Alireza Aghasi

Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results in a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization, and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the way for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which, used in the proposed manner, provide flexibility in presenting a larger class of shapes with fewer terms. Also they provide a “narrow-banding” advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography, and diffuse optical tomography.

A geometric approach to joint inversion with applications to contaminant source zone characterization

This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inverse problems, the main motivation here is to characterize subsurface contaminant source zones by processing down-gradient hydrological data and cross-gradient electrical resistance tomography observations. Inspired by Newtons method for multi-objective optimization, we present an iterative inversion scheme in which descent steps are chosen to simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixel-based inversion over the whole domain. Instead, we employ a parametric level set technique that substantially reduces the number of underlying parameters, making the inversion computationally tractable. The performance of the technique is examined and discussed through the reconstruction of source zone architectures that are representative of dense non-aqueous phase liquid (DNAPL) contaminant release in a statistically homogenous sandy aquifer. In these examples, the geometric configuration of the DNAPL mass is considered along with additional information about its spatial variability within the contaminated zone, such as the identification of low and high saturation regions. Comparison of the reconstructions with the true DNAPL architectures highlights the superior performance of the model-based technique and joint inversion scheme.

Convex Cardinal Shape Composition

We propose a new shape-based modeling technique for applications in imaging problems. Given a collection of shape priors (a shape dictionary), we define our problem as choosing the right dictionary elements and geometrically composing them through basic set operations to characterize desired regions in an image. This is a combinatorial problem solving which requires an exhaustive search among a large number of possibilities. We propose a convex relaxation to the problem to make it computationally tractable. We take some major steps towards the analysis of the proposed convex program and characterizing its minimizers. Applications vary from shape-based characterization, object tracking, optical character recognition, and shape recovery in occlusion, to other disciplines such as the geometric packing problem.

Sensitivity Calculations for Poisson's Equation via the Adjoint Field Method

Adjoint field methods are both elegant and efficient for calculating sensitivity information required across a wide range of physics-based inverse problems. In this letter, we provide a unified approach to the derivation of such methods for problems whose physics are provided by Poissons equation. Unlike existing approaches in the literature, we consider in detail and explicitly the role of general boundary conditions in the derivation of the associated adjoint-field-based sensitivities. We highlight the relationship between the adjoint field computations required for both gradient decent and Gauss-Newton approaches to image formation. Our derivation is based on standard results from vector calculus coupled with transparent manipulation of the underlying partial different equations, thereby making the concepts employed in this letter easily adaptable to other systems of interest.

Flat-Top Footprint Pattern Synthesis Through the Design of Arbitrary Planar-Shaped Apertures

The problem of generating a flat-top main beam with an arbitrary footprint for array elements placed in an arbitrary planar aperture is considered in this paper. Some simplifying properties of the Bessel functions, encourages the general framework of the paper to encompass patterns produced by circular aperture and eventually generalize it to arbitrary aperture geometries. In this regard two synthesis methods are presented. The first method is based on the use of the Rayleigh quotient to obtain constant phase array patterns, hence, a class of generally linear phase patterns can be considered. The second approach is based on power pattern synthesis where there is no restriction on the phase of the pattern, hence, it provides us with greater flexibility. The nonlinear problem is appropriately modeled and formulated for amenable performance. These two new methods can exhibit a significant reduction in the number of unknown parameters, and high flexibility in shaping the desired main beam by arbitrary lattice geometry.

Environmental Remediation and Restoration: Hydrological and Geophysical Processing Methods

Remediation and restoration of sites contaminated by hazardous organic chemicals has become an increasingly critical problem as the dissolution and transport of these compounds by groundwater threatens many aquifers providing drinking water to populations worldwide. The characterization of these sites prior to remediation and the monitoring of the restoration progress rely on the processing of data provided by geophysical sensing technology and direct groundwater measurements collected over sparsely distributed wells. Many of the tools and techniques developed within the context of geophysical signal processing (GSP) have a role to play in this domain. In this article, we discuss environmental restoration and remediation (ERR) problems and review the state of the art in the modeling and inversion of geoelectrical and groundwater concentration data, including a discussion of recent work in the area of joint inversion. While methods such as electrical impedance (EI) tomography are well known within the GSP community, the models and inverse methods associated with subsurface flow and transport of contaminants represent a relatively new area. We conclude with an overview of opportunities for collaborative activities among signal processors, geophysicists, and environmental engineers.

Flat-top power patterns of arbitrary footprint produced by arrays of arbitrary planar geometry

This paper presents a novel technique for synthesis of flat-top power patterns with desired footprints, generated by array elements positioned on an arbitrary planar geometry. Traditionally, this problem had been considered as a protracted complex nonlinear optimization problem specially when the number of array elements is large for obtaining a desired and detailed pattern. Using the formulation of patterns generated by circular apertures the problem is converted to a collocation problem which not only reduces the number of unknowns but also generalizes the method to any arbitrary planar aperture. Derivation of the closed form formulae specially for the the nonlinear problem simplifies implementation of the method.

Synthesis of planar arrays with arbitrary geometry for flat-top footprint patterns

This paper presents a new synthesis algorithm to produce flat-top main beams with arbitrary footprint for array elements placed in an arbitrary planar geometry. The general framework of the paper would encompass the patterns produced by circular aperture distribution. It is shown that for patterns generated by circular apertures some efficient simplifying facts based on properties of Bessel functions can be applied. The method shows a high performance in generating optimally flat-top patterns with detailed geometry footprints.

High-Order Regularized Regression in Electrical Impedance Tomography

We present a novel approach for the inverse problem in electrical impedance tomography based on regularized quadratic regression. Our contribution introduces a new formulation for the forward model in the form of a nonlinear integral transform, that maps changes in the electrical properties of a domain to their respective variations in boundary data. Using perturbation theory the transform is approximated to yield a high-order misfit unction which is then used to derive a regularized inverse problem. In particular, we consider the nonlinear problem to second-order accuracy, hence our approximation method improves upon the local linearization of the forward mapping. The inverse problem is approached using Newtons iterative algorithm and results from simulated experiments are presented. With a moderate increase in computational complexity, the method yields superior results compared to those of regularized linear regression and can be implemented to address the nonlinear inverse problem.

Extracting the Principal Shape Components via Convex Programming

We present a general method for extracting a region from an image (or 3D object) that can be expressed, or approximated, by taking unions and set differences from a collection of template shapes in a dictionary. We build on recent work that shows how this geometric problem can be recast in the language of linear algebra, with set operations on shapes translated into linear combinations of vectors, and solved using convex programming. This paper presents a set of sufficient conditions for which this convex program returns the “correct” shape. These conditions are robust in that they can account for the shapes that have indistinct boundaries, or model mismatch between the shapes in the dictionary and the target region in the image. We also present two different methods for solving the convex extraction program. The first method simply recasts the problem as a linear program, while the second uses the alternating direction method of multipliers with a series of easily computed proximal operators. We present a number of numerical experiments that use the framework to perform image segmentation, optical character recognition, and find multi-resolution geometrical descriptions of 3D objects.