SCIENCE

Full-wave computed tomographyPart 2: Resolution limits

A.D. Seagar, B.E., Ph.D., T.S. Yeo, M.Eng., 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, Biomedical engineering, Tomography, Imaging schemes

Abstract: Invoking our recently developed exact inverse theory, we derive limits on the resolution of individualspatial frequency components of a reconstructed image, set by imperfections of measurement. These areminimum limits, only achievable when one knows how to invert data without adulterating the underlyingtheory with strong assumptions (sweeping approximations are implicit in most practical inversion algorithms).Resolution limits on imaging with conservative fields are also derived, but (recognising the physical constraintson the formation of useful images) for situations simple enough that data inversion is easily effected withoutintroducing approximations.

1 Introduction

This is the second in a series of papers devoted to extend-ing the principles and practice of computed tomography(CT). Because our approach goes beyond the simple theo-retical concepts which underpin conventional X-ray andultrasonic CT systems, we call it full-wave CT. The firstpaper [1] treats the fundamental theory underlying theformation of clean images of cross-sections through bodies.The term 'clean' implies that any reconstructed image is afaithful representation of what would be seen if the bodycould be sliced open through the cross-section of interest.It is shown in Reference 1 that the interactions with pon-derable bodies of many species of emanation (a graphicterm describing a physical process used to probe a body orsense any of its properties) are characterised by a canonicalpartial differential equation of second-order. We herederive from this equation, whose form is recalled in Section2 together with other necessary preliminaries, certain basiclimits on the resolution of full-wave CT systems.

Our first concern in this paper is with the effects ofobservational errors on the faithfulness of images formedfrom measurements of wave-like emanations. Because ourresults, which are reported in Sections 3 and 4, are derivedfrom the exact inverse theory developed in Section 5 ofReference 1, they are more far reaching than conventionalconstraints on the resolution of imaging systems [2].

In Section 3 of Reference 1 it is pointed out that conser-vative fields are described by the aforementioned canonicalequation, thereby implying that such fields can be used inimaging systems. We cannot expect too much in the way ofresolution because the effective wavelength is infinite.Images can nevertheless be reconstructed [3], althoughone must be careful not to attempt overly close analogieswith conventional CT systems [4]. It is therefore appropri-ate to treat the resolution of conservative field imagingsystems as a special case; this is done in Section 5. InSection 6 we explain how the best possible resolution canbe accurately assessed in any particular physical applica-tion. The absolute resolution limits are set by measure-ment uncertainties, such as noise and imprecise locations

Paper 3342A (S9), first received 18th October 1983 and in revised form 4th May1984

Dr. Seagar is with the Medical Physics & Clinical Engineering Department, Shef-field University, Royal Hallamshire Hospital, Glossop Road, Sheffield S10 2JF,England, and Mr. Yeo and Prof. Bates are with the Electrical & Electronic Engin-eering Department, University of Canterbury, Christchurch, New Zealand

of apparatus. These limits can only be attained in practicewhen a manageable, exact inverse theory is available.Sweeping approximations are unavoidable in many appli-cations because the construction of an exact inverse theoryis usually so difficult and complicated [1]. These approx-imations are often the major limit on the attainableresolution.

2 Preliminaries

The canonical equation referred to in Section 1 describesthe behaviour of monochromatic emanations, representedby the scalar function ij/ = i//(p; cj), k), where k is the wavenumber and (p; </>) are the cylindrical polar co-ordinates ofan arbitrary point P in the plane of a particular cross-section of a body. The equation, which is eqn. 1 of Refer-ence 1, is

+ Zc2AiA = 0 (1)

It is postulated that the emanations can be sensed at anypoint Q on the circle, here called the measurement circle,of radius r (centred at O) circumscribing the cross-section(see Fig. 1). Space is assumed to be homogeneous outside

Fig. 1 Co-ordinate systems for full-wave CTThe cross-section of the body lies inside the measurement circle, on which is thearbitrary point Q. An arbitrary point in the plane of the cross-section is P

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

the cross-section so that the generalised constitutive par-ameter A takes on an appropriate constant value for p > r.On the other hand, A can vary arbitrarily with p, (f) and kfor p <r.

