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Abstracts and Talk Materials
Imaging from Wave Propagation
October 17-21, 2005


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Gaik Ambartsoumian (Texas A & M University)

Image Reconstruction in Thermoacoustic Tomography

Thermoacoustic tomography (TCT or TAT) is a new and promising method of medical imaging. It is based on a so-called hybrid imaging technique, where the input and output signals have different physical nature. In TCT a radiofrequency (RF) electromagnetic pulse is sent through the biological object triggering an acoustic wave measured on the edge of that object. The obtained data is then used to recover the RF absorption function.

The poster addresses several problems of image reconstruction in thermoacoustic tomography. The presented results include injectivity properties of the related spherical Radon transform, its range description, reconstruction formulas and their implementation as well as some other results.


Michael Andre (University of California - San Diego)

Ultrasound Breast Tomography with Full-Wave Non-Linear Inverse Scattering

The fundamentals of medical ultrasound imaging have not changed since its inception 60 or more years ago. 180-degree pulse-echo backscatter is used for image formation without accounting for refraction, diffraction, multiple scattering, etc. Various forms of ultrasound computed tomography that incorporate the transmitted wave component have been proposed and investigated for many years with mixed success. These methods apply approximations of inverse scattering tomography: time-of-flight, Born, Rytov, Diffraction Tomography, etc. Techniscan Medical Systems (Salt Lake City, Utah) and the University of California, San Diego are beginning pre-clinical evaluation of a new system for breast imaging that applies novel inverse scattering methods to provide a unique method for calculating ultrasound characteristics of speed and attenuation of sound traveling through human tissue. We have developed an efficient inversion method for the coefficients of the partial differential equation that governs wave propagation in the human breast. The procedure is based on nonlinear minimization, fast computation of the forward problem and analytic computational formulas for actions of the Jacobian of the forward operator and its Hermitian adjoint. The goal of the development is to provide quantitative, high-resolution two- and three-dimensional ultrasonic imaging combined with unique information about tissue properties at sub millimeter resolution in an effort to improve diagnosis of breast cancer. Details of the imaging system design and the inversion method will be summarized. Sample images from human subjects and preliminary results in 26 patients with known breast masses will be presented.


D. Gregory Arnold (Air Force Research Laboratory)

VISUAL-D: Backhoe challenge problem

AFRL/SNA has several data sets and challenge problems for imaging. Specifically, I will highlight the data that was recently made available at IMA. This includes a radar "Backhoe Data Dome," adaptive learning of MSTAR data, a 3-D Ladar ATR Challenge problem, Video EO and IR data for feature-aided tracking, and Thru-wall data. All of these data sets have been publicly released and are also available from https://www.sdms.afrl.af.mil/main.php.


Paul E. Barbone (Boston University)

Progress in Quantitative Biomechanical Imaging

Joint work with Michael S. Richards, Nachiket H. Gokhale, Carlos Rivas Aroni, Ricardo Leiderman, Jeffrey C. Bamber, and Assad A. Oberai.

It is widely recognized that tissue pathologies often change biomechanical properties. For instance, neoplastic tissue is typically highly vascularized, contains abnormal concentrations of extracellular proteins (i.e. collagen, proteoglycans) and has a high interstitial fluid pressure compared to most normal tissues. These differences in tissue microstructure effectively change a tissues response to mechanical stimuli. Our work focuses on noninvasively measuring and thereby imaging in vivo distributions of the biomechanical properties of soft tissues. The intended short term application of our work is the detection and diagnosis of breast cancer and other soft tissue pathologies. Our efforts include the development and computational implementation of mathematical models to describe soft tissue behavior, developing novel ultrasound techniques to accurately measure vector displacements of tissue deformation, the analysis of inverse problems associated with quantitative inference of material properties from measured displacements, and development of algorithms to solve those inverse problems. We present a combined ultrasound and image registration technique to quantitatively measure tissue response to mechanical manipulation. We further present several different mathematical models describing tissue responses for different experimental stimuli. Some of these models are motivated by microstructural considerations. Where possible, these model parameters are compared to values determined by independent mechanical testing.


Bob Bonneau (Air Force Research Laboratory)

A waveform strategy for detection of targets in multiplicative clutter

Classical detection theory for sensing relies on fixed target illumination and independent identically distributed noise for target and clutter characterization. Unfortunately in many cases such as sensing and communications in urban or atmospheric scenarios, we encounter much more complex clutter and target conditions due to scattering from multiple sources. As a result we must envision a new sensing scheme to combat noise not handled by classical sensing methods. We will therefore develop framework whereby we can use the waveform to manage the physical scattering process such that our return statistics conform to the classical detection assumption such as idependent identically distributed data. We will first describe how to use a well known set of waveforms in the context of a given aperture and how these waveforms propagate when we consider the wideband multifrequency nature of the waveform. We will then show how the scattering process and the model for the interaction of the waveform with the target and the environment model occurs. Finally we will show the propagation process of the returned waveform and how we can use our waveform selection to manage the data statistics of the target and noise. Finally we will conclude with a detection example of this process.


Liliana Borcea (Rice University)

Theoretical and computational aspects of statistically stable adaptive coherent interferometric imaging in random media

jointly with George Papanicolaou (Stanford) and Chrysoula Tsogka (U. Chicago)

I will discuss a robust, coherent interferometric approach for array imaging in cluttered media, in regimes with significant multipathing of the waves by the inhomogeneities in clutter. In such scattering regimes, the recorded traces at the array have long and noisy codas and classic imaging methods give unstable results. Coherent interferometry is essentially a very efficient statistical smoothing technique that exploits systematically the spatial and temporal coherence in the data to obtain stable images.

I will describe in some detail the resolution of this method for two types of cluttered media: (1) isotropic, weakly scattering clutters, where waves are scattered mostly forward and (2) layered, strongly fluctuating clutters, where back scattering is strong. I will show that in spite of such opposite wave scattering regimes, coherent interferometry behaves equally well, which indicates its wide applicability.

In coherent interferometry, there is a delicate balance between having stable and sharp images and achieving the optimal resolution depends on our knowledge of the clutter dependent spatial and temporal decoherence parameters. I will explain briefly how we can estimate these parameters efficiently during the image formation process, as we do in adaptive coherent interferometry.


