Formula: see text] is usually not straightforward computationally because there is no known analytic description of [Formula: see text] for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone [Formula: see text] for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Σ(2) (m) ⊂ Ω(2) (m) for m ≥ 1, using the subset [Formula: see text] of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a non-negative linear least-squares (NNLS) optimization problem with the complexity that equals the number of generators in Σ(2) (m). Finally, we experimentally validate the proposed approach and we present an application for computing 2m(th)-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images.
... A few works have attempted to reduce the scan time using compressed sensing for DSI [3,4], however, the acquisition time is still too long for clinical applications. Consequently, other imaging schemes have been proposed, namely, Hybrid Diffusion Imaging (HYDI) [5], Diffusion Propagator Imaging (DPI) [6], Diffusion Kurtosis Imaging (DKI) [7], spherical polar Fourier basis [8] and high-order tensor models [9,10], all of which fall under the category of multi-shell imaging (MSI). Each of these techniques captures different aspect of the underlying tissue geometry. ...
Diffusion magnetic resonance imaging (dMRI) is an important tool that allows non-invasive investigation of neural architecture
of the brain. The data obtained from these in-vivo scans provides important information about the integrity and connectivity of neural fiber bundles in the brain. A multi-shell
imaging (MSI) scan can be of great value in the study of several psychiatric and neurological disorders, yet its usability
has been limited due to the long acquisition times required. A typical MSI scan involves acquiring a large number of gradient
directions for the 2 (or more) spherical shells (several b-values), making the acquisition time significantly long for clinical
application. In this work, we propose to use results from the theory of compressive sampling and determine the minimum number
of gradient directions required to attain signal reconstruction similar to a traditional MSI scan. In particular, we propose
a generalization of the single shell spherical ridgelets basis for sparse representation of multi shell signals. We demonstrate its efficacy on several synthetic and in-vivo data sets and perform quantitative comparisons with solid spherical harmonics based representation. Our preliminary results
show that around 20-24 directions per shell are enough for robustly recovering the diffusion propagator.
... Diffusion within biological structures can be non-Gaussian, for example, when bifurcating or intersecting white-matter fiber bundles occur in the brain. In particular, the 4th order tensor model has recently been studied by Barmpoutis et al. [4], Moakher [5] and others for representing not only diffusivity , but also the orientation distribution function. Tensor interpolation is valuable in tractography and registration algorithms, or any application which may require computation of tensor values between voxels in the image lattice. ...
We present a tensor field interpolation method based on tensor-valued Bezier patches. The control points of the patch are determined by imposing physical constraints on the interpolated field by constraining the divergence and curl of the tensor field. The method generalizes to Cartesian tensors of all orders. Solving for the control points requires the solution of a sparse linear system. Results are presented for order 2 and 4 tensors and comparisons are made with linear interpolation and a previously proposed subdivision scheme.
... A few works have attempted to reduce the scan time using compressed sensing for DSI [3,4], however, the acquisition time is still too long for clinical applications. Consequently, other imaging schemes have been proposed, namely, Hybrid Diffusion Imaging (HYDI) [5], Diffusion Propagator Imaging (DPI) [6], Diffusion Kurtosis Imaging (DKI) [7], spherical polar Fourier basis [8] and high-order tensor models [9,10], all of which fall under the category of multi-shell imaging (MSI). Each of these techniques captures different aspect of the underlying tissue geometry. ...
Diffusion magnetic resonance imaging (dMRI) is an important tool that allows non-invasive investigation of neural architecture of the brain. The data obtained from these in-vivo scans provides important information about the integrity and connectivity of neural fiber bundles in the brain. A multi-shell imaging (MSI) scan can be of great value in the study of several psychiatric and neurological disorders, yet its usability has been limited due to the long acquisition times required. A typical MSI scan involves acquiring a large number of gradient directions for the 2 (or more) spherical shells (several b-values), making the acquisition time significantly long for clinical application. In this work, we propose to use results from the theory of compressive sampling and determine the minimum number of gradient directions required to attain signal reconstruction similar to a traditional MSI scan. In particular, we propose a generalization of the single shell spherical ridgelets basis for sparse representation of multi shell signals. We demonstrate its efficacy on several synthetic and in-vivo data sets and perform quantitative comparisons with solid spherical harmonics based representation. Our preliminary results show that around 20-24 directions per shell are enough for robustly recovering the diffusion propagator.