We are concerned with two types of situation in thispaper, one quite general and the other rather specialised.The first involves wave-like emanations, which are charac-terised by the wave number taking on arbitrary finitevalues, i.e.

0 < k < oo (2)

The second situation concerns conservative fields, whichare characterised by

and

/c2A = -(2V2 In (a) + (V In (a)) • (V In (<x)))/4 (3)

as is confirmed by inspection of eqn. 34 of Reference 1.It is pointed out in Section 7 of Reference 1 that it is

likely to be impracticable, and may even be impossible, toreconstruct, from observed values of a conservative fieldrecorded on the measurement circle, the form of a (k2A(p;(j), 0)) which varies arbitrarily with p and (j> for p < r. Onthe other hand, one can expect to achieve successful recon-structions for classes of piecewise constant media, as willbe shown in a later paper. Such reconstructions are com-paratively straightforward when the shapes of thepiecewise constant regions are given a priori. In Section 5of this paper the discussion is restricted to circular regionsbecause they serve as well as any others as bases for thedetermination of resolution limits.

The many possible physical meanings of the parametersA = A(p; (f>, k) and a = a(p; </>) are listed and discussed insome detail in Section 3 of Reference 1.

Suppose the body is successively excited in a largenumber of independent ways. For the /th, say, of theseexcitations, the emanations existing in the plane of thecross-section (see Fig. 1) are conveniently represented bythe scalar function

•A/, m{p> k) (4)

where the \\)x m(p, k) are here called the /th Fourier coeffi-cients.

In order to take advantage of the exact theory devel-oped in Section 5 of Reference 1, it is necessary to intro-duce the full-wave impedances;

i, m = Zt, Jk) = (# , , Jr, k)/dpWlt Jr, k) (5)

This is as far as it is appropriate to take our preliminarydiscussion, because different forms for the i//h m(p, k) areneeded for the two types of situation mentioned above.

3 Effects of measurement errors onrepresentations of wave-like emanations

This Section is concerned with emanations having thecharacter of wave motion, for which eqn. 2 applies. It isnecessary to express the Fourier coefficients i/f, m(p, k) dif-ferently inside and outside the measurement circle. In theformer case, an appropriate expression is

oo

<A/, m(p> *) = E #/, m, H(k)Jm(alt m> „ p/r) for p < r (6)n = 1

where JJ •) identifies the Bessel function of the first kind

of order m. The eigenvalues a, m „, which depend on k forthe reasons given in Section 5 of Reference 1, are discussedat some length later in this Section.

Since, as is stated in Section 2, space is homogeneousoutside the cross-section, no generality is lost by requiringthat

A = 1 for p > r (7)

which implies that, when eqns. 1 and 4 are compared forpoints P outside the cross-section, a general expression forthe mth Fourier coefficient is

^i.miP' k) = At Jk)JJkp) + R,Jk)H(2\kp)

for p > r (8)

where H(2\ •) identifies the Hankel function of the secondkind of order m. If, as is here appropriate, the Ax Jk) arechosen to define the incident wave, the Rt Jk) characterisethe reflections from the cross-section. We call the latter thereflection coefficients. Whatever incident waves areactually used, the data can be rearranged in any appropri-ate way. The choice adopted in Section 5 of Reference 1 isequivalent to requiring that di^, Jp k)/dp = 0 when p = rfor all values of m except /. For simplicity it is more appro-priate here to force a similar condition on the At Jk), i.e.

At Jk) = Slm (9)

where Slm is the Kronecker delta (unity for / = m and zerootherwise).

It is convenient in practice to measure reflected (orscattered) emanations. Provided enough such measure-ments are made, it is straightforward to estimate theRt Jk) from the data. It is then possible to reconstruct i/̂ (r;6, k) and d\j/{r; 6, k)/dp on the measurement circle. TheFourier coefficients of the latter two quantities imme-diately give the full-wave impedances defined by eqn. 5.

On introducing the abbreviated notations

,t {(k) = Zi and (10)

and identifying a total derivative by a prime, it is seen fromeqns. 5, 8 and 9 that

R{H\2\kr))

and'

Z,. Jk) = k(H™Xkr))/H%\kr) formal

(11)

(12)

It is worth noting that the reflection coefficients affect onlythose full-wave impedances for which m = I.