Brett Borden (Naval Postgraduate School)

Microwave imaging of airborne targets

An important problem (perhaps the most important problem to the Department of Defense) in modern remote sensing is that of correctly identifying potential targets at great distances and in all kind of weather. Because of their ability to see through clouds and in the absence of ambient radiation, active radar systems are usually required for this task. The practical differences between ground and airborne targets allow the airborne case to focus more on actual imaging (as opposed to clutter rejection) and we will review the current methods within this simpler context. Open problems will be discussed.

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Mujdat Cetin (Massachusetts Institute of Technology)

Sparsity-Driven Feature-Enhanced Imaging

same abstract as the regular talk


Mujdat Cetin (Massachusetts Institute of Technology)

Sparsity-driven feature-enhanced imaging

We present some of our recent work on coherent image reconstruction. The primary application that has driven this work has been synthetic aperture radar, although we have extended our approach to other modalities such as ultrasound imaging as well. One of the motivations for our work has been the increased interest in using reconstructed images in automated decision-making tasks. The success of such tasks (e.g. target recognition in the case of radar) depends on how well the computed images exhibit certain features of the underlying scene. Traditional coherent image formation techniques have no explicit means to enhance features (e.g. scatterer locations, object boundaries) that may be useful for automatic interpretation. Another motivation has been the emergence of a number of applications where the scene is observed through a sparse aperture. Examples include wide-angle imaging with unmanned air vehicles (UAVs), foliage penetration radar, bistatic imaging, and passive radar imaging. When traditional image formation techniques are applied to these sparse aperture imaging problems, they often yield high sidelobes and other artifacts that make the image difficult to interpret. We have developed a mathematical foundation and associated algorithms for model-based, feature-enhanced imaging to address these challenges. Our framework is based on a regularized reconstruction of the scattering field, which combines an explicit mathematical model of the data collection process with non-quadratic functionals representing prior information about the nature of the features of interest. In particular, the prior information we exploit is that the underlying signals exhibit some form of sparsity. We solve the challenging optimization problems posed in our framework by computationally efficient numerical algorithms that we have developed. The resulting images offer improvements over conventional images in terms of visual and automatic interpretation of the underlying scenes. We also discuss a number of open research avenues inspired by this work.


Herve Chauris (Ecole des Mines de Paris)

Seismic velocity analysis: in time or depth domain?

jointly with Gilles Lambare (Ecole des Mines de Paris)

Seismic velocity analysis is a crucial step needed to obtain consistent images of the subsurface. Several new methods appeared in the last 10 years, among them Slope Tomography and Differential Semblance Optimization. We want to discuss here the link between these a priori different methods.

Slope Tomography is formulated in the prestack unmigrated time domain and uses not only time information picked on seismic gathers, but also associated slopes that better constrain the inversion scheme. On the other side, Differential Semblance Optimization is formulated in the depth migrated domain where adjacent images are compared to obtain a final consistent image of the subsurface.

We analyse these two types of methods to show that they are in fact equivalent from a theoretical point of view despite the different formulation.


Giulio Ciraolo (Università di Firenze)

Wave propagation in optical waveguides with imperfections

The problem of electromagnetic wave propagation in a 2-D infinite optical waveguide will be presented. We give a description on how to construct a solution to the electromagnetic wave propagation problem in a 2-D and 3-D rectilinear optical waveguide. Numerical simulations will also be shown. Furthermore, in the 2-D case, we will present a mathematical framework which allows us to study waveguides with imperfections. In this case, some numerical result concerning the far field of the solution and the coupling between guided modes will be shown.


Shakti K. Davis (University of Wisconsin - Madison)

Ultrawideband microwave breast cancer detection: beamforming for 3-D MRI-derived numerical phantoms

Microwave imaging has the potential to be a highly sensitive modality for breast cancer detection due to the dielectric-properties contrast that exists between malignant and normal breast tissue at microwave frequencies. One microwave imaging approach is to transmit ultrawideband (UWB) microwave pulses into the breast, record the scattered fields, and use radar methods such as beamforming to detect and localize significant scatterers such as tumors. We previously proposed a beamforming technique and demonstrated its accuracy and robustness for tumor detection using 2-D MRI-derived numerical breast phantoms (Davis, et. al, JEMWA, 17(2):357-381, 2003) and simple 3-D physical phantoms (Li, et. al, IEEE T-MTT, 52(8):1856-1865, 2004). In this poster we extend our investigation to 3-D MRI-derived numerical breast phantoms. These anatomically realistic breast phantoms represent a prone patient with an antenna array surrounding the breast. Small (< 1 cm) tumors are added by elevating the dielectric properties in a region to represent a specified malignant-to-normal tissue contrast. We solve for backscattered fields at each antenna position using the FDTD-method and construct a 3-D image of scattered energy in the breast using our beamforming technique. The resulting images exhibit localized high-energy peaks within a few mm of the true tumor locations as expected. This work represents our first successful demonstration of detecting and localizing very small tumors in 3-D MRI-derived numerical breast models.


Maarten De Hoop (Purdue University)

Analysis of 'wave-equation' imaging of reflection seismic data with curvelets

in collaboration with Gunther Uhlmann and Hart Smith

In reflection seismology one places sources and receivers on the Earth's surface. The source generates waves in the subsurface that are reflected where the medium properties vary discontinuously; these reflections are observed in all the receivers. The data thus obtained are commonly modeled by a scattering operator in a single scattering approximation: the linearization is carried out about a smooth background medium, while the scattering operator maps the (singular) medium contrast to the scattered field observation. In seismic imaging, upon applying the adjoint of the scattering operator, the data are mapped to an image of the medium contrast.

We discuss how multiresolution analysis can be exploited in representing the process of `wave-equation' seismic imaging. The frame that appears naturally in this context is the one formed by curvelets. The implied multiresolution analysis yields a full-wave description of the underlying seismic inverse scattering problem on the one the hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The analysis presented here relies on the factorization of the seismic imaging process into Fourier integral operators associated with canonical transformations.

The approach and analysis presented in this talk aids in the understanding of the notion of scale in the data and how it is coupled through imaging to scale in - and regularity of - the background medium. In this framework, background media of limited smoothness can be accounted for. From a computational perspective, the analysis presented here suggests an approach that requires solving for the geometry on the one hand and solving a matrix Volterra integral equation on the other hand. The Volterra equation can be solved by recursion - as in the computation of certain multiple scattering series; this process reveals the curvelet-curvelet interaction in seismic imaging. The extent of this interaction can be estimated, and is dependent on the Hölder class of the background medium.


