Julio Martin Duarte-Carvajalino

Learning to Sense Sparse Signals: Simultaneous Sensing Matrix and Sparsifying Dictionary Optimization

Sparse signal representation, analysis, and sensing have received a lot of attention in recent years from the signal processing, optimization, and learning communities. On one hand, learning overcomplete dictionaries that facilitate a sparse representation of the data as a liner combination of a few atoms from such dictionary leads to state-of-the-art results in image and video restoration and classification. On the other hand, the framework of compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical Shannon-Nyquist Theorem. The samples used in CS correspond to linear projections obtained by a sensing projection matrix. It has been shown that, for example, a nonadaptive random sampling matrix satisfies the fundamental theoretical requirements of CS, enjoying the additional benefit of universality. On the other hand, a projection sensing matrix that is optimally designed for a certain class of signals can further improve the reconstruction accuracy or further reduce the necessary number of samples. In this paper, we introduce a framework for the joint design and optimization, from a set of training images, of the nonparametric dictionary and the sensing matrix. We show that this joint optimization outperforms both the use of random sensing matrices and those matrices that are optimized independently of the learning of the dictionary. Particular cases of the proposed framework include the optimization of the sensing matrix for a given dictionary as well as the optimization of the dictionary for a predefined sensing environment. The presentation of the framework and its efficient numerical optimization is complemented with numerous examples on classical image datasets.

Hierarchical topological network analysis of anatomical human brain connectivity and differences related to sex and kinship

Modern non-invasive brain imaging technologies, such as diffusion weighted magnetic resonance imaging (DWI), enable the mapping of neural fiber tracts in the white matter, providing a basis to reconstruct a detailed map of brain structural connectivity networks. Brain connectivity networks differ from random networks in their topology, which can be measured using small worldness, modularity, and high-degree nodes (hubs). Still, little is known about how individual differences in structural brain network properties relate to age, sex, or genetic differences. Recently, some groups have reported brain network biomarkers that enable differentiation among individuals, pairs of individuals, and groups of individuals. In addition to studying new topological features, here we provide a unifying general method to investigate topological brain networks and connectivity differences between individuals, pairs of individuals, and groups of individuals at several levels of the data hierarchy, while appropriately controlling false discovery rate (FDR) errors. We apply our new method to a large dataset of high quality brain connectivity networks obtained from High Angular Resolution Diffusion Imaging (HARDI) tractography in 303 young adult twins, siblings, and unrelated people. Our proposed approach can accurately classify brain connectivity networks based on sex (93% accuracy) and kinship (88.5% accuracy). We find statistically significant differences associated with sex and kinship both in the brain connectivity networks and in derived topological metrics, such as the clustering coefficient and the communicability matrix.

Comparative Study of Semi-Implicit Schemes for Nonlinear Diffusion in Hyperspectral Imagery

Nonlinear diffusion has been successfully employed over the past two decades to enhance images by reducing undesirable intensity variability within the objects in the image, while enhancing the contrast of the boundaries (edges) in scalar and, more recently, in vector-valued images, such as color, multispectral, and hyperspectral imagery. In this paper, we show that nonlinear diffusion can improve the classification accuracy of hyperspectral imagery by reducing the spatial and spectral variability of the image, while preserving the boundaries of the objects. We also show that semi-implicit schemes can speedup significantly the evolution of the nonlinear diffusion equation with respect to traditional explicit schemes

Multiscale Representation and Segmentation of Hyperspectral Imagery Using Geometric Partial Differential Equations and Algebraic Multigrid Methods

A fast algorithm for multiscale representation and segmentation of hyperspectral imagery is introduced in this paper. The multiscale/scale-space representation is obtained by solving a nonlinear diffusion partial differential equation (PDE) for vector-valued images. We use algebraic multigrid techniques to obtain a fast and scalable solution of the PDE and to segment the hyperspectral image following the intrinsic multigrid structure. We test our algorithm on four standard hyperspectral images that represent different environments commonly found in remote sensing applications: agricultural, urban, mining, and marine. The experimental results show that the segmented images lead to better classification than using the original data directly, in spite of the use of simple similarity metrics and piecewise constant approximations obtained from the segmentation maps.

Task-Driven Adaptive Statistical Compressive Sensing of Gaussian Mixture Models

A framework for adaptive and non-adaptive statistical compressive sensing is developed, where a statistical model replaces the standard sparsity model of classical compressive sensing. We propose within this framework optimal task-specific sensing protocols specifically and jointly designed for classification and reconstruction. A two-step adaptive sensing paradigm is developed, where online sensing is applied to detect the signal class in the first step, followed by a reconstruction step adapted to the detected class and the observed samples. The approach is based on information theory, here tailored for Gaussian mixture models (GMMs), where an information-theoretic objective relationship between the sensed signals and a representation of the specific task of interest is maximized. Experimental results using synthetic signals, Landsat satellite attributes, and natural images of different sizes and with different noise levels show the improvements achieved using the proposed framework when compared to more standard sensing protocols. The underlying formulation can be applied beyond GMMs, at the price of higher mathematical and computational complexity.

