Tensor splines for interpolation and approximation of DT-MRI with applications to segmentation of isolated rat hippocampi

In this paper, we present novel algorithms for statistically robust interpolation and approximation of diffusion tensors-which are symmetric positive definite (SPD) matrices-and use them in developing a significant extension to an existing probabilistic algorithm for scalar field segmentation, in order to segment diffusion tensor magnetic resonance imaging (DT-MRI) datasets. Using the Riemannian metric on the space of SPD matrices, we present a novel and robust higher order (cubic) continuous tensor product of B-splines algorithm to approximate the SPD diffusion tensor fields. The resulting approximations are appropriately dubbed tensor splines. Next, we segment the diffusion tensor field by jointly estimating the label (assigned to each voxel) field, which is modeled by a Gauss Markov measure field (GMMF) and the parameters of each smooth tensor spline model representing the labeled regions. Results of interpolation, approximation, and segmentation are presented for synthetic data and real diffusion tensor fields from an isolated rat hippocampus, along with validation. We also present comparisons of our algorithms with existing methods and show significantly improved results in the presence of noise as well as outliers.     

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.     

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.     

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.     

A new method to derive white matter conductivity from diffusion tensor MRI

We propose a new algorithm to derive the anisotropic conductivity of the cerebral white matter (WM) from the diffusion tensor MRI (DT-MRI) data. The transportation processes for both water molecules and electrical charges are described through a common multicompartment model that consists of axons, glia, or the cerebrospinal fluid (CSF). The volume fraction (VF) of each compartment varies from voxel to voxel and is estimated from the measured diffusion tensor. The conductivity tensor at each voxel is then computed from the estimated VF values and the decomposed eigenvectors of the diffusion tensor. The proposed VF algorithm was applied to the DT-MRI data acquired from two healthy human subjects. The extracted anisotropic conductivity distribution was compared with those obtained by using two existing algorithms, which were based upon a linear conductivity-to-diffusivity relationship and a volume constraint, respectively. The present results suggest that the VF algorithm is capable of incorporating the partial volume effects of the CSF and the intravoxel fiber crossing structure, both of which are not addressed altogether by existing algorithms. Therefore, it holds potential to provide a more accurate estimate of the WM anisotropic conductivity, and may have important applications to neuroscience research or clinical applications in neurology and neurophysiology.      

Bilateral filtering of diffusion tensor magnetic resonance images.

We extend the well-known scalar image bilateral filtering technique to diffusion tensor magnetic resonance images (DTMRI). The scalar version of bilateral image filtering is extended to perform edge-preserving smoothing of DT field data. The bilateral DT filtering is performed in the Log-Euclidean framework which guarantees valid output tensors. Smoothing is achieved by weighted averaging of neighboring tensors. Analogous to bilateral filtering of scalar images, the weights are chosen to be inversely proportional to two distance measures: The geometrical Euclidean distance between the spatial locations of tensors and the dissimilarity of tensors. We describe the noniterative DT smoothing equation in closed form and show how interpolation of DT data is treated as a special case of bilateral filtering where only spatial distance is used. We evaluate different recent DT tensor dissimilarity metrics including the Log-Euclidean, the similarity-invariant Log-Euclidean, the square root of the J-divergence, and the distance scaled mutual diffusion coefficient. We present qualitative and quantitative smoothing and interpolation results and show their effect on segmentation, for both synthetic DT field data, as well as real cardiac and brain DTMRI data.     

A variational model and graph cuts optimization for interactive foreground extraction

In this paper, an interactive segmentation method is proposed, which is based on an improved Chan–Vese model, i.e. multiple piecewise constant model with geodesic active contour. The k-means method is used to learn the models of the foreground and background, which are the optimal piecewise constant approximation of the original image according to the input seeds clue by the user. Based on the piecewise constant models of the foreground and background, the multiple piecewise constant with a geodesic active contour energy function can be minimized by effective graph cuts algorithm, and the minimum cuts can be used to partition the image into the foreground and background. Numerical experiments demonstrate the superior performance of the proposed interactive foreground extraction method based on the improved Chan–Vese model compared to the original Chan–Vese model by simple user interaction.     

图像分割中的交叉熵和模糊散度算法

本文将交叉熵和模糊散度应用于图像分割中,提出了中最优灰度值选取算法,其一是基于均匀分布假设的最小交叉熵算法,其二是利用后难概率的最大类间交叉熵算法,其三是类间最大模糊散度的改进算法,其四是最小模糊散度算法,针对图像阈什化分割的要求,在后两种算法中构造一种新的模糊录改度函数,本文采用均匀测试和开头测试比较各算法的性能,利用多种类型测试 是到的分割结果,显示了所筛算法的有效性和通用性。     

Image segmentation and selective smoothing by using Mumford-Shah model.

