SCIENCE

Full-wave computed tomographyPart 3: Coherent shift-and-add imagingR.A. Minard, B.Sc, B.S. Robinson, B.E., Ph.D., and Prof. R.H.T. Bates,

D.Sc.(Eng.), C.Eng., F.I.P.E.N.Z., Fel.I.E.E.E., F.R.S.N.Z., F.I.E.E.

Indexing terms: Computer applications, Image processing

Abstract: The theory of coherent shift-and-add imaging is summarised, is extended to be more generally applic-able than in Parts 1 and 2, and is illustrated with the results of computer simulations and laboratory experi-ments. 2-dimensional images of collections of cylindrical objects (immersed in a water tank into which plasticcontainers filled with animal offal can be introduced) are reconstructed by shift-and-add (in both its elementaryand extended forms) from wideband ultrasonic signals scattered over a wide angle.

1 Introduction

There is wide scientific and technical interest in the meansof compensating for distortions introduced by propagationmedia. Much of the research into signal processing forcommunication systems and for medical imaging is a partof this. Also related to it are the astronomers' endeavoursto overcome their 'seeing problem', which is caused by thewarping of wavefronts on their passage through the earth'satmosphere. The growing application of 'speckle pro-cessing' methods in optical astronomy [1] has promptedus to inquire whether similar techniques might be usefullyinvoked when attempting to improve ultrasonic imagingresolution. We have already reported results [2] of astraightforward analogue of our optical shift-and-add pro-cedure [3, 4], and also of an experimental demonstration[5] of a previously conjectured [6] stochastic imagingmethod. Our concern here is with an improved processingstrategy that promises to widen very significantly the classof collections of ultrasonic scatterers which can be suc-cessfully imaged by shift-and-add. Whereas the mainconcern in earlier papers [2, 4, 5] was with images whichwere essentially 1-dimensional, the emphasis here is on 2-dimensional images.

This paper is the third in a series devoted to 'full-wavecomputed tomography', (CT) which is our terminology forthe data processing required to form 'clean' images ofcross-sections through bodies. We are chiefly concernedwith situations for which the simple assumptions, on whichstandard tomographic systems (e.g. ultrasonic B-scan,X-ray CT) are based [7], are inadequate. Cross-sectionsare of course 2-dimensional, which is why we concentrateon 2-dimensional imaging techniques. The concept of aclean image is precisely defined in the two previous papersin this series [8, 9], which deal with certain fundamentalsof image formation.

We summarise the basic theory of shift-and-add [2-4]in Section 2 before introducing the extension which is themain business of this paper. We outline our experimentalset-up and the associated data reduction in Sections 3 and4, respectively; extra details can be found elsewhere [10,11]. Sections 5 and 6 are devoted to the results of experi-ments performed in our computational and ultrasoniclaboratories. The significance of these results is assessed inSection 7.

Paper 3545A, first received 4th January and in revised form 14th September 1984Prof. Bates and Mr. Minard are, and Dr. Robinson was formerly, with the Depart-ment of Electrical and Electronic Engineering, University of Canterbury, Christ-church 1, New Zealand. Dr. Robinson is now with the Biodynamics Research Unit,Mayo Clinic, Rochester, Minnesota 55905, USA

2 Theory of shift-and-add imaging

Shift-and-add imaging is an instance of speckle processing,which can itself be regarded as a species of blind deconvol-ution [1]. Only certain statistical properties of the mecha-nism causing the image distortion are known a priori.Consequently, one cannot obtain a faithful restorationfrom a single blurred image, but must average appropri-ately over a number of independently blurred images (eventhe stochastic method [5, 6] referred to in Section 1,although it requires only one recorded image, operates ona sequence of pseudo-blurred images generated within thecomputer). A critical feature of speckle processing is thatthe best possible resolution of the restored image is set bythe effective width of the average of the autocorrelations ofthe point spread functions (PSF), which are the cause ofthe blurring of individual recorded images. It is appropri-ate, therefore, to think of the recorded images as beingcomposed of pixels whose effective spatial extents corre-spond to this resolution. It thus accords with our physicalintuition to introduce the 'delta notation' b{x — £) to iden-tify a 'just-resolved' point in a recorded image at the parti-cular point in image space labelled by the position vector£. An arbitrary point is labelled by the position vector x,which has Cartesian co-ordinates x and y.

