SCIENCE

Full-wave computed tomographyPt. 1 : Fundamental theory

Prof. R.H.T. Bates, D.Sc.(Eng.), C.Eng., F.I.P.E.N.Z., F.A.C.P.S.M.,Fel.I.E.E.E., F.R.S.N.Z., F.I.E.E

Indexing terms: Computer applications, Biomedical engineering, Mathematical techniques, Tomography,Inverse theory

Abstract: The definition of computed tomography (CT) is generalised to the formation, from data gatheredusing any appropriate physical process, of a clean image of a cross-section of a body. The theory is based on acanonical partial differential equation of second-order, so that the existence of reflection, refraction, scatteringand diffusion is recognised implicitly from the outset. It is shown that the mathematical physics of manytechnically important physical processes can be reduced to canonical form. The general CT problem is formu-lated, and a new exact solution is developed. The significance of this solution is assessed, as is its relation toestablished approximate solutions.

1 Introduction

The spectacular successes of X-ray computed (orcomputer-assisted) topography (now generally abbreviatedto CT) in medical diagnostic radiology has emphasised tothe engineering science community the practical use aswell as the theoretical importance of forming high-qualityimages of the interiors of bodies. The remarkable apparentfreedom from artefacts of the latest NMR (nuclear mag-netic resonance) images [1] highlights this further. A hostof imaging procedures and data inversion techniques, invirtually all major fields of technical science, are now beingput into a CT context [2]. It has even been suggested thatCT of the earth should be attempted with high-energy neu-trinos [3].

Images of the best quality are obtained only in applica-tions which permit very considerable simplification of theunderlying physics [2]. Medical ultrasonics illustrates thisnicely. By far the most useful modality is the conventionalpulse-echo B-scan, which nowadays rivals X-ray CT andNMR for image quality. Its image reconstruction algo-rithms are based, however, on elementary sonar (or radar)concepts. While attempts at accounting for diffractioneffects in transmission tomography (the direct ultrasonicanalogue of X-ray CT) are showing promise in simple situ-ations [4], it seems safe to state that satisfactory practicalresults have so far only been obtained when one canapproximate wave motion by bundles of rays, and alsoassume that ray curvature is everywhere negligible.

It is important to make a clear distinction between thekinds of image that are provided by a conventionalimaging system (for example microscope, binoculars, etc.)and a CT system. The latter displays images that are'clean', in the sense that any feature in a clean image isuncontaminated by other features separated from it bymore than the effective resolution of the system [2]. Notethat a conventional imaging system cannot generate aclean image because wave motion brought to focus at apoint must converge before reaching it and diverge afterpassing it. Each such point therefore contaminates the restof the image throughout the double cone which containsthe converging and diverging rays.

This is the first in a series of papers devoted to speciesof CT which can provide clean images when the conven-

Paper 3339A (S9), first received 19th July 1983 and in revised form 26th March 1984The author is with the Department of Electrical & Electronic Engineering, Uni-versity of Canterbury, Christchurch, New Zealand

tional simplifcations no longer apply. Approximations stillhave to be made, but they are much less restrictive thanthose on which established CT image-reconstruction algo-rithms are based. Because most of these algorithms gener-ate two-dimensional images, and because it is so mucheasier to visualise in two as opposed to three dimensions,and because any three-dimensional image can be built upby stacking two-dimensional images on top of each other,all of the papers will be concerned with two-dimensionalimaging problems. It should be noted that this implies,strictly, that either the bodies being imaged vary negligiblyin the direction perpendicular to the 'imaging plane' or thephysical processes producing the data from which theimages are formed exhibit no tendency to wander out ofthis plane. Even though these are restrictive constraints,the physical situations treated in this series of papers willbe found to have practical significance. Furthermore, theproblems to be posed in the papers will be seen to be ade-quately challenging in their two-dimensional forms. Onehopes, of course, that once the two-dimensional difficultieshave been properly disposed of, it will then prove possibleto develop useful three-dimensional treatments.