It is impossible to be certain that the measuring appar-atus moves on an exactly circular path around the body.Furthermore, it is impossible to determine the wavenumber k precisely. It is convenient to lump these uncer-tainties together and characterise them approximately bythe quantity Sr, here called the radius error, which is to bethought of as an error in the estimate of the radius of themeasurement circle. In most practical applications involv-ing wave motions which probe bodies, the dimensions ofthe latter are typically large compared with the wave-length. Consequently, br can often be expected to be com-parable with the wavelength. Here, in fact, we assume that

kbr > n (13)

Measurement errors, denoted here by 3Rt and SRt Jk), inthe observed reflection coefficients are, of course, inevita-ble. The radius error and the measurement errors aretogether the cause of errors 3Zt in-the computed estimates

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984 617

of the Z,. Note from eqn. 12 that measurement errors haveno effect on <5Z, Jk) for m±l.

The Fourier coefficients i]/lt m(p, k) are only wavelike onthe boundary of the cross-section for those values of m forwhich kr > \m\, because the imaginary part H(^\kr)becomes large and positive, while its real part falls rapidlyto zero once kr/\m\ falls significantly below unity [5].Since r is here taken to be large compared to the wave-length, it is seen that kr is appreciably greater than \m\ formost of the Fourier coefficients that carry useful, mea-sureable information concerning the cross-section. Theasymptotic formulas [5]

JJx) ^ (2/nx)il2 cos (x - mn/2 - n/4)

and

H(2\x) ~ (2/nx)112 exp (-j(x - mn/2 - n/4))

(14)

(15)

describe the dominant large-argument behaviours of thesefunctions (i.e. for \x\ appreciably greater than |m | ) . Com-parison of eqn. 12 with eqn. 15 suggests that\(8Zlm{kr))/kZlJkr) dr\ is appreciably less than unity,when m ^ I, for most \m\ < kr. This is confirmed by Fig. 2

kr/lml

Fig. 2 Behaviour of zn

Parameter = m

which shows zm, plotted against kr/\m\ for four values ofm, calculated from the right-hand side of eqn. 12, where

= \{dZUm{kr)ld{kr))/ZuJkr)\ (16)

The larger \m\ is, the more rapidly zm tends to zero withincreasing kr/\m\. This ensures that zm must be small formost values of \m\ < kr, when kr is itself appreciable(more than ten, say).

We can conclude therefore that impedance errors areonly appreciable when m = /. The effect on bZx of bR{ isfound by replacing Zx and Rx in eqn. 11 by (Z, + bZ{) and(Rx + bRx), respectively. On retaining only those termswhich are of first order in the errors, the quantity sx,defined by

(17)

(18)

is found to be given by

x | Jt(kr) + Rt H(2\kr) | nkr

where the forms [5] of Bessel's equation and the Wrons-kian of Jt( •) and H\2)( •) have been invoked. So, for mostvalues of | / | less than kr, it is seen from eqns. 18, 14 and15 that

It is only possible to reconstruct the form of A(p; 0, k) ifthe incident wave penetrates right through the body,implying that most of the | R/1 can be expected to be atleast moderately small compared with unity. It follows thateqn. 19 can usually be rewritten as

e, ~ 2/1 cos (2kr) + 2Rt exp (-j2kr)\ (20)

The inequality emphasises that the expected value of e,spans a wide range because |cos (2kr)\ can assume anyvalue between zero and unity. Consequently, when Rx issmall, the maximum and minimum values of ex are givenclosely by

£,(maximum) ~ 1/1 Rx \

and

^(minimum) cz 2 (21)

which is confirmed by the curves shown in Fig. 3, whichhave been calculated from the right-hand side of eqn. 18for a single value of /. The agreement is very close for Rx =0.01, reasonably close for Rx = 0.1 and usefully close whenRx = 0.3. Even when Rx is as large as 0.6, the estimatesprovided by eqns. 21 are within factors of two of theircorrect values. Fig. 4 illustrates the fact that the behaviour

1.0

0.60

0.30

o~i6~"

0.01

Fig. 3 Behaviour ofe, for I = 1

Parameter = R,

r\ !\A /

A

A3 A'

\ i\ !X

/

/ \ / V/ v /\\ \ \

20kr

21

e, =* 2/ | cos (2kr) + 2(1 + Rt)Rt exp (-j2kr)\ (19)

Fig. 4 Behaviour ofe, for R, = 0.1

Parameter = /The decrease in peak height with increasing / is due to eqns. 21 only being satisfiedwhen kr/\m\ is large enough

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

of s, is similar for all values of /, provided kr/\l\ is largeenough.

