Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing

Let $v$ be a vector field in a bounded open set $G\subset {\mathbb {R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i, i=1,...,n,$ that are independent and uniformly distributed in $G.$ The problem is to estimate the integral curve of the differential equation $$\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_0\in G,$$ starting at a given point $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $\Gamma\subset G.$ We develop an estimation procedure based on a Nadaraya--Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface $\Gamma\subset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $\Gamma$. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.Comment: Published in at http://dx.doi.org/10.1214/009053607000000073 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Expectation Propagation for Poisson Data

The Poisson distribution arises naturally when dealing with data involving counts, and it has found many applications in inverse problems and imaging. In this work, we develop an approximate Bayesian inference technique based on expectation propagation for approximating the posterior distribution formed from the Poisson likelihood function and a Laplace type prior distribution, e.g., the anisotropic total variation prior. The approach iteratively yields a Gaussian approximation, and at each iteration, it updates the Gaussian approximation to one factor of the posterior distribution by moment matching. We derive explicit update formulas in terms of one-dimensional integrals, and also discuss stable and efficient quadrature rules for evaluating these integrals. The method is showcased on two-dimensional PET images.Comment: 25 pages, to be published at Inverse Problem

Procrustes analysis for diffusion tensor image processing

There is an increasing need to develop processing tools for diffusion tensor image data with the consideration of the non-Euclidean nature of the tensor space. In this paper Procrustes analysis, a non-Euclidean shape analysis tool under similarity transformations (rotation, scaling and translation), is proposed to redefine sample statistics of diffusion tensors. A new anisotropy measure Procrustes Anisotropy (PA) is defined with the full ordinary Procrustes analysis. Comparisons are made with other anisotropy measures including Fractional Anisotropy and Geodesic Anisotropy. The partial generalized Procrustes analysis is extended to a weighted generalized Procrustes framework for averaging sample tensors with different fractions of contributions to the mean tensor. Applications of Procrustes methods to diffusion tensor interpolation and smoothing are compared with Euclidean, Log-Euclidean and Riemannian methods

Spatio-temporal wavelet regularization for parallel MRI reconstruction: application to functional MRI

Parallel MRI is a fast imaging technique that enables the acquisition of highly resolved images in space or/and in time. The performance of parallel imaging strongly depends on the reconstruction algorithm, which can proceed either in the original k-space (GRAPPA, SMASH) or in the image domain (SENSE-like methods). To improve the performance of the widely used SENSE algorithm, 2D- or slice-specific regularization in the wavelet domain has been deeply investigated. In this paper, we extend this approach using 3D-wavelet representations in order to handle all slices together and address reconstruction artifacts which propagate across adjacent slices. The gain induced by such extension (3D-Unconstrained Wavelet Regularized -SENSE: 3D-UWR-SENSE) is validated on anatomical image reconstruction where no temporal acquisition is considered. Another important extension accounts for temporal correlations that exist between successive scans in functional MRI (fMRI). In addition to the case of 2D+t acquisition schemes addressed by some other methods like kt-FOCUSS, our approach allows us to deal with 3D+t acquisition schemes which are widely used in neuroimaging. The resulting 3D-UWR-SENSE and 4D-UWR-SENSE reconstruction schemes are fully unsupervised in the sense that all regularization parameters are estimated in the maximum likelihood sense on a reference scan. The gain induced by such extensions is illustrated on both anatomical and functional image reconstruction, and also measured in terms of statistical sensitivity for the 4D-UWR-SENSE approach during a fast event-related fMRI protocol. Our 4D-UWR-SENSE algorithm outperforms the SENSE reconstruction at the subject and group levels (15 subjects) for different contrasts of interest (eg, motor or computation tasks) and using different parallel acceleration factors (R=2 and R=4) on 2x2x3mm3 EPI images.Comment: arXiv admin note: substantial text overlap with arXiv:1103.353

Joint Estimation and Smoothing of Clinical DT-MRI with a Log-Euclidean Metric

Diffusion tensor MRI is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data has to be acquired rapidly at the detriment of the image quality. We propose a new variational framework that specifically targets low quality DT-MRI. The hypothesis of an additive Gaussian noise on the images leads us to estimate the tensor field directly on the image intensities. To further reduce the influence of the noise, we optimally exploit the spatial correlation by adding to the estimation an anisotropic regularization term. This criterion is easily optimized thanks to the use of the recently introduced Log-Euclidean metrics. Results on real clinical data show promising improvements of fiber tracking in the brain and we present the first successful attempt, up to our knowledge, to reconstruct the spinal cord