Dirac, Faraday, Born and medical imaging

S. Leeman, MSc, DPhilProf. V.C. Roberts, MSc, PhDD.A. Seggie, PhD

Indexing terms: Image processing, Algorithms, Biomedical applications, Computer applications, Mathematical techniques

Abstract: Most modern medical imaging methodsrely on the analysis and processing of acquireddata to reconstruct a desired tomographic image,generally via digital methods. A unified view ofthese techniques is developed here, using classicalapproaches associated with Dirac, Faraday andBorn. A formalism due to Dirac is shown toprovide a unifying basis for standard reconstruc-tion-from-projection algorithms, as well as topoint the way towards new methods. An unusualapproach towards enhancing backprojectedimages is based on Faraday's work, while anapproximation first proposed by Born is shown tolink imaging methods utilising scattered radiation,to those based on a linear ray hypothesis.

1 Introduction

A feature of virtually all of the many newly developedapproaches towards medical imaging is that the desiredimage is not obtained by a direct display of the measureddata, but emerges only as a result of considerable furthermanipulation or mathematical analysis of large sets ofmeasurements. Because such processing is generallyachieved via digital techniques, modern medical imagingoften relies on the availability of dedicated, and not insig-nificant, computer power.

It may be argued that there are essentially two differ-ent classes of medical imaging techniques. The first com-prises those methods where some (medically acceptable)radiation is allowed to penetrate into a selected region ofinterest within the patient. The radiation is modified byinteractions before it emerges from the body, and mea-surements of the resultant field outside the patient,together with an accurate knowledge of the input field,provide the data set from which an image must bereconstructed. Clearly, the end point of this class ofmethods is a mapping of some parameter which charac-terises the interaction(s) of the probing radiation withinthe human body. The second class of methods attemptsto map the distribution of some source of radiation dis-persed throughout a region of interest, again from mea-surements conducted outside the patient. The radiation

Paper 5152A (S9), first received 16th October and in revised form 11thDecember 1986.Dr. Leeman and Prof. Roberts are with King's College School of Medi-cine & Dentistry, Department of Medical Engineering and Physics,Dulwich Hospital, East Dulwich Grove, London SE22 8PT.Dr. Seggie is with University College London, Department of Phonetics& Linguistics, 4 Stephenson Way, London NW1 2HE, United Kingdom

source may either be artificially induced, as in nuclearmedicine techniques, or may be a consequence of naturalactivity within the body (e.g., thermographic imaging).

A major problem besetting both approaches is that themeasurement of the emergent radiation at any given(external) field point cannot be directly related, ingeneral, to a specific (internal) interaction site or sourcepoint. However, by dint of painstaking and imaginativeexperimental ( = hardware) design, as well as by carefulchoice of radiation employed, it has long been possible tostrongly restrict the range of internal locations fromwhich contributions to the value of an external field pointmeasurement may arise. Thus, to choose but a singleexample from many, an external 'point' X-ray source,suitably filtered, or otherwise devised to emit radiation ina predominantly narrow energy band only, may be usedin conjunction with appropriately collimated detectors toproduce highly informative images, for which each imagepoint may be quite accurately interpreted as relating tothe total attenuation suffered by the X-ray beam alongsome well defined ray-like unique and relatively restrictedpathway through the patient. However, ascribing theimage information to a specific 'point' within a region ofinterest, calls for both the judgment of an experiencedinterpreter as well as, in all likelihood, the amalgamationof information gleaned from more than one image alone.It is the exploitation of the latter, by means of advancedcomputer methods, that finds its elegant expression in thewell known computerised tomographic imaging methods.

It is not intended to review all such imaging methodshere, or to discuss the physical principles on which theyare based, or to describe the often remarkably ingenioushardware configurations and experimental techniqueswhich facilitate their implementation. Rather, an attemptis made to show, in a general way, that the mathematicaltechniques presently used in computerised medicalimaging may be related, in quite unexpected ways, toclassical analysis methods extensively used in otherbranches of physics. In this way, many apparent diversethreads of thought may be brought together, to indicatethe underlying unity of all such medical imagingmethods, as well as to suggest new avenues for investiga-tion.

