Alexander V. Mamonov

Automated Polyp Detection in Colon Capsule Endoscopy

Colorectal polyps are important precursors to colon cancer, a major health problem. Colon capsule endoscopy is a safe and minimally invasive examination procedure, in which the images of the intestine are obtained via digital cameras on board of a small capsule ingested by a patient. The video sequence is then analyzed for the presence of polyps. We propose an algorithm that relieves the labor of a human operator analyzing the frames in the video sequence. The algorithm acts as a binary classifier, which labels the frame as either containing polyps or not, based on the geometrical analysis and the texture content of the frame.We assume that the polyps are characterized as protrusions that are mostly round in shape. Thus, a best fit ball radius is used as a decision parameter of the classifier. We present a statistical performance evaluation of our approach on a data set containing over 18 900 frames from the endoscopic video sequences of five adult patients. The algorithm achieves 47% sensitivity per frame and 81% sensitivity per polyp at a specificity level of 90%. On average, with a video sequence length of 3747 frames, only 367 false positive frames need to be inspected by an operator.

Quantitative photoacoustic imaging in the radiative transport regime

The objective of quantitative photoacoustic tomography (QPAT) is to reconstruct optical and thermodynamic properties of heterogeneous media from data of absorbed energy distribution inside the media. There have been extensive theoretical and computational studies on the inverse problem in QPAT, however, mostly in the diffusive regime. We present in this work some numerical reconstruction algorithms for multi-source QPAT in the radiative transport regime with energy data collected at either single or multiple wavelengths. We show that when the medium to be probed is non-scattering, explicit reconstruction schemes can be derived to reconstruct the absorption and the Gruneisen coefficients. When data at multiple wavelengths are utilized, we can reconstruct simultaneously the absorption, scattering and Gruneisen coefficients. We show by numerical simulations that the reconstructions are stable.

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

We introduce an inversion algorithm for electrical impedance tomography (EIT) with partial boundary measurements in two dimensions. It gives stable and fast reconstructions using sparse parameterizations of the unknown conductivity on optimal grids that are computed as part of the inversion. We follow the approach in Borcea et al (2008 Inverse Problems 24 035013) and Vasquez (2006 PhD thesis Rice University, Houston, TX, USA) that connects inverse discrete problems for resistor networks to continuum EIT problems, using optimal grids. The algorithm in Borcea et al (2008 Inverse Problems 24 035013) and Vasquez (2006 PhD Thesis Rice University, Houston, TX, USA) is based on circular resistor networks, and solves the EIT problem with full boundary measurements. It is extended in Borcea et al (2010 Inverse Problems 26 045010) to EIT with partial boundary measurements, using extremal quasi-conformal mappings that transform the problem to one with full boundary measurements. Here we introduce a different class of optimal grids, based on resistor networks with pyramidal topology, that is better suited for the partial measurements setup. We prove the unique solvability of the discrete inverse problem for these networks and develop an algorithm for finding them from the measurements of the Dirichlet to Neumann map. Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in Borcea et al (2010).

A model reduction approach to numerical inversion for a parabolic partial differential equation

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magneticfield. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss‐Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

Point source identification in nonlinear advection–diffusion–reaction systems

We consider a problem of identification of point sources in time-dependent advection–diffusion systems with a nonlinear reaction term. The linear counterpart of the problem in question can be reduced to solving a system of nonlinear algebraic equations via the use of adjoint equations. We extend this approach by constructing an algorithm that solves the problem iteratively to account for the nonlinearity of the reaction term. We study the question of improving the quality of source identification by adding more measurements adaptively using the solution obtained previously with a smaller number of measurements.

Direct, Nonlinear Inversion Algorithm for Hyperbolic Problems via Projection-Based Model Reduction

We estimate the wave speed in the acoustic wave equation from boundary measurements by constructing a reduced-order model (ROM) matching discrete time-domain data. The state-variable representation of the ROM can be equivalently viewed as a Galerkin projection onto the Krylov subspace spanned by the snapshots of the time-domain solution. The success of our algorithm hinges on the data-driven Gram--Schmidt orthogonalization of the snapshots that suppresses multiple reflections and can be viewed as a discrete form of the Marchenko--Gelfand--Levitan--Krein algorithm. In particular, the orthogonalized snapshots are localized functions, the (squared) norms of which are essentially weighted averages of the wave speed. The centers of mass of the squared orthogonalized snapshots provide us with the grid on which we reconstruct the velocity. This grid is weakly dependent on the wave speed in traveltime coordinates, so the grid points may be approximated by the centers of mass of the analogous set of squared orthogonalized snapshots generated by a known reference velocity. We present results of inversion experiments for one- and two-dimensional synthetic models.

