Elham Sakhaee

Isosurface Visualization of Data with Nonparametric Models for Uncertainty

The problem of isosurface extraction in uncertain data is an important research problem and may be approached in two ways. One can extract statistics (e.g., mean) from uncertain data points and visualize the extracted field. Alternatively, data uncertainty, characterized by probability distributions, can be propagated through the isosurface extraction process. We analyze the impact of data uncertainty on topology and geometry extraction algorithms. A novel, edge-crossing probability based approach is proposed to predict underlying isosurface topology for uncertain data. We derive a probabilistic version of the midpoint decider that resolves ambiguities that arise in identifying topological configurations. Moreover, the probability density function characterizing positional uncertainty in isosurfaces is derived analytically for a broad class of nonparametric distributions. This analytic characterization can be used for efficient closed-form computation of the expected value and variation in geometry. Our experiments show the computational advantages of our analytic approach over Monte-Carlo sampling for characterizing positional uncertainty. We also show the advantage of modeling underlying error densities in a nonparametric statistical framework as opposed to a parametric statistical framework through our experiments on ensemble datasets and uncertain scalar fields.

Volumetric data reduction in a compressed sensing framework

In this paper, we investigate compressed sensing principles to devise an in‐situ data reduction framework for visualization of volumetric datasets. We exploit the universality of the compressed sensing framework and show that the proposed method offers a refinable data reduction approach for volumetric datasets. The accurate reconstruction is obtained from partial Fourier measurements of the original data that are sensed without any prior knowledge of specific feature domains for the data. Our experiments demonstrate the superiority of surfacelets for efficient representation of volumetric data. Moreover, we establish that the accuracy of reconstruction can further improve once a more effective basis for a sparser representation of the data becomes available.

A Statistical Direct Volume Rendering Framework for Visualization of Uncertain Data

With uncertainty present in almost all modalities of data acquisition, reduction, transformation, and representation, there is a growing demand for mathematical analysis of uncertainty propagation in data processing pipelines. In this paper, we present a statistical framework for quantification of uncertainty and its propagation in the main stages of the visualization pipeline. We propose a novel generalization of Irwin-Hall distributions from the statistical viewpoint of splines and box-splines, that enables interpolation of random variables. Moreover, we introduce a probabilistic transfer function classification model that allows for incorporating probability density functions into the volume rendering integral. Our statistical framework allows for incorporating distributions from various sources of uncertainty which makes it suitable in a wide range of visualization applications. We demonstrate effectiveness of our approach in visualization of ensemble data, visualizing large datasets at reduced scale, iso-surface extraction, and visualization of noisy data.

Gradient-based sparse approximation for computed tomography

Limited-data Computed Tomography (CT) presents challenges for image reconstruction algorithms and has been an active topic of research aiming at reducing the exposure to X-ray radiation. We present a novel formulation for tomo-graphic reconstruction based on sparse approximation of the image gradients from projection data. Our approach leverages the interdependence of the partial derivatives to impose an additional curl-free constraint on the optimization problem. The image is then reconstructed using a Poisson solver. The experimental results show that, compared to total variation methods, our new formulation improves the accuracy of reconstruction significantly in few-view settings.

Sparse partial derivatives and reconstruction from partial Fourier data

Signal reconstruction from the smallest possible Fourier measurements has been a key motivation in the compressed sensing research. We present an approach that exploits the interdependency and structural sparsity of partial derivatives for lowering the sampling rates necessary for accurate reconstruction. Our experiments show that for signals that are sparse in the gradient domain our proposed method significantly outperforms the existing approaches including the total variation (TV) based CS reconstruction.

