R-3 Repository :: Browsing by Author "Borcea, Liliana"

Browsing by Author "Borcea, Liliana"

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A variational study of the electrical impedance tomography problem
(2002) Gray, Genetha Anne; Borcea, Liliana; Zhang, Yin
This research is focused on the numerical solution of the inverse conductivity problem, widely known as electrical impedance tomography (EIT). The EIT problem is concerned with imaging electrical properties, such as conductivity (sigma) and permittivity (epsilon), in the interior of a body given measurements of d.c. or a.c. voltages and currents at the boundary. Given complete and perfect knowledge of the boundary data, the EIT problem is known to have a unique solution. In practice however, the data is noisy and incomplete. Hence, satisfactory solutions of the nonlinear ill-posed EIT problem are difficult to obtain. In this thesis, we introduce a family of variational formulations for the EIT problem which we show to have advantages over the popular output least squares approach. Output least squares seeks sigma and/or epsilon by minimizing the voltage misfit at the boundary measured in the L 2 norm. In contrast, the variational methods ensure that the measured data is being fit in a more natural norm, which is not the L 2 norm. These methods also introduce some natural regularization. Through extensive numerical experimentation, we compare the performance of our variational formulations with one another and with the standard least squares algorithm. Using the same data, we demonstrate that our variational algorithms produce better images without significantly increasing computational cost.
Autofocus for Synthetic Aperture Radar
(2013-06-05) Gallardo Palacios, Ricardo; Borcea, Liliana; Symes, William W.; Yin, Wotao
In this thesis, I compare the performance of three different autofocus techniques for Synthetic Aperture Radar (SAR). The focusing is done by estimating phase errors in SAR data. The first one, the Phase Gradient Autofocus, is the most popular in the industry, it has been around for more than 20 years and it relies on the redundancy of the phase error in the SAR images. The second one, the Entropy-based minimization, uses measurements of image sharpness to focus the images and it has been available for about 10 years. The last, the Phase-space method, uses the Wigner transform and the ambiguity function of the SAR data to estimate the phase perturbations and it was recently introduced. Additionally, I develop a criteria for filtering the data for the cases in which the Phase-space method does not capture the entirety of the error.
Coherent Interferometric Imaging, Time Gating and Beamforming
(2010-12) Borcea, Liliana; Garnier, Josselin; Papanicolaou, George; Tsogka, Chrysoula
Coherent interferometric imaging is based on the backpropagation of local space-time cross correlations of array data and was introduced in order to improve images when the medium between the array and the object to be imaged is inhomogeneous and unknown [Borcea et al., Inverse Problems, 21 (2005), 1419]. Although this method has been shown to be effective and is well founded theoretically, the coherent interferometric imaging function is computationally expensive and therefore difficult to use. In this paper we show that this function is equivalent to a windowed beamformer energy function, that is, a quadratic function that involves only time gating and time delaying signals in emission and in reception. In this form the coherent interferometric imaging can be implemented efficiently both in hardware and software, that is, at a computational cost that is comparable to the usual beamforming and migration imaging methods. We also revisit the trade-off between enhanced image stability and loss of resolution in coherent interferometry from the point of view of its equivalence to a windowed beamformer energy imaging function.
Consulting in Applied Mathematics
(1998-07) Borcea, Liliana
This report contains a description of four projects brought to the attention of the Consulting Course, CAAM 513 in the Department of Computational and Applied Mathematics at Rice University. The enclosed reports reflect the work done by Genetha Gray, Nathan Hillson, Shannon Walsh and the instructor Liliana Borcea, in the Spring semester of 1998.
Filtering random layering effects for imaging and velocity estimation
(2008) Gonzalez del Cueto, Fernando; Borcea, Liliana
Imaging compactly supported reflectors in highly heterogeneous media is a challenging problem due to the significant interaction of waves with the medium which causes considerable delay spread and loss of coherence. The imaging problem consists in finding the support of small reflectors using recorded echoes at an array of sensors. The thesis considers the case of randomly layered media, in which significant multiple scattering by the layered structures and quick loss of coherence is observed. These strong, backscattered echoes from the layers can overwhelm the weaker coherent signals due to the compactly supported reflectors. This signal-to-noise problem must be addressed to image effectively. Using techniques routinely used in exploration seismology, filters (layer annihilators) are designed to remove the primary reflections of the stronger layered features in the medium. However, it observed that these filters also remove the incoherent signal that is due to the fine, random layers. The main achievement of this thesis is the theoretical and numerical analysis of this phenomenon. Additionally, the applicability of the layer annihilators for velocity estimation is presented.
First Order Signatures and Knot Concordance
(2012-09-05) Davis, Christopher; Cochran, Tim D.; Harvey, Shelly; Borcea, Liliana
Invariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.
