Approximating symmetric positive semi-definite tensors of even order A. Barmpoutis, J. Ho and B. C. Vemuri 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 P^2m_0 of 2m^th- order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on P^2m_0 is usually not straightforward computationally because there is no known analytic description of P^2m_0 for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone P^2m_0 for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Sigma_2m < Omega_2m for m >= 1, using the subset Omega_2m < P^2m_0 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 leastsquares (NNLS) optimization problem with the complexity that equals the number of generators in Sigma_2m. Finally, we experimentally validate the proposed approach and we present an application for computing 2mth-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images.

Diffusion Kurtosis Imaging: Robust Estimation from DW-MRI using Homogeneous Polynomials A. Barmpoutis and J. Zhuo Several tensor-based models have been presented in literature for parameterizing the water diffusion in Diffusion-Weighted MRI datasets, namely Diffusion Tensor Imaging (DTI), Generalized Tensor Imaging (GTI), and Diffusion Kurtosis Imaging (DKI). In this paper we use homogeneous trivariate polynomials to show that GTI is a special case of DKI for single angular shell acquisitions, and then we employ the theory for imposing positive semi-definite (PSD) constraints to GTIs in order to performrobust estimation of the DKI parameters. We propose a novel framework for DKI estimation that simultaneously imposes constraints to the diffusivity function, diffusion tensor and diffusion kurtosis. These three constraints are parameterized explicitly as a set of linear systems that can be efficiently solved using the non-negative least squares technique. The robustness of our framework is demonstrated using synthetic and real data from a human brain.

A unified framework for estimating diffusion tensors of any order with symmetric positive-definite constraints A. Barmpoutis and B. C. Vemuri Cartesian tensors of various orders have been employed for either modeling the diffusivity or the orientation distribution function in Diffusion-Weighted MRI datasets. In both cases, the estimated tensors have to be positive-definite since they model positive-valued functions. In this paper we present a novel unified framework for estimating positive-definite tensors of any order, in contrast to the existing methods in literature, which are either order-specific or fail to handle the positive-definite property. The proposed framework employs a homogeneous polynomial parametrization that covers the full space of any order positive-definite tensors and explicitly imposes the positive-definite constraint on the estimated tensors. We show that this parametrization leads to a linear system that is solved using the non-negative least squares technique. The framework is demonstrated using synthetic and real data from an excised rat hippocampus.

Symmetric Positive-Definite Cartesian Tensor Orientation Distribution Functions (CT-ODF) Y. T. Weldeselassie, A. Barmpoutis and S. Atkins In this paper we present a novel method for estimating a field of orientation distribution functions (ODF) from a given set of Diffusion- Weighted MR images. In our technique the ODF is modeled by Cartesian tensor basis using a parametrization that explicitly enforces the positive definite property to the computed ODFs. The computed Cartesian tensors, dubbed Cartesian Tensor-ODFs (CT-ODFs), are symmetric positive definite tensors whose coefficients can be efficiently estimated by solving a linear system with non-negative constraints. Furthermore, we show how to use our method for converting higher-order diffusion tensors to CT-ODFs, which is an essential task since the maxima of higher-order tensors do not correspond to the underlying fiber orientations. We quantitatively evaluate our method using simulated DW-MR images as well as a real brain dataset from a post-mortem porcine brain. The results conclusively demonstrate the superiority of the proposed technique over several existing multi-fiber reconstruction methods.

Regularized Positive-Definite Fourth-Order Tensor Field Estimation from DW-MRI A. Barmpoutis, M. S. Hwang, D. Howland, J. R. Forder and B. C. Vemuri 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.

Non-Lambertian Reflectance Modeling and Shape Recovery for Faces using Anti-Symmetric Tensor Splines R. Kumar, A. Barmpoutis, A. Banerjee, B. C. Vemuri Modeling illumination effects and pose variations of a face is of fundamental importance in the field of facial image analysis. Most of the conventional techniques that simultaneously address both of these problems work with the Lambertian assumption and thus, fall short of accurately capturing the complex intensity variation that the facial images exhibit or recovering their 3D shape in presence of specularities and cast shadows. In this paper we present a novel anti-symmetric tensor spline based framework for facial image analysis. We show that using this framework, facial apparent BRDF field can be accurately estimated while seamlessly accounting for cast shadows and specularities. Further, using local neighborhood information, the same framework can be exploited to recover the 3D shape of the face (to handle pose variation). We quantitatively validate the accuracy of the anti-symmetric tensor spline model using a more general continuous mixture of single lobed spherical functions. We demonstrate the effectiveness of our technique by presenting extensive experimental results for face relighting, 3D shape recovery and face recognition using the Extended Yale B and CMU PIE benchmark datasets.

Groupwise Registration and atlas construction of 4th-order tensor fields using the R+ Riemannian metric A. Barmpoutis, and B. C. Vemuri Registration of Diffusion-Weighted MR Images (DW-MRI) can be achieved by registering the corresponding 2nd-order Diffusion Tensor Images (DTI). However, it has been shown that higher-order diffusion tensors (e.g. order-4) outperform the traditional DTI in approximating complex fiber structures such as fiber crossings. In this paper we present a novel method for unbiased group-wise non-rigid registration and atlas construction of 4th-order diffusion tensor fields. To the best of our knowledge there is no other existing method to achieve this task. First we define a metric on the space of positive-valued functions based on the Riemannian metric of real positive numbers (denoted by R+). Then, we use this metric in a novel functional minimization method for non-rigid 4th-order tensor field registration. We define a cost function that accounts for the 4th-order tensor re-orientation during the registration process and has analytic derivatives with respect to the transformation parameters. Finally, the tensor field atlas is computed as the minimizer of the variance defined using the Riemannian metric. We quantitatively compare the proposed method with other techniques that register scalar-valued or diffusion tensor (rank-2) representations of the DWMRI. This comparison is achieved using synthetic data and atlas construction results are depicted for real human hippocampal data sets.

Adaptive kernels for multi-fiber reconstruction A. Barmpoutis, B. Jian, and B. C. Vemuri In this paper we present a novel method for multi-fiber reconstruction given a diffusion-weighted MRI dataset. There are several existing methods that employ various spherical deconvolution kernels for achieving this task. However the kernels in all of the existing methods rely on certain assumptions regarding the properties of the underlying fibers, which introduce inaccuracies and unnatural limitations in them. Our model is a non trivial generalization of the spherical deconvolution model, which unlike the existing methods does not make use of a fix-shaped kernel. Instead, the shape of the kernel is estimated simultaneously with the rest of the unknown parameters by employing a general adaptive model that can theoretically approximate any spherical deconvolution kernel. The performance of our model is demonstrated using simulated and real diffusion-weighed MR datasets and compared quantitatively with several existing techniques in literature. The results obtained indicate that our model has superior performance that is close to the theoretic limit of the best possible achievable result.

Information theoretic methods for diffusion-weighted MRI analysis A. Barmpoutis and B. C. Vemuri 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.

Extracting Tractosemas from a displacement probability field for tractography in DW-MRI A. Barmpoutis, B. C. Vemuri, D. Howland and J. R. Forder 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 robustness of our model in estimating accurate fiber orientations is validated via experiments on synthetic and real diffusion-weighted magnetic resonance (DW-MRI) datasets from an isolated rat hippocampus and a cats spinal cord.

Beyond the Lambertian Assumption: A generative model for Apparent BRDF fields of Faces using Anti-Symmetric Tensor Splines A. Barmpoutis, R. Kumar, B. C. Vemuri and A. Banerjee Human faces are neither exactly Lambertian nor entirely convex and hence most models in literature which make the Lambertian assumption, fall short when dealing with specularities and cast shadows. In this paper, we present a novel anti-symmetric tensor spline (a spline for tensor-valued functions) based method for the estimation of the Apparent BRDF (ABRDF) field for human faces that seamlessly accounts for specularities and cast shadows. Furthermore, unlike other methods, it does not require any 3D information to build the model and can work with as few as 9 images. In order to validate the accuracy of our anti-symmetric tensor spline model, we present a novel approximation of the ABRDF using a continuous mixture of single-lobed spherical functions. We demonstrate the effectiveness of our anti-symmetric tensor-spline model in comparison to other popular models in the literature, by presenting extensive results for face relighting and face recognition using the Extended Yale B database.

Multi-Fiber Reconstruction from DW-MRI using a Continuous Mixture of von Mises-Fisher Distributions R. Kumar, A. Barmpoutis, B. C. Vemuri, P. R. Carney and T. H. Mareci In this paper we propose a method for reconstructing the Diffusion Weighted Magnetic Resonance (DW-MR) signal at each lattice point using a novel continuousmixture 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.

Fast displacement probability profile approximation from HARDI using 4th-order tensors A. Barmpoutis, B. C. Vemuri and J. R. Forder 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 4th 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.

