A discussion of two wavefront aberration correction procedures

ULTRASONIC IMAGING 14,

387-397

(1992)

A DISCUSSION OF TWO WAVEFRONT ABERRATION CORRECTION PROCEDURES

Bernard D. Steinberg

Valley Forge ResearchCenter The Moore Schoolof ElectricalEngineering University of Pennsylvania Philadelphia,PA 19104

This review paper discussesthe basic properties of two adaptive signal processing procedures for dealing with weak scattering in a phased array transducer system. A fundamentalimprovementin the lateral resolutionof ultrasonicecho scannerswill result if the weight vector of a largephasedarray transducercan be modified to accountfor distortion in the propagationmedium. Lateral resolution in most tissueis limited to a few mm by wavefrontdistortion-induced sound-speedvariations. One important wavefront-distortion source is scatteringfrom local speedvariations within large and reasonablyhomogeneoustissuebeds suchas the liver. Scatteringdispersessomeenergyfrom the beamand perturbsthe wavefront, thereby distorting the image and limiting the resolution to the scaleof the distortion. Often, suchscatteringis weak, meaningthat mostof the energy in the beamis unscattered.The total field at the receiving transduceris the vector sumof the unscatteredand scatteredfields. In weak scattering the unscatteredfield is dominant and the resultantfield can be treated as the unscatteredfield plus a perturbation. The net effect is primarily a distorted phasefront,while the ampitudeor modulusof the wavefront remainsreasonablyintact. Refraction and strong scatteringaffect the wavefront moreseverelyand arelessresponsiveto thesealgorithms. Key words: Aberration correction;adaptivebeamforming;dominantscattereralgorithm; refraction; scattering; spatialcorrelationalgorithm; wavefront distortion.

INTRODUCTION The distortion induced by weak scattering of ultrasound energy is primarily in the wavefront phase. Such distortion can be modeled by a random phase screen, having appropriatestatistical properties,locatedin the apertureplane. The generic term usedin this paperfor wavefront aberrationcorrection is adaptivebeamforming(ABF). Two ABF methods for determiningthe natureof the distortion from wavefront measurements arediscussedin this paper. In each,phaseinformation is extracted from the complex samplesof the radiation field. The measurementsare made at each transducerelement. The systemthen alters the weight vector of the phasedarray accordingly to compensatefor the distortion. In essence,the equivalent of a correcting lensis createdand insertedinto the signalpath that includessource, distorting medium and transducerarray. ABF methodsare effective when the distortion is causedby weak scattering. The result is nearly diffraction-limited lateml resolution, given by AS = R3L/L,where h is wavelength, R is depth of penetration and L is transducersize. 0161-7346/92 387

$5.00

Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

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A more severe cause of wavefront distortion is refraction at interfaces between tissue beds having different acoustic speeds. Refraction causes beam bending and beam splitting. On-going in vivo measurements at our laboratory indicate that it is particularly serious in complicated soft tissue such as the breast [ 11. Multiple beams are produced from a single initial beam, and these subbeams can arrive at the receiving transducer from different directions and thereby interefere. Coherent wave interference results in amplitude distortion of the wave&it in addition to phase distortion (Fig. 1). The imagery produced is severely distorted, often with as many artifacts or false sources as the number of subbeams. Distortion is avoided when the transducer is sufficiently small so as not to “see” more than one beam, but this solution leads to poor lateral resolution. In addition, scattering need not be weak. Large spatial derivitives in the acoustic refractive index can give rise to strong diffracted waves [2]. These waves interact with the primary wave and distort the wavefront amplitude as well as the phase. Strong scattering is far

distance alongreceiving array (mm) (a)

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1 12.O

distance alongreceivingarray(mm) (b) Fig. 1 In vivo transmissionmeasurement of (a) amplitudeand (b)phaseof wavefront through breastacrosslargereceiving array. Sourceis single3 MHz transducerelementat distanceof 12cm. Note jaggedamplitudeprofile. which can not be accountedfor either by differential attenuationin tissueor by weak scattering. Smoothprofile (dashed)through baby oil (without breast) shownforcomparison. From [27].