Reference to eqns. 25-28 of Reference 1 (setting thequantity b introduced in the previous paper equal to r)confirms that the eigenvalues introduced in eqn. 5 of thispaper must satisfy

, n) = rZ,, Jk)J m(a^ m. „) (22)

Radius errors and measurement errors lead to erroneousvalues for the a, m n, which, in turn, cause the represent-ation for the field inside the cross-section to be inconsis-tent. This is only serious for those Fourier coefficients forwhich m = I, because of the result embodied in eqn. 16 andbecause the reflection coefficients do not affect the imped-ances when m =/= /. Eqn. 22 reduces for m = / to

aJ\(a) = rZxJx(a)

where eqn. 10 has been invoked and a, , „ has been writtenas a for convenience.

On replacing a and Zx in eqn. 23 by (a + da) and (Z,+ SZX), respectively, and retaining only terms of first order

in da and bZx, we find that the quantity Y\X , defined by

= \(5a/a)/(3Zx/Zx)\ (24)

is given by

-a2(JX«))2)l (25)

The Bessel function Jx(a) varies as (a/2)' ' ' / |/ | ! whena <̂ | /1. It increases monotonically until a is a little lessthan (| /1 +2) , after which it executes damped oscillations,asymptotically approaching the form described by eqn. 14.When a is significantly less than | /1, the a-dependence ofeqn. 23 effectively vanishes, with Zx being given approx-imately by l/r. Since Zx can, in general, be expected todiffer significantly from l/r, the smallest value of a (i.e. thelowest eigenvalue a, , x) satisfying eqn. 23 can be expectedto be close to | /1. Higher eigenvalues are thus usuallylarge enough compared with | /1 that the asymptotic eqn.14 can be invoked to approximate the Bessel functions ineqn. 23 to a useful level of accuracy. Furthermore, theterms in a2 in the denominator of the right-hand side ofeqn. 25 dominate the term in I2. It follows that:

- |(cos(2a))/2a| (26)

Fig. 5 shows rj, as a function of (real) a for four values of /,calculated from the right-hand side of eqn. 25. It is clearthat rjx only exceeds l/2a by any significant amount when ais comparable with or less than | /1.

0.5

\\\

\\

,2iiI

\\\ \ / \ /

\\\

• \5

\

\

\

\

\ 8

\

\

\

10

Fig. 5 Behaviour oft],The parameter = /* I/2a

4 Resolution limits for full-wave CT

The inverse approach introduced in Section 5 of Reference1 allows A to be reconstructed arbitrarily faithfully, inprinciple. Quite apart from the computational problemsnoted in Reference 1, measurement errors and the radiuserror necessarily limit the fineness of the detail which canactually be resolved within the cross-section. We nowexamine the effect on A of errors 8ax m „ in the eigenvalues.Remember that these stem from 5r and SRX, as explainedin Section 3.

It is instructive to examine the effect on \jjl of retainingthe correct values of kr and of the expansion coefficientsBi, m, n(k) but replacing the a, m „ by their erroneous values,which arise because of the measurement errors. On repla-cing a,, M. „ by (a,, m> „ + <5aJt m> „), retaining only terms of first

(23) order in (5a, m „, and recalling from Section 3 that (5a, m „ isnegligible except when m = /, we find from eqns. 4 and 6that

dip, = (p/r) exp (jl4>) £ BnJlcLnp/r) dan (27)

where Bu, n(k) and a, , „ have been written for convenienceas Bn and an, respectively. It follows from eqn. 24 that San

is proportional to (t]l an) for all but perhