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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 Gaussia...
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... relies on the minimization of an energy functional derived from the linearized Stejskal- Tanner equation [57] while ensuring to remain in S + (3). An axial slice of the resulting DT image is presented in the first row of figure 12 together with a 3D surface modeling the spinal cords. ...
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... data-set is well suited to evaluate the robustness to the initialization of our segmentation framework as well as to demonstrate the importance of the Riemannian framework to achieve good segmentation results. The second, third and fourth rows of figure 12 illustrate the evolution of the segmentation process, using the geodesic distance, for 3 very different initializations: One large sphere and one small sphere centered at the cord crossing, and one small sphere placed at one end of a cord. These three examples yield the same final result, thus experimentally showing the non-dependence of our method on the initialization. ...
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... is not suprising and proves that the associated statistics do not constitute accurate descriptors of the tensors distribution. On the other side, the statistics computed with the geodesic distance make it possible to perform the desired segmentation, as presented on figure 20. This is a very interesting result since the superior part of the corona radiata is partially recovered. ...
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... is indeed well-known that the corpus callosum merges with association and projection fibers as its gets toward the cortex. We can see on figure 20 that the tapetum, the posterior region of the corona radiata and a part of the superior longitudinal fasciculus are extracted since they fuse with the splenium of the corpus callosum. The posterior limb of the internal capsule (essentially the corticospinal tract) is equally segmented since it intersects with the corpus callosum and with the superior longitudinal fasciculus in the region of the centrum semiovale. ...
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... Segmentation can be followed by tracking methods to have the tract segments with their inside streamlines. The methods developed based on geometric flow (Guo et al., 2008;Jonasson et al., 2005), template matching (Eckstein et al., 2009), statistical surface evolution Lenglet et al., 2006), Markov Random Field optimization (Bazin et al., 2011), and k-NN classification (Ratnarajah & Qiu, 2014) are among the direct methods. Generally, the limited quality of direct methods causes them to be used less than ROI-based and clustering-based methods; however, the promising performance of the recently proposed direct methods caught renewed attentions to them. ...
Quantitative analysis of white matter fiber tracts from diffusion Magnetic Resonance Imaging (dMRI) data is of great significance in health and disease. For example, analysis of fiber tracts related to anatomically meaningful fiber bundles is highly demanded in pre-surgical and treatment planning, and the surgery outcome depends on accurate segmentation of the desired tracts. Currently, this process is mainly done through time-consuming manual identification performed by neuro-anatomical experts. However, there is a broad interest in automating the pipeline such that it is fast, accurate, and easy to apply in clinical settings and also eliminates the intra-reader variabilities. Following the advancements in medical image analysis using deep learning techniques, there has been a growing interest in using these techniques for the task of tract identification as well. Recent reports on this application show that deep learning-based tract identification approaches outperform existing state-of-the-art methods. This paper presents a review of current tract identification approaches based on deep neural networks. First, we review the recent deep learning methods for tract identification. Next, we compare them with respect to their performance, training process, and network properties. Finally, we end with a critical discussion of open challenges and possible directions for future works.
... Researchers in [24] address the problem of segmentation of white matter structures from DT-MRI by using SPD matrices. This work models the movement of water molecules. ...
The diffusion tensor magnetic resonance imaging (DT-MRI) is a non-invasive and effective technique that ables us to study the micro-structural integrity of white matter of fibers and detecting tumors or anomalies in living tissues. Image segmentation in DT-RMI is used to identify and separate a tissue in different regions which preserve similar properties. \(K\) -means and deep learning algorithms can be applied for this purpose, being the second class the most used currently. Despite actually deep learning algorithms are the most frequently approach in image segmentation, $K$-means algorithm performed an important role in a recent past, carrying implicitly lots of sophisticated mathematical results. Originally, $K$-means algorithm was proposed to cluster a dataset in subsets with similar characteristics. Our interesting consists in empirically analyzing this technique and their variants under both Euclidean and Riemannian setting when applied to image segmentation in DT-MRI. Pixels in the bidimensional case and voxels in the tridimensional one can be respectively represented by 2-by-2 and 3-by-3 symmetric positive definite matrices (SPD) in DT-RMI. Since the set of \(n\) -by-$n$ SPD matrices can be seen as both a non-pointed convex cone in the vector space of $n$-by-$n$ symmetric matrices as well as a Hadamard manifold, centroids are well-posed in both sense, for any positive integer $n$. Thus, we developed a comparative study on the influence of centroids under Euclidean and Riemannian settings on the image segmentation by using the $K$-means algorithm and its variants.