2 Projection and backprojection

In the following, some basic ideas in the theory of imagereconstruction from complete sets of projections arereviewed. A much more complete treatment, as well asreferences to the very extensive literature on the subject,may be found, for example, in the treatise by Herman [8]and the review by Bates, Garden and Peters [9].

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987 101

It is convenient to refer the entity which is desired tobe mapped on a point-by-point basis, as the density. Inimaging with X- or y-rays, for example, the density mayrefer in some techniques to the appropriate tissue attenu-ation coefficient; in other approaches, it may refer to thetissue electron density or atomic number distribution.

In most examples of practical interest, it is possible torestrict, by hardware design, the source or interactionsites of the measured radiation to a selected planethrough the patient (object). Thus, for simplicity, it is suf-ficient to consider only a two-dimensional scalar densityfunction D{r), with r denoting spatial location in theplane of interest. It is convenient to express r in terms ofa Cartesian co-ordinate system [X, Y~\ located within thisplane: a particular orientation of the system is assumed,such that the X-axis points along the direction indicatedby the unit vector n.

A notion of fundamental importance is that of the pro-jection of D{r) (== D{x, y)) onto the direction n. Formally,this is defined as

duJ0( | r - r' | u)

x) - J - , dy'D(x, y')

Note that the projection is a one-dimensional function,and conforms closely to the notion of the (parallel,straight ray) 'shadow' of the density function. Each direc-tion n will determine a different projection, so that thefunctional form of P depends on n. It is for this reasonthat n appears in the notation for the projection: it is nota variable in the usual sense.

The importance of the projection concept for medicalimaging lies in the circumstance that it is this feature ofthe object density that may be accurately measured, for avery wide variety of radiations and experimental configu-rations.

By the well known properties of the Dirac delta func-tion [1], the projection may be written as

P(n;x)= | dx' r dy'D(x',y')S(x-x')J - oo J - oo

= drD(r)d{x — r • n) as x' = r' • n

The importance of expressing P in this concise form liesin the useful circumstance that n now appears as a simplevariable in the argument of the delta function.

An interesting entity, associated with any projection, isits backprojection. This is a two-dimensional density,formed by smearing out the values of P(n; x) along thestraight line paths perpendicular to the n-direction.Clearly, the backprojection BP(n; r) is related to theappropriate projection via

BP(n; r) = P(n; r n)

The backprojection bears, in general, little resemblanceto the original density, but something approaching theobject distribution may be recovered by simply summingall the backprojections, to form the so-called back-projected image, IBP(r),

IBM = I BP(n; r)

dr'D(r')S(r n - r' n)

= (1/2TT) \ dO I dr'D{r') \ du

J0 J J - oox exp ({i\r — r'\u cos 9}

= \dr'D{r')l\r-r'\

(1)

where ® denotes the convolution operation. In this deri-vation, both a representation for and documented inte-gral of the Bessel function of zero order Jo have beenused [1], as well as the well known representation for theDirac delta function

(2)3(t) = (1/2TT) du exp {iuf}

The backprojected image is of some interest, as it maybe formed directly by analogue methods [2], but it isclear that a troublesome deconvolution procedure willhave to be implemented before high-resolution images ofD(r) can be obtained. The usual interpretation of eqn. 1 isthat the backprojected image is a somewhat smoothedversion of the density, and that conventional deconvolu-tion techniques should be applied. A somewhat differentinterpretation, suggesting different avenues to search foran effective deconvolution approach, is posited below.