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more serious than weak scattering. refraction.

ABERRATION

CORRECTION

It has the same basic degrading effect upon imaging as

The ABF algorithms discussed in this paper, as well the more general classes of which they are the simplest examples, are phasefront-aberration correction algorithms. They do not fare well when amplitude distortion is significant. Refraction and strong scattering are candidates for stronger algorithms.

BACKGROUND Resolution gains have been reported in clinical ultrasonic instruments through an increase of aperture size and the use of higher transducer frequencies, both for spatial resolution [3] and contrast resolution [3,4]. Lateral resolution is the minimum angular or lateral separation between two approximately equal strength targets or sources at which they can be separately distinguished. Contrast resolution, or target dynamic range, is proportional to the level of the sidelobes associated with the beamforming process [5,6]; it is important in distinguishing between tissue areas with moderately differing backscatter strengths, or small cysts and tumors. Resolution is directly dependent on the beam formation process. Every commercial instrument currently available assumes a constant speed of sound throughout the field of view. In reality, the velocities range from 1410 ms-1 in fat to above 1600 ms-1 for muscle and certain tumors. In an attempt to achieve higher resolution, transducer apertures have become larger and transducer frequencies have become higher. However, the erroneous assumption of constancy of propagation speed introduces an upper limit to improvements in spatial and contrast resolution. The larger the aperture, the more likely that diverse tissue beds will be crossed by sound waves from the individual transducer elements. The higher the transducer frequency the shorter the wavelength will be, which increases beam degradation for a given rms propagation-time error. As a consequence, proper image formation is degraded. The first observable effect of wavefront aberration is a rise in the sidelobes. With increasing aberration, the mainlobe widens and reduces in intensity. Eventually sidelobes increase to the point where no effective beam formation occurs at all. Given a phase-aberrated wavefront in which the aberration can be modelled as an uncorrelated random process having variance 02 (in sq rad), the expected value of the sidelobe level floor of an N-element phased array is o2 /N [6,7]. This floor is -26 dB for a 100-element array and rms phasefront errors of l/4 rad. When the standard deviation of the normalized amplitude errors is also equal to l/4, the floor rises to -23 dB. These levels are unacceptably high for they would preclude seeing small cysts in the ultrasound image. Thus, it is not enough to simply increase the aperture size; it is necessary to increase the useful or effective size as well. A way to address this nonuniform velocity problem is to use phased arrays with ABF, i.e., adaptive time-delay control to each element. In phased array systems, an increase in aperture size results in a proportional increase in the number of signal processing channels and hence the cost of the instrument. However, without wavefrontaberration compensation, an increase in aperture size tends to result in an ever-decreasing improvement in resolution. The point is rapidly reached where additional size increases cost but neither lateral or contrast resolution.

ADAPTIVEBEAMFORMING In wavefront-aberration correction based upon adaptive beamforming, the signals detected at the array elements are independently delayed to compensate for variations in arrival times. Adaptive control of the weight vector in a phased array is an active subject of current