It should be kept in mind, however, that the experi-ments which are to be reported in later papers will bearranged so that there is negligible variation of the wavemotion, or whatever kind of emanation is employed toprobe the body which it is desired to image, in directionsperpendicular to the planes of the images. The emanationsare assumed to satisfy linear partial differential equationsof second-order, thereby permitting one to account forseveral significant physical effects largely neglected in mostprevious treatments of CT. This is why the term 'full-wave'is included in the title of the planned series of papers, eventhough some of them will deal with emanations that areessentially nonwavelike. The unifying principle is thereconstruction of clean images using algorithms based on acanonical formulation to which the mathematical physicsof all the different emanations can be reduced. Therestriction to linear equations of second-order excludescertain species of emanation which it may (one hopes)eventually prove feasible to bring into a more general CTcontext.

Section 2 sets the scene and defines the general CTproblem. The reduction to canonical form of a wide rangeof physical processes is carried out in Section 3. Thecentral difficulty associated with the general CT problem isidentified in Section 4. A new exact solution to thisproblem (which could also have been called the general

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profile inversion problem) is developed in Section 5, andSection 6 discusses established approximate inverse theo-ries and points out where they break down. The signifi-cance of the approach adopted in this paper is assessed inSection 7.

While the results obtained in Section 3 could probably,after much searching, be found scattered throughout theliterature, the way they are organised here can be claimedto be original as well as useful. The inverse solutionannounced in Section 5 is new. It has been developed froma previous treatment [5], in which the author failed to rec-ognise the central result: that the determinant introducedin eqn. 33 depends upon two variable parameters, ratherthan upon a single parameter as the author thought in1975.

2 General CT problem

A cross-section of a static body lies within the 'circum-scribing circle' Fc, of radius a, shown in Fig. 1. An experi-menter directs certain emanations at the body, from

Fig. 1 Geometry and co-ordinate systems for the general CT problemNote that, in the text, the co-ordinate perpendicular to the p, 0-plane is denotedby 2

outside the 'observation circle' To, of radius b. The emana-tions that have interacted with the body (i.e. they haveeither passed through the body or have penetrated into itand been reflected or scattered) are detected by sensorssituated on Fo. For the reasons given in Section 1, boththe body and the emanations are assumed to vary negligi-bly in the z-direction, which is perpendicular to the cross-section.

The word 'emanation' is invoked [2] because it seems tobe the best portmanteau term covering wave motion, parti-cle beams, radiation and conservative fields, the mathe-matical physics of all of which can be reduced to aconvenient canonical form (as is shown in Section 3.)

An arbitrary point P inside or on Fo has the polar co-ordinates p and 4>. An arbitrary point Q on Fo is identifiedby the polar angle 6. The emanations are represented bythe scalar function ^{p; 0, t), where the convention isadopted of placing a semicolon, as opposed to a comma, infront of an angular variable (inside parentheses denotingfunctional dependence). One of the conveniences of thetwo-dimensional formulation is that a wide class of ema-nations can be characterised by scalar functions (seeSection 3.)

The generalised CT problem is here posed as: given theform of the emanations applied to the body, and the mea-surements made on Fo, reconstruct the distribution(throughout the interior of Fc) of the constitutive par-ameters of the body.

When setting out to solve an inverse problem, one firsthas to decide whether to approach it via the time or fre-quency domains. The former is usually easier conceptuallybecause one can build up a picture of sequential inter-actions between the emanations and the body. Eventhough there is rarely any significant difficulty in construc-ting a general flowchart for evaluating the interaction ateach instant and estimating the consequent evolution ofthe emanations [6], it is usually far from clear how toensure numerical stability.