Anthony J. Devaney (Northeastern University)

Wave Based Coherent 3D Microscopy

A new form of coherent optical microscopy is described that generates images (reconstructions) of two and three-dimensional, penetrable scattering objects computationally from sets of measured digital holograms of scattered field data collected in a suite of scattering experiments. The microscope uses the technique of "phase shifting holography" (explained in the talk) to compute both the amplitude and phase of the coherently scattered field data which is then input into wave based inversion algorithms to generate the image of the object. For 2D (thin) objects the inversion algorithm is based on a coherent back propagation of field data collected in single experiment and, unlike conventional microscopy, yields both the amplitude as well as phase of the 2D sample. For thick (3D) objects the object is embedded in an index matching fluid and a suite of scattering experiments is required using dirent incident illuminating waves. In this case the object reconstruction is generated using a "generalized" filtered back propagation (FBP) algorithm that allows for multiple scattering between the object and index matching bath. The generalized FBP algorithm is described and examples using simulated and real data are presented.


Eric Dussaud (Rice University)

Velocity Analysis in the Presence of Uncertainty

Velocity Analysis resolves relatively long scales of earth structure, typically wavelengths larger than 500m. Migration produces images with length scales (wavelengths) on the order of 10's of m. In between these two scale regimes lies another, corresponding roughly to structures between 60 to 300m in extent, in which the resolution of velocity analysis is uncertain and the energy of images is small to non-existent. This work aims at assessing the impact on velocity analysis of uncertainty at these intermediate length scales, using ideas on time reversal and imaging in randomly inhomogeneous media developed by Papanicolaou and colleagues, in combination with velocity estimation methods of differential semblance type.


David A. Garren (Science Applications International Corp. (SAIC))

Image Preconditioning for a SAR Image Reconstruction Algorithm for Multipath Scattering

Recent analysis has resulted in an innovative technique for forming synthetic aperture radar (SAR) images without the multipath ghost artifacts that arise in traditional methods. This technique separates direct-scatter echoes in an image from echoes that are the result of multipath, and then maps each set of reflections to a metrically correct image space. Current processing schemes place the multipath echoes at incorrect (i.e., ghost) locations due to fundamental assumptions implicit in conventional array processing. Two desired results are achieved by use of this Image Reconstruction Algorithm for Multipath Scattering (IRAMS). First, the intensities of the ghost returns are reduced in the primary image space, thereby improving the relationship between the image pattern and the physical distribution of the scatterers. Second, a higher dimensional image space that enhances the intensities of the multipath echoes is created which offers the potential of dramatically improving target detection and identification capabilities. This paper develops techniques in order to precondition the input images at each level and each offset in the IRAMS architecture in order to reduce multipath false alarms.


James F. Greenleaf (Mayo Clinic /Foundation) http://www.mayo.edu/ultrasound/

Estimating mechanical tissue properties with vibro-acoustography and vibrometry

Detecting pathology using the "stiffness" of the tissue is more that 2000 years old. Even today it is common for surgeons to feel lesions during surgery that have been missed by advanced imaging methods. Palpation is subjective and limited to individual experience and to the accessibility of the tissue region to touch. It appears that a means of noninvasively imaging elastic modulus (the ratio of applied stress to strain) may be useful to distinguish tissues and pathologic processes based on mechanical properties such as elastic modulus. The approaches to date have been to use conventional imaging methods to measure the mechanical response of tissue to mechanical stress. Static, quasi-static or cyclic stresses have been applied. The resulting strains have been measured using ultrasound or MRI and the related elastic modulus has been computed from viscoelastic models of tissue mechanics. Recently we have developed a new ultrasound technique that produces speckle free images related to both tissue stiffness and reflectivity. This method, termed "Ultrasound Stimulated Vibro-acoustography" (Science 280:82-85, April 3, 1998; Proc Natl Acad Sci USA 96:6603-6608, June 1999), uses ultrasound radiation pressure to produce sound vibrations from a small region of the tissue that depend in part on the elastic characteristics of the tissue. The method can detect micro-calcification within breasts, and promises to provide high quality images of calcification within arteries. In addition, vibro-acoustography can detect mechanical defects in certain prostheses such as artificial mitral and aortic valves. Extensions of the method include vibrometry, in which motion of an object is detected with laser vibrometry or an accelerometer, and shear wave detection, in which the resulting shear waves within objects such as arteries or tissue are detected with Doppler or MRI.

Keywords : vibrometry, stiffness, ultrasound, acoustic, shear waves


Murthy N. Guddati (North Carolina State University)

Towards Effective Seismic Imaging in Anisotropic Elastic Media

[joint work with A.H. Heidari]

A critical ingredient in high-frequency imaging is the migration operator that back-propagates the surface response to the hidden reflectors. Migration is often performed using one-way wave equations (OWWEs) that allow wave propagation in a preferred direction while suppressing the propagation in the opposite direction. OWWEs are typically obtained by approximating the factorized full-wave equation; this process is well-developed for the acoustic wave equation, but not for elastic wave equations, especially when the material is anisotropic. Furthermore, existing elastic OWWEs are computationally expensive. For these reasons, in spite of the existence of strongly coupled elastic waves, seismic migration is performed routinely using acoustic OWWEs, naturally resulting in significant errors in the image.

With the ultimate goal of developing accurate and efficient imaging algorithms for anisotropic elastic media, we develop new approximations of elastic OWWEs. Named the arbitrarily wide-angle wave equations (AWWEs), these approximations appear to be effective for isotropic as well as anisotropic media. The implementation of AWWE-migration in isotropic (heterogeneous) elastic media is complete, while further work remains to be done to incorporate the effects of anisotropy. This poster outlines (a) the basic idea behind AWWEs, (b) the implementation of AWWE-migration along with some results, and (c) future challenges related to using AWWEs for imaging in anisotropic elastic media.


Thorsten Hohage (George August University Goettingen)

A Newton-Type Method for 3D Inverse Obstacle Scattering Problems

We consider the inverse problem to reconstruct the shape of an obstacle from measurements of scattered fields. The forward problem is solved by a wavelet boundary element method. For the inverse problem we use a preconditioned Newton method. Particular emphasis is put on the reconstruction of non star-shaped obstacles.


E. Mckay Hyde (Rice University)

Fast, High-Order Integral Equation Methods for Scattering by Inhomogeneous Media

Integral equation methods for the time-harmonic scattering problem are attractive since the radiation condition at infinity is automatically satisfied (no absorbing boundary condition is required), only the scattering obstacle itself needs to be discretized, and the integral operator is compact, leading to better conditioned linear systems than for differential operators. However, there has been limited success in developing integral equation methods which are both efficient and high-order accurate.

We will present recent work on integral equation methods that are both efficient (O(N log N) complexity) and high-order accurate in computing the time-harmonic scattering by inhomogeneous media. The efficiency of our methods relies on the use of fast Fourier transforms (FFTs) while the high-order accuracy results from systematic use of partitions of unity, regularizing changes of variables, and Fourier smoothing of the refractive index.


Olha Ivanyshyn (Graduieerten Kolleg)

Nonlinear Integral Equations in Inverse Obstacle Scattering

We present a novel solution method for inverse obstacle scattering problems for time-harmonic waves based on a pair of nonlinear and ill-posed integral equations for the unknown boundary that arises from the reciprocity gap principle. This integral equations can be solved by linearization, i.e., by regularized Newton iterations. We present a mathematical foundation of the method and illustrate its feasibility by numerical examples.


Michael Klibanov (University of North Carolina - Charlotte)

Uniqueness, stability and numerical methods for some inverse and ill-posed Cauchy problems

Some new results concerning global uniqueness theorems and stability estimates for coefficient inverse problems will be presented. In addition, the presentation will cover some new and previous results about the stability of the Cauchy problem for hyperbolic equations with the data at the lateral surface. This problem is almost equivalent with the inverse problem of determining initial conditions in hyperbolic equations. Therefore, stability estimates for this Cauchy problem actually imply refocusing of time reversed wave fields. Our recent numerical studies confirming this statement will be presented. In addition, a globally convergent algorithm for a class of coefficient inverse problems will be discussed. The main tool of all these studies is the method of Carleman estimates.


Patrick Lailly (Institut Francais du Petrole)

Towards 3D least-squares inversion of prestack seismic data

(work in collaboration with Y. Pion (IFP), Jerome Le Rousseau and Thierry Gallouet (Univ. Aix-Marseille 1)

Migration is the standard tool used in seismic imaging. It consists in applying to the data the adjoint of the linearized forward map. 3D least-squares inversion would consist in solving a huge linear system, a tremendous task at first glance. However physical intuition and numerical evidence (i. e. visualization of the Hessian) indicate that solving this linear system should not be that difficult. By doing so, we expect to improve the spatial resolution and to remove the acquisition footprint.


Patrick Lailly (Institut Francais du Petrole)

The insidious effects of fine-scale heterogeneity in reflection seismology

Joint work with Florence Delprat-Jannaud.

Geophysicists are quite aware of the important troubles that can be met when the seismic data are contaminated by multiple reflections. The situation they have in mind is the one where multiple reflections are generated by isolated interfaces associated with high impedance contrasts. We here study a more insidious effect of multiple scattering, namely the one associated with fine scale heterogeneity. Our numerical experiments show that the effect of such multiple scattering can be far from negligible. As a consequence, it can lead standard imaging techniques (based on high-frequency analysis for wave propagation) to complete failure. The parameters that control the importance of the phenomenon are the depth of the target and the heterogeneity of the overburden. The dynamic theory of homogenization, unfortunately available only in 1D, allows us to better understand the role of the seismic frequency band: the multiple scattering phenomenon is all the more important as we deal with high frequencies. This leads to an interesting consequence: we can take advantage of a super-resolution phenomenon; namely, in situations where multiple scattering is important, we can expect a higher resolution than the one given by the classical Rayleigh criterion.

References Delprat-Jannaud, F. and Lailly, P., 2004. The insidious effects of fine-scale heterogeneity in reflection seismology. Journal of Seismic Exploration, 13: 39-84. Bamberger, A., Chavent, B. and Lailly, P., 1979. About the stability of the inverse problem in the 1D wave equation, application to the interpretation of seismic profiles, Journal of Applied Mathematics and Optimization, 5: 1-47.


Jerome Le Rousseau (University Aix Marseille I)

Convergence of approximations of solutions to first-order pseudodifferential wave equations with products of Fourier integral operators

An approximation of the solution to a hyperbolic equation with a damping term is introduced. It is built as the composition of Fourier integral operators (FIO). We prove the convergence of this approximation in the sense of Sobolev norms as well as for the wavefront set of the solution. We apply the introduced method to numerically image seismic data.


Jerome Le Rousseau (University Aix Marseille I)

Convergence of products of Fourier integral operators to solutions to first-order pseudodifferential wave equations; Application to seismic imaging

An approximation of the solution to a hyperbolic equation with a damping term is introduced. It is built as the composition of Fourier integral operators (FIO). We prove the convergence of this approximation in the sense of Sobolev norms as well as for the wavefront set of the solution. We apply the introduced method to numerically image seismic data.


Hstau Liao (University of Minnesota)

Direct Reconstruction-Segmentation, as Motivated by Electron Microscopy

Quite often in electron microscopy it is desired to segment the reconstructed volumes of biological macromolecules, whose 3D structural inference is crucial for the understanding of biological functions. We propose approaches that directly produce a label (segmented) image from the tomograms (projections).

Knowing that there are only a finitely many possible labels and by postulating Gibbs priors on the underlying distribution of label images, it is possible to recover the unknown image from only a few noisy projections.


Hao Ling (University of Texas - Austin)

Multiple scattering and microDoppler effects in radar imaging and target recognition

Synthetic aperture radar (SAR) and inverse synthetic aperture radar (ISAR) systems have long been used by the radar community for imaging air, sea and ground targets. The standard radar imaging algorithms used in these systems are based on the single-scattering, point-scatterer model of the target. When the actual target scattering is well approximated by this simple model, the resulting high-resolution imagery reveals useful geometrical features of the target for classification and identification. However, sensor data collected from real targets often contain higher order effects. For instance, strong multiple scattering can occur in a real target with reentrant structures and inlet cavities. Further, a real target being imaged by a radar sensor is often engaged in dynamic maneuvers where the target does not remain a rigid body. Some examples include the flexing and vibration of the target frame and moving parts on the target such as scanning antennas, moving wheels and treads. These motions give rise to Doppler features after the standard radar processing and have been referred to as the microDoppler phenomenon. When these higher order effects are present, the resulting target imagery contains artifacts due to the mismatch between the imaging model and the actual data. More importantly, these features contain useful information about the motion of the moving components and the interior characteristics of the target, and should be better exploited for target recognition. In this talk, I will discuss our ongoing research in: (i) the extraction, understanding and modeling of these phenomena, and (ii) the exploitation of the resulting models to achieve better imaging and recognition performance.