3 Reconstruction from projections via Dirac

The range of applications of reconstruction from projec-tion algorithms in medical imaging can hardly be exag-gerated. Three analytic methods (the central slicetheorem, the filtered backprojection technique and theradon transform) have been the foci of particular atten-tion [8, 9]. Given the validity of any one of the methods,the other two may be naturally derived, as they are, afterall, merely different expressions of the same decomposi-tion ; namely, the expression of the object density in termsof its projections. Conventionally, the radon transform[10] has been perceived to be the underlying basis for allthese techniques; in the following, it is suggested that theDirac delta function, by virtue of its manifold of repre-sentations, underpins a particularly straightforward, butpowerful, approach towards deriving reconstructionalgorithms independently and relatively simply. Althoughno new reconstruction algorithms are derived here bythis previously unexplored approach, it is felt that thetechnique obviates some of the more abstract mathemati-cal manipulations necessary when following the conven-tional route, via the radon transform [8, 10].

A fundamental property of the Dirac delta functionenables the following identity to be written for thedensity

•ID(R) = drD(r)d(r - R) (3)

On invoking the representation eqn. 2, the following formis derived:

D(R) = [ 1 / 4 T I 2 ] j drD{r) \ dk exp {i(r - R) • k)

Noting that

TOO

exp {ir • k] = I dx exp {ixk}d(x — r • n)J-ca

102 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987

with k = nk, allows the density to be written as

D(R) = [1/4TT2] dk exp {-iR • k] °° dx

x exp {ixk} drD(r)d(x — r • n)

This may be symbolically condensed to

D{R) = FT-Rl{FTk{P{n;x)}}

where FT denotes the Fourier transform operation

FTk{f} = [1/2*] dxf{x) exp {ikx}J— oo

and with F T " 1 denoting the inverse transform:

FT~l{FT{f}}=f

It now follows that

FTk{D(R)} = FTk{P(n; x)} (4)

This is the central slice theorem: the (two-dimensional)Fourier transform of the density function assumes thesame values along the line in the direction n through theorigin in Fourier space, as the (one-dimensional) Fouriertransform of the projection of that density onto the n-direction. In this way, in practical medical imaging, theFourier transform of the density in a plane through thepatient may be built up from the Fourier transforms ofthe (measured) projections in that plane. The actual two-dimensional image follows by inverse (two-dimensional)Fourier transformation.

To derive this fundamental theorem, only the mostbasic of the properties of the delta function have beenused. Indeed, the theorem follows almost immediately onincorporating eqn. 2 into the formalism.

Another representation for the delta function may becontrived as follows:

f4n2d(r -R)= du exp {i(r - R) • u)

J

= \ dd \ dkk exp {i{r - R) • nk}Jo Jof* f00

= \ d<f> \ dK | K | exp {i{r - R) • NK}Jo J-oo

(5)

where the area element du has been expressed first, forcomparison, in the familiar standard polar form (u = nk,with direction of n specified by the angle 6), and then,equally validly, in terms of a vector K whose magnitudeK ranges over ( —oo, +oo), and with direction specifiedby the unit vector N making an angle (j> with some refer-ence direction. For consistency, </> lies in the range (0, n)only.

Eqn. 3 may now be written in 'A"-space', to give

4n2D{R) = | d<t> I dK \ K| exp { - i K R • N} \ dxJo J-oo J-oo

x exp {ixK} drD(r)S(x - r • N)

= d(f> \ dK\K\Jo J-oo

x exp {-iR • NK}FTK{P(N; x)}

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987

This last expression embodies the filtered backprojectionmethod for reconstructing a density from its projections.The following steps are involved; form a projection, filterit by transforming to Fourier K-space and multiplying by\K\, transform back to /?-space (as indicated by the K-integration), then, finally, form the backprojected image(as indicated by the (^-integration) from the modified pro-jections.

On remembering the properties of the derivative of thedelta function [1], it is straightforward to rewrite eqn. 5as

4n2S(r -R) = i \ d<f> \ dK \ dx sgn (K)Jo J-oo J-oo

x 5'(x - r • N) exp {i(x - r • N)K}

where sgn (K) = +1 for K > 0, - 1 for K < 0, 0 forK = 0 and where ' denotes differentiation with respectto x.

