DTI segmentation using an information theoretic tensor dissimilarity measure

In recent years, diffusion tensor imaging (DTI) has become a popular in vivo diagnostic imaging technique in Radiological sciences. In order for this imaging technique to be more effective, proper image analysis techniques suited for analyzing these high dimensional data need to be developed. In this paper, we present a novel definition of tensor "distance" grounded in concepts from information theory and incorporate it in the segmentation of DTI. In a DTI, the symmetric positive definite (SPD) diffusion tensor at each voxel can be interpreted as the covariance matrix of a local Gaussian distribution. Thus, a natural measure of dissimilarity between SPD tensors would be the Kullback-Leibler (KL) divergence or its relative. We propose the square root of the J-divergence (symmetrized KL) between two Gaussian distributions corresponding to the diffusion tensors being compared and this leads to a novel closed form expression for the "distance" as well as the mean value of a DTI. Unlike the traditional Frobenius norm-based tensor distance, our "distance" is affine invariant, a desirable property in segmentation and many other applications. We then incorporate this new tensor "distance" in a region based active contour model for DTI segmentation. Synthetic and real data experiments are shown to depict the performance of the proposed model.     

A unified computational framework for deconvolution to reconstruct multiple fibers from diffusion weighted MRI

Diffusion magnetic resonance imaging (MRI) is a relatively new imaging modality which is capable of measuring the diffusion of water molecules in biological systems noninvasively. The measurements from diffusion MRI provide unique clues for extracting orientation information of brain white matter fibers and can be potentially used to infer the brain connectivity in vivo using tractography techniques. Diffusion tensor imaging (DTI), currently the most widely used technique, fails to extract multiple fiber orientations in regions with complex microstructure. In order to overcome this limitation of DTI, a variety of reconstruction algorithms have been introduced in the recent past. One of the key ingredients in several model-based approaches is deconvolution operation which is presented in a unified deconvolution framework in this paper. Additionally, some important computational issues in solving the deconvolution problem that are not addressed adequately in previous studies are described in detail here. Further, we investigate several deconvolution schemes towards achieving stable, sparse, and accurate solutions. Experimental results on both simulations and real data are presented. The comparisons empirically suggest that nonnegative least squares method is the technique of choice for the multifiber reconstruction problem in the presence of intravoxel orientational heterogeneity.     

Diffeomorphic image registration of diffusion MRI using spherical harmonics

Nonrigid registration of diffusion magnetic resonance imaging (MRI) is crucial for group analyses and building white matter and fiber tract atlases. Most current diffusion MRI registration techniques are limited to the alignment of diffusion tensor imaging (DTI) data. We propose a novel diffeomorphic registration method for high angular resolution diffusion images by mapping their orientation distribution functions (ODFs). ODFs can be reconstructed using q-ball imaging (QBI) techniques and represented by spherical harmonics (SHs) to resolve intra-voxel fiber crossings. The registration is based on optimizing a diffeomorphic demons cost function. Unlike scalar images, deforming ODF maps requires ODF reorientation to maintain its consistency with the local fiber orientations. Our method simultaneously reorients the ODFs by computing a Wigner rotation matrix at each voxel, and applies it to the SH coefficients during registration. Rotation of the coefficients avoids the estimation of principal directions, which has no analytical solution and is time consuming. The proposed method was validated on both simulated and real data sets with various metrics, which include the distance between the estimated and simulated transformation fields, the standard deviation of the general fractional anisotropy and the directional consistency of the deformed and reference images. The registration performance using SHs with different maximum orders were compared using these metrics. Results show that the diffeomorphic registration improved the affine alignment, and registration using SHs with higher order SHs further improved the registration accuracy by reducing the shape difference and improving the directional consistency of the registered and reference ODF maps.     