Recently, Chan and Vese developed an active contour model for image segmentation and smoothing by using piecewise constant and smooth representation of an image. Tsai et al. also independently developed a segmentation and smoothing method similar to the Chan and Vese piecewise smooth approach. These models are active contours based on the Mumford-Shah variational approach and the level-set method. In this paper, we develop a new hierarchical method which has many advantages compared to the Chan and Vese multiphase active contour models. First, unlike previous works, the curve evolution partial differential equations (PDEs) for different level-set functions are decoupled. Each curve evolution PDE is the equation of motion of just one level-set function, and different level-set equations of motion are solved in a hierarchy. This decoupling of the motion equations of the level-set functions speeds up the segmentation process significantly. Second, because of the coupling of the curve evolution equations associated with different level-set functions, the initialization of the level sets in Chan and Vese's method is difficult to handle. In fact, different initial conditions may produce completely different results. The hierarchical method proposed in this paper can avoid the problem due to the choice of initial conditions. Third, in this paper, we use the diffusion equation for denoising. This method, therefore, can deal with very noisy images. In general, our method is fast, flexible, not sensitive to the choice of initial conditions, and produces very good results.     

Automatic segmentation of coronary arteries in CT imaging in the presence of kissing vessel artifacts

In this paper, we present a novel two-step algorithm for segmentation of coronary arteries in computed tomography images based on the framework of active contours. In the proposed method, both global and local intensity information is utilized in the energy calculation. The global term is defined as a normalized cumulative distribution function, which contributes to the overall active contour energy in an adaptive fashion based on image histograms, to deform the active contour away from local stationary points. Possible outliers, such as kissing vessel artifacts, are removed in the postprocessing stage by a slice-by-slice correction scheme based on multiregion competition, where both arteries and kissing vessels are identified and tracked through the slices. The efficiency and the accuracy of the proposed technique are demonstrated on both synthetic and real datasets. The results on clinical datasets show that the method is able to extract the major branches of arteries with an average distance of 0.73 voxels to the manually delineated ground truth data. In the presence of kissing vessel artifacts, the outer surface of the entire coronary tree, extracted by the proposed algorithm, is smooth and contains fewer erroneous regions, originating in kissing vessel artifacts, as compared to the initial segmentation.     

基于SVM能量模型的改进主动轮廓图像分割算法研究

为克服经典主动轮廓模型曲线内外区域能量定义在复杂目标与背景分布情况下的不足,本文将高效的支持向量机有监督学习分类器引入基于Mumford-shah模型的主动轮廓图像分割算法中,提出了基于SVM能量模型的改进主动轮廓图像分割方法.该方法首先利用支持向量机的分类结果对于封闭曲线的内外区域分别构造了一种新的图像能量表示方法,因为分割过程充分利用了有监督学习策略,使得本文提出的算法具有更高的稳定性和更加广泛的适用范围,特别是对目标灰度分布不均或存在多纹理的目标也可以得到较好的分割结果.分割时,首先利用SVM实现粗分割得到目标初始轮廓,然后利用改进的Mumford-shah主动轮廓模型进行精确分割,采用粗分割策略一方面可以大大提高分割速度,另一方面也可以提高了算法的自动化程度.对比实验结果表明本文提出的算法具有更大灵活性和更好的分割性能.     

The elliptical cone of uncertainty and its normalized measures in diffusion tensor imaging

Diffusion tensor magnetic resonance imaging (DT-MRI) is capable of providing quantitative insights into tissue microstructure in the brain. An important piece of information offered by DT-MRI is the directional preference of diffusing water molecules within a voxel. Building upon this local directional information, DT-MRI tractography attempts to construct global connectivity of white matter tracts. The interplay between local directional information and global structural information is crucial in understanding changes in tissue microstructure as well as in white matter tracts. To this end, the right circular cone of uncertainty was proposed by Basser as a local measure of tract dispersion. Recent experimental observations by Jeong et al. and Lazar et al. that the cones of uncertainty in the brain are mostly elliptical motivate the present study to investigate analytical approaches to quantify their findings. Two analytical approaches for constructing the elliptical cone of uncertainty, based on the first-order matrix perturbation and the error propagation method via diffusion tensor representations, are presented and their theoretical equivalence is established. We propose two normalized measures, circumferential and areal, to quantify the uncertainty of the major eigenvector of the diffusion tensor. We also describe a new technique of visualizing the cone of uncertainty in 3-D.     

Trimmed-likelihood estimation for focal lesions and tissue segmentation in multisequence MRI for multiple sclerosis

We present a new automatic method for segmentation of multiple sclerosis (MS) lesions in magnetic resonance images. The method performs tissue classification using a model of intensities of the normal appearing brain tissues. In order to estimate the model, a trimmed likelihood estimator is initialized with a hierarchical random approach in order to be robust to MS lesions and other outliers present in real images. The algorithm is first evaluated with simulated images to assess the importance of the robust estimator in presence of outliers. The method is then validated using clinical data in which MS lesions were delineated manually by several experts. Our method obtains an average Dice similarity coefficient (DSC) of 0.65, which is close to the average DSC obtained by raters (0.66).     