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research in several fields [5],[8-241. Phase aberrations induced by spatial inhomogeneities are 1 major problem in all coherent imaging systems, whether ultrasound, sonar, or radar. The problem also has been studied in incoherent imaging systems such as astronomical telescopy For many years [9,11,13,16,23]. Methods of adaptively phase-correcting the aperture weight vector of coherent systems have been developed and demonstrated in experimental equipment by our group and have beenshownto be valuable tools for self-calibrating large, distributed md distortedmicrowave imaging systems[5,12,14,20,21]. The earliestA!bF self-calibrating principlefor microwave imagingwasdescribedin 1973[8] anddemonstratedin 1975[lo]. A meansfor self-coheringa multiple mirror optical telescopewaspublishedin 1974by Muller et %l.[8]. In the early 1980’s,self-calibrationwasindependentlydiscoveredin radio astronomy 1s a means of self-cohering huge microwave receiving arrays [ 13,16,23]. ABF was introducedinto ultrasoundin the samedecade[17-19,241. ABF corrects the phaseaberrationsvia feedback to the beamforming circuitry or by modifying stored received signal data in a digital beamforming system. The sourceof the Feedbackis either the image itself or measurementstaken from the individual receiving zhannels. The processcompensates for phasefronterrors introducedby the intervening tissue andto any inadvertently introducedby the instrumentitself. Two successfulalgorithmstested with high resolutionmicrowave data aredescribedin this paper. Eachis the simplestmember of a broad classof algorithms. The two classesare describedin chapters8 and9 of a recent book [5]. The first class, while exceedingly simple and robust in a radar environment, is expectedto have but limited utility in ultrasound. It dependsuponthe presenceof a smallbut strongly reflecting scattererin one or more depth cells in the field of view of the beam. This classincludes the ultrasound algorithm proposedin [25]. These methodsare exceedingly successfulin imaging manmadetargets with microwaves [5,12,14,20] but their utility in ultrasoundis questionable. Imagesobtained with ultrasoundmay have dominant scatterers from specularsurfaces,but often the phaseacrossthe imagefrom sucha scatterervaries with viewing angle becausethe scattererreflects from different points of the surface. Hence, the applicability of this classof algorithmis limited. The secondclass,which dependsupon the secondorder statisticsof the distortion in the wavefmnt, is a morelikely candidate. This classincludesthe algorithmsproposedin [ 191and ~241.

DOMINANT SCATTERER ALGORITHM (DSA) The DSA is the most basicadaptive beamformingalgorithm. It requiresthat the scene beingimagedcontainsan outstandingscatterer,one significantly strongerthan its surroundings, in order to focus the array [12,14,25]. The received data set is searchedfor this dominant scattererand its echo phaseis measuredat each element at the center frequency of the signal spectrum. This set of phasesis usedto correct for the distortion and to focus a beamon the dominant scatterer.The sameset of phasecorrections compensatesfor aberrationsalong the bearingto the dominantscatterer,and is approximately correct in the generalangular vicinity of that scatterer. This algorithm anda family derived from it aredescribedin detail in Chapter8 of [5]. There it is also shown to work successfully with a group of only moderately strong scatterersprovided that thay are locatedat different distances. Let a transducerinsonify a volume that containsamongits scatterersa reflector that hasa large ultrasoundcrosssection,is physically smallandwhosereradiation field approximatesthat of a point sourceradiating in free space. The reradiationwavefront is initially spherical(Fig. 2) but, with distance,developserrorsbecauseof the weak scattering. The amplitudesof the field measuredat the receiving transducer(which may or may not be colocatedwith the transmitter)are substantially correct notwithstanding the scattered field. The phases of the- complex measurements may be useless,however, becausescattering-inducederrors assmall asthe order of one-half wavelength are enough to render all phaseinformation meaningless. The DSA cophasesall the signalsenteringthe system,after which the appropriatetime delaysnecessaryfor imagingcan thenbe applied.

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Tissue

R (Depth in tissue)

Element Number

--- _

_

_

_

-- -- -- --

-

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RO

b

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-

-

-

-

-

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-

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Conjugate of column vector of nearly constant amplitudes is weight vector of array

;-

Fig. 2 The dominantscattereralgorithm. Ro is the depth from which the echoeshave nearly constantamplitude. The complexconjugateof the columnvector at Ro is the aberration-correctionweight vector of the array.

The key aspectsof the procedure are searchingthe data for indications of a point source, testing the quality of the data for use in deriving the self-calibration function, and deducing the proper compensationweight vector. Digital beamformingfollowing coherent quadraturedemodulationis the systemmodel upon which the figure is based. The received echo waves are sampledand storedwith the format shownin the lower portion. The abscissa is echodepthor time of arrival. The ordinateis antennaelementnumber.The systemexamines vectors (shown as vertical column vectors for simplicity) of complex echo strength from successivedepth cells to determinewhich cell, if any, appearsto contain a point source, as evidencedby uniformity of amplitudeof the responseacrossthe antennaarray, i..e., constancy