Enhancement of fiber orientation distribution reconstruction in diffusion-weighted imaging by single channel blind source separation

In diffusion-weighted imaging (DWI), reliable fiber tracking results rely on the accurate reconstruction of the fiber orientation distribution function (fODF) in each individual voxel. For high angular resolution diffusion imaging (HARDI), deconvolution-based approaches can reconstruct the complex fODF and have advantages in terms of computational efficiency and no need to estimate the number of distinct fiber populations. However, HARDI-based methods usually require relatively high b-values and a large number of gradient directions to produce good results. Such requirements are not always easy to meet in common clinical studies due to limitations in MRI facilities. Moreover, most of these approaches are sensitive to noise. In this study, we propose a new framework to enhance the performance of the spherical deconvolution (SD) approach in low angular resolution DWI by employing a single channel blind source separation (BSS) technique to decompose the fODF initially estimated by SD such that the desired fODF can be extracted from the noisy background. The results based on numerical simulations and two phantom datasets demonstrate that the proposed method achieves better performance than SD in terms of robustness to noise and variation in b-values. In addition, the results show that the proposed method has the potential to be applied to low angular resolution DWI which is commonly used in clinical studies.     

基于贝叶斯约束统计框架的DT-MRI脑白质纤维追踪成像

弥散张量磁共振成像(DT-MRI)的脑白质纤维追踪成像利用脑白质水分子弥散构成的弥散张量信息追踪脑白质纤维束并无创重建其3维结构图像。针对现有追踪方法一般以局部体素的弥散张量为主要追踪依据,缺乏对纤维结构、弥散度等人体解剖结构和生理机能的综合考量的缺陷,该文基于贝叶斯理论框架综合分析追踪路径与各体素弥散张量方向和纤维束几何结构相关性,并使用弥散度和追踪纤维角度对两者进行约束,获得各步追踪方向的概率密度分布,通过Markov Chain Monte Carlo采样确定其追踪方向进行追踪成像,通过多次追踪获得具有统计意义的3维结果。最后利用文中方法在合成弥散张量数据上进行了成像仿真,在真实脑部DT-MRI数据上进行了成像实验。仿真和实验结果表明,该方法能实现预期的脑白质纤维追踪成像,比现有追踪成像方法结果更可靠,可重复性更强。     

A reproducing kernel hilbert space approach for q-ball imaging

Diffusion magnetic resonance (MR) imaging has enabled us to reveal the white matter geometry in the living human brain. The Q-ball technique is widely used nowadays to recover the orientational heterogeneity of the intra-voxel fiber architecture. This article proposes to employ the Funk-Radon transform in a Hilbert space with a reproducing kernel derived from the spherical Laplace-Beltrami operator, thus generalizing previous approaches that assume a bandlimited diffusion signal. The function estimation problem is solved within a Tikhonov regularization framework, while a Gaussian process model allows for the selection of the smoothing parameter and the specification of confidence bands. Shortcomings of Q-ball imaging are discussed.     

On Approximation of Orientation Distributions by Means of Spherical Ridgelets

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.      

DTI segmentation by statistical surface evolution

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). A DTI produces, from a set of diffusion-weighted MR images, tensor-valued images where each voxel is assigned with a 3 x 3 symmetric, positive-definite matrix. This second order tensor is simply the covariance matrix of a local Gaussian process, with zero-mean, modeling the average motion of water molecules. As we will show in this paper, the definition of a dissimilarity measure and statistics between such quantities is a nontrivial task which must be tackled carefully. We claim and demonstrate that, by using the theoretically well-founded differential geometrical properties of the manifold of multivariate normal distributions, it is possible to improve the quality of the segmentation results obtained with other dissimilarity measures such as the Euclidean distance or the Kullback-Leibler divergence. The main goal of this paper is to prove that the choice of the probability metric, i.e., the dissimilarity measure, has a deep impact on the tensor statistics and, hence, on the achieved results. We introduce a variational formulation, in the level-set framework, to estimate the optimal segmentation of a DTI according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. We must also respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the DTI. We show how to express all the statistical quantities for the different probability metrics. We validate and compare the results obtained on various synthetic data-sets, a biological rat spinal cord phantom and human brain DTIs.     