It may be shown [3] that, for some function/(x), thefollowing holds:

I dxf(x)/x = -{i/l) \dx\dy sgn (y)f(x) exp {ixy}

where $ denotes the Cauchy principal value integral. It isnow a simple matter to arrive at the representation

2n2S(r -R) = dxd'(x - r • N)/(x - R • N)

Substitution into eqn. 3 leads to the Radon transformrelationship:

2K2D(R) = dxP'{N; -R N)

The various analytical reconstruction-from-projectionalgorithms have been shown here to be independentlyrecoverable via classical techniques pioneered by Dirac.In particular, these reconstruction algorithms (includingthe radon transform) are seen to be directly associatedwith different representations of the Dirac delta function.Although the derivations have been for two-dimensionaldensities, it is clear that the methods are immediatelyapplicable to three (and even higher!) dimensions as well.The implication of this result is that other reconstructionalgorithms may be discovered by examining appropriaterepresentations of the Dirac delta function.

4 Faraday and the backprojected image

The backprojected image is derived from the density bythe convolution operation shown in eqn. 1. Convolutionis usually associated with filtering techniques, but it isinteresting to observe that there is a more directly physi-cal interpretation. It is clear that the density D(r) isrelated to the backprojected image IBP(r) in precisely thesame way as an electric charge density p{r) is related toits electrostatic potential U(r). Thus [4]

U(r)= \dr'p{r')/\r-r'\

This corresponds exactly to the expression in eqn. 1, withthe correspondences

U+->IBP

Consequently, as far as formal mathematical manipula-tions are concerned, there is no difference between hand-ling the relationship between the backprojected image

103

and the object density, to handling that between the elec-trostatic potential and the electrostatic charge density!This also suggests that simple imaging analogues of evensomewhat complicated problems in electrostatics may beeasily constructed. Of course, a similar analogue may bemade in terms of the relationship between a mass densityand its gravitational potential, but electrostatic fields aremuch more easily measured and manipulated in a labor-atory environment. The extensive framework of knowl-edge accumulated about the electrostatic field, originallypioneered by Faraday, may now be probed to uncoverproperties of the backprojected image. This is amammoth task, and only some early results are reportedhere.

A much used, and fairly easily proven, result in thetheory of the electrostatic field [4] is

= -4nd(R) (6)

where R is some location vector in a three-dimensionalspace. Application of eqn. 6 to eqn. 1 leads to the ana-logue of the Poisson equation of classical electrostatics[4]:

Eqn. 7 provides the basis of an exact, and deceptivelysimple, technique for recovering the density from thebackprojected image. However, a problem arises becauseeqn. 7 holds in three dimensions only. Thus, even if thedensity is to be mapped in a single plane only, a wholefamily (both in- and out-of-plane) of densely packedbackprojected images would nonetheless be required forthe full (three-dimensional) contribution of the V2 oper-ator to be evaluated.

In a strictly two-dimensional context, D(r) may beregarded as the analogue of a surface-charge density.Unfortunately, even in this case, some knowledge of theout-of-plane behaviour of IBP(r) is needed. This arisesbecause

V2 = d2/dx2 + d2/dy2 + d2/dz2

and only the x- and ^-components of the operator can beevaluated from the two-dimensional backprojected imageitself.

Two remedies suggest themselves, if it is desired thatthe procedure be implemented with only a single back-projected image. One is to estimate the out-of-planevalues of IBP(r) by some technique such as analytic con-tinuation; for this approach, there are bound to be anumber of powerful methods waiting to be discovered inthe classical electrostatics literature. Another, more ele-mentary, but naive, approach is to disregard all suchdimensional difficulties, and to simply apply the two-dimensional operator, d2/dx2 + d2/dy2, to the availablebackprojected image. While this is obviously not an exacttechnique, it should provide a reasonable estimate of thedensity in a plane, at least in some cases.

Another interesting result may be established by calcu-lating the field associated with the potential IBP(r). It maybe shown from eqn. 1, using the type of vector algebracommon in electrostatic field theory [4], that

= <p t x dlD{r)/\R- r\ drVD(r)/\R-r\

(8)

Here, V is the two-dimensional gradient operator, R andr are two-dimensional vectors (in the image plane), D(r) is

assumed bounded by the contour C, with dl the elementalcontour line element, t is a unit vector perpendicular tothe image plane, and x denotes vector multiplication.

Eqn. 8 shows that, provided the density is relativelysmoothly varying within its (sharp) boundary C, then the(two-dimensional) gradient of the backprojected imagetakes on a maximal value on the density boundary. Thus,the gradient operator provides a convenient approachtowards using backprojected images for obtaining out-lines of relatively smoothly varying density, particularlywhen the latter is piecewise continuous, with regions ofhigh density relatively isolated from one another. Thisresult may be of interest for radiotherapy planning with(cheap?) analogue tomographic systems with only rela-tively crude processing capabilities.

5 Born and inverse scatter imaging

Implicit in most of the reconstruction-from-projectionimaging techniques, is the notion that the probing radi-ation propagates in a ray (geometric optics) fashion, sothat the density may be summed along well defined path-ways (line integrals) to form the projection. In thiscontext, the wave nature of the radiation is not overtlymanifest, and may be disregarded to good approx-imation. For certain radiations, however, the wavelengthis of the same order, or larger, than the tissue structureswith which interaction occurs, and significant scatteringresults. Under these conditions, the ray approximation isno longer valid. This is a particular problem whenattempting to do computer-based medical imaging withultrasound. However, methods may be devised whichincorporate measurements of scattered radiation intoalgorithms which reconstruct maps of the scatteringinteraction density from the data. Such approaches maybe referred to as inverse scatter imaging techniques.While these methods have not yet been advanced to thestage of being pressed into routine clinical use, they areregarded by many as the most likely route whereby thenext generation of clinical ultrasound scanners will beaccessed. A good overview of the various approaches tothe problem, with comprehensive references, has beengiven by Wade [11].

We have suggested that these methods may be regard-ed as consisting of three basic components [5]: (a) anunderlying physical model which specifies the propaga-tion and scattering of the wave in the object; (b) a dataacquisition configuration which may be chosen to mini-mise eventual artefacts in the final image and (c) a com-putational model which is essentially the algorithmwhereby the desired mapping is recovered from the mea-sured data set. In practice, most effort has been devotedto the last aspect of the problem.

We have chosen to further classify the ultrasoundmethods, in particular, into three major groupings,according to their data acquisition configurations: (a)through transmission methods, which may be regardedas utilising the forward-scattered field but are actuallyimplemented as conventional reconstruction-from-projection methods, (b) reflectivity tomography methods,which utilise the backscattered field only, and whosereconstruction algorithms are very close to those basedon ray (but not necessarily straight-line path) consider-ations, (c) diffraction tomography techniques, whichmeasure the angle-scattered fields, and for which a hostof new, unfamiliar, reconstruction algorithms have arisen.

Only diffraction tomography methods are consideredhere. Despite the consistent designation of these methods

104 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987

as tomographic, they are all, indeed, essentially three-dimensional imaging techniques. The computationalmodels employed in this context fall into essentially twomain groups: inverse Fourier techniques and (somewhatcontroversial and very computer-intensive) successiveapproximation schemes. It is demonstrated below that itis the classical Born approximation of scattering theorythat allows inverse Fourier techniques to be expressed asanalogues of the filtered-backprojection and central-slicealgorithms of conventional reconstruction-from-projection methods. Moreover, the Born approximationacts as a unifying basis which underpins the essentialidentity of all diffraction tomography techniques.

It is sufficient to work within the framework of a rela-tively simple physical model; namely, the inhomogeneousHelmholtz equation. This wave equation is appropriatefor describing both acoustic and electromagnetic wavepropagation in inhomogeneous media, under a variety ofconditions of relevance to medical imaging. A linearlypropagating wave \j/(r), of circular frequency co, is thusassumed to propagate through the inhomogeneousmedium according to

k2\p{r) = D{r)\Kr) (9)

As before, r denotes the spatial location vector; the con-stant magnitude of the wave vector k is equal to co/c,where c may be taken to be the mean wave velocity inthe scattering region. The scattering interaction densityhas been denoted by D(r), and a mapping of this functionconstitutes the desired image. The actual functional formof the scattering density, and its dependence on the physi-cal properties of the scattering medium, will vary mark-edly with the type of radiation field, but the generalformulation eqn. 9 is sufficiently accurate for many pur-poses.