Nonlinear Seismic Imaging via Reduced Order Model Backprojection

We introduce a novel nonlinear seismic imaging method based on model order reduction. The reduced order model (ROM) is an orthogonal projection of the wave equation propagator operator on the subspace of the snapshots of the solutions of the wave equation. It can be computed entirely from the knowledge of the measured time domain seismic data. The image is a backprojection of the ROM using the subspace basis for the known smooth kinematic velocity model. The implicit orthogonalization of solution snapshots is a nonlinear procedure that differentiates our approach from the conventional linear methods (Kirchhoff, RTM). It allows for the removal of multiple reflection artifacts. It also enables us to estimate the magnitude of the reflectors similarly to the true amplitude migration algorithms.

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

We propose a discrete approach for solving an inverse problem for the two-dimensional Schrodinger equation, where the unknown potential is to be determined from the Dirichlet to Neumann map. In the continuum, the problem for absorptive potentials can be transformed with the Liouville identity into a conductivity inverse problem. Its discrete analogue is to find a resistor network matching the measurements, and is well understood. Here we use a discrete Liouville identity to transform its solution to that of Schrodingers problem. The discrete Schrodinger potential given by the discrete Liouville identity can be used to reconstruct the potential in the continuum in two ways. First, we can obtain a direct but coarse reconstruction by interpreting the values of the discrete Schrodinger potential as averages of the continuum Schrodinger potential on a special sensitivity grid. Second, the discrete Schrodinger potential may be used to reformulate the conventional nonlinear output least squares formulation of the inverse Schrodinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between discrete Schrodinger potentials. This results in a better behaved optimization problem converging in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

Study of noise effects in electrical impedance tomography with resistor networks

We present a study of the numerical solution of the two dimensional nelectrical impedance tomography problem, with noisy measurements of nthe Dirichlet to Neumann map. The inversion uses parametrizations of nthe conductivity on optimal grids. The grids are optimal in the sense nthat finite volume discretizations on them give spectrally accurate napproximations of the Dirichlet to Neumann map. The approximations are nDirichlet to Neumann maps of special resistor networks, that are nuniquely recoverable from the measurements. Inversion on optimal ngrids has been proposed and analyzed recently, but the study of noise neffects on the inversion has not been carried out. In this paper we npresent a numerical study of both the linearized and the nonlinear ninverse problem. We take three different parametrizations of the nunknown conductivity, with the same number of degrees of freedom. We nobtain that the parametrization induced by the inversion on optimal ngrids is the most efficient of the three, because it gives the nsmallest standard deviation of the maximum a posteriori estimates of nthe conductivity, uniformly in the domain. For the nonlinear problem nwe compute the mean and variance of the maximum a posteriori estimates nof the conductivity, on optimal grids. For small noise, we obtain that nthe estimates are unbiased and their variance is very close to the noptimal one, given by the Cramer-Rao bound. For larger noise we nuse regularization and quantify the trade-off between reducing the nvariance and introducing bias in the solution. Both the full and npartial measurement setups are considered.

A Nonlinear Method for Imaging with Acoustic Waves Via Reduced Order Model Backprojection

We introduce a novel nonlinear imaging method for the acoustic wave equation based on model order reduction. The objective is to image the discontinuities of the acoustic velocity, a coefficient of the scalar wave equation from the discretely sampled time domain data measured at an array of transducers that can act as both sources and receivers. We treat the wave equation along with transducer functionals as a dynamical system. A reduced order model (ROM) for the propagator of such system can be computed so that it interpolates exactly the measured time domain data. The resulting ROM is an orthogonal projection of the propagator on the subspace of the snapshots of solutions of the acoustic wave equation. While the wavefield snapshots are unknown, the projection ROM can be computed entirely from the measured data. The image is obtained by backprojecting the ROM. Since the basis functions for the projection subspace are not known, we replace them with the ones computed for a known smooth kinematic velocity model. A crucial step of ROM construction is an implicit orthogonalization of solution snapshots. It is a nonlinear procedure that differentiates our approach from the conventional linear imaging methods (Kirchhoff migration and reverse time migration - RTM). It resolves all the dynamical behavior captured by the data, so the error from the imperfect knowledge of the velocity model is purely kinematic. This allows for almost complete removal of multiple reflection artifacts, while simultaneously improving the resolution in the range direction compared to conventional RTM.