Fiber continuity: an anisotropic prior for ODF estimation

The accurate and reliable estimation of fiber orientation distributions, based on diffusion-sensitized magnetic resonance images is a major prerequisite for tractography algorithms or any other derived statistical analysis. In this work, we formulate the principle of fiber continuity (FC), which is based on the simple observation that the imaging of fibrous tissue implies certain expectations for the measured images. From this principle we derive a prior for the estimation of fiber orientation distributions based on high angular resolution diffusion imaging (HARDI). We demonstrate on simulated, phantom, and in vivo data the superiority of the proposed approach. Further, we propose another application of the FC principle, named FC flow, a method to resolve complex crossing regions solely based on diffusion tensor imaging (DTI). The idea is to infer directional information in crossing regions from adjacent anisotropic areas.     

Limits on the accuracy of 3-D thickness measurement in magnetic resonance images--effects of voxel anisotropy

Measuring the thickness of sheet-like thin anatomical structures, such as articular cartilage and brain cortex, in three-dimensional (3-D) magnetic resonance (MR) images is an important diagnostic procedure. This paper investigates the fundamental limits on the accuracy of thickness determination in MR images. We defined thickness here as the distance between the two sides of boundaries measured at the subvoxel resolution, which are the zero-crossings of the second directional derivatives combined with Gaussian blurring along the normal directions of the sheet surface. Based on MR imaging and computer postprocessing parameters, characteristics for the accuracy of thickness determination were derived by a theoretical simulation. We especially focused on the effects of voxel anisotropy in MR imaging with variable orientation of sheet-like structure. Improved and stable accuracy features were observed when the standard deviation of Gaussian blurring combined with thickness determination processes was around square root of 2/2 times as large as the pixel size. The relation between voxel anisotropy in MR imaging and the range of sheet normal orientation within which acceptable accuracy is attainable was also clarified, based on the dependences of voxel anisotropy and the sheet normal orientation obtained by numerical simulations. Finally, in vitro experiments were conducted using an acrylic plate phantom and a resected femoral head to validate the results of theoretical simulation. The simulated thickness was demonstrated to be well-correlated with the actual in vitro thickness.     

Super-resolution registration using tissue-classified distance fields

We present a method for registering the position and orientation of bones across multiple computed-tomography (CT) volumes of the same subject. The method is subvoxel accurate, can operate on multiple bones within a set of volumes, and registers bones that have features commensurate in size to the voxel dimension. First, a geometric object model is extracted from a reference volume image. We use then unsupervised tissue classification to generate from each volume to be registered a super-resolution distance field--a scalar field that specifies, at each point, the signed distance from the point to a material boundary. The distance fields and the geometric bone model are finally used to register an object through the sequence of CT images. In the case of multiobject structures, we infer a motion-directed hierarchy from the distance-field information that allows us to register objects that are not within each other's capture region. We describe a validation framework and evaluate the new technique in contrast with grey-value registration. Results on human wrist data show average accuracy improvements of 74% over grey-value registration. The method is of interest to any intrasubject, same-modality registration applications where subvoxel accuracy is desired.     

A Diffusion Tensor Imaging Tractography Algorithm Based on Navier–Stokes Fluid Mechanics

We introduce a fluid mechanics based tractography method for estimating the most likely connection paths between points in diffusion tensor imaging (DTI) volumes. We customize the Navier–Stokes equations to include information from the diffusion tensor and simulate an artificial fluid flow through the DTI image volume. We then estimate the most likely connection paths between points in the DTI volume using a metric derived from the fluid velocity vector field. We validate our algorithm using digital DTI phantoms based on a helical shape. Our method segmented the structure of the phantom with less distortion than was produced using implementations of heat-based partial differential equation (PDE) and streamline based methods. In addition, our method was able to successfully segment divergent and crossing fiber geometries, closely following the ideal path through a digital helical phantom in the presence of multiple crossing tracts. To assess the performance of our algorithm on anatomical data, we applied our method to DTI volumes from normal human subjects. Our method produced paths that were consistent with both known anatomy and directionally encoded color images of the DTI dataset.      