Uttam K. Majumder (Air Force Research Laboratory)

Wideband Through-The-Wall Radar Imaging Experimentations

The Center for Advanced Communications (CAC) at Villanova University along with Air Force Research Laboratory (AFRL) has conducted several preliminary experimentations on through-the-wall imaging and collected real data on different settings behind the wall using a newly-integrated RF instrumentation suite. The full-polarization, 2D aperture data measurements are taken using an Agilent network analyzer, Model ENA 5071B, implementing a step frequency waveform over a 2-3 GHz frequency range. The imaging room is a typical computer lab that has been lined with radar absorbing material. Three different arrangements of the room's contents are considered: empty scene, calibration scene, and populated scene. The empty scene allows measurement of the noise/clutter background and supports coherent subtraction with the other two scenes. The calibration scene contains isolated reflectors that may be used to determine a fully-polarimetric radiometric calibration solution for the experimental system. The populated scene contains a number of common objects such as a phone, computer, tables, chair and filing cabinet and a jug of saline solution. Data was collected each scene with and without a wall. The wall is composed of plywood and gypsum board on a wood frame. The antennas are mounted on a 2D scanner that moves the antennas along and adjacent to the wall and is controlled by the network analyzer. Two additional antennas are fixed to the scanner frame and act as bistatic receivers.


Gary Margrave (University of Calgary)

A compactly supported aproximate wavefield extrapolator for seismic imaging

Seismic imaging in highly heterogeneous media requires an adaptive, robust, and efficient wavefield extrapolator. The homgeneous medium wavefield extrapolator has no spatial adaptivity but the locally homogeneous approximate extrapolator (LHA) is a highly accurate Fourier integral operator that adapts rapidly in space. Efficient application of either wavefield extrapolator is complicated by the fact that they have impulse responses that are not compactly supported, though they decay rapidly. Simple locaization methods, such as windowing, result in compactly supported approximations that are unstable in a recursive marching scheme. I present an analysis of this instability effect and a localization scheme that can design compactly supported approximate extrapolators that are sufficiently stable for hundreds of marching steps. I illustrate the method with seismic images from the Marmousi synthetic dataset.


Anna Mazzucato (Pennsylvania State University) http://www.math.psu.edu/mazzucat/, Lizabeth Rachele (Rensselaer Polytechnic Institute)

Unique determination of the travel time from dynamic boundary measurements in anisotropic elastic media

We microlocally decouple the system of equations for anisotropic elastodynamics (in 3 dimensions) following a result of M. Taylor. We then show that the dynamic Dirichlet-to-Neumann map uniquely determines the travel time through a bounded elastic body for any wave mode that has disjoint light cone. We apply this result to cases of transversely isotropic media with rays that are geodesics with respect to Riemannian metrics, and conclude that certain material parameters are uniquely determined up to diffeomorphisms that fix the boundary. We have shown that material parameters of general anisotropic elastic media may be uniquely determined by the Dirichlet-to-Neumann map only up to pullback by diffeomorphisms fixing the boundary.


Jodi L. Mead (Boise State University)

Regularization and Prior Error Distributions in Ill-posed Problems

We will examine the validity of parameter estimates in ill-posed problems when errors in data and initial parameter estimates are from normal and non-normal distributions. Given appropriate initial parameter estimates and the data error covariance matrix, the covariance matrix for errors in initial parameter estimates can be recovered and highly accurate parameter estimates can be found. This approach allows the regularization to be varied with each parameter.


Scott Morton (Amerada Hess Corporation)

Problems in Sub-salt Imaging due to Layered-Earth Assumptions

The standard approach to seismic imaging is rife with limitations due to the assumption that the earth is approximately a layered medium. Unfortunately much of the current petroleum exploration in the Gulf of Mexico is around or beneath salt bodies which have complex 3-D shapes. We illustrate several problems attributable to the layered-earth approach in the standard model building process, state-of-the-art imaging algorithms and available data interpretation tools used in sub-salt imaging.


Randy Moses (Ohio State University)

Model-Based Imaging and Feature Extraction for Synthetic Aperture Radar

We present recent work on imaging and reconstruction of objects from radar backscatter measurements taken over wide aspect angles. Radar backscattering is a function of several variables, including location, (complex-valued) amplitude, polarization, and the aspect (azimuth and elevation) of the interrogating sensor. This high-dimensional data is often displayed as a projection onto a two-dimensional image. As next-generation radar systems become increasingly diverse and capable, the assumptions and algorithms that have been used for traditional imaging need to be reconsidered. We present imaging techniques that accommodate limited persistence of scattering centers on objects, and use these techniques to develop two-dimensional object reconstructions from wide-aperture radar measurements. We also consider physical models of the scattering behavior of canonical shapes, and present results on scattering feature estimation from radar backscatter data or from complex-valued radar imagery. We discuss applications in object reconstruction, visualization, and recognition using these techniques.


Jennifer Lynn Mueller (Colorado State University)

Imaging Cardiac Activity by the D-bar Method for Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is an imaging technique that uses the propagation of electromagnetic waves through a medium to form an image. In medical EIT, current is applied through electrodes on the surface of the body, the resulting voltages are measured on the electrodes, and the inverse conductivity problem is solved numerically to reconstruct the conductivity distribution in the interior. Here results are shown from EIT data taken on electrodes placed around the circumference of a human chest to reconstruct a 2-D cross-section of the torso. The images show changes in conductivity during a cardiac cycle made from the D-bar reconstruction algorithm based on the 1996 uniqueness proof of A. Nachman [Ann.Math. 143].


Wim Mulder (Shell Research)

Two-way wave-equation migration

Joint with R.-E. Plessix.

The goal of seismic surveying is the determination of the structure and properties of the subsurface. Oil and gas exploration is restricted to the upper 5 to 10 kilometers. Seismic data are usually recorded at the earth's surface as a function of time. Creating a subsurface image from these data is called migration.