... where q is the reciprocal space vector, S(q) is the DW-MRI signal value associated with vector q, S 0 the zero gradient signal and r is the displacement vector. Through there are various methods available in literature for obtaining the water displacement probability, in our implementation we use the method proposed in [5] for its effectiveness. Once the water displacement probability has been obtained, fiber orientations can be recovered by finding the maxima of either a radial iso-surface of P (r) or the radial integral of P (r). ...
Multi-fiber reconstruction has attracted immense attention lately in the field of diffusion weighted MRI analysis. Several mathematical models have been proposed in literature but there is still scope for improvement. The key issues of importance in multi-fiber reconstruction are, fiber detection accuracy, robustness to noise and computational efficiency. To this end, we propose a novel mathematical model for representing the MR signal attenuation in the presence of multiple fibers at a single voxel and estimate the parameters of this model given the diffusion weighted MRI data. Our model for the diffusion MR signal consists of a continuous mixture of Hyperspherical von Mises-Fisher distributions. Being a continuous mixture, our model does not require the specification of the number of mixture components. We present a closed form expression for this continuous mixture that leads to a computationally efficient implementation. To validate our model we present extensive results on both synthetic and real data (human and rat brain) and demonstrate that even in presence of noise, our model clearly outperforms the state-of-the-art methods in fiber orientation estimation while maintaining a substantial computational advantage.
In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.
Diffusion magnetic resonance imaging (dMRI) is an important tool that allows non-invasive investigation of the neural architecture of the brain. Advanced dMRI protocols typically require a large number of measurements for accurately tracing the fiber bundles and estimating the diffusion properties (such as, FA). However, the acquisition time of these sequences is prohibitively large for pediatric as well as patients with certain types of brain disorders (such as, dementia). Thus, fast echo-planar imaging (EPI) acquisition sequences were proposed by the authors in [6, 16], which acquired multiple slices simultaneously to reduce scan time. The scan time in such cases drops proportionately to the number of simultaneous slice acquisitions (which we denote by R). While preliminary results in [6, 16] showed good reproducibility, yet the effect of simultaneous acquisitions on long range fiber connectivity and diffusion measures such as FA, is not known. In this work, we use multi-tensor based fiber connectivity to compare data acquired on two subjects with different acceleration factors (R = 1, 2, 3). We investigate and report the reproducibility of fiber bundles and diffusion measures between these scans on two subjects with different spatial resolutions, which is quite useful while designing neuroimaging studies.
The average diffusion propagator (ADP) obtained from diffusion MRI (dMRI) data encapsulates important structural properties of the underlying tissue. Measures derived from the ADP can be potentially used as markers of tissue integrity in characterizing several mental disorders. Thus, accurate estimation of the ADP is imperative for its use in neuroimaging studies. In this work, we propose a simple method for estimating the ADP by representing the acquired diffusion signal in the entire q-space using radial basis functions (RBF). We demonstrate our technique using two different RBF’s (generalized inverse multiquadric and Gaussian) and derive analytical expressions for the corresponding ADP’s. We also derive expressions for computing the solid angle orientation distribution function (ODF) for each of the RBF’s. Estimation of the weights of the RBF’s is done by enforcing positivity constraint on the estimated ADP or ODF. Finally, we validate our method on data obtained from a physical phantom with known fiber crossing of 45 degrees and also show comparison with the solid spherical harmonics method of Descoteaux et al. (Med Image Anal 2010). We also demonstrate our method on in-vivo human brain data.