Error propagation framework for diffusion tensor imaging via diffusion tensor representations

An analytical framework of error propagation for diffusion tensor imaging (DTI) is presented. Using this framework, any uncertainty of interest related to the diffusion tensor elements or to the tensor-derived quantities such as eigenvalues, eigenvectors, trace, fractional anisotropy (FA), and relative anisotropy (RA) can be analytically expressed and derived from the noisy diffusion-weighted signals. The proposed framework elucidates the underlying geometric relationship between the variability of a tensor-derived quantity and the variability of the diffusion weighted signals through the nonlinear least squares objective function of DTI. Monte Carlo simulations are carried out to validate and investigate the basic statistical properties of the proposed framework.     

An efficient interlaced multi-shell sampling scheme for reconstruction of diffusion propagators

In this paper, we propose an interlaced multi-shell sampling scheme for the reconstruction of the diffusion propagator from diffusion weighted magnetic resonance imaging (DW-MRI). In standard multi-shell sampling schemes, sample points are uniformly distributed on several spherical shells in q-space. The distribution of sample points is the same for all shells, and is determined by the vertices of a selected polyhedron. We propose a more efficient interlaced scheme where sample points are different on alternating shells and are determined by the vertices of a pair of dual polyhedra. Since it samples more directions than the standard scheme, this method offers increased angular discrimination. Another contribution of this work is the application of optimal sampling lattices to q-space data acquisition and the proposal of a model-free reconstruction algorithm, which uses the lattice dependent sinc interpolation function. It is shown that under this reconstruction framework, the body centered cubic (BCC) lattice provides increased accuracy. The sampling scheme and the reconstruction algorithms were evaluated on simulated data as well as rat brain data collected on a 600 MHz (14.1T) Bruker imaging spectrometer.     

Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry

In this paper, we present a new formulation for recovering the fiber tract geometry within a voxel from diffusion weighted magnetic resonance imaging (MRI) data, in the presence of single or multiple neuronal fibers. To this end, we define a discrete set of diffusion basis functions. The intravoxel information is recovered at voxels containing fiber crossings or bifurcations via the use of a linear combination of the above mentioned basis functions. Then, the parametric representation of the intravoxel fiber geometry is a discrete mixture of Gaussians. Our synthetic experiments depict several advantages by using this discrete schema: the approach uses a small number of diffusion weighted images (23) and relatively small b values (1250 s/mm2), i.e., the intravoxel information can be inferred at a fraction of the acquisition time required for datasets involving a large number of diffusion gradient orientations. Moreover our method is robust in the presence of more than two fibers within a voxel, improving the state-of-the-art of such parametric models. We present two algorithmic solutions to our formulation: by solving a linear program or by minimizing a quadratic cost function (both with non-negativity constraints). Such minimizations are efficiently achieved with standard iterative deterministic algorithms. Finally, we present results of applying the algorithms to synthetic as well as real data.     

A Continuous STAPLE for Scalar, Vector, and Tensor Images: An Application to DTI Analysis

The comparison of images of a patient to a reference standard may enable the identification of structural brain changes. These comparisons may involve the use of vector or tensor images (i.e., 3-D images for which each voxel can be represented as an ${BBR}^N$ vector) such as diffusion tensor images (DTI) or transformations. The recent introduction of the Log-Euclidean framework for diffeomorphisms and tensors has greatly simplified the use of these images by allowing all the computations to be performed on a vector-space. However, many sources can result in a bias in the images, including disease or imaging artifacts. In order to estimate and compensate for these sources of variability, we developed a new algorithm, called continuous STAPLE, that estimates the reference standard underlying a set of vector images. This method, based on an expectation-maximization method similar in principle to the validation method STAPLE, also estimates for each image a set of parameters characterizing their bias and variance with respect to the reference standard. We demonstrate how to use these parameters for the detection of atypical images or outliers in the population under study. We identified significant differences between the tensors of diffusion images of multiple sclerosis patients and those of control subjects in the vicinity of lesions.      