Seismic data are band-limited with frequencies in the range from about 10 to 60 Hz. As a result, they are mainly generated by short-range variations in the subsurface impedance, the product of velocity and density. The common approach towards migration is the construction of a reflection-free background velocity model from the apparent travel times from source to receiver. In this background model, the migration algorithm maps the data amplitudes to the impedance contrasts that generated them. Single scattering is implicitly assumed.

The wave propagation in the background model is usually described by an approximation to the wave equation to keep the required computer time down to months. Ray tracing used to be a popular choice, but is gradually taken over by one-way (paraxial or parabolic) wave equation approximations. We have investigated the use of the acoustic wave equation, which will we call the two-way wave equation in order to distinguish it from the widely used one-way wave equation. The two-way approach provides a more accurate description of wave propagation than the one-way method, particularly near underground structures that have steep interfaces. The one-way equation requires considerably less computer time in 3D, but in 2D the one-way and two-way methods compete.

Migration algorithms can be derived from the least-squares error that measures the difference between observed and modeled data. The gradient of this functional with respect to the model parameters is a migration image. This gradient can be used to minimize the error, but leads to a problem that is nonlinear in the model parameters and has many local minima. Gradient-based optimization algorithms will only provide meaningful results if the initial model is close to the global minimum. Because migration is computational very costly, global searches are not an option.

An alternative is to return to the classic approach, where migration serves to map the impedance contrasts without changing the wave propagation model. This can be achieved in the context of the two-way wave equation by assuming that the contrasts are small perturbations, leading to a linearization with respect to the model parameters. This is the well-known Born approximation.

A disadvantage of the two-way method is that it models all waves, not only reflections. This may produce artifacts in the images. We will discuss ways to remove them. In the nonlinear approach, these artifacts can actually be used to update the background model. Also, multiple reflections can be included in the minimization. In general, however, the least-squares functional is not very well suited to determine the background model and an alternative cost functional needs to be sought.

Examples on synthetic and real data will serve as illustrations.


Frank Natterer (Universitaet Muenster) http://wwwmath.uni-muenster.de/math/u/natterer/

Adjoint method in time domain ultrasound tomography

We model ultrasound tomography by the wave equation. Adjoint methods can be used for the inversion. Unfortunately, due to the large number of sources, adjoint methods are very time consuming. By preprocessing of the data (wavefront synthesizing, plane wave stacking), adjoint methods can be sped up by orders of magnitude. We analyse the preprocessed data in Fourier domain. We present numerical results for the Salt Lake City breast phantom and for the Marmousi data.


Clifford Nolan (University of Limerick)

Radar imaging from multiply scattered waves

We consider imaging the earth's topography using synthetic aperture RADAR (SAR), as well as real aperture RADAR (RAR). We use a simple scalar wave model for the radio waves. Instead of the common approach of singly-scattered waves, we consider the situation where a reflecting 'wall' is located in the vicinity of the region of interest (ROI). We will show how it is possible to take advantage of scattering between the wall and object(s) in the ROI to improve on the convention imaging methods. An obvious benefit of such a situation is the improved angular resolution available from this kind of data.

Our approach is based on microlocal analysis, which is often considered an opaque subject area to the uninitiated. However, the simplicity of the experimental set up in SAR and RAR makes for a very straightforward application of microlocal tools.


Roman G. Novikov (University de Nantes)

Some recent developments of the analytical reconstruction techniques for inverse scattering and inverse boundary value problems in multidemension

Some recent developments of the analytical reconstruction techniques for inverse scattering and inverse boundary value problems in multidemension Abstract: This poster presents (the abstracts and references of) the recent works 1. R.G.Novikov, The d-bar approach to approximate inverse scattering at fixed energy in three dimensions, International Mathematics Research Papers 2005:6 (2005) 287-349; 2. R.G.Novikov, Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential, Inverse Problems 21 (2005) 257-270.

These works give some new developments of the analytical reconstruction techiques for inverse scattering and inverse boundary value problems in multidimension.


Roman G. Novikov (University de Nantes)

The partial- approach to approximate inverse scattering at fixed energy in three dimensions.

See pdf file.


George C. Papanicolaou (Stanford University)

A resolution theory for stable imaging in clutter

I will present a qualitative, model free theory for imaging in clutter with coherent interferometry. Coherent interferometry is a smoothed form of Kirchhoff or travel time migration that is implemented adaptively in order to optimize the bias-variance tradeoff in the image quality, as it is being formed. I will show the results of numerical simulations that illustrate the theory. This is joint work with L. Borcea and C. Tsogka.

(Lecture Materials)


Rene-Edouard Plessix (Shell Research)

Iterative solver for the wave equation in the frequency domain

Joint work with Wim Mulder.

To retrieve the long and short spatial frequencies of the velocity model from seismic data, several authors have proposed to work in the frequency-domain. The data are inverted per frequency going from the low to the high. This approach has been used for long offset data in two dimensional space. It relies on the solution of the wave equation in the frequency domain (Helmholtz equation). Whereas in two dimensional space, a direct solver of the frequency-domain wave equation provides an efficient method, in three dimensional space, this approach is not feasible because the linear system becomes too large. This difficulty may be overcome with an iterative solver for the Helmholtz equation. During his Ph. D work, Y. Erlangga has studied an iterative approach based on a preconditioned bicgstab (conjugate-gradient type) method. The efficiency of the method depends on the preconditioner. It was proposed to use a damped wave equation as a preconditioner and to approximate the inverse of the damped equation with a multigrid method. Strong damping is required for the preconditioner, otherwise the multigrid method does not convergence. Two-dimensional examples show that this approach is robust and that the number of iterations depends linearly on the frequency when the number of grid points per wavelength is kept constant. Thus, this approach provides a sub-optimal solution. In the poster, several numerical examples will be presented to assess the efficiency of the iterative approach. Its relevance for migration in two and three dimensions and for inversion algorithms will also be discussed.