It is instructive to consider briefly a type of situationwhich can, in fact, be handled straightforwardly in the timedomain. Suppose that the interior of Fc is empty apartfrom isolated regions whose linear dimensions are smaller,and whose separations are greater, than the inherentresolution of the envisaged CT system. The interaction ofthe emanations, and their consequent evolution, can thenbe simply evaluated at a sequence of distinct instantswithin each region. One might, of course, argue that thisapproach should merely be taken to its obvious 'limit'when the constitutive parameters differ from zero through-out (most of) the interior of Fc (so that the aforesaidregions merge into each other). Computational algorithmsare necessarily 'discrete', however, and nobody yet seemsto have summoned up enough intellectual courage andenergy to develop a thoroughly satisfactory multidimen-sional time-domain inverse numerical procedure.

Regretfully, then, a frequency-domain approach isadopted in this paper and in those to follow. Instead of thetemporally evolving emanation *P(p; </>; t), its temporalFourier transform \jj = ij/(p; $, k) is studied, where thewave number k is (as usual) proportional to temporal fre-quency (it is necessary to emphasise the word 'temporal'because the goal of CT is to form images, which are oftenconveniently characterised by their 'spatial' frequencycontent). One advantage of the frequency-domain scalarfunction is that it permits conservative fields, such as (low-frequency) electric currents, stationary magnetic fields andstreamlined fluid flow, to be straightforwardly incorpo-rated into a unified treatment of CT {k is identically zerofor such fields).

The frequency-domain scalar function is here taken tosatisfy the 'canonical' second-order partial differentialequation

V2ij/ + k2Aij/ = 0 (1)

where A = A(p; (j), k) is the generalised constitutive par-ameter. In the following Section it is shown how A relatesto various physical constitutive parameters of importancein engineering science.

3 Reduction of physical processes to canonicalform

It is convenient to start with what is here called 'standardform', which is defined by

V2/ + (V£) • V/ + kzTf = 0 (2)

where/ = /(p; 4>, k), £ = £(p; <f>, k) and T = T(p; 0, k) areall scalar functions of space and frequency.

It is shown later in this Section that the mathematicalphysics of many important physical processes can bereduced to standard form. They can be further reduced to

1EE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984 611

canonical form, which is defined by eqn. 1, by introducingthe substitution

/ = ( A e x p ( - L / 2 ) (3)

It is then seen that eqn. 2 reduces to eqn. 1, with

A = Y - (2V2L + (V£) • (VZ))/4/c2 (4)

Solving the CT problem at one frequency, i.e. for a singlevalue of k, provides explicit information only about A. Ingeneral, however, H and Y possess different ^-dependences(as shown below), so that solutions for a range of values ofk permit one to estimate the variations throughout theinterior of Tc of individual constitutive parameters.

The following derivations of the relationships of L andY to the constitutive parameters figuring in particularphysical processes are based on the convenient catalogueof mathematical physics collected in chapter 2 of the trea-tise by Morse and Feshbach [7].

3.1 ElectromagneticsFirst consider £-polarised fields, for which the electric fieldE is everywhere directed parallel to the z-axis. Its z-component is here denoted by / Substitute the twoMaxwell equations:

V x H = (joe + o)E and V • E = 0

into the curl of the third Maxwell equation:

V x E = —jcopH

(5)

(6)

and note that E • Ve = 0 because, by definition, there is nocomponent of Ve in the z-direction. It follows that:

= — In (p) and Y = (e — ja/co)p/p0 e0 (7)

where p = p(p; (j>) is the permeability, e = s(p; (f>) is thepermittivity, a = a(p; (j)) is the conductivity, co is theangular frequency and p0 and e0 are the free-space valuesof p and e. For H-polarised fields (H entirely in the z-direction), Y remains the same but Z = — In (e —ja/co).

3.2 Conservative fieldsFirst consider electric currents of low enough frequencythat free charge and 'displacement' can be neglected,implying that 'conservation of charge-current' reduces toV • J = 0, where J is the electric conduction currentdensity. Since the resulting electric field is effectively con-servative, it can be written as the gradient of a potential.The latter is conveniently denoted b y / It follows from thegeneralised Ohm's Law, J = oE, that

£ = In and Y = 0 (8)

For magnetostatics, electrostatics and irrotational fluidflow, o is replaced in eqn. 8 by /i, e and on, respectively,where m, — m,{p\ </>) is the mass density of the fluid.

3.3 AcousticsLinear wave motion in a fluid (liquid or gas, with thewavelength long compared to the greatest separationbetween adjacent particles constituting the fluid) isdescribed by

= — V(p — aV • v) and jcop = — KV • v (9)

where v is the acoustic particle velocity, p is the acousticpressure, a = (X + 4*7/3), where X is the coefficient of expan-sive friction and rj is the coefficient of viscosity, and K is thecompressibility modulus. Remember that all these quan-tities are functions of p and cf), with v and p depending alsoupon k. Remember also that V x v = 0 for acoustics, by

definition. On setting

/ = ( 1 +jcoot/K)p

it follows that

L = In (1/m) and Y = •mc2/(K + jcoot)

(10)

(11)

where c is the acoustic wave speed in any convenient 'refer-ence fluid', say tap water.

3.4 Elastic wave motionThe mathematical physics only reduces to the standardform of eqn. 2 when the smallest effective spatial variationof any of the physical constitutive parameters (i.e. the massdensity •m, the compression modulus (X + 2p/3) and theshear modulus p) is negligible compared to the wavelength.The standard form is then identical to the canonical formof eqn. 1. For longitudinal wave motion,/is the velocitypotential and A = mc2/(X + 2p). For transverse wavemotion, / is the z-component (the only component for the'two-dimensional' situations envisaged in this paper) of thevector potential and A = <mc2p. Note that c is the wavespeed in any convenient reference fluid.

3.5 PercolationTaking/ to be the mass density of a liquid flowing througha porous solid, its behaviour can be described by

jcof= - V • J and J= -a2Vf

where J is the mass flow vector anddiffusion coefficient. It follows that:

= In {a2) and /c2Y = -j

(12)

= a{p\ (p) is the

(13)

Any other diffusive motion (for example, heat and variousrandom walk processes) are described by eqns. 13. Non-relativistic quantum mechanics (i.e. the Schroedingerequation) can be classified as a percolative process: it isalready in canonical form, with

/c2A = -%(a)h/2n + V)mn2/h2 (14)

where h is Planck's constant, m is the mass of the particlewhose dynamics are being considered and V = V(p; (f>) isthe wave mechanical potential.

4 Dimensionality difficulty

Consider the CT problem as posed in Section 2. It isappropriate to denote the emanations applied to the bodyby ifrapp = i>apP(P'i 0> k). The body perturbs the appliedemanations, so that the total emanations, represented bythe scalar function \jj introduced in Section 2, are conve-niently expressed as

•A = ^app + </Vr (15)

where i)/per = ij/per(p; 4>, k) represents the perturbed part ofthe total emanations.

There is no unique decomposition of i/f into \j/app andil/per. It is perhaps intuitively most satisfying if \j/app rep-resents the 'initial emanation' on its passage through theinterior of Fc, so that ij/per consists of all reflections andany retarded part of the applied emanations. In the lan-guage of wave motion, such a ipapp represents the wave-front: it can be distorted, attenuated and slowed down, butit is the first emanation to appear at each point P. Thisdecomposition is particularly appropriate when ray-opticalapproximations are invoked (see Section 6). Unfortunately,however, there is no known way whereby eqn. 15, with thisdecomposition understood, can be combined with eqn. 1to provide an exact integral representation for ^ .