The problem considered here is the recovery of an orig-inal (or true, as we prefer to call it) image f(x) from anensemble of a (reasonably) large number (M say) of record-ed images sm(x), all of which are heavily blurred (theinteger index m runs from 1 to M). The true image, as wellas each sm(x), is to be thought of as being made up of afinite number of pixels, adjacent ones being separated fromeach other, in both the x and y directions, by the effectivewidth of the 'delta function' 5{x) introduced in the previousparagraph. The centres of these pixels are identified by theposition vectors £,, where the integer index / takes on asmany values as are needed (note that the £, form a rec-tangular array in image space).

The recorded images are called 'speckle images' byanalogy with the short exposures that serve as the rawdata for optical astronomical speckle interferometry [1].The with speckle image is expressed in the form

sm(x)=f(x)Qhm(x) (1)

where O denotes convolution; hm(x) is the PSF character-ising the point spread invariant (or isoplanatic, usingoptical terminology) part of the blurring; and cm{x), whichwe call the 'contamination', includes every imperfectionpresent in sm(x) apart from the purely isoplanatic distor-tion represented by hm(x). We think it worthwhile empha-

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sising that speckle processing cannot be successful whencm(x) completely swamps the convolution on the right-hand side of eqn. 1. We rely on there being a significantisoplanatic component in the total distortion.

We are interested in imaging with coherent wave fields(e.g. ultrasound), so that all quantities appearing in eqn. 1should be regarded as complex amplitudes, i.e. each onepossesses a phase as well as a magnitude at each pixel. It isto be understood that the way in which the sm(x) arerecorded and the manner of their blurring are such that allof the hm(x) belong to the same statistical ensemble, but arestatistically independent. The contaminations are alsotaken to be statistically independent of each other and ofeach PSF. The property of the averaged autocorrelationsof the hm(x), mentioned in the first paragraph of thisSection, is expressed by

<hm(x) * h*(x)}m = 5(x) (2)

where * denotes correlation and the angular bracketsdenote an average (the integer index, over which theaverage is taken, is identified by the subscript on theclosing angular bracket). The asterisk on the second hm

denotes the complex conjugate.It is convenient to choose the origin of image space such

that it coincides with the 'brightest' pixel in the true image(i.e. the point in the image having greatest magnitude):

| / ( 0 ) | = max (3)

If/(x) possesses more than one 'brightest point', any one ofthem is chosen to coincide with the origin of image space.

Taking explicit account of the finite resolution(mentioned in the first paragraph of this Section) permitsus to usefully approximate the convolution appearing ineqn. 1 by a summation. Denoting the complex amplitudeof hm(x) at the /th pixel £, by /im , we see that

f(x) O hm(x) — Y;hm,i f(x — %i) (4)

We need notation for the brightest of the hm , and for thepoint in image space where it lies:

V n m ) I = m a x \hmti\ wi th hm<L(m) = h(xm) (5)

Note that xm is shorthand for £L(m).The implication of eqn. 4 is that each speckle image

consists of many replicas of f(x). These replicas are dis-placed from each other and are differently weighted. Thegoal of shift-and-add imaging is to identify the brightestreplica in each speckle image and to superimpose it on itsfellows in the other speckle images. So, what we try to dois first shift sm(x) by ( — xm), then normalise it by dividingby hmL(m), and finally add it to the other speckle imageswhich have been similarly processed. It is convenient todivide through by M at the end. Referring to eqns. 1, 3, 4and 5 therefore shows that the ideal shift-and-add imageft{x) can be expressed as

fix) =(sm(x + xm))

"m, L(m)

It follows from eqns. 1, 3, 4, 5 and 6 that

fix) =f(x) + r(x)

(6)

(7)

where the remainder r(x) is the average of all the contami-nations and of all the fainter replicas of f(x), Rememberthat only the brightest replica in each sm(x) is reinforced bythe averaging process defined on the right-hand side ofeqn. 6. Since the fainter replicas are (effectively) randomly

phased and positioned, and because the contaminationsare statistically independent, r(x) can be expected to cancelout as M—>co, or at worst reduce to a smooth back-ground.

In practice, there is of course no unambiguous means ofdetermining xm and hmL(m) by inspection of sm(x).However, the former often corresponds to the brightestpoint xm in sm(x), i.e.

with |sm(xm)| =max |sm(x)| (8)

Furthermore, when eqn. 8 holds, sm(xm) can be expected tobe a reasonable approximation to (f(0)hmL(m)). Theseassumptions are (effectively) invoked in previous reports ofour shift-and-add experiments [2-5; see also 8.7 & 8.9 ofReference 1]. The elementary shift-and-add image fe(x),which corresponds exactly to these assumptions, is heredefined by

fJLx) =(sm(x + xj)

sm(xm)(9)

An almost equivalent image, which is the one we haveactually made more use of in the past, is what we here callthe simple shift-and-add image fsa(x). It is defined by

fjx) =(\sm(xj\sm(x

(10)

It makes little difference in practice whether fe(x) orfsa(x) isinvoked. From this point we shall use the notation fa(x) todenote either of them. When /(0) stands out sharply fromthe rest of f(x), we find that/fl(x) is indeed a faithful rep-resentation of J{x). When | /(0) | is not much larger thanthe rest of | / ( JC) | , however, the brightest points in somespeckle images correspond to parts of/(jt) different from/(0), thereby generating false detail, which we refer to as'ghosts [2-6, 11, 13]. This has suggested to us that thespeckle images should be subjected to a preliminary trans-formation, designed to permit the brightest version of/(0)in each sm(x) to be more surely identified. Before describinga particularly promising transformation, we note from eqn.1 that

<5m(x) * s*(x)}m = (f(x) * /•(*)) O <hjx) * hZ(x)}m + c(x)

(11)

because of the commutative properties of convolution andcorrelation. The quantity c(x) is to be thought of as a com-posite contamination, it comprises the average of the auto-correlations of the cm(x) plus the averagedcrosscorrelations of the cjx) and the (f(x) Q hm(x)). Invok-ing eqn. 2 then indicates that eqn. 11 reduces to

* /*(*) C(X)

which shows that the average of the autocorrelations of thespeckle images provides an estimate of the autocorrelationof the true image (resolved, it must be remembered, onlydown to the effective width of the average of the autocor-relations of the PSFs). We can expect c(x), like r(x), to besmall. The linear dimensions of the region of image space,within which (f(x) * f*(x)) has significant value, are gener-ally twice those of a 'box' b(x) which just encloses f(x).Note that we can only deduce the size, and not the posi-tion, of the box from inspection of <sm(x) * s*(jt)>m. Toimplement our extension of shift-and-add (describedbelow), we construct a larger box b(xj whose linear dimen-sions are 10% bigger than those of b(x). The point of thislarger box is that it ensures we include all ofj(x) when wecarry out the processing defined by eqn. 12 below.

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We now argue that by correlating the box with themagnitude of either the elementary or the simple shift-and-add image we identify the part of fa(x) which correspondsmost closely to/(jc). On defining

\fa(x)\b(x+y)da(x) = max \\\fa(x)\

x b(x + x') da(x) (12)

where do(x) is the element of area in the region A of imagespace occupied by fa(x) (and also by all the sm(x)), it isassumed that

is the best estimate of/(jc) which can be obtained fromeither elementary or simple shift-and-add. We now postu-late that ym, which is defined by

1/CvJ * sm(ym)\ = max \J{x) * sm(x)\ (14)

is in general a better estimate of xm than xm. It follows thatthe modified shift-and-add image fmod(x), which is definedby the right-hand side of either eqn. 9 or eqn. 10 with xmreplaced by ym, should be an improvement on/a(x).

The above procedure is iteratively continued by repla-cing fa(x) in eqn. 12 by fmod(x). This gives new versions off(x) and theym, from eqns. 13 and 14, so that a new/mod(jc)is generated etc. The iterations are stopped when the differ-ences between successive versions of fmod{x) are less thansome prescribed threshold, or when any appropriate meanof such differences exhibits no further decrease.