As with all linear differential equations, it is possible toformally express the Helmholtz equation in its integralformulation:

dr'G{r, r')D(r'W(r') (10)

where ij/0 denotes the incident field (i.e., the wave thatwould exist in the absence of the scattering interaction),and G denotes the Green's function which embodies theboundary conditions appropriate for the scatteringproblem,

G(r, r') = e x p {ik\r-r'\ } / 4 T T \ r - r ' \

It is convenient to assume that the scattering region Vfills a volume contained within a bounding surface B.

In general, ip0 may be considered known or accuratelymeasured, and the form of the integral eqn. 10 suggeststhat the field may be considered everywhere to be thesum of the incident field and a scattered field ij/s(r) with

Mr) -i dr'G(r, (11)

In practice, it is a set of measurements of ij/s outside thescattering region, bounded by B, that constitutes a physi-cally meaningful data set, and it seems immediatelyobvious from eqn. 11 that this is insufficient to recoveraccurate values of the density D within the scatteringregion. The basic problem is clearly that, at the very least,a knowledge of the field within V will be needed to inverteqn. 11 to give an expression for the desired D in terms ofa measured ij/s.

One way round these difficulties is to invoke the (first)Born approximation, which states that, when the scat-tering is not too strong, then the scattered field may beexpressed as

Mr) = f dr'G{r, r')D(r>0(r')Jv

For the case that the incident wave is a plane wavedirected along the unit vector n, i.e., ij/0{r) = exp {ikn • r},the Born approximation becomes

Mr) = (1/4TT) dr' exp{ik\r-r'\}Jv

x D(r') exp {ikn • * • ' } / \ r - r ' \ (12)

Up to now, our analysis has followed a more or less con-ventional path. At this point, however, we introduce aparticularly powerful and succinct notation, which allowsus to derive a new approach which makes the essentialunity of the ray-path and diffraction-tomographyapproaches immediately apparent. We write eqn. 12 sym-bolically as

with

A(i-)

and

9(r)

= A®g

= D{r) exp {ikn • r)

== exp {ikr}/r

Assume now that a function g l(r) may be found, suchthat

9®9~l = 1

Then, it is clear from eqn. 12 that

A(r) = ij/J(r) (x) g ~ l(r) (13)

Provided that g ~x can be derived, and provided that theBorn approximation is a good one, then the right-handside of eqn. 13 may be computed from the measuredvalues of the scattered field, to obtain A and hence D. Afurther simplification is obtained by noting that

(14)

(15)

A(r) = dr'A(r')6(r - r')

= dr'^BP{r')d{r - r')

where if/Bp = ij/s®g Ms known as the backpropagated(scattered) field. By writing different representations forthe ^-function, the algorithms for reconstruction fromprojections may once again be recovered. Hence, onemethod for carrying out diffraction tomography wouldbe to carry through a filtered backprojection algorithmon the projections of the backpropagated scattered wave.The necessary projections may be obtained by(numerically) backpropagating the measured scatteredfield values. This is the essence of the filtered back-propagation technique [6]: it depends rather critically onthe validity of the Born approximation.

We propose that another approach would be toforward propagate the measured scattered field into theasymptotic region, i.e., very distant from the scatteringregion V. Alternatively, the measurements of the scat-tered field may be considered to be performed only atvery far distances from V (far-field measurement). In thefar field, the scattered wave behaves as an outgoing

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987 105

spherical wave, modulated in amplitude and phase by thescattered amplitude Q which has the form [7]

(16)Q(0, 4>) = (1/4TT) drD{r) exp {-ik(m-n) • r}

where n is the unit vector in the direction of propagationof the incident plane wave, and m is a unit vector indicat-ing the direction (6, 4>) with respect to n, in which themeasurement of the far field is taken.