Walter Richardson Jr. (University of Texas - San Antonio)

Texture discrimination, nonlinear filtering, and segmentation in mammography

There are two primary signs used by the radiologist to detect lesions. The first is mass: a benign neoplasm is smoothly marginated whereas a malignancy is characterized by an indistinct border which becomes more spiculated with time. The second sign is microcalcification. An essential ingredient of these indicators is texture, used by the radiologist in many subtle ways to discriminate between normal and cancerous tissue. The irregular boundaries of suspect lesions suggest that they could be identified by their local fractal signature. Any real image is corrupted by some noise and it is necessary to prefilter the data. Results are presented for two edge-enhancing filters: the Weighted Majority - Minimum Range filter and the mean-curvature dependent PDE filter of Morel. Once the image has been filtered/transformed, the Mumford-Shah approach is used for segmentation.


Partha S. Routh (Boise State University) http://cgiss.boisestate.edu/~routh

Appraisal analysis in geophysical inverse problem: Tool for image interpretation and survey design

Joint work with Doug Oldenburg.

Image appraisal in geophysical inverse problem can provide insight into the resolving capability and uncertainty of estimates. Although a rigorous approach to solve nonlinear appraisal analysis is still lacking but several methods have been proposed in the past such as linearized Backus-Gilbert analysis, funnel function method and nonlinear Backus-Gilbert formulation where forward problem can be expressed as scattering series. In this talk I will discuss appraisal analysis and how it can be used for image interpretation and survey design. In image interpretation the goal is to quantify what part of model can be resolved by the data and what parts are consequence of regularization operator? In survey design the objective is to determine optimal survey parameters, such as the position of sources/receivers and possibly frequencies in EM experiments, that would provide better' model resolution in a region of interest. For both of these problems we examine the resolution measure called point spread function. The point spread function quantifies how an impulse in the true model is observed in the inversion result and, hence, the goal is to adjust the survey parameters so that the point spread function is as delta-like as possible. This problem is solved as a nonlinear optimization problem with constraints on the parameters. Examples from ray-based tomography and controlled source electromagnetics will be presented.


Tom L. Scofield (Calvin College)

Local Tikhonov regularization in n dimensions

Many ill-posed linear integral equations are solved using standard Tikhonov regularization. When solutions have "edges", as is usually the case in the image deblurring problem, this procedure generally carries with it a choice between capturing the near-discontinuities found at edges at the expense of introducing oscillations in regions that should be smooth, or preserving smooth regions but oversmoothing edges. More recently, local Tikhonov regularization methods have been introduced, attempting to make this choice a local rather than global one. We prove the convergence of such methods in R^n for general n. We also carry out a discrete numerical implementation of such methods and provide examples in 1 and 2 dimensions of results using both these methods and standard Tikhonov regularization.


Jakob J. Stamnes (University of Bergen)

Imaging using coherently and diffusely scattered radiation

The talk will be divided in two parts. The first part will be devoted to diffraction tomography based on the scalar wave equation to obtain images of the refractive-index distribution of an object embedded in a homogeneous medium, with emphasis on experimental verifications of this technique in applications using light or ultrasound. The second part of the talk will be devoted to imaging of objects embedded in turbid media, based on radiative transfer theory. Here the emphasis will be on passive optical remote sensing from satellite for identifying and mapping algae distributions in the ocean, as well as on the use of light for diagnosis of skin abnormalities, such as skin cancer.


Jakob J. Stamnes (University of Bergen)

Imaging of physiological properties of human skin from spectral reflectance data

Joint with K.P. Nielsen, M. Biryulina, G.Ryzhikov, K. Stamnes, and L. Zhao.

We present a new method, based on inverse radiative transfer modelling, for retrieving physiological parameters of human skin tissue from multi-spectral reflectance data. Whereas previous attempts of such retrievals have been based either on empirical formulas or simplified, inaccurate forward models, such as the Kubelka-Munk theory, our forward model is based on the discrete-ordinate solution of the radiative transfer equation, which is both fast and accurate. Examples are given of retrievals based on simulated reflectance data or in-vivo measurements.


Wolfgang Stefan (Arizona State University)

Signal restoration through deconvolution applied to deep mantle seismic probes

We present a method of signal restoration to improve the signal to noise ratio, sharpen seismic arrival onset, and act as an empirical source deconvolution of specific seismic arrivals. The method is used on the shear wave time window containing SKS and S, whereby using a Gaussian PSF produces more impulsive, narrower, signals in the wave train. The resulting restored time series facilitates more accurate and objective relative travel time estimation of the individual seismic arrivals. Clean and sharp reconstructions are obtained with real data, even for signals with relatively high noise content. Reconstructed signals are simpler, more impulsive, and narrower, which allows highlighting of some details of arrivals that are not readily apparent in raw waveforms.


Chris Stolk (University of Twente)

Shot-geophone migration for seismic data

We consider the problem of determining earth properties from seismic data, i.e. measurements with broadband acoustic waves using sources and receivers at the surface. For current data processing methods this is considered as a partially linearized inverse problem, where data is modelled by linearization about a smooth background medium, with a medium perturbation that contains only high-frequency components. Both the background, and the perturbation are to be estimated from the data. Reconstructing the high-frequency perturbation is an imaging problem, for which so called migration methods are used, that are based on geometrical wave propagation in the background medium.

This talk is about establishing whether a choice of background medium is consistent with the data. A criterion for this, needed in the estimation of the background model, is given by the so called semblance principle that must be satisfied by migrated data, and that express internal consistency of redundant data, given the background medium.

This talk focuses on the class of shot-geophone migration schemes. We show that shot-geophone migrated data satisfies an appropriate semblance principle, even in complex background velocities (that lead to the presence of conjugate points). The latter is not the case for binwise migration schemes, in particular Kirchhoff schemes, that form the alternative to shot-geophone migration.


Mark Stuff (General Dynamics Advanced Information Systems)

Using invariant theory to obtain estimates of unknown shape and motion, and imaging moving objects in 3D from single aperture Synthetic Aperture Radar

When a moving object is imaged with conventional synthetic aperture radar (SAR) the result is a displaced smear. This is due to the extra information the objectmotion is imparting to the radar return. When a sensor collects data from a moving extended object, estimation of the direction vectors from the object to the sensor is often essential to the extraction of useful information from the sensor data. If the object or the sensor moves as result of uncontrolled or unknown forces, simple parametric models for the angular motions often rapidly loose fidelity. So, even if the object can be modeled parametrically, nonparametric motion estimates are desirable.