Imaging in cluttered acoustic waveguides
(2008) Issa, Leila; Borcea, Liliana
We consider an inverse problem for the acoustic scalar wave equation in a cluttered waveguide. The problem is to find the location of sources or scatterers, given measurements of the pressure at a remote array of transducers. The sound speed in the waveguide fluctuates rapidly due to the presence of small inhomogeneities. These fluctuations, not known in detail, are viewed as clutter and modeled as a random process. We consider the regime of weak fluctuations, O (ε), whose effect on the wave-field accumulates after long propagation distances, O (1/ε 2 ). This recorded field is backpropagated using the homogeneous Green's function to search points where an image is formed. The multiple scattering due to clutter may lead to significant loss of coherence in the data, which in turn can cause instability and loss of resolution in the images. Parameters such as frequency band, depth of the waveguide, number of modes and aperture play a major role in the resolution and stability of the resulting images.
Inverse source problems for time-dependent radiative transport
(2014-03-20) Acosta Valenzuela, Sebastian; Borcea, Liliana; Riviere, Beatrice M.; Symes, William W.; Hardt, Robert M.; Alonso, Ricardo J
In the first part of this thesis, I develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small. Then, using duality arguments, I show that the solvability of the inverse problem leads to exact controllability of the transport field. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery. The second portion of the work is dedicated to the simultaneous recovery of a source of the form "s(t,x,d) f(x)" (with "s" known) and an isotropic initial condition "u0(x)", using the single measurement induced by these data. This result is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. More precisely, based on exact boundary controllability, I derive a system of equations for the unknown terms "f" and "u0". The system is shown to be of Fredholm type if "s" satisfies a certain positivity condition. This condition requires that the radiation visits the region over which "f" is to be recovered. I show that for generic term "s" and weakly absorbing media, the inverse problem is well-posed.
On the approximation of the Dirichlet to Neumann map for high contrast two phase composites
(2013-09-16) Wang, Yingpei; Borcea, Liliana; Riviere, Beatrice M.; Gorb, Yuliya; Symes, William W.; Hardt, Robert M.
Many problems in the natural world have high contrast properties, like transport in composites, fluid in porous media and so on. These problems have huge numerical difficulties because of the singularities of their solutions. It may be really expensive to solve these problems directly by traditional numerical methods. It is necessary and important to understand these problems more in mathematical aspect first, and then using the mathematical results to simplify the original problems or develop more efficient numerical methods. In this thesis we are going to approximate the Dirichlet to Neumann map for the high contrast two phase composites. The mathematical formulation of our problem is to approximate the energy for an elliptic equation with arbitrary boundary conditions. The boundary conditions may have highly oscillations, which makes our problems very interesting and difficult. We developed a method to divide the domain into two different subdomains, one is close to and the other one is far from the boundary, and we can approximate the energy in these two subdomains separately. In the subdomain far from the boundary, the energy is not influenced that much by the boundary conditions. Methods for approximation of the energy in this subdomain are studied before. In the subdomain near the boundary, the energy depends on the boundary conditions a lot. We used a new method to approximate the energy there such that it works for any kind of boundary conditions. By this way, we can have the approximation for the total energy of high contrast problems with any boundary conditions. In other words, we can have a matrix up to any dimension to approximate the continuous Dirichlet to Neumann map of the high contrast composites. Then we will use this matrix as a preconditioner in domain decomposition methods, such that our numerical methods are very efficient to solve the problems in high contrast composites.
On the approximation of the Dirichlet to Neumann map for high contrast two phase composites and its applications to domain decomposition methods
(2014-08-01) Wang, Yingpei; Borcea, Liliana; Riviere, Beatrice M.; Symes, William W.; Hardt, Robert M.
An asymptotic approximation of the Dirichlet to Neumann (DtN) map of high contrast composite media with perfectly conducting inclusions that are close to touching is presented. The result is an explicit characterization of the DtN map in the asymptotic limit of the distance between the inclusions tending to zero. The approximation of DtN map is applied directly to nonoverlapping domain decomposition methods as preconditioners in order to obtain more computational efficiency.
On the parameterization of ill-posed inverse problems arising from elliptic partial differential equations
(2006) Guevara Vasquez, Fernando; Borcea, Liliana
Electric impedance tomography (EIT) consists in finding the conductivity inside a body from electrical measurements taken at its surface. This is a severely ill-posed problem: any numerical inversion scheme requires some form of regularization. We present inversion schemes that address the instability of the problem by proper sparse parametrization of the unknown conductivity. To guide us, we consider a consistent finite difference approach to an inverse Sturm-Liouville problem arising in EIT for layered media. The method first solves a model reduction problem for the differential equation where the reduced model parameters are essentially averages of the conductivity over the cells of a grid depending on the conductivity. Fortunately the dependence is weak. Thus one can efficiently estimate conductivity averages by using the grid for a reference conductivity. This simple inversion method converges to the true solution as the number of measurements increases. We analyze the sensitivity of the reduced model parameters to small changes in the conductivity, and introduce a Newton-type iteration to improve the reconstructions of the simple inversion method. As an added bonus, our method can benefit from a priori information if available. We generalize both methods to the 2D EIT problem by considering finite volumes discretizations of size determined by the measurement precision, but where the node locations are to be determined adaptively. This discretization can be viewed as a resistor network, where the resistors are essentially averages of the conductivity over grid cells. We show that the model reduction problem of finding the smallest resistor network (of fixed topology) that can predict meaningful measurements of the Dirichlet-to-Neumann map is uniquely solvable for a broad class of measurements. We propose a simple inversion method that, as in the simple method for the inverse Sturm-Liouville problem, is based on an interpretation of the resistors as conductivity averages over grid cells, and an iterative method that improves such reconstructions by using sensitivity information on the changes in the resistors due to small changes in the conductivity. A priori information can also be incorporated to the latter method.