... This dMRI technique is called diffusion tensor imaging (DTI). Some practical techniques for estimating the diffusion tensors and population mean of diffusion tensors accurately can be found in Wang & Vemuri (2004), Chefd'Hotel et al. (2004), Fletcher & Joshi (2004, Alexander (2005), Zhou et al. (2008), Lenglet et al. (2006), andDryden et al. (2009). DTI has been the de facto non-invasive dMRI diagnostic imaging technique of choice in the clinic for a variety of neurological ailments. ...
The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this paper, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of $N \times N$ symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and is widely used in applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an analytic form and show that it is asymptotically optimal within a large class of estimators including the MLE, which is the sample Fr\'echet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data experiments. Furthermore, we apply the shrinkage estimator to perform statistical inference in diffusion magnetic resonance imaging problems.
... Surface normal vectors on the unit sphere have been widely used in graphics [48], and probability density functions, as well as covariance matrices, are common in both machine learning and computer vision [45,16]. In neuroimaging, which is a key focus of our paper, the structured measurement at a voxel of an image may capture water diffusion [6,53,36,31,2,13] or local structural change [25,59,32]. The latter example is commonly known as the Cauchy deformation tensor (CDT) [32] and has been utilized to achieve improvements over brain imaging methods such as tensor-based morphometry [37,43,4]. ...
Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are {\em sequential manifold-valued measurements}. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fr\'{e}chet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.
... Surface normal vectors on the unit sphere have been widely used in graphics [48], and probability density functions, as well as covariance matrices, are common in both machine learning and computer vision [45,16]. In neuroimaging, which is a key focus of our paper, the structured measurement at a voxel of an image may capture water diffusion [6,53,36,31,2,13] or local structural change [25,59,32]. The latter example is commonly known as the Cauchy deformation tensor (CDT) [32] and has been utilized to achieve improvements over brain imaging methods such as tensor-based morphometry [37,43,4]. ...
Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are sequential manifold-valued measurements. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fréchet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.
... Many Riemannian structures have been introduced on the manifold of SPD matrices depending on the problem and showing significantly different results from one another on statistical procedures such as the computation of barycenters or the principal component analysis. Non exhaustively, we can cite Euclidean metrics, power-Euclidean metrics [7], log-Euclidean metrics [8], which are flat; affine-invariant metrics [5,6,9] which are negatively curved; the Bogoliubov-Kubo-Mori metric [10] whose curvature has a quite complex expression. ...
... Observation We denote M = SP D n the manifold of SPD matrices and N = dim M = n(n+1) 2 . The (A)ffine-invariant metric g A on SPD matrices [5,6,9], i.e. ...
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.
... In this context, the affine-invariant metric [2,8,3] was introduced to give an invariant computing framework under affine transformations of the variables. This metric endows the manifold of SPD matrices with a structure of a Riemannian symmetric space. ...
... The affine-invariant metric [2,8,3] and the polar-affine metric [12] are different but they both provide a Riemannian symmetric structure to the manifold of SPD matrices. Moreover, both claim to be very naturally introduced. ...
Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the action of the affine transformations was suggested as a principle. Another concept is based on symmetries. However, the affine-invariant metric and the recently proposed polar-affine metric are both invariant and symmetric. Comparing these two cousin metrics leads us to introduce much wider families: power-affine and deformed-affine metrics. Within this continuum, we investigate other principles to restrict the family size.
... Modeling of covariance matrices builds on important geometrical concepts and the medical image analysis community has made significant progress in terms of mathematical descriptions and practical applications motivated by data in diffusion tensor imaging (Pennec, 1999(Pennec, , 2006Moakher, 2005;Arsigny et al., 2006Arsigny et al., /2007Lenglet et al., 2006;Fletcher and Joshi, 2007;Fillard et al., 2007;Schwartzman et al., 2008;Dryden et al., 2009). The underlying geometry is called Lie group theory and it appears when we consider the covariance matrices as elements in a non-linear space. ...
