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The linear sampling method and the MUSIC algorithm
 Inverse Problems
, 2001
"... This article gives a short tutorial on the MUSIC algorithm and the linear sampling method of [3], and explains how the latter is an extension of the former. 1 MUSIC MUSIC is an abbreviation for MUltiple SIgnal Classification [10]; we see below why the name is appropriate. 1.1 The basics of MUSIC MUS ..."
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Cited by 37 (1 self)
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This article gives a short tutorial on the MUSIC algorithm and the linear sampling method of [3], and explains how the latter is an extension of the former. 1 MUSIC MUSIC is an abbreviation for MUltiple SIgnal Classification [10]; we see below why the name is appropriate. 1.1 The basics of MUSIC MUSIC is essentially a method of characterizing the range of a selfadjoint operator. Suppose is a selfadjoint operator with eigenvalues ¡£¢¥¤¦¡¨§©¤����� � , and 1 corresponding ��¢�����§������� � eigenvectors. Suppose the ¡�����¢���¡�����§�������� eigenvalues are all zero, so that the ������¢���������§������� � vectors span the null space of. Alterna¡�����¢���¡�����§������� � tively, could merely be very small, below the noise level of the system represented by; in this case we say that the ������¢���������§������� � vectors span the noise subspace of. We can form the projection onto the noise subspace; this projection is given explicitly by � � � �� � (1) where the superscript � denotes transpose, the bar denotes complex conjugate, and � � � is the linear functional that maps a vector � to the inner product �� � � ���� �. The (essential) range of, meanwhile, is spanned by the ��¢�����§������������� � vectors. The key idea of MUSIC is this: because is selfadjoint, we know that the noise subspace is orthogonal to the (essential) range. Therefore, a � vector is in the range if and only if its projection onto the noise subspace is zero, i.e., if
Selective acoustic focusing using timeharmonic reversal mirrors
 SIAM J. Appl. Math
"... Abstract. A mathematical study of the focusing properties of acoustic fields obtained by a timereversal process is presented. The case of timeharmonic waves propagating in a nondissipative medium containing soundsoft obstacles is considered. In this context, the socalled D.O.R.T. method (decompo ..."
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Abstract. A mathematical study of the focusing properties of acoustic fields obtained by a timereversal process is presented. The case of timeharmonic waves propagating in a nondissipative medium containing soundsoft obstacles is considered. In this context, the socalled D.O.R.T. method (decomposition of the timereversal operator in French) was recently proposed to achieve selective focusing by computing the eigenelements of the timereversal operator. The present paper describes a justification of this technique in the framework of the far field model, i.e., for an ideal timereversal mirror able to reverse the far field of a scattered wave. Both cases of closed and open mirrors, that is, surrounding completely or partially the scatterers, are dealt with. Selective focusing properties are established by an asymptotic analysis for small and distant obstacles.
Cognitive Theory
 International Organization,” International Studies Quarterly
, 1991
"... This series is divergent, therefore we may be able to do something with it. ..."
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Cited by 9 (1 self)
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This series is divergent, therefore we may be able to do something with it.
Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors., in "Inverse Problems
"... using time reversal mirrors ..."
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Optimal Waveform Design for Array Imaging
, 2007
"... We introduce and analyze several algorithms for optimal illumination in array imaging. We consider time reversal and Kirchhoff migration imaging in homogeneous media, in regimes where the signaltonoise ratio is high (infinite). Extensions to coherent interferometric imaging in clutter are describe ..."
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Cited by 4 (2 self)
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We introduce and analyze several algorithms for optimal illumination in array imaging. We consider time reversal and Kirchhoff migration imaging in homogeneous media, in regimes where the signaltonoise ratio is high (infinite). Extensions to coherent interferometric imaging in clutter are described briefly. We show with numerical simulations that the optimal illumination algorithms image selectively closely spaced point scatterers and extended scatterers with considerably better resolution than without the optimization. We analyze the imaging algorithms in the Fraunhofer diffraction regime for small and extended scatterers. Using the prolate spheroidal wave functions we also derive analytic expressions of optimal illuminations for imaging strips. 1
Acoustic TimeReversal Mirrors in the Framework of OneWay Wave Theories
"... We investigate the implications of directional wavefield decomposition with a view to time reversibility. In particular, we discuss how wavefield decomposition preserves the reciprocity theorem of timeconvolution type but looses the reciprocity theorem of timecorrelation type. As a consequence, a ..."
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We investigate the implications of directional wavefield decomposition with a view to time reversibility. In particular, we discuss how wavefield decomposition preserves the reciprocity theorem of timeconvolution type but looses the reciprocity theorem of timecorrelation type. As a consequence, a perfect `timereversal mirror' in the framework of oneway wave theory does not exist: We find that on the wavefront set (`classical limit') a timereversal mirror can retrofocus the wavefield to its originating source, but that nonperfectly retrofocusing lowerorder distributions contribute to the process as well. These distributions can be attributed to `evanescent' wave constituents but are not negligible; we will study them explicitly. As a peculiarity, we discuss how a Schrodingerlike equation can be obtained out of the (exact) frequencydomain oneway wave equation. This involves an approximation  known in ocean acoustics and exploration seismology as the `parabolic equation' approximation  that restores timereversibility.
FeatureEnhancing Inverse Methods for LimitedView Tomographic Imaging Problems
, 2003
"... In this paper we overview current efforts in the development of inverse methods which directly extract targetrelevant features from a limited data set. Such tomographic imaging problems arise in a wide range of fields making use of a number of different sensing modalities. Drawing these problem are ..."
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In this paper we overview current efforts in the development of inverse methods which directly extract targetrelevant features from a limited data set. Such tomographic imaging problems arise in a wide range of fields making use of a number of different sensing modalities. Drawing these problem areas together is the similarity in the underlying physics governing the relationship between that which is sought and the data collected by the sensors. After presenting this physical model, we explore its use in two classes of featurebased inverse methods. Microlocal techniques are shown to provide a natural mathematical framework for processing synthetic aperture radar data in a manner that recovers the edges in the resulting image. For problems of diffusive imaging, we describe our recent efforts in parametric, shapebased techniques for directly estimating the geometric structure of an anomalous region located against a perhaps partiallyknown background.
Houston,
"... 1 The problem of optimal illumination for selective array imaging of small and not well separated scatterers in clutter is considered. The imaging algorithms introduced are based on the Coherent Interferometric (CINT) imaging functional, which can be viewed as a smoothed version of traveltime mig ..."
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1 The problem of optimal illumination for selective array imaging of small and not well separated scatterers in clutter is considered. The imaging algorithms introduced are based on the Coherent Interferometric (CINT) imaging functional, which can be viewed as a smoothed version of traveltime migration. The smoothing gives statistical stability to the image but it also causes blurring. The tradeoff between statistical stability and blurring is optimized with an adaptive version of CINT. The algorithm for optimal illumination and for selective array imaging uses CINT. It is a constrained optimization problem that is based on the quality of the image obtained with adaptive CINT. The resulting optimal illuminations and selectivity improve significantly the resolution of the images, as can be seen in the numerical simulations presented in the paper.