ODVBA: optimally-discriminative voxel-based analysis

Gaussian smoothing of images prior to applying voxel-based statistics is an important step in voxel-based analysis and statistical parametric mapping (VBA-SPM) and is used to account for registration errors, to Gaussianize the data and to integrate imaging signals from a region around each voxel. However, it has also become a limitation of VBA-SPM based methods, since it is often chosen empirically and lacks spatial adaptivity to the shape and spatial extent of the region of interest, such as a region of atrophy or functional activity. In this paper, we propose a new framework, named optimally-discriminative voxel-based analysis (ODVBA), for determining the optimal spatially adaptive smoothing of images, followed by applying voxel-based group analysis. In ODVBA, nonnegative discriminative projection is applied regionally to get the direction that best discriminates between two groups, e.g., patients and controls; this direction is equivalent to local filtering by an optimal kernel whose coefficients define the optimally discriminative direction. By considering all the neighborhoods that contain a given voxel, we then compose this information to produce the statistic for each voxel. Finally, permutation tests are used to obtain a statistical parametric map of group differences. ODVBA has been evaluated using simulated data in which the ground truth is known and with data from an Alzheimer's disease (AD) study. The experimental results have shown that the proposed ODVBA can precisely describe the shape and location of structural abnormality.     

A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, D, in each voxel within an imaging volume. We propose a new normal distribution, p(D) alpha exp(-1/2 D: A: D), which describes the variability of D in an ideal DT-MRI experiment. The scalar invariant, D : A : D, is the contraction of a positive definite symmetric, fourth-order precision tensor, A, and D. A correspondence is established between D: A: D and the elastic strain energy density function in continuum mechanics--specifically between D and the second-order infinitesimal strain tensor, and between A and the fourth-order tensor of elastic coefficients. We show that A can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic fourth-order tensor, we derive an explicit analytic expression for p(D), and for the distribution of the three eigenvalues of D, p(gamma1, gamma2, gamma3), which are confirmed by Monte Carlo simulations. We show how A can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that A be an isotropic fourth-order tensor. This condition ensures that the statistical properties of D (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize p(D) to treat normal random tensor variables that are of third- (or higher) order. We expect that this new distribution, p(D), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.     

Partial-volume Bayesian classification of material mixtures in MRvolume data using voxel histograms

The authors present a new algorithm for identifying the distribution of different material types in volumetric datasets such as those produced with magnetic resonance imaging (MRI) or computed tomography (CT). Because the authors allow for mixtures of materials and treat voxels as regions, their technique reduces errors that other classification techniques can create along boundaries between materials and is particularly useful for creating accurate geometric models and renderings from volume data. It also has the potential to make volume measurements more accurately and classifies noisy, low-resolution data well. There are two unusual aspects to the authors' approach. First, they assume that, due to partial-volume effects, or blurring, voxels can contain more than one material, e.g., both muscle and fat; the authors compute the relative proportion of each material in the voxels. Second, they incorporate information from neighboring voxels into the classification process by reconstructing a continuous function, ρ(x), from the samples and then looking at the distribution of values that ρ(x) takes on within the region of a voxel. This distribution of values is represented by a histogram taken over the region of the voxel; the mixture of materials that those values measure is identified within the voxel using a probabilistic Bayesian approach that matches the histogram by finding the mixture of materials within each voxel most likely to have created the histogram. The size of regions that the authors classify is chosen to match the sparing of the samples because the spacing is intrinsically related to the minimum feature size that the reconstructed continuous function can represent     

一种三维数据场多表面显示方法

本文提出了一种基于体绘制的三维数据场多表面显示方法.首先,采用灰度梯度加权提取出三维数据场中不同物质间的边界,根据显示的需要只对这些边界上的体元赋予相应的阻光度并进行光亮度合成计算,因而可大大减少计算量,提高显示的速度;将边界上的体元作为不同物质的混合体,采用与方向有关的三线性插值来计算视线方向与体素内等值面的交点,根据交点的法向量进行光照效应计算以提高显示图像的质量;最后用投影成像法显示最终的图像.本文对医学CT图像做了相关的实验,取得了较好的效果.