Resistor Network Approaches to the Numerical Solution of Electrical Impedance Tomography with Partial Boundary Measurements
(2009) Mamonov, Alexander Vasilyevic; Borcea, Liliana
Electrical Impedance Tomography (EIT) problem consists of finding the electric conductivity inside a conductive body from simultaneous measurements of electric potential and current at its boundary. The EIT problem with partial boundary measurements corresponds to the case, where only certain portions of the boundary are available for measuring the potentials and currents. A deep connection exists between the EIT problem in continuum and the discrete inverse problem of recovering the conductances of resistors in a resistor network. The connection comes in the form of special finite volume discretizations on carefully chosen grids, known as optimal grids. The optimal grids allows us to use the existing theory of inverse problems for resistor networks to compute an approximation to the solution of the continuum EIT problem. This was done recently in the case of full boundary measurements. The main goal of our work is to generalize the existing results to the case of partial measurement settings. Two alternative approaches are presented: one is based on (quasi-) conformal mappings of a well-studied conductivity problem in a unit disk, and the other is based on a previously unstudied pyramidal graph topology. We establish the existence and uniqueness of the solutions of discrete problems for both settings. Then we use these discrete results to obtain numerical approximations to the solution of the continuum EIT problem with partial measurements.
Resistor networks and optimal grids for the numerical solution of electrical impedance tomography with partial boundary measurements
(2010) Mamonov, Alexander Vasilyevich; Borcea, Liliana
The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids, that are computed as part of the problem. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly using discrete Fourier transform and rational interpolation with continued fractions. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems This is the first study of the optimal grids for the case, where reduction to one dimension is not possible. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show several advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct both smooth and discontinuous conductivities including those of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods. Finally, our method does not require any form of artificial regularization via penalty terms. However, our method allows for such regularization to incorporate prior information in the solution.
Synthetic Aperture Radar Imaging and Motion Estimation via Robust Principal Component Analysis
(arXiv, 2012-08-22) Borcea, Liliana; Callaghan, Thomas; Papanicolaou, George
We consider the problem of synthetic aperture radar (SAR) imaging and motion estimation of complex scenes. By complex we mean scenes with multiple targets, stationary and in motion. We use the usual setup with one moving antenna emitting and receiving signals. We address two challenges: (1) the detection of moving targets in the complex scene and (2) the separation of the echoes from the stationary targets and those from the moving targets. Such separation allows high resolution imaging of the stationary scene and motion estimation with the echoes from the moving targets alone. We show that the robust principal component analysis (PCA) method which decomposes a matrix in two parts, one low rank and one sparse, can be used for motion detection and data separation. The matrix that is decomposed is the pulse and range compressed SAR data indexed by two discrete time variables: the slow time, which parametrizes the location of the antenna, and the fast time, which parametrizes the echoes received between successive emissions from the antenna. We present an analysis of the rank of the data matrix to motivate the use of the robust PCA method. We also show with numerical simulations that successful data separation with robust PCA requires proper data windowing. Results of motion estimation and imaging with the separated data are presented, as well.
Variationally Constrained Numerical Solution of Electrical Impedance Tomography
(2002-10) Borcea, Liliana; Gray, Genetha Anne; Zhang, Yin
We propose a novel, variational inversion methodology for the electrical impedance tomography problem, where we seek electrical conductivity σ inside a bounded, simply connected domain Ω, given simultaneous measurements of electric currents I and potentials V at the boundary. Explicitly, we make use of natural, variational constraints on the space of admissible functions σ, to obtain efficient reconstruction methods which make best use of the data. We give a detailed analysis of the variational constraints, we propose a variety of reconstruction algorithms and we discuss their advantages and disadvantages. We also assess the performance of our algorithms through numerical simulations and comparisons with other, well established, numerical reconstruction methods.
Wave propagation in randomly layered media with an application to time reversal
(2005) Gonzalez del Cueto, Fernando; Borcea, Liliana
We describe the propagation of acoustic waves through randomly layered media over distances much larger than the typical wavelength of a pulse that is emitted from a point source. The layered medium is modeled by a smooth reference background modulated by fast random small-scale variations. Using asymptotic methods, we arrive to the O'Doherty-Anstey (ODA) formula which describes the coherent part of the pulse in a deterministic expression up to a small random time correction. An application on time-reversal is presented, where a pulse is sent through the medium, recorded in a small window, time-reversed, and then sent back towards the source. The striking phenomenon of enhanced refocusing occurs, where the randomness in the medium actually improves the spatial refocusing around the initial source.

Browsing by Author "Borcea, Liliana"

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