3 Experimental arrangements

The experiments reported in Sections 5 and 6 are con-cerned with imaging scatterers which are insonified in awater tank. Each scatterer is cylindrical in the sense thatits axial length is much greater than any of its cross-sectional dimensions. Its axis is perpendicular to an inci-dent ultrasonic beam (which is contrived to have, as nearly

Rx

Tx

Fig. 1 Geometry of the scattering experimentsThe dashed circle SR represents the scattering region and the dashed circular arcMC, which stretches from 0i to 62 (these two angles define the measurement range),represents the measurement circle. The solid circle Bl and the truncated annuli B2and B3 represent mylar 'buckets' which can be filled with offal. The actual experi-mental parameters were: T = 13 cm, R = 25 cm, 0, = —60°, 62 = 60°, r = 5 mm,diameter of Bl = 8 cm, inner diameters of B2 & B3 = 8 cm & 40 cm, outer diam-eters of B2 & B3 = 16 cm & 48 cm

as possible, the form of a plane wave), thereby ensuringthat the set-up is essentially the same for any planethrough the scatterers perpendicular to their axes. Thegeometry can therefore be regarded as 2-dimensional (asindicated in Section 1).

Fig. 1 shows the juxtaposition of an insonifying trans-ducer, the scattering region SR (wherein the scatterersreside) and the arc of the measuring circle MC at any pointP on which a receiving transducer can be positioned. Theabbreviations Tx (for transmitter) and Rx (for receiver)identify the two transducers. Cartesian co-ordinates x andy, with origin O at the centre of SR, are erected as shown.An arbitrary point in the 'scattering target' (i.e. the collec-tion of individual scatterers placed within SR) is denotedby(2-

The circle and the two truncated annuli, labelled Bl, B2and B3, respectively, positioned between SR and MC, rep-resent 'buckets' made of mylar in which we can place bio-logical tissue (e.g. animal offal). This is a very realistic wayof introducing propagation distortion. We emphasise herethat distortion is isoplanatic (by definition) when theamplitude and phase errors, which are introduced into thefield, scattered from each point Q in SR to any point P onMC, vary with P but not with Q. Inspection of thegeometry of Fig. 1 then confirms that the distortion mustbecome increasing nonisoplanatic as the buckets B3, B2and Bl are filled in turn with tissue. Our previous experi-ence [2, 11] suggests that the distortion should be mainlyisoplanatic when only B3 is filled, but is likely to be signifi-cantly nonisoplanatic when Bl is filled.

Tx emits linear FM sweeps of 26.2 ms duration from 1.7MHz to 3.4 MHz. A single sweep is emitted for each posi-tion P adopted by Rx. We choose the distance T largeenough to ensure that the incident insonifying beam isessentially plane by the time it reaches SR. Measurementsare made with Rx at several positions, spanning what wecall the measurement range (0l5 62). The values of 6 corre-sponding to the positions P adopted by Rx are spaced uni-formly in sin (9), for a reason made clear in Section 4.

The actual values of the experimental parameters arelisted in the caption to Fig. 1.

4 Data reduction

The species of speckle processing of interest here arose inoptical astronomical contexts [1]. Because the earth'satmosphere is a fluctuating propagation medium, one canrecord ensembles of statistically independent speckleimages by taking short exposures (typically of 10 msduration) at intervals of a few 100 ms. In many ultrasonicimaging applications (especially medical ones), whereas thepropagation media introduces severe distortion, theychange only slowly with time. In such cases, we cannotgather independently distorted data by recording sequen-tially. Our previous experience has shown, however, thatthe propagation distortion in closely spaced, compara-tively narrow frequency bands often exhibits a significantdegree of effective independence [2, 11]. Consequently, wetransmit a wide band of ultrasonic frequencies andseparate the received signals into a number of much narro-wer bands. The information gathered in each narrow bandis then processed (in the manner outlined below) to gener-ate each member of our ensemble of speckle images.

The received signals are separated into 128 contiguousnarrow bands. The number 128 is chosen because it iscomputationally convenient and provides large enoughensembles for our purposes. It is appropriate to think ofthe received signal at P in each narrow band as the value

52 IEE PROCEEDINGS, Vol. 132, Pt. A, No. 1, JANUARY 1985

of the scattering pattern in that band for the angle 9. Wewrite the scattering pattern for the mth narrow band asSJocJ, where

oem = sin (0) (15)

where Xm is the wavelength in water, at the temperature ofthe ultrasonic laboratory, at the centre of the mth narrowband.