In one example of such a problem, a direct approach to estimating all the unknowns leads to difficult nonlinear optimization problems. But a characterization of the shape of the object, using the right choice of geometric invariants, can decouple the problem, temporarily isolating the object shape estimation from the motion estimation. This facilitates the extraction of nonparametric motion estimates both by subdividing the parameter space, and by enabling parts of the problem to be solved using linear methods.

If the motion is rich enough there should be a possibility of forming a 3D image of the object. This involves understanding the way the radar data is arranged in phase space. The data lies on a convoluted surface that occupies three dimensions rather than the two dimensional plane used in conventional SAR. To achieve three dimensional images the data must be extrapolated from the surface into a volume. In this complex space, there is a great deal of structure and therefore the possibility of extrapolating to a volume of data.


William W. Symes (Rice University) http://www.trip.caam.rice.edu/txt/bios/symes/william_symes.html

Nonlinear inverse scattering and velocity analysis

Migration velocity analysis ("MVA") can be viewed as a solution method for the linearized ("Born") inverse scattering problem, in its reflection seismic incarnation. MVA is limited by the single scattering assumption - for example, it misinterprets multiply scattered waves - but it is capable of making large changes in the model, and moving estimated locations of scatterers by many wavelengths. The salient features of MVA is its use of an extended (nonphysical) scattering model. Nonlinear least squares inversion ("NLS"), on the other hand, incorporates whatever details of wave physics are built into its underlying modeling engine. However success appears to require that the initial estimate of wave velocity (in an iterative solution method) be "accurate to within a wavelength", i.e. have kinematic properties very close to that of the optimal model.

This poster will describe a nonlinear extended scattering model and a related optimization formulation of inverse scattering. I will present the results of some preliminary numerical explorations which suggest that this approach may combine the global nature of MVA with the capacity of NLS to accomodate nonlinear wave phenomena.


Fons Ten Kroode (Shell Research)

On the dynamics of interbed multiples

Interbed multiples form a class of multiples in seismic data characterized by the property that all reflection points lie in the subsurface. This sets them apart from surface multiples, which have at least one reflection point at the surface of the earth.

For surface multiples there is a well established procedure to predict them from the data, i.e. without any a-priori knowledge of the subsurface. This procedure is firmly based on the wave equation and is exact from a theoretical point of view.

For interbed multiples the situation is much less satisfactory. In 1997 Art Weglein published an algorithm to predict them from the data. This algorithm is clearly a generalization of the surface related case, but its derivation is not. In fact, the algorithm initially came without a formal proof. I have tried to fill that gap in a 2001 paper, by providing a derivation based on weak scattering and asymptotics. This derivation demonstrated that the kinematics of Wegleinxs algorithm are correct, but at the same time left open the question of the dynamics. Since then I have obtained results for the dynamics by replacing the weak scattering assumption by the Kirchhoff scattering assumption.

In the presentation I will explain how to obtain prediction algorithms for interbed multiples under the weak scattering and Kirchhoff scattering assumptions.


Gunther A. Uhlmann (University of Washington)

Travel time tomography, boundary rigidity and electrical impedance tomography

In inverse boundary problems one attempts to determine the properties of a medium by making measurements at the boundary of the medium. In the lecture we will concentrate on two inverse boundary problems, Electrical Impedance Tomography and Travel Tomography in anisotropic media. These problems arise in medical imaging, geophysics and other fields. We will also discuss a surprising connection between these two inverse problems.

Travel Time Tomography, consists in determining the index of refraction or sound speed of a medium by measuring the travel times of waves going through the medium. In differential geometry this is known as the the boundary rigidity problem. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points of the boundary of a compact Riemannian manifold with boundary. The inverse boundary problem consists in determining the Riemannian metric from the boundary distance function.

Calderön's inverse boundary problem consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. This inverse problem is also called Electrical Impedance Tomography (EIT). The boundary information is

Calderön's inverse boundary problem consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. This inverse problem is also called Electrical Impedance Tomography (EIT). The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map.

A connection between these two inverse problems has led to a solution of the boundary rigidity problem in two dimensions for simple Riemannian metrics. We will also discuss a reconstruction method in two dimensions for the sound speed from first arrival times of waves.


Frank Wuebbeling (Universitat Munster)

Marching schemes for inverse acoustic scattering problems

The solution of time-harmonic inverse scattering problems usually involves solving the Helmholtz equation many times. On the other hand, these boundary value problems with radiation condition at infinity are notoriously hard to solve. In the context of inverse scattering, however, boundary value problems can be rewritten as initial value problems.

We develop an efficient marching scheme for computing a filtered version of the solution of the initial value problem for the Helmholtz equation in 2D and 3D. Stability and error estimates are developped, a numerical example is given.


Yuan Xu (University of Oregon)

A new reconstruction algorithm for Radon data

A new reconstruction algorithm for Radon data is introduced. We call the new algorithm OPED as it is based on Orthogonal Polynomial Expansion on the Disk. OPED is fundamentally different from the filtered back projection (FBP) method. It allows one to use fan geometry directly without the additional procedures such as interpolation or rebinning. It reconstructs high degree polynomials exactly and converges unifomly for smooth functions without the assumption that functions are band-limited. Our initial test indicates that the algorithm is stable, provides high resolution images, and has a small global error. Working with the geometry specified by the algorithm and a new mask, OPED could also lead to a reconstruction method working with reduced x-ray dose.


Can Evren Yarman (Rensselaer Polytechnic Institute)

Exponential radon transform inversion based on harmonic analysis of the Euclidean motion group

This paper presents a new method for the exponential Radon transform inversion based on harmonic analysis of the Euclidean motion group (M(2)). The exponential Radon transform is modified to be formulated as a convolution over M(2). The convolution representation leads to a block diagonalization of the modified exponential Radon transform in the Euclidean motion group Fourier domain, which provides a deconvolution type inversion for the exponential Radon transform. Numerical examples are presented to show the viability of the proposed method.


Hongkai Zhao (University of California - Irvine)

A direct imaging algorithm for extended targets using active arrays

We present a direct imaging algorithm for both the location and geometry of extended targets. Our algorithm is based on a physical factorization of the response matrix of an active array. A resolution and noise level based thresholding is used for regularization. Our algorithm is extremely simple and efficient since no forward solver or iterations are needed. Multiple-frequencies can be used to improve the stability of our algorithm. We demonstrate the efficiency and roubustness with respect to both measurement noise and random background.

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