Tensor splines for interpolation and approximation of DT-MRI with applications to segmentation of isolated rat hippocampi A. Barmpoutis, B. C. Vemuri, T. M. Shepherd, and J. R. Forder 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 DT-MRI data sets. 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. Tensor Spline DEMO:[Click here]

Exponential Tensors: A framework for efficient higher-order DT-MRI computations A. Barmpoutis, and B. C. Vemuri In Diffusion Tensor Magnetic Resonance Image (DT-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the 3D volume image. These tensors are symmetric positive definite matrices and the appropriate constraints required in algorithms for processing them makes these algorithms complex and significantly increases their computational complexity. In this paper we present a novel parameterization of the diffusivity function using which the positive definite property of the function is guaranteed without any increase in computation. This parameterization can be used for any order tensor approximations; we present Cartesian tensor approximations of order 2, 4, 6 and 8 respectively, of the diffusivity function all of which retain the positivity property in this parameterization without the need for any explicit enforcement. Furthermore, we present an efficient framework for computing distances and geodesics in the space of the coefficients of our proposed diffusivity function. Distances & geodesics are useful for performing interpolation, computation of statistics etc. on high rank positive definite tensors. We validate our model using real diffusion weighted MR data from excised, perfusion-fixed rat optic chiasm.

Symmetric Positive 4th Order Tensors & their Estimation from Diffusion Weighted MRI A. Barmpoutis, B. Jian, B. C. Vemuri and T. M. Shepherd 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. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert's theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus. Note: This is not the journal version. Click here for the latest version.

A novel DTI method for analyzing the diffusion of water in retina A. Barmpoutis, S. Chandra, J. R. Forder and B. C. Vemuri In this paper, we present a novel DTI method for analyzing the diffusion of water in retina. Studies on DTI datasets from mouse eyes have shown that the diffusivity in the layer of retina has an organized structure. In this work, we analyze this structure by evaluating a quantitative measure using the following method. We represent each diffusion tensor of the retinal layer by a multivariate Gaussian probability, whose isosurfaces have the same orientations as the primary eigenvector of the corresponding diffusion tensor. The weighted sum of these probabilities formulates a mixture of Gaussians. In our experimental results we show that this mixture of Gaussians takes its global maximum value near the center of the eye (focal point). In our experiments we used 3D DTI datasets acquired from three normal mouse eyes, and an additional one eye that was subjected to focal laser treatment, resulting in disruption of the BRB. In the latter dataset the global maximum of the probability is far from the center of the eye, as a result of the changes in the diffusion properties.

Orthonormal Basis Latice Neural Networks A. Barmpoutis, and G. X. Ritter Lattice based neural networks are capable of resolving some difficult non-linear problems and have been successfully employed to solve real-world problems. In this paper a novel model of a lattice neural network (LNN) is presented. This new model generalizes the standard basis lattice neural network (SB-LNN) based on dendritic computing. In particular, we show how each neural dendrite can work on a different orthonormal basis than the other dendrites. We present experimental results that demonstrate superior learning performance of the new Orthonormal Basis Lattice Neural Network (OB-LNN) over SB-LNNs.

Smart Dust: Monte Carlo Simulation of Self-Organised Transport J. Barker, and A. Barmpoutis Smart dust is envisaged as swarms of miniature communication/sensor devices useful for remote monitoring in space exploration. With diameters and densities comparable to sand particles the behaviour of passive dust will be identical to the movement of airborne sand. Here we examine algorithms for the adaptive shape change of smart dust modes that permits a change in drag coefficient depending on whether or not the random motion is in a favourable direction. Monte Carlo simulations are reported for swarms of smart dust devices transporting in the wind-dominated environment of the Martian landscape. It is concluded that relatively simple shape changing algorithms, activated through an electro-active polymer sheath, will permit self-organised transport over large distances.

Face 3D Pose Estimation Using a Generic 3D Face Model and Facial Feature Extraction A. Barmpoutis, N. Nikolaides, and I. Pittas In this paper an algorithm that processes a video sequence for humean face 3D pose estimation, is presented. The procedure that is followed is described briefly below. In the beginning, facial features are extracted for each frame. Afterwards, these are used in order to make an initial estimation of their position in 3D space. The results produced are optimized, either by taking into consideration anthropometric features, or by using a 3D model of the human face. In that way, the initial prediction is greately improved and the resulting accuracy is more than satisfactory. Such rechniques could process videos displaying news, journalists, actors or even people in general, or even be used in object-based techniques for video coding (eg. MPEG-4), in machine vision applications and in human computer interaction enviroments.