Our aim is to image the scatterers within SR. The beamfrom Tx causes the scatterers to vibrate, thereby inducingin them reradiating sources which give rise to the scatteredfield. These sources are frequency dependent, implying thatan appropriate wavelength compensation must be intro-duced before they can be related to the true image. Thereis also the question of multiple scattering, which is takenup briefly in Section 7. We denote the reradiating sourcedensity in the mth narrow band by gm(x, y) = gm(x).

The transducer Tx is designed to illuminate SR with aplane wave the spatial variation of the latter is expressedby exp (—j2ny/?,m) in the mth narrow band, where Xm is thewavelength in the medium embedded in SR. This mediumis animal tissue when the bucket Bl is filled, and is waterotherwise. The mean ultrasonic refractive index of tissue isclose to that of water [11] and we approximated Xm by Xm

when reducing all the data for the experiments reported inSections 5 and 6; the error introduced by this assumptionmust increase with the linear dimensions of SR, and waysof compensating for it would be worth studying (however,we found it unnecessary to take account of this error wheninterpreting the experimental results presented in thispaper). We postulate that the main frequency dependenceof gm(x) can be removed by multiplying it by exp (j2ny/2.m)because this cancels out the phase variation due to the illu-mination by Tx. This does not compensate in any way forthe inevitable attenuation of the ultrasonic beam on itspassage through SR, but this is relatively unimportantbecause variations of magnitude with m have much lesseffect than variations of phase on averaging operations ofthe kind defined in eqns. 6, 9, 10 and 11. We thereforeassert that the quantity

gm(x) exp =f(x, y) =f(x) (16)

is appropriately associated with the true image (as intro-duced in Section 2).

The distance R is such that MC is effectively in the farfield of SR at 3.4 MHz. In the special case when all scat-terers reside on the y-axis (this corresponds to the 1-dimensional imaging situation, illustrated in Section 5), itfollows that gJO, y) and Sm(ccm) are ideally (i.e. in theabsence of propagation distortion, reception errors and

noise) related by Fourier transformation [14]. Thisexplains why Rx is positioned at points P spaced uni-formly in sin (0); refer back to Section 3. In the presence ofthe inevitable propagation and measurement imperfec-tions, we hold that the Fourier transform of a measuredSm(am) gives gm (0, y) convolved with a PSF, plus contami-nation due to the nonisoplanatic manifestations of the saidimperfections [2, 11].

When the scatterers are distributed throughout SR, weimage them along lines parallel to the y-axis. A separate1-dimensional transform is needed for each line. We notethat the difference between the lengths of the rays OP andQP is (x cos (0) + y sin (0)), in the far field. So, to form theimage in the mth narrow band along the line distance xfrom the y-axis, we multiply Sm(am) by exp (—j2nx(}m((xm))and then take its Fourier transform with respect to am,where

= OAJ cos (0) = - (aj2)1 '2 (17)

Since MC extends from 9X to 92, the limits on the Fourierintegral are aml and am2, where

«»i = (1MJ sin (0J and am2 = (1/AJ sin (02) (18)

We therefore write the mth image, for any point (x, y)within SR, as

x exp {-j2n[xpjctj + da (19)

Since each of the measured Sm(am) includes the effects ofpropagation/measurement imperfections, the am(x) havethe character of speckle images. Each am(x) is a blurredand contaminated version of gm(x) rather than f(x). Toobtain an ensemble of blurred versions of f(x) we need tomultiply each am(x) by the same factor which gm(x) ismultiplied by in eqn. 16. So, we define our mth speckleimage by

sm(x) = sm(x, y) = am(x, y) exp (j2ny/AJ

1-dimensional images

(20)

This Section is solely concerned with 1-dimensionalimages, for which we use the notations introduced in thepreceding Sections, except for replacing the position vectorx with the Cartesian co-ordinate y. Computer-generateddata are employed to illustrate the algorithms described inSection 2. With the aid of a standard pseudorandomnumber routine we generate PSFs which are convolvedwith true images to produce speckle images. As Figs. 2b

10r

0.5

10r

0.5

10r

0.5

^ J U A N A-10

a b eFig. 2 1-dimensional computer simulation of elementary shift-and-add. Image magnitudes are displayed

0yfmm

10

IEE PROCEEDINGS, Vol. 132, Pt. A, No. 1, JANUARY 1985 53

and 3b show, we ensure the blurring is severe enough toobscure the true image in each speckle image. Throughoutthis Section,/a(y) is actually/e(>>).