612 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984

There is, in fact, a way of decomposing \jj which doeslead to an exact integral representation for \j/per. This is thevolume source, or polarisation source [8], formulationwhich provides rather less physcial insight but is fully com-patible with eqn. 1. In this formulation the decompositionof \j/ into \jjapp and \jjper is effected by stating that

= 0

and (16)

The freedom from approximation is confirmed by referringto eqn. 15 and the noting that the sum of the two eqns. 16is identical to eqn. 1. The left-hand sides of the two eqns.16 indicate that both \jjapp and \J/per can be thought of aspropagating everywhere as though they are in free space.The effect of the medium is exactly accounted for by theright-hand side of the second equation. Equivalent sources,the so-called polarisation sources whose density is(/c2(A — l)ij/), exist wherever the medium differs from freespace (the emanations are said to polarise the medium[8]). This does not accord well with one's physical intu-ition, but it has the advantage that the second of the eqns.16 can be exactly re-expressed in the form of an integralrepresentation for ij/per, as is explained by Jones [9], forexample. Evaluating ipper on Fo gives

(A(p; <£, k) - 1)o jo

x ij/(p; cf), k)H(2\kr)p d<f> dp (17)

where r = (b2 + p2 - 2bp cos (6 - <p))1/2 and H(2\ •)denotes the Hankel function of the second kind of order m.

The perturbed emanations are observed on F o , so thateqn. 17 characterises what is measured. Eqn. 17 is an inte-gral equation, which is (hopefully) to be solved for theunknown quantities, i.e. A(p; 0, k) and \jj{p; 0, k), in theintegrand. The measured data are seen to be two-dimensional in the sense that they are characterised by twovariable parameters, 6 and k. The unknowns are three-dimensional, however, because the behaviour of ij/(p; </>, k)is not observed inside Tc. It is worth noting that A(p; </>, k)is a two-dimensional rather than a three-dimensionalunknown, because its /c-dependence is here understood tobe given a priori. In any particular physical situation, therelevant part of Section 3 can be invoked to express A as asum of terms each having a simple ^-dependence.

It is seen therefore that the general CT problem is 'ill-posed' in the sense that if/ must be 'continued' in some wayfrom Fo back into the interior of F c . Such continuationprocedures tend to be significantly sensitive to errors inmeasured data. The 'dimensionality difficulty' noted in thepreceding paragraph, is highlighted by expanding if/per inthe following trigonometrical Fourier series:

£ AJb, k)H{2\kb) exp

(18)

where, for each value of k, each Am(b, k) is obtainedfrom the measured ipper by multiplying it by(/2(exp (—jmd))/nk2H{2)(kb)) and integrating with respectto 0 from 0 to 2n. The factors H(2\kb) are included in thesummation in eqn. 18 merely as a convenient normal-isation. The reason for their introduction becomes appar-ent when the right-hand side of the next equation isexamined. It is now appropriate to invoke the additiontheorem for functions which are solutions to Bessel's equa-

tion [10], taking note that b is greater than p for any pointP either on or inside F c . When applied to the Hankel func-tion in the integral in eqn. 17, this addition theorem gives

H%\kr)= £ H<2\kb)Jm(kp) exp (jm(6 - (19)

where Jm( •) denotes the Bessel function of the first kind oforder m. Note the factors H(2\kb) in the summation in eqn.19 and recall the previous remarks about the same factorsin eqn. 18. Substituting eqns. 18 and 19 into eqn. 17 trans-forms both sides of the latter into trigonometrical Fourierseries in 6. Equating corresponding terms in these seriesgives

AJb, k) = AJk)

(20)

o Jo

x exp (—jm(f))p d(f) dp

for all integers m.Inspection of eqn. 20 reveals that nothing is gained by

making measurements on more than one circle, becausethe inegral is independent of b. The dimensionality diffi-culty persists because Am(k) varies with m and k, whereasthe dependence of \j/(p; <j), k) upon all three of its par-ameters is unknown.

Either one must try to solve eqn. 1 exactly or approx-imations must be invoked. The first alternative is followedin Section 5, while the second is discussed in Section 6.