Leveraging EAP-Sparsity for Compressed Sensing of MS-HARDI in

Compressed Sensing (CS) for the acceleration of MR scans has been widely investigated in the past decade. Lately, considerable progress has been made in achieving similar speed ups in acquiring multi-shell high angular resolution diffusion imaging (MS-HARDI) scans. Existing approaches in this context were primarily concerned with sparse reconstruction of the diffusion MR signal S(q) in the q-space. More recently, methods have been developed to apply the compressed sensing framework to the 6-dimensional joint (k, q)-space, thereby exploiting the redundancy in this 6D space. To guarantee accurate reconstruction from partial MS-HARDI data, the key ingredients of compressed sensing that need to be brought together are: (1) the function to be reconstructed needs to have a sparse representation, and (2) the data for reconstruction ought to be acquired in the dual domain (i.e., incoherent sensing) and (3) the reconstruction process involves a (convex) optimization. In this paper, we present a novel approach that uses partial Fourier sensing in the 6D space of (k, q) for the reconstruction of P(x, r). The distinct feature of our approach is a sparsity model that leverages surfacelets in conjunction with total variation for the joint sparse representation of P(x, r). Thus, our method stands to benefit from the practical guarantees for accurate reconstruction from partial (k, q)-space data. Further, we demonstrate significant savings in acquisition time over diffusion spectral imaging (DSI) which is commonly used as the benchmark for comparisons in reported literature. To demonstrate the benefits of this approach,.we present several synthetic and real data examples.

({\mathbf {k}},{\mathbf {q}})

In a few-view or limited-angle computed tomography (CT), where the number of measurements is far fewer than image unknowns, the reconstruction task is an ill-posed problem. We present a spline-based sparse tomographic reconstruction algorithm where content-adaptive patch sparsity is integrated into the reconstruction process. The proposed method leverages closed-form Radon transforms of tensor-product B-splines and non-separable box splines to improve the accuracy of reconstruction afforded by higher order methods. The experiments show that enforcing patch-based sparsity, in terms of a learned dictionary, on higher order spline representations, outperforms existing methods that utilize pixel-basis for image representation as well as those employing wavelets as sparsifying transform.

-Space

We present a spline-based sparse tomographic reconstruction framework. The proposed method utilizes the closed-form analytical Radon transform of B-splines and box splines of any order and integrates the (transform-domain) sparsity of the image into the reconstruction algorithm. Our experiments show that the synergy of sparse reconstruction together with higher order basis functions (e.g., cubic B-splines) improves the accuracy of the reconstruction. This gain can also be exploited for reducing the number of projection angles in the data acquisition.

Learning Splines for Sparse Tomographic Reconstruction

Sparse signal recovery from limited and/or degraded samples is fundamental to many applications, such as medical imaging, remote sensing, astronomical and seismic imaging. Discrete wavelet transform (DWT) has been commonly used for sparse representation of signals; nevertheless, due to its shift-variant nature, pseudo-Gibbs artifacts are present in the recovered signals. Using the redundant shift-invariant wavelet transform (SWT) is the ideal solution to obtain shift invariance; however, high redundancy factor of SWT limits its application in practical settings. We propose a dictionary splitting approach for sparse recovery from incomplete data, which leverages the ideas of cycle spinning in combination with Bregman splitting. The proposed method significantly improves the conventional signal reconstruction with DWT, offers the advantages of SWT, and overcomes high redundancy factor of SWT. We solve parallel sparse recovery problems with orthogonal dictionaries (DWT and its permuted versions), while we impose consistency between the results by updating the recovered image at each iteration. Our experiments demonstrate that few shifts are sufficient to achieve reconstruction accuracy as high as recovery with SWT, and significantly reduces its computational cost and redundancy factor.

A spline framework for sparse tomographic reconstruction

Tomographic reconstruction from limited X-ray data is an ill-posed inverse problem. A common Bayesian approach is to search for the maximum a posteriori (MAP) estimate of the unknowns that integrates the prior knowledge, about the nature of biomedical images, into the reconstruction process. Recent results on the Bayesian inversion have shown the advantages of Besov priors for the convergence of the estimates as the discretization of the image is refined. We present a spline framework for sparse tomographic reconstruction that leverages higher-order basis functions for image discretization while incorporating Besov space priors to obtain the MAP estimate. Our method leverages tensor-product B-splines and box splines, as higher order basis functions for image discretization, that are shown to improve accuracy compared to the standard, first-order, pixel-basis. Our experiments show that the synergy produced from higher order B-splines for image discretization together with the discretization-invariant Besov priors leads to significant improvements in tomographic reconstruction. The advantages of the proposed Bayesian inversion framework are examined for image reconstruction from limited number of projections in a few-view setting.