1.0

of the speckle images. On invoking the 1-dimensional ver-sions of eqns. 12 and 13, we obtain the \f(y)\ shown inFig. 3d. Fig. 3e shows the first version of | fmod{y) \. Note

0.5

0.5 -

-10 -10 K)

c y,mm /

Fig. 3 1-dimensional computer simulation of extension to shift-and-add. Image magnitudes are displayedaf(y), b s75O0> cfa(y) =fe(y), df(y), e first version of f^J^y), /third version of

Fig. 2a shows a true image whose brightest point standsout sufficiently sharply for us to expect/a(y) to mimic f(y)faithfully. Fig. 2c shows | fa(y) |, in which the ghost ampli-tudes are small, as anticipated.

The true image shown in Fig. 3a is rudimentary, but ithighlights the crucial deficiency of both elementary andsimple shift-and-add, i.e. their inability to distinguishbetween points of comparable brightness. In fact, each ofthe peaks in f(y) is selected by the algorithm as the shift-and-add reference as often as the other, on average. One ofthe side peaks (the right-hand one in this instance) isalways larger than the other because of the random nature

that even this first version is a distinct improvement onfa(y). The amplitude of the ghost reduces at each iteration,as is illustrated by Fig. 3/which shows the third version of

6 2-dimensional images

The results reported in this Section are illustrated by grey-tone images. Because the dynamic range of any such imageis relatively small, we always choose a threshold (expressedas a fraction £ of the brightest pixel in the image) belowwhich all pixel values are displayed as black. The values of

54 IEE PROCEEDINGS, Vol. 132, Pt. A, No. 1, JANUARY 1985

Fig. 4 2-dimensional images of hexagonal target; x and y axes aredirected across and up the paper

Image magnitudes are displayed:a the target consisting of 7 wires, the central one, of copper, is 0.183 mm diameter;the others, of nylon, are 0.125 mm diameterb s75(.x, y)cfsa(x, y) for waterdfjx, y) when B3 filled with offalefjx, y) when B2 filled with offal.e = 0.15 for images b, c, d and e

e adopted for Figs. 4, 5, 6 and 7 are listed in the captionsto these figures.

We begin this Section by demonstrating the per-formance of simple shift-and-add when it operates on mea-sured data. A scattering target, whose brightest point issharply defined, is shown in Fig. 4a. We call this the hex-agonal target; its physical characteristics are listed in thecaption to Fig. 4. A typical speckle image, formed fromdata obtained when the bucket B3 was filled with offal, isshown in Fig. 4b. Note that the form of the scatteringtarget is completely obscured. The three versions of|/Sfl(x, y)\ shown in Figs. 4c, 4d and 4e confirm that theshift-and-add principle can be effective under varyingdegrees of isoplanatism. The images appear increasinglydegraded, of course, as the blurring becomes less iso-planatic; Fig. 4e is significantly messier than Fig. 4c, butall seven of the 'wires' are reconstructed in their correctjuxtapositions in all three images.

Fig. 5 demonstrates that a naturally occurring bio-logical structure can be capable of serving as a shift-and-add reference. It also suggests that shift-and-add may be aviable (and perhaps more easily implementable) alternativeto the ingenious approach to resolution improvement thatD.E. Robinson and his colleagues have developed in thecontext of compound B-scan imaging [15, 16]. A bovineartery selected from the offal used to fill our mylar buckets

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was mounted in the water tank as a cylindrical scatterer.Fig. 5a is a sketch of the cross-section of the artery. Fig. 5bshows | fsa(x, y) |. Fig. 5c shows Fig. 5a superimposed uponFig. 5b. It is clear that a point on the inner wall of theartery is acting as a shift-and-add reference. The parts ofthe artery which are actually imaged (determined by thearc of MC traversed by Rx) are reminiscent of the high-lights displayed by B-scans (refer to Fig. 4a of Reference16); however, we emphasise that the inner diameter of theartery is clearly resolved in our Figs. 5b, c. Fig. 5d shows\fsa(x> y)\ when a copper wire was placed within SRtogether with the artery. The shift-and-add reference isseen to be the same as in Figs. 5b, c. This surprised usbecause we had anticipated that the wire would be thestronger scatterer.