5 Exact canonical solution

An experimenter is free to generate an arbitrary set ofapplied emanations with which to excite the body sequen-tially. For the /th of these, it is convenient to denote thetotal emanations inside and on Fo by \jj(l) = ij/il)(p; </>, k). Itis also convenient to define

and

B,,m(/c)exp(/m0)

9, k)/dp = t Bltm(k)exp(jm6)

(21)

(22)

Since the experimenter knows what emanations are inci-dent upon the body, the form of \jjapp can be taken as giveneverywhere outside F c . So, both \j/app and dij/app/dp areknown on F o . Furthermore, the experimenter can measure\j/per and di//per/dp on Fo . This means that all of the Bt Jk)and all of the Bt m(k) can be deduced from the measure-ments and from the experimenter's prior knowledge.

It is now convenient to require that the particular set ofemanations applied by the experimenter should be suchthat, for all / and m,

Bl,M = 5lm (23)

where <5/m is the Kronecker delta (equal to unity when/ = m and to zero when / ^ m). It is worth emphasisingthat, given eqn. 23, the Bt Jk) are determined by the inter-action of the emanations with the body, so that no a priorispecification of the B, m(k) can be allowed.

It is appropriate to expand \j/ll\p; 4>, k) in the form

£ £m = — oo n= 1

>n(p, k) exp

(24)

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984 613

In order to match this expansion to the data, i.e. theBt m(k) and the Bt m(k), it is convenient to introduce the'full-wave impedances':

Zlt Jk) = B,. Jk)/Blt Jk) = Zlk) for m = /

= 0 formal (25)

where eqn. 23 has been invoked. It follows from eqns. 21and 22 that each of the \j/u m> n{p, k) must satisfy

( # , , m> n(b, k)ldp)Mlt m, n(b, k) = Zlf Jk) (26)

which can be ensured by setting

*i.m,Jj>,k) = Jj!*i,m.nPlb) (27)

where a,t m> „ = a,, m> n(/c) and

<*i,«. „ J'J&i, «, „) = bZu m Jm(ctlt m> „) (28)

where the prime denotes the derivative.Given the Z, m deduced from the measurements as out-

lined above, the a,,m>n can be obtained by numericallysolving the transcendental eqns. 28. In fact, eqn. 25 showsthat, except when m = /, the a; lff> „ are the zeros of J'J •).The theory of Dini series [10] confirms that, for any fixedintegers / and m, the \j/t m n(p, k) are orthogonal within0 < p <b:

dt = NI, (29)

where the values of the constants N, m> „ can be abstracteddirectly from Watson's formulas.

In any particular physical situation, the generalised con-stitutive parameter A(p; 0, k), introduced in eqn. 1, can bepartitioned into the sum of a number of terms with simple,specified dependence upon k. The actual number of termsand their /c-dependences are found by appealing to theappropriate part of Section 3. It is sufficient for the pur-poses of this section to take A(p; <p, k) = A(p; </>), which itis worth emphasising accords with the great majority ofinvestigations of inverse problems in field and potentialtheory [11]. It is necessary to introduce an expansion ofthe form

(30)m = — oo n = 1

where, for each m, the Am n(t) are any suitable functionswhich are independent (but not necessarily orthogonal)within 0 < t < 1.

Substituting eqn. 30 into eqn. 1, multiplying through by((exp {—jm<fi))l2n) and integrating with respect to <f> from 0to 2n, and multiplying through by (pJJat m „ p/b)) andintegrating with respect to p from 0 to b, shows that

m' = — oo i, n' = 1

l, i, m, m', n, n'

where it follows from eqn. 29 that

I, i, m, m', n, ri *l, i, m, m', n, n W

= (kb)

(32)

The essential point to recognise about eqn. 31 is that, forany given /, there is an infinity of eqns. 31, each of which ischaracterised by particular values of m and n. So, eqn. 31can be thought of as a system of linear, algebraic, homoge-

nous equations for the Blmn(k). Since the system ishomogenous, it only has a solution when the determinant,noted here by D,(/c), of the coefficients of the Bu m n(k) iszero:

Dt(k) = 0 (33)

The dimensionality difficulty noted in Section 4 arisesbecause the unknown field inside To is of higher dimen-sionality than the data. The remarkable aspect of eqn. 33 isthat it removes the dimensionality difficulty by eliminatingthis unknown field. The latter is characterised by theBi,m,n(k)- The elements of Dt(k) depend on the constantsCmn which characterise the body that it is desired toimage, and on the a, m n which depend only on the mea-sured data.