In this paper, we review the state of the art in diffusion magnetic resonance imaging (dMRI) and we present current trends in modelling the brain's tissue microstructure and the human connectome. dMRI is today the only tool that can probe the brain's axonal architecture in vivo and non-invasively, and has grown in leaps and bounds in the last two decades since its conception. A plethora of models with increasing complexity and better accuracy have been proposed to characterise the integrity of the cerebral tissue, to understand its microstructure and to infer its connectivity. Here, we discuss a wide range of the most popular, important and well-established local microstructure models and biomarkers that have been proposed from these models. Finally, we briefly present the state of the art in tractography techniques that allow us to understand the architecture of the brain's connectivity.
The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic
data sets of order 2, 3 and 4.
Recently various mathematical models have been proposed to model the signal attenuation obtained from Diffusion Weighted Magnetic Resonance Imaging (DW-MRI). Though effective to various extents, almost all of the existing methods involve model parameters which are abstract mathematical quantities without any tangible connection to physical quantities (e.g. the b-value, gradient pulse duration, pulse separation etc.) involved in the DW-MRI acquisition process. To address this disconnect, in this paper, we present a multi-compartmental model which uses a physical model for restricted diffusion in the cylindrical geometry as the constituent basis function for multi-fiber reconstruction. Through extensive experiments on synthetic data we establish the superiority of the proposed method over the state-of-the-art techniques in terms of fiber orientation detection accuracy. We also present detailed results using human and rat brain data and demonstrate that our method leads to meaningful multi-fiber reconstruction even in the case of real data.
Many recent high angular resolution diffusion imaging (HARDI) reconstruction techniques have been introduced to infer an orientation distribution function (ODF) of the underlying tissue structure. These methods are more often based on a single-shell (one b-value) acquisition and can only recover angular structure information contained in the ensemble average propagator (EAP) describing the three-dimensional (3D) average diffusion process of water molecules. The EAP can thus provide richer information about complex tissue microstructure properties than the ODF by also considering the radial part of the diffusion signal. In this paper, we present a novel technique for analytical EAP reconstruction from multiple q-shell acquisitions. The solution is based on a Laplace equation by part estimation between the diffusion signal for each shell acquisition. This simplifies greatly the Fourier integral relating diffusion signal and EAP, which leads to an analytical, linear and compact EAP reconstruction. An important part of the paper is dedicated to validate the diffusion signal estimation and EAP reconstruction on real datasets from ex vivo phantoms. We also illustrate multiple q-shell diffusion propagator imaging (mq-DPI) on a real in vivo human brain and perform a qualitative comparison against state-of-the-art diffusion spectrum imaging (DSI) on the same subject. mq-DPI is shown to reconstruct robust EAP from only several different b-value shells and less diffusion measurements than DSI. This opens interesting perspectives for new q-space sampling schemes and tissue microstructure investigation.
A magnetic resonance imaging method is presented for quantifying the degree to which water diffusion in biologic tissues is non-Gaussian. Since tissue structure is responsible for the deviation of water diffusion from the Gaussian behavior typically observed in homogeneous solutions, this method provides a specific measure of tissue structure, such as cellular compartments and membranes. The method is an extension of conventional diffusion-weighted imaging that requires the use of somewhat higher b values and a modified image postprocessing procedure. In addition to the diffusion coefficient, the method provides an estimate for the excess kurtosis of the diffusion displacement probability distribution, which is a dimensionless metric of the departure from a Gaussian form. From the study of six healthy adult subjects, the excess diffusional kurtosis is found to be significantly higher in white matter than in gray matter, reflecting the structural differences between these two types of cerebral tissues. Diffusional kurtosis imaging is related to q-space imaging methods, but is less demanding in terms of imaging time, hardware requirements, and postprocessing effort. It may be useful for assessing tissue structure abnormalities associated with a variety of neuropathologies.
This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffusion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this technique's ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.
We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace-Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk-Radon transform which approximates the ODF. We prove a new corollary of the Funk-Hecke theorem to obtain this simplification. Then, we show that the Laplace-Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace-Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method.