In order to illustrate our extension to shift-and-add, wenow present images of a uniformly spaced grid consistingof eight identical wires. Figs. 6 and 7 relate, respectively, tothe grid being parallel to the y-axis and inclined at 45° toit. Figs. 6a and la are true images of the wire grid, whosephysical characteristics are listed in the caption to Fig. 6.Figs. 6b, c and d and 1b, c and d, show images reconstruct-ed from data measured when the bucket Bl was filled withoffal.

Figs. 6b and 1b show typical speckle images. Note thatthe blurring is so severe that the forms of the true images

55

Fig. 5 2-dimensional images of a bovine artery in water as a scatteringtarget

Image magnitudes are displayed:a sketch of a cross-section through artery, internal & external mean diameter are 1.3mm and 4.0 mmbfj^x, y) of the artery by itselfc superposition of a upon bdfsjtx, y) of the artery plus a copper wire of 0.183 mm diameterCo-ordinate axes as for Fig. 4. e = 0.05 for images b and d

Fig 6 2-dimensional images of wire grid, positioned along the y-axis,with Bl filled with offal. Image magnitudes are displayedImage magnitudes are displayed:a the grid itself, 8 copper wires, each of 0.183 mm diameter spaced by 1.1 mmbs15(x,y)cfjix, y)d fourth version of/m0<i(x, y)Co-ordinate axes as for Fig. 4. £ = 0.3 for images b, c and d

are obscured. Figs. 6c and 1c show the parts of | fsa(x, y) \that exceed the thresholds indicated in the respectiveFigure captions (note that when these images are closelyexamined on a high-dynamic-range display, many 'appar-ent' wires can be seen amid the background contami-nation, but they could not be unambiguously recognised

56 IEE PROCEEDINGS, Vol. 132, Pt. A, No. 1, JANUARY 1985

wires from the top are coalesced. Only six of the wires areapparent in Fig Id, due probably to the intensity of theincident beam being somewhat lower at the outer wires ofthe grid. We feel these results suggest that our extension toshift-and-add is potentially capable of significantly improv-ing the faithfulness of images reconstructed from measureddata.

Fig. 7 2-dimensional images of inclined wire grid, with Bl filled withoffalImage magnitudes are displayed:a the grid itself, with the same characteristics as for Fig. 6b sis(x, y)c/JLx, y)d fourth version o{fmoj,x, y).Co-ordinate axes as for Fig. 4. e = 0.13 for images b, c and d

without prior knowledge of their existence). When thealgorithm introduced in the final two paragraphs ofSection 2 is invoked, the wires stand out with increas-ing clarity after each iteration. To illustrate this, we showin Figs. 6d and Id the respective fourth versions ofI fmoAx-> y) I • All eight of the wires are reconstructed in Fig.6d. Taking the close spacing of the wires into account, thisis an encouraging result, even though the second and third

7 Conclusions

The chief attraction of the shift-and-add principle is itsextreme simplicity. In its basic form, however, it has thedrawback of requiring the true image to possess a sharplydefined brightest point. The extension introduced inSection 2 promises to relax this constraint appreciably.The results presented in Figs. 3, 6 and 7 support this con-tention.

All of the 2-dimensional images presented in this paperfavour our view that shift-and-add can be regarded as ageneralised CT technique (refer to the second paragraph ofSection 1), because of the considerable reduction (displayedby the shift-and-add images) in the artefacts which blemishthe individual narrowband images.

The scattering targets which we have so far studiedcould perhaps be considered rather rudimentary. Never-theless, the results are sufficiently encouraging for us to feelthe shift-and-add principle should be pushed as far as itcan go, because it seems to be much the simplest of theschemes suggested for realising diffraction-limitedresolution in the presence of severe and significantly non-isoplanatic propagation distortion.