Functional brain connectivity is the co-occurrence of brain activity in different areas during resting and while doing tasks. The data of interest are multivariate timeseries measured simultaneously across brain parcels using resting-state fMRI (rfMRI). We analyze functional connectivity using two heteroscedasticity models. Our first model is low-dimensional and scales linearly in the number of brain parcels. Our second model scales quadratically. We apply both models to data from the Human Connectome Project (HCP) comparing connectivity between short and conventional sleepers. We find stronger functional connectivity in short than conventional sleepers in brain areas consistent with previous findings. This might be due to subjects falling asleep in the scanner. Consequently, we recommend the inclusion of average sleep duration as a covariate to remove unwanted variation in rfMRI studies. A power analysis using the HCP data shows that a sample size of 40 detects 50% of the connectivity at a false discovery rate of 20%. We provide implementations using R and the probabilistic programming language Stan.
... Following a prior work where DTs were identified as elements of R 6 , 58 which ignores the algebraic properties of symmetric and positive-definite matrices, the concept of a tensor-variate normal distribution 59 perspective and developed a computational framework to DT processing with a particular emphasis on interpolation, regularization and restoration of noisy tensor fields. 63 Finally, Lenglet et al. 64 relied on the information geometry of the space of multivariate normal distributions with zero mean to tackle similar theoretical questions, while focusing on the estimation and level-set-based segmentation 65 of DTI data. A related and recent work, which focuses on the DT estimation problem, 66 formulates a Bayesian framework for the simultaneous estimation and regularization of DTI data, while accounting for Rician noise in diffusion-weighted data and providing error bounds. ...
Computational methods are crucial for the analysis of diffusion magnetic resonance imaging (MRI) of the brain. Computational diffusion MRI can provide rich information at many size scales, including local microstructure measures such as diffusion anisotropies or apparent axon diameters, whole-brain connectivity information that describes the brain's wiring diagram and population-based studies in health and disease. Many of the diffusion MRI analyses performed today were not possible five, ten or twenty years ago, due to the requirements for large amounts of computer memory or processor time. In addition, mathematical frameworks had to be developed or adapted from other fields to create new ways to analyze diffusion MRI data. The purpose of this review is to highlight recent computational and statistical advances in diffusion MRI and to put these advances into context by comparison with the more traditional computational methods that are in popular clinical and scientific use. We aim to provide a high-level overview of interest to diffusion MRI researchers, with a more in-depth treatment to illustrate selected computational advances.
... Researchers in the field of medical imaging, use such matrix-valued images to segment areas of the scanned brain into regions that exhibit similar behaviors, to detect injuries or chronic illnesses. Overviews of such segmentation algorithms are presented in Lenglet et al. (2006) and Barmpoutis et al. (2007). There are two main approaches to do such segmentation: (i) Convert the DTI to a scalar/vector valued image based on DT properties such as fractional anisotropy or mean diffusivity Shepherd et al. (2006) and apply well-established edge-based or level-set based image segmentation algorithms. ...
... Through this dataset, we can show the scalability of our proposed clustering algorithm and demonstrate that it's generic enough to be applied to different fields, with no modification. The Fractional Anisotropy (FA) value of the tensor field is used to evaluate the performance of our algorithm, as it is a popular quantity used to process and analyze DTIs (Lenglet et al., 2006). We used the fanDTasia Matlab Toolbox (Barmpoutis et al., 2007) and the accompanying tutorial (Barmpoutis, 2010) to generate a 32 × 32 synthetic DTI, shown in Figure 12. ...
In this work, we tackle the problem of transform-invariant unsupervised learning in the space of Covariance matrices and applications thereof. We begin by introducing the Spectral Polytope Covariance Matrix (SPCM) Similarity function; a similarity function for Covariance matrices, invariant to any type of transformation. We then derive the SPCM-CRP mixture model, a transform-invariant non-parametric clustering approach for Covariance matrices that leverages the proposed similarity function, spectral embedding and the distance-dependent Chinese Restaurant Process (dd-CRP) (Blei and Frazier, 2011). The scalability and applicability of these two contributions is extensively validated on real-world Covariance matrix datasets from diverse research fields. Finally, we couple the SPCM-CRP mixture model with the Bayesian non-parametric Indian Buffet Process (IBP) - Hidden Markov Model (HMM) (Fox et al., 2009), to jointly segment and discover transform-invariant action primitives from complex sequential data. Resulting in a topic-modeling inspired hierarchical model for unsupervised time-series data analysis which we call ICSC-HMM (IBP Coupled SPCM-CRP Hidden Markov Model). The ICSC-HMM is validated on kinesthetic demonstrations of uni-manual and bi-manual cooking tasks; achieving unsupervised human-level decomposition of complex sequential tasks.