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of the amplitude of the wavefront characterizes a point source. The cell with the smallest echo amplitude-variance is chosen. Experience has shown that the process is almost always successful if the normalized echo amplitude-variance does not exceed 0.12 [5]. Although the phase across the wavefront of a point reflector is also constant, the measured phases appear random because of the medium-induced distortion in the wavefront. When the signal waveform is wideband, the optimum weight vector that corrects this distortion consists of a set ?f vernier time delays that correctly match the distorted shape of the wavefront. A more practical weight vector is the complex conjugate of the column vector of data in the reference range bin, where the phase @n of the nth component is the desired time delay 2n times the angular center frequency 00 of the signal spectrum, i.e., $n = WOXn. Substitution of phase weighting for time delay compensation is satisfactory when the Bto product (B = signal bandwidth, t() = sampling interval) is sufficiently small. Analysis shows that the mainlobe of the array pattern can be reconstructed with phase weights when Bto is in the neighborhood of i to f, while maintaining suitable sidelobe control demands that the product not exceed the range h to $ [S, Chap. 31. The radiation pattern of the array after compensation is approximately the spatial matched filter response to the azimuthal or transverse complex reflection profile of :he reference reflector [ 141. That is, if r(l3) is the reflection coefficient of the reference reflector where 8 is the angle from the normal to the array, the compensated radiation pattern f(0) approximately equals r*(0). This algorithm is practical, simple, powerful and very robust. Its utility derives from two of its characteristics. The first is that the reradiation field from the reference reflector need be only four or more d.B greater than the sum of all the other coincident fields for a satisfactory main beam :o be ,,constructed. The second is that the scanning or imaging process is highly tolerant to gross :rrors in the phasefront. The upper images in figure 3 are examples of the use of this algorithm in microwave Imaging of airplanes (201. The Boeing 727 and the Lockheed L-1011 were each flying approximately 3 km from our laboratory. Severe aperture plane errors resulted from the use of highly distorted phased arrays. The DSA is used for self-calibration. The resolution is 1 m in :ach dimension. At the bottom are images formed prior to self-calibration. The differences are obvious. The targets can be recognized in the DSA images whereas the nonadaptive system provides useless jumbles of pixels.

SPATIAL CORRELATION

ALGORITHM

(SCA)

In some portions of ultrasonic scenes, the scattering exhibits characteristics of a random process, and the input sequence to the transducer aperture is a random, white-noise-like signal colored by the beam. The DSA is inappropriate in this case. A more powerful set of algorithms, sensitive to the spatial correlation properties of the ultrasonic wave, often sotves this problem and is able to self-calibrate under these conditions [ 15,21,22]. This family is described in detail in Chap. 9 of [5]. These algorithms capture the phase distortion information from the spatial correlation function of the ultrasonic wave. The means by which they do so are described below in the context of the simplest algorithm in this class. Attia showed that when the angular distribution of the scatterers causing the backscattered radiation field is statistically homogeneous, and the radiation pattern of the transmitter is symmetrical about its optical axis (which is the common case), then the autocorrelation function (acf) of the wavefront is real, except for a linear phase component associated with the pointing direction of the beam [l&21). Themeasured-aef devekvps an imaginary component when propagation anomalies, geometric distortions within the array, impedance mismatches from receiver channel to channel or other causes of phase error exist~in the signals received by the array elements. The algorithm provides a means for deriving a

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(a). g

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2555-

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CROSS-RANGE DMENSKN AT DiSTANCE OF FUSELAGE CENTER (METERS) (c)

W-T

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AZMJTH-&GlE

~LLFIAiANSl

Fig. 3 The dominantscattereralgorithm correcting badly distortedhigh resolutionradar data. The airplanesarea Boeing727 anda Lockheed-1011 flying approximately3 km from the Valley Forge laboratory. The uselessbottom imagesareformed from the samedata setswithout adaptivebeamforming. From [20].