Estimation of the CSA-ODF using bayesian compressed sensing of multi-shell HARDI

Diffusion MRI provides important information about the brain white matter structures and has opened new avenues for neuroscience and translational research. However, acquisition time needed for advanced applications can still be a challenge in clinical settings. There is consequently a need to accelerate diffusion MRI acquisitions.

Adapted statistical compressive sensing: Learning to sense gaussian mixture models

A framework for learning sensing kernels adapted to signals that follow a Gaussian mixture model (GMM) is introduced in this paper. This follows the paradigm of statistical compressive sensing (SCS), where a statistical model, a GMM in particular, replaces the standard sparsity model of classical compressive sensing (CS), leading to both theoretical and practical improvements. We show that the optimized sensing matrix outperforms random sampling matrices originally exploited both in CS and SCS.

A Framework for Linear and Non-Linear Registration of Diffusion-Weighted MRIs Using Angular Interpolation

Registration of diffusion-weighted magnetic resonance images (DW-MRIs) is a key step for population studies, or construction of brain atlases, among other important tasks. Given the high dimensionality of the data, registration is usually performed by relying on scalar representative images, such as the fractional anisotropy (FA) and non-diffusion-weighted (b0) images, thereby ignoring much of the directional information conveyed by DW-MR datasets itself. Alternatively, model-based registration algorithms have been proposed to exploit information on the preferred fiber orientation(s) at each voxel. Models such as the diffusion tensor or orientation distribution function (ODF) have been used for this purpose. Tensor-based registration methods rely on a model that does not completely capture the information contained in DW-MRIs, and largely depends on the accurate estimation of tensors. ODF-based approaches are more recent and computationally challenging, but also better describe complex fiber configurations thereby potentially improving the accuracy of DW-MRI registration. A new algorithm based on angular interpolation of the diffusion-weighted volumes was proposed for affine registration, and does not rely on any specific local diffusion model. In this work, we first extensively compare the performance of registration algorithms based on (i) angular interpolation, (ii) non-diffusion-weighted scalar volume (b0), and (iii) diffusion tensor image (DTI). Moreover, we generalize the concept of angular interpolation (AI) to non-linear image registration, and implement it in the FMRIB Software Library (FSL). We demonstrate that AI registration of DW-MRIs is a powerful alternative to volume and tensor-based approaches. In particular, we show that AI improves the registration accuracy in many cases over existing state-of-the-art algorithms, while providing registered raw DW-MRI data, which can be used for any subsequent analysis.

Sparse Bayesian Inference of White Matter Fiber Orientations from Compressed Multi-resolution Diffusion MRI

The RubiX [1] algorithm combines high SNR characteristics of low resolution data with high spacial specificity of high resolution data, to extract microstructural tissue parameters from diffusion MRI. In this paper we focus on estimating crossing fiber orientations and introduce sparsity to the RubiX algorithm, making it suitable for reconstruction from compressed (under-sampled) data. We propose a sparse Bayesian algorithm for estimation of fiber orientations and volume fractions from compressed diffusion MRI. The data at high resolution is modeled using a parametric spherical deconvolution approach and represented using a dictionary created with the exponential decay components along different possible directions. Volume fractions of fibers along these orientations define the dictionary weights. The data at low resolution is modeled using a spatial partial volume representation. The proposed dictionary representation and sparsity priors consider the dependence between fiber orientations and the spatial redundancy in data representation. Our method exploits the sparsity of fiber orientations, therefore facilitating inference from under-sampled data. Experimental results show improved accuracy and decreased uncertainty in fiber orientation estimates. For under-sampled data, the proposed method is also shown to produce more robust estimates of fiber orientations.

Fast multiscale regularization and segmentation of hyperspectral imagery via anisotropic diffusion and algebraic multigrid solvers

This paper presents an algorithm that generates a scale-space representation of hyperspectral imagery using Algebraic Multigrid (AMG) solvers. The scale-space representation is obtained by solving with AMG a vector-valued anisotropic diffusion equation, with the hyperspectral image as its initial condition. AMG also provides the necessary structure to obtain a hierarchical segmentation of the image. The scale space representation of the hyperspectral image can be segmented in linear time complexity. Results in the paper show that improved segmentation is achieved. The proposed methodology to solve vector PDEs can be used to extend a number of techniques currently being developed for the fast computation of geometric PDEs and its application for the processing of hyperspectral and multispectral imagery.