It is convenient to introduce the notation

ks = km

kj = kn

in order to write eqn. 16 as

This shows that a relationship of the Fourier transformtype exists between Q and D. It may be inverted to yield

D(r) = n dfiQ(fi) exp {ifi • r}

Hence, £>(#•) can be computed if the scattering amplitudecan be measured for all values of ft. Indeed, the variousways in which measurements may be devised to access allof the ji-space are, in reality, different implementations ofthe diffraction-tomography concept, and not, as oftenclaimed, new types of scatter-imaging techniques.

It is interesting to observe that eqn. 16 indicates thatthe measured data (Q, in this case) can provide informa-tion about the Fourier transform of the desired density.This is to be compared with eqn. 4, which conveys muchthe same idea, but in a rather different context. Thus, theclassical approach of Born may be construed, in thissense, to link the methods of inverse scatter imaging tothose of conventional (linear-ray) reconstruction fromprojections.

6 Conclusions

It has been shown that the three fundamental analyticinversion algorithms for reconstruction-from-projectionimaging methods may be independently derived on thebasis of manipulations involving only fundamentalproperties of the Dirac delta function. The different algo-

rithms are related directly to different representations forthe delta function, and it can be suggested that this is thekey to uncovering novel reconstruction techniques. Cer-tainly, the approach appears to be didactically straight-forward, and simpler than the usual derivation whichproceeds, ultimately, from the properties of the Radontransformation.

An association with Faraday's work was made byinterpreting the backprojected image as the electrostaticpotential of a charge with the same spatial distributionand strength as the object density function. The resultsand methods of electrostatic field theory were invoked tosuggest new approaches towards recovering either anexact mapping, or, at least, some important features, ofthe object density.

The Born approximation was shown to be essential forrelating some of the inversion algorithms of imaging withscattered radiation to those based on ray-path propaga-tion. A new derivation of diffraction-tomographymethods was given, and a forward propagation techniquewas proposed, which simplifies some of the mathematicalcomplexities associated with conventional approaches. Inthis context, Born's work acts as a unifying link betweenthe diffraction and geometric optics approximations towave propagation in inhomogeneous media.

7 References

1 BRACEWELL, R.: 'The Fourier transform and its applications'(McGraw-Hill, 1978, 2nd edn.)

2 GMITRO, A.F., GRIEVENKAMP, J.E., SWINDELL, W.,BARRETT, H.H., CHIU, M.Y., and GORDON, S.K.: 'Opticalcomputers for reconstructing objects from their X-ray projections,Opt. Eng., 1980,19, (3), p. 260

3 KURSUNOGLU, B.: 'Modern quantum theory' (Freeman, 1962)4 JACKSON, J.D.: 'Classical electrodynamics' (John Wiley, 1975, 2nd

edn.)5 LEEMAN, S., and ROBERTS, V.C.: 'Inverse scatter imaging', in

BERKTAY, H.O. (Ed.): 'Ultrasound in medicine' (Institute ofAcoustics, 1986)

6 DEVANEY, A.J.: 'A filtered backpropagation algorithm for diffrac-tion tomography', Ultrason. Imaging, 1982,14, (4), p. 336

7 LEEMAN, S.: 'Impediography revisited', Acoust. Imaging, 1980, 9,p. 513

8 HERMAN, G.T.: 'Image reconstruction from projections: the fun-damentals of computerised tomography' (Academic Press, 1980)

9 BATES, R.H.T., GARDEN, K.L., and PETERS, T.M.: 'Overview ofcomputerised tomography with emphasis on future developments',Proc. IEEE, 1983, 71, (3), p. 356

10 HELGASON, S.: 'The radon transform' (Birkhauser, 1980)11 WADE, G.: 'Ultrasonic imaging by reconstructive tomography',

Acoust. Imaging, 1980, 9, p. 379

106 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987