compensating weight vector based on the phasesof the measuredunit-lag correlation coefficient acrossthe receiving array. Let the N antennaelementsdeliver signalsen to the signalprocessor.Thesesignalsare in error becauseof the problemsmentionedabove. Let the sumof the phaseerrors in each channelbe groupedinto a singlenumberbn and let the amplitudeerrors be negligible. Define en = enexp(-gn) as error-free values that would have been obtained had there been no propagationanomalies.The componentsof the proper compensationweight vector are wn = exp(-jgn). Any procedurethat disclosesthe fin solvesthe problem. The unit-lag spatialcorrelationcoefficient of the error-freesamplesis Rn[l] = E(ene$t )

(1)

which, by [14,20], is presumedto be real. E is the expectation operator and the asterisk

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means complex conjugate. The correlation coefficient of the measured signals is Ml1

= Wend11 = EIehi~levXPn = Mllexpj(Pn - Pn+O

- Pn+l)) (2)

and its phase argRn[ll=

Pn - Pn+l

(3)

is the difference between the phase errors in adjacent channels. Measurementof Eq.(3) presentssomedifficulty unlessthe radiation field is incoherent (created by a large number of independent sources,as in astronomy). Then Eq.(2), and therefore Eq. (3), can be estimated successfully from the time average of enen%1, as is regularly done in aperture synthesisradio telescopes[15]. In ultrasound, radar and other coherentprocesses,however, the backscatteredfield is rarely incoherentbecauseechoesfrom successivetransmissionsare substantially identical. Attia’s thesis showed that, when the sourcedistribution is statisticallyhomogeneous both laterally andin range,measureddatafrom anensembleof rangecells canbe averagedto properly estimateEq.(2). Let the subscriptk, k = 1,2,...K, designatethe range ceil and n the element number. The finite sampleestimateof Eq.(2) is finll] =kE en,kcn:l,k k=l .K = KZ eir,keti:l,k cxpj@n - Pn+l) k=l = &i[llevi(h

- Pn+l)

(4)

wherefin[ 1] is a finite-sampleestimateof Eq.( 1) and therefore, for large K, may be assumed to be real. The argumentof the measuredvalue fl n[l] is the estimate of the phaseerror difference betweenadjacentantennaelements.Thus,if we setpl = 0 (any onephaseerror may be setto an arbitrary value), the estimatedvaluesof the phaseerrorsarep^n+1 = $n - arg fin[l] , or pi = 0,62 = -arg l%l[l] , F3 = $2 - arg a2[1] , etc.,

(5)

and the problemis solved. This logic forms the basis for the algorithm in [21], which has proven highly successfulin experimentalradar imagingat VFRC [5]. It hasthe samebasisasthe algorithm usedin [ 191for ultrasonicphaseaberrationcorrection in a weakly scatteringmedium,the liver [26]. It can alsobe usedto someextent when refraction is present,provided that it isnot too severe. Figure 4 showstwo in vivo casesfrom female breasts{2g]. Measurementswere at 3 MHz. In both tests, two adjacent sources, spaced by 1.5 mm, were imaged. In (a) measurementwas through the breastof a 66 year old postmenopausalwoman. Glandular tissuehad beenlargely replacedby fat, which is relatively bomogenous.In this case,the SCA converted the distorted (dashed)image, which falsely showsthree sources,to a proper twosource image. Sidelobes are very poor, however. In (b), in which the subject was a premenopausal woman, SCA (solid) imageis only slightly better than the uselessuncorrected image. Phaseerror information residesnot only in the unit-lag correlation coefficient but also in higher order lags. Multiple-lag estimation of the errors introduces redundancy in measurementand therefore can improve the estimationaccuracy [5, Chap.91. Although not

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Image inmm (b)

Image inmm (4

Fig. 4 The unit-lag spatialcorrelationalgorithmappliedin vivo to (a) post-and (b) premenopausalfemalebreast. The sourcebeingimagedis anadjacentpair of transducer elementsspacedby 1.5mm. Frequencyis 3 MHz. In (a) the glandulartissuewas largely replacedby fat. The SCA convertedthe distorted(dashed)image,which falsely showsthree sources,to a proper two-sourceimage. Sidelobesarevery poor, however. In (b), in which the breast wasdense, the SCA (solid) imageis only slightly better than the uselessuncorrectedimage. From [28].