The technical significance of the solution to the generalCT problem introduced in this Section is assessed inSection 7.

6 Exact versus approximate theories

The enormous complication of the computational conse-quences of the solution developed in the preceding Sectionis characteristic of exact inverse theories. For instance, vir-tually nothing is known of the sensitivity to measurementerrors of the Gelfand-Levitan procedure, which has been inexistence for some 30 years [12]. All this emphasises whyapproximate approaches have almost always been adoptedin practice.

There are two well established approximate theories.The first, known as the Born [11] or Rayleigh-Gans [9]approximation, is based on the two eqns. 16 together withthe assumption that there is negligible error in replacingthe unknown \(/(p; cf>, k) in eqn. 17 by the known \jjapp{p',<j), k). There is no longer any dimensionality difficultybecause, as already noted in Sections 4 and 5, the k-dependence of A(p; (f>, k) is given a priori. When thisapproximation is valid, as in X-ray crystallography [13] orin standard radar (and some sonar) applications, it givesaccurate results in practice. It relies, however, on the incre-mental phase shift across a diameter of Fc being appre-ciably less than n/2. As is obvious, the assumption breaksdown badly in many practical situations involving electro-magnetic, seismic and acoustic wave motion.

The second of the well known approximate theoriesinvolves replacing wave motion by rays. It then makessense to think of \j/app as representing the intial wavefronton its passage through the body. As intimated in thesecond paragraph of Section 4, this is a more 'physical'point of view than the polarisation source formulationbecause it accounts for refraction and the effects of varia-tions of wave speed within the body. Unfortunately, thereis as yet no proven way of accounting for ray bending ininverse problems [2]. When the rays are known to bestraight, excellent images can be formed, as is evinced bythe spectacular results obtained with conventional X-rayCT.

The Rayleigh-Gans approximation can be markedlyimproved by incorporating ray-optical ideas into it. TheRytov approximation, and an extension of it [14], thenresults. When \jjapp is a plane wave, or a cylindrical waveoriginating from a point appreciably removed from To, theform of i//per in directions close to that of the forward pro-gression of \\iapp is predicted much more faithfully than bythe Rayleigh-Gans approximation. However, the Rytovapproximations do not seem to predict wide-angle scat-tering any more accurately.

614 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984

An inverse method can only be considered of generalpractical use when it has been brought to the point whereit can provide a faithful image of an object immersed in amedium whose constitutive parameters exhibit appreciablespatial variations and which has many other comparableobjects immersed in it. It is not enough to form images ofisolated objects suspended in uniform media. The purposeof later papers in the series, of which this is the first, will beto introduce techniques which show promise of satisfyingthe above criterion. It is worth pointing out that Devaney'srecent back-propagation approach [15] may eventuallydevelop into a viable full-wave CT algorithm, in the senseintroduced in this paper.

7 Conclusions

This paper shows (in Sections 2 and 3) how the mathemati-cal physics of a wide range of important physical processescan be cast into a single canonical form, which is appropri-ate for a generalised approach to CT.

The way in which the central 'dimensionality difficulty'(which afflicts all inverse problems) arises is explained inSection 4. The exact method introduced in Section 5 over-comes this difficulty, but with the unavoidable intro-duction of so much complication that it can be readilyappreciated why the established approximate approaches(discussed in Section 6) have such restricted practicalapplication.