The simple correction, introduced in eqn. 16, for thevariation with m of gm(x) only compensates for the phase ofthe incident beam on its initial passage through SR; but ittakes no account of any multiple reflections. This may notmatter, however, because we have recently presented theo-retical arguments, supported by experimental results, thatshift-and-add processing tends to reject all signals otherthan those scattered directly from the incident beam [17].This implies that shift-and-add permits the Rayleigh-Gans(or Born) approximation to be invoked even when the levelof multiply reflected signals is appreciable.

8 Acknowledgments

We acknowledge the provision of experimental and com-putational facilities, and the awards of postgraduateresearch scholarships to R.A. Minard and B.S. Robinson,from the University Grants Committee of New Zealand.We thank Dr. Jerry Zelenka, of Science Applications Inc.,for suggesting after a seminar by R.H.T. Bates in Tucson,Arizona, that some form of correlation processing mightbe usefully incorporated into shift-and-add.

9 References

1 BATES, R.H.T.: 'Astronomical speckle imaging', Phys. Rep., 1982, 90,pp. 203-297

2 BATES, R.H.T., and ROBINSON, B.S.: 'Ultrasonic transmissionspeckle imaging', Ultrason. Imaging, 1981, 3, pp. 378-394

3 BATES, R.H.T., and CADY, F.M.: 'Towards true imaging by wide-band speckle interferometry', Opt. Commun., 1980, 32, pp. 365-369

4 BATES, R.H.T., HUNT, B.R, ROBINSON, B.S., FRIGHT, W.R.,and GOUGH, P.T.: 'Aspects of speckle interferometric imaging'. IEEConf. Publ. 214, 1982, pp. 164-168

5 BATES, R.H.T., and ROBINSON, B.S.: 'A stochastical imaging pro-cedure', in ASH, E.A., and HILL, C.R. (Eds.) 'Acoustical Imaging Vol.12' (Plenum Publishing Corporation, 1982), pp. 185-191

6 BATES, R.H.T.: 'A stochastic image restoration procedure', Opt.Commun., 1976, 19, pp. 240-244

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7 BATES, R.H.T., GARDEN, K.L., and PETERS, T.M.: 'Overview ofcomputerized tomography with emphasis on future developments',Proc. IEEE, 1983, 71, pp. 356-372

8 BATES, R.H.T.: 'Full-wave computed tomography. Part 1: Funda-mental theory', IEE Proc. A, 1984,131, (8), pp. 610-615

9 SEAGAR, A.D., YEO, T.S., and BATES, R.H.T.: 'Full-wave com-puted tomography. Part 2: Resolution limits', ibid., 1984, 131, (8), pp.616-622

10 ROBINSON, B.S., and BATES, R.H.T.: 'Wideband ultrasonic diffrac-tion measurements', Australas. Phys. & Eng. Sci. Med., 1980, 3, pp.233-238

11 ROBINSON, B.S.: 'Speckle processing for ultrasonic imaging'. Ph.D.Thesis, Engineering Library, University of Canterbury, Christchurch,New Zealand, 1982

12 BRACEWELL, R.N.: 'Fourier transform and its applications'(McGraw-Hill, New York, 1978, 2nd edn.)

13 BATES, R.H.T., and FRIGHT, W.R.: 'Towards imaging with aspeckle-interferometric optical synthesis telescope', Mon. Not. R.Astron. Soc, 1982,198, pp. 1017-1031

14 JORDAN, E.C., and BALMAIN, K.G.: 'Electromagnetic waves andradiating systems' (Prentice-Hall, 1968, 2nd edn.)

15 ROBINSON, D.E., CHEN, F., and WILSON, L.S.: 'Measurement ofvelocity of propagation from ultrasonic pulse-echo data', UltrasoundMed. & Biol., 1982, 8, pp. 413-420

16 ROBINSON, D.E., CHEN, C.F., and WILSON, L.S.: 'Image match-ing for pulse echo measurement of ultrasonic velocity', Image &Vision Computing, 1983, 1, pp. 145-152

17 BATES, R.H.T., and MINARD, R.A.: 'Compensation for multiplereflection', IEEE Trans., 1984, SU-31, (4), pp. 330-336

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