intuitively obvious,the ultrasoundproceduredescribedin [24] is a form of multilag correlation correction. The link is found in Muller’s seminalpaper [9] and in a subsequentpaper by Hamakeret al. [1 I]. Reference[9] showedthat the integral of the squareof the intensity in the imageplane of a telescopecontains suitableinformation to self-calibratethe instrument. Let I(u,v) be the imageintensity and J = II2(u,v)dudv

(6)

Muller showedthat maximizing J by adjusting time delaysfully corrects the image. Reference[lo] showedthat Muller’s integral can be expressedasan algebraicfunction of the spatial or transverseacf of the wavefront entering a telescope. Thus (in one dimensionfor simplicity), J = I 12(u)du = I 12(u)exp(jkxu)du],~

(7)

where k = 27tlh is the wavenumber. I and R, the acf of the wavefront measuredin the aperture, form a Fourier transform pair, symbolized by I <--> R. Similarly, 12 <--> R*R where * meansconvolution. Therefore J = R*Rlx=u, which is an algebraicfunction of all the lag products of the acf. Consequently, maximizing J is a multilag, spatial correlation operation. Lastly, a special case of Muller’s theorem relates directly to the ultrasound procedure described in [24]. In [9], he showedthat maximizing Jl = I I(u,v)Iu(u,v)dudv where IOis a maskor window of the imagingneighborhoodalsofully correctsthe image. When the correlation coefficients become small or drop to zero for any reason, an alternateprocedureis required. Significant distortionof the amplitudeof the wavefront signals trouble in the application of correlation-basedalgorithms. Strong refraction and/or scattering within the propagation medium causes such distortion. Evidence from our in vivo measurementproject at the Hospital of the University of Pennsylvaniaindicatesthat this is the casein the female humanbreast[ 11.

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SUMMARY Adaptive beamforming self-calibrates large phased arrays to compensate for wave propagation anomolies and/or geometric distorion in the array. Various algorithms have been applied successfully in radar and radio astronomy. ABF has also been shown to work in optical astronomy to counteract variations in the optical refractive index of the atmosphere. Success has been primarily achieved in those cases in which the wavefront distortion-inducing phenomenon can be modeled as a thin, random phase screen close to the receiving transducer array. Weak scattering, which distorts the wavefront primarily in the phase, responds well to existing ABF algorithms. Relatively homogeneous tissue with local perturbations in sound speed, such as the liver, is an example. The spatial correlation class of algorithms is suitable in this case. Strong scattering and refraction, on the other hand, distort the amplitude of the wavefront as well as the phase. The dense, premenopausal female breast is an example. Stronger algorithms are required for such situations.

ACKNOWLEDGMENTS This work was generously supported by the Army Research Office, the Rome Air Development Center, now Rome Lab, of the USForce, and Interspec Inc., Ambler PA. The author wishes to acknowledge in particular the many invaluable discussions with Dr. Kai Thomenius, Dir. of Research of Interspec.

REFERENCES [I] Zhu, Q., and Steinberg, B. D., Large-transducer measurements of wavefront distortion in the female breast, accepted for publication in Ultrasonic Imaging. Also Zhu, Q., Valley Forge Research Center rep. UP-VFRC-lOa(Univ. Penn., Phila., PA, 1991). [2] Morse, P. M. and Ingard, K. Uno, Theoretical Princeton New Jersey, 1986).

Acoustics,

(Princeton

Univ. Press,

[3] Mash&, S. H., Computed Sonography, in Ultrasound Annual 1985, (R.C. Sanders and M.C. Hill, Eds., Raven Press, New York , 1985) [4] von Ramm, 0. T. and Smith, S., Beam steering with linear arrays, iEEE Trans. Biomed. Eng. BME-30, 438452 (1983). [5] Steinberg, B. D. and Subbaram, H., Microwave (John Wiley and Sons, New York, 1991).