The chief use of the exact solution introduced inSection 5 is in helping to establish the fundamental limi-tations on CT. Although a later paper is to be devoted tothis, it is nevertheless appropriate to make some generalcomments here. It is well known in nonrelativisticquantum-mechanical inverse scattering theory that a cylin-drically (or spherically) symmetric potential can be recon-structed from the scattering amplitudes at one energy (i.e.one frequency) for all partial waves, or at all energies forone partial wave [12]. Eqn. 33 makes clear why this mustbe so. When A(p; 0) can be written as A(p), it is of courseone-dimensional, so that Dt(k) need only be given for eithervariable / or variable k. It is equally clear that one can onlyhope to reconstruct an arbitrary A(p; (f>) if D,(k) is given forboth / and k variable. It follows that there are strongrestrictions on the kinds of image that can be successfullyreconstructed when data are available only for a singlevalue of k, as is the case, for instance, when CT is attempt-ed with conservative fields; for the latter, k = 0. It followsfrom eqns. 4 and 8 that

k2A = - (2V2 In (a) + (V In (<r)) • V In (a))/4 (34)

Those of the later papers to be concerned with reconstruc-tion algorithms will introduce inverse methods interme-diate in accuracy between the approximate techniquesdiscussed in Section 6 and the exact approach introducedin Section 5.

8 Acknowledgments

In developing the ideas presented here I have benefittedfrom many discussions with G.C. McKinnon and R.P.Millane. I am also indebted to R.A. Minard, B.S. Robin-son, A.D. Seagar and T.S. Yeo, who are to co-author laterpapers in this series.

9 References

1 HINSHAW, W.S., and LENT, A.H.: 'An introduction to NMRimaging: from the Block equation to the imaging equation', Proc.IEEE, 1983, 71, pp. 338-350

2 BATES, R.H.T., GARDEN, K.L., and PETERS, T.M.: 'Overview ofcomputerised tomography with emphasis on future developments',ibid., 1983,71, pp. 356-372

3 DE RUJULA, A., GLASHOW, S.L., WILSON, R.R., andCHARPAK, G.: 'Neutrino exploration of the earth'. Harvard Uni-versity, Cambridge, MA, USA, Report HUTP-83/A019, 1983

4 GREENLEAF, J.F.: 'Computerized tomography with ultrasound',Proc. IEEE, 1983, 71, pp. 330-337

5 BATES, R.H.T.: 'Global solution to the scalar inverse scatteringproblem', J. Phys. A, 1975, 8, pp. L8O-L82

6 MILLANE, R.P., and BATES, R.H.T.: 'Inverse methods for branchedducts and transmission lines', IEE Proc. F, Commun., Radar & SignalProcess., 1982, 129, (1), pp. 45-51

7 MORSE, P.M., and FESHBACH, H.: 'Methods of theoreticalphysics' (McGraw-Hill, New York, 1953), Chap. 2

8 BATES, R.H.T., and NG, F.L.: 'Polarisation-source formulation ofelectromagnetism and dielectric-loaded waveguides', Proc. IEE, 1972,119, (11), pp. 1568-1574

9 JONES, D.S.: 'The theory of electromagnetism' (MacMillan, NewYork, 1964), Sections 1.34 & 6.13

10 WATSON, G.N.: 'Theory of Bessel functions' (Cambridge UniversityPress, 1958, 2nd edn.), Chaps. 11 & 18

11 COLIN, L.: 'Mathematics of profile inversion'. NASA TechnicalMemorandum TM X-62-150, 1972

12 CHADAN, K, and SABATIER, P.C.: 'Inverse problems in quantumscattering theory' (Springer-Verlag, New York, 1977)

13 RAMACHANDRAN, G.N., and SRINIVASAN, R.: 'Fouriermethods in crystallography' (Wiley, New York, 1970)

14 BATES, R.H.T., BOERNER, W.M., and DUNLOP, G.R.: 'Anextended Rytov approximation and its significance for remote sensingand inverse scattering', Opt. Commun., 1976,18, pp. 421-423

15 DEVANEY, A.J.: 'A filtered backpropagation algorithm for diffrac-tion tomography', Ultrason. Imaging, 1982, 4, pp. 336-350

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