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Register a new userLow Rank plus Sparse Decomposition of ODFs for Improved Detection of Group-level Differences and Variable Correlations in White Matter
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A novel approach is presented for group statistical analysis of diffusion weighted MRI datasets through voxelwise Orientation Distribution Functions (ODF). Recent advances in MRI acquisition make it possible to use high quality diffusion weighted protocols (multi-shell, large number of gradient directions) for routine in vivo study of white matter architecture. The dimensionality of these data sets is however often reduced to simplify statistical analysis. While these approaches may detect large group differences, they do not fully capitalize on all acquired image volumes. Incorporation of all available diffusion information in the analysis however risks biasing the outcome by outliers. Here we propose a statistical analysis method operating on the ODF, either the diffusion ODF or fiber ODF. To avoid outlier bias and reliably detect voxelwise group differences and correlations with demographic or behavioral variables, we apply the Low-Rank plus Sparse (L + S) matrix decomposition on the voxelwise ODFs which separates the sparse individual variability in the sparse matrix S whilst recovering the essential ODF features in the low-rank matrix L. We demonstrate the performance of this ODF L + S approach by replicating the established negative association between global white matter integrity and physical obesity in the Human Connectome dataset. The volume of positive findings agrees with and expands on the volume found by TBSS, Connectivity based fixel enhancement and Connectometry. In the same dataset we further localize the correlations of brain structure with neurocognitive measures such as fluid intelligence and episodic memory. The presented ODF L + S approach will aid in the full utilization of all acquired diffusion weightings leading to the detection of smaller group differences in clinically relevant settings as well as in neuroscience applications.
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Data processing constitutes a critical component of high-contrast exoplanet imaging. Its role is almost as important as the choice of a coronagraph or a wavefront control system, and it is intertwined with the chosen observing strategy. Among the data processing techniques for angular differential imaging (ADI), the most recent is the family of principal component analysis (PCA) based algorithms. PCA serves, in this case, as a subspace projection technique for constructing a reference point spread function (PSF) that can be subtracted from the science data for boosting the detectability of potential companions present in the data. Unfortunately, when building this reference PSF from the science data itself, PCA comes with certain limitations such as the sensitivity of the lower dimensional orthogonal subspace to non-Gaussian noise. Inspired by recent advances in machine learning algorithms such as robust PCA, we aim to propose a localized subspace projection technique that surpasses current PCA-based post-processing algorithms in terms of the detectability of companions at near real-time speed, a quality that will be useful for future direct imaging surveys. We used randomized low-rank approximation methods recently proposed in the machine learning literature, coupled with entry-wise thresholding to decompose an ADI image sequence locally into low-rank, sparse, and Gaussian noise components (LLSG). This local three-term decomposition separates the starlight and the associated speckle noise from the planetary signal, which mostly remains in the sparse term. We tested the performance of our new algorithm on a long ADI sequence obtained on beta Pictoris with VLT/NACO. Compared to a standard PCA approach, LLSG decomposition reaches a higher signal-to-noise ratio and has an overall better performance in the receiver operating characteristic space. (abridged).
Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data. This model is the foundation of robust principle component analysis (Candes et al., 2011) (Chandrasekaran et al., 2009), and popularized by dynamic-foreground/static-background separation (Bouwmans et al., 2016) amongst other applications. Compressed sensing, matrix completion, and their variants (Eldar and Kutyniok, 2012) (Foucart and Rauhut, 2013) have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that $mtimes n$ matrices that can be expressed as the sum of a rank-$r$ matrix and a $s$-sparse matrix can be recovered by computationally tractable methods from $mathcal{O}(r(m+n-r)+s)log(mn/s)$ linear measurements. More specifically, we establish that the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided $p/mn$, $s/p$, and $r(m+n-r)/p$ reman fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operators RICs are sufficiently small. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/static-background separation, and multispectral imaging.
Objective: Interventional MRI (i-MRI) is crucial for MR image-guided therapy. Current image reconstruction methods for dynamic MR imaging are mostly retrospective that may not be suitable for i-MRI in real-time. Therefore, an algorithm to reconstruct images without a temporal pattern as in dynamic imaging is needed for i-MRI. Methods: We proposed a low-rank and sparsity (LS) decomposition algorithm with framelet transform to reconstruct the interventional feature with a high temporal resolution. Different from the existing LS based algorithm, we utilized the spatial sparsity of both the low-rank and sparsity components. We also used a primal dual fixed point (PDFP) method for optimization of the objective function to avoid solving sub-problems. Intervention experiments with gelatin and brain phantoms were carried out for validation. Results: The LS decomposition with framelet transform and PDFP could provide the best reconstruction performance compared with those without. Satisfying reconstruction results were obtained with only 10 radial spokes for a temporal resolution of 60 ms. Conclusion and Significance: The proposed method has the potential for i-MRI in many different application scenarios.
We consider the problem of estimating high-dimensional covariance matrices of a particular structure, which is a summation of low rank and sparse matrices. This covariance structure has a wide range of applications including factor analysis and random effects models. We propose a Bayesian method of estimating the covariance matrices by representing the covariance model in the form of a factor model with unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low rank component combined with a Bayesian lasso method for the sparse component estimation. Simulation studies show that our method can recover the rank as well as the sparsity of the two components respectively. We further extend our method to a graphical factor model where the graphical model of the residuals as well as selecting the number of factors is of interest. We employ a hyper-inverse Wishart prior for modeling decomposable graphs of the residuals, and a Bayesian graphical lasso selection method for unrestricted graphs. We show through simulations that the extended models can recover both the number of latent factors and the graphical model of the residuals successfully when the sample size is sufficient relative to the dimension.
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