Imaging Techniques, Chapters 8 and 9

[6] Steinberg, B. D., Principles of Aperture and Array System Design. (John Wiley and Sons, New York, 1976). [7] Taheri, S. H. and Steinberg, B. D., Tolerances in self-cohering antenna arrays of arbitrary geometry, IEEE Trans. Antennas Prop. AP-24,733-739 (1976). [8] Steinberg, B. D., Design approach for a high resolution microwave imaging radio camera, J. Franklin Inst. , 296,415432 (1973). [9] Muller, R. A. and Buffington, A., Real-time correction of atmospherically degraded telescope images through image sharpening, J. Opt. Sot. Am. 64, 1200- 1210 (1974).

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[lo] Powers, E. N. Adaptive Arrays for Microwave Electrical Engr., University of Pennsylvania, 1974).

CORRECTION

Imaging, Ph.D. Dissertation

(Dept. of

[ll] Hamaker, J. P., Sullivan, J. D. and Noordam, J. E. Image sharpness, Fourier optics, and redundant-spacing interferometry, J. Opt. Sot. Am. 67, 1122- 1123 (1977). [12] Steinberg, B. D., Radar imaging from a distorted array: the radio camera algorithm and experiments, IEEE Trans. Antennas Prop. AP-29,740-748 (1981). [13] Readhead, A. C. S. Radio astronomy by very long baseline interferometry, 53-61 (1982). [14] Steinberg, B. D. Microwave Imaging with Large Antenna Arrays: Principles and Techniques (John Wiley and Sons, New York, 1983).

Sci. Am., 6,

Radio Camera

[15] Attia, E. H., Phase Synchronizing Large Antenna Arrays Using the Spatial Correlation Properties of Radar Clutter, Ph.D. Dissertation, (Dept. of Electrical Engr., University of Pennsylvania, 1984). [ 161 Thompson, A. R., Moran, J. M. and Swenson, Jr.,G.W., Interferometry in Radio Astronomy (John Wiley and Sons, Inc., New York, 1986).

and Synthesis

[17] Trahey, G. E. and Smith, S. W., Properties of acoustical speckle in the presence of phase aberration Part I, first order statistics, UZtrasonic Imaging IO, 12-28 (1988). [18] Smith, S. W., Trahey, G. E., Hubbard, S. M. and Wagner, R. F., Properties of acoustical speckle in the presence of phase aberration Part II: correlation lengths Ultrasonic Imaging I, 29-51 (1988). [19] Flax, S. W. and O’Donnell, M., Phase aberration correction using signals from point reflectors.and diffuse scatterers: basic principles, IEEE Trans. Ultrason. Ferroelec. and Freq. Cont. 35,758~767 (1988). [20] Steinberg, B. D., Microwave

imaging of aircraft, Proc. IEEE 76, 1578-1592 (1988).

[21] Attia, E. H. and Steinberg, B. D., Self-cohering large antenna arrays using the spatial correlation properties of radar clutter, lEEE Trans. Antennas Prop. AP-37,30-38 (1989). [22] Subbaram, H. M. and Steinberg, B. D., Scene independent self-calibration array antennas, IEEE Trans. Antennas and Prop. (to be published).

of phased

[23] Comwell, T. J., The application of closure phase to astronomical imaging, Science 245, 263-269 (1989). [24] Trahey, G. E., Zhao, D., Miglin, J. A. and Smith, S. W., Experimental results with a real-time adaptive ultrasonic imaging system for viewing through distorting media, IEEE Trans. Ultrason. Ferroelec. Freq. Cont. 37,418-427 (1990). [25] M. Fink, Time reversal of ultrasonic fields - part I: basic priciples, Ultrason. Ferroelec. Freq. Contr., 39, 555-566, (1992).

IEEE Trans.

[26] O’Donnell, M., and Flax, S.W., Phase aberration measurements in medical ultrasound: human studies, Ultrasound Imaging 10, l- 11 (1988). [27] Pauls, R. J., Research in High Resolusion Ultrasound Mammograpy, Dissertation, (Dept. of Electical Engr., University of Pennsylvania, 1992).

Ph,D

[28] Zhu, Q., Large-Transducer Measurements of Ultrasonic Wavefront Distortion in the Female Breast, Ph.D Dissertation (Dept. of Bioengr., University of Pennsylvania, 1992).

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