Image segmentation based on multi-region multi-scale local binary fitting and Kullback–Leibler divergence

The inhomogeneity of intensity and the noise of image are the two major obstacles to accurate image segmentation by region-based level set models. To provide a more general solution to these challenges and address the difficulty of image segmentation methods to handle an arbitrary number of regions, we propose a region-based multi-phase level set method, which is based on the multi-scale local binary fitting (MLBF) and the Kullback–Leibler (KL) divergence, called KL–MMLBF. We first apply the multi-scale theory and multi-phase level set framework to the local binary fitting model to build the multi-region multi-scale local binary fitting (MMLBF). Then the energy term measured by KL divergence between regions to be segmented is incorporated into the energy function of MMLBF. KL–MMLBF utilizes the between-cluster distance and the adaptive kernel function selection strategy to formulate the energy function. Being more robust to the initial location of the contour than the classical segmentation models, KL–MMLBF can deal with blurry boundaries and noise problems. The results of experiments on synthetic and medical images have shown that KL–MMLBF can improve the effectiveness of segmentation while ensuring the accuracy by accelerating this minimization of this energy function and the model has achieved better segmentation results in terms of both accuracy and efficiency to analyze the multi-region image.     

Unsupervised segmentation based on robust estimation and color active contour models.

One of the most commonly used clinical tests performed today is the routine evaluation of peripheral blood smears. In this paper, we investigate the design, development, and implementation of a robust color gradient vector flow (GVF) active contour model for performing segmentation, using a database of 1791 imaged cells. The algorithms developed for this research operate in Luv color space, and introduce a color gradient and L2E robust estimation into the traditional GVF snake. The accuracy of the new model was compared with the segmentation results using a mean-shift approach, the traditional color GVF snake, and several other commonly used segmentation strategies. The unsupervised robust color snake with L2E robust estimation was shown to provide results which were superior to the other unsupervised approaches, and was comparable with supervised segmentation, as judged by a panel of human experts.     

Ultrasound kidney segmentation with a global prior shape

In this paper, we focus on segmentation of ultrasound kidney images. Unlike previous work by using trained prior shapes, we employ a parametric super-ellipse as a global prior shape for a human kidney. The Fisher–Tippett distribution is employed to describe the grey level statistics. Combining the grey level statistics with a global character of a kidney shape, we propose a new active contour model to segment ultrasound kidney images. The proposed model involves two subproblems. One subproblem is to optimize the parameters of a super-ellipse. Another subproblem is to segment an ultrasound kidney image. An alternating minimization scheme is used to optimize the parameters of a super-ellipse and segment an image simultaneously. To segment an image fast, a convex relaxation method is introduced and the split Bregman method is incorporated to propose a fast segmentation algorithm. The efficiency of the proposed method is illustrated by numerical experiments on both simulated images and real ultrasound kidney images.     

Local morphology fitting active contour for automatic vascular segmentation

In this paper, we propose an active contour model using local morphology fitting for automatic vascular segmentation on 2-D angiogram. The vessel and background are fitted to fuzzy morphology maximum and minimum opening, separately, using linear structuring element with adaptive scale and orientation. The minimization of the energy associated with the active contour model is implemented within a level set framework. As in the current local model, fitting the image to local region information makes the model robust against the inhomogeneous background. Moreover, selective local estimations for fitting that are precomputed instead of updated in each contour evolution makes the evolution of level set robust again initial location compared to the current local model. The results on synthetic image and real angiogram compared with other methods are presented. It is shown that the proposed method can achieve automatic and accurate segmentation of vascular angiogram.     

A robust hybrid active contour model based on pre-fitting bias field correction for fast image segmentation

An array of existing active contour models is prone to suffering from the deficiencies of poor anti-noise ability, initialization sensitivity, and slow convergence. In order to handle these problems, a robust hybrid active contour method based on bias correction is proposed in this research paper The energy functional is formulated through incorporating the adaptive edge indicator function and level set formulation driven by bias field correction. The adaptive edge indicator function, which is formulated based on image gradient information, is utilized to detect object boundaries and accelerate the segmentation in the homogeneous region. The level set formulation is constructed based on an improved criterion function, in which bias field information is considered. Specifically, the bias field distribution is approximated through the local mean gray value algorithm as a prior. Moreover, a new regularized function is proposed so as to maintain the stability of curve evolution. The segmentation process is implemented by the optimized energy function and the novel regularized term. Compared to previous active contour models, the modified active contour method can yield more precise, stable, and efficient segmentation results on some challenging images.     

Integrating local distribution information with level set for boundary extraction

This paper presents a general object boundary extraction model for piecewise smooth images, which incorporates local intensity distribution information into an edge-based implicit active contour. Unlike traditional edge-based active contours that use gradient to detect edges, our model derives the neighborhood distribution and edge information with two different region-based operators: a Gaussian mixture model (GMM)-based intensity distribution estimator and the Hueckel operator. We propose the local distribution fitting model for more accurate segmentation, which incorporates the operator outcomes into the recent local binary fitting (LBF) model. The GMM and the Hueckel model parameters are estimated before contour evolution, which enables the use of the proposed model without the need for initial contour selection, i.e., the level set function is initialized with a random constant instead of a distance map. Thus our model essentially alleviates the initialization sensitivity problem of most active contours. Experiments on synthetic and real images show the improved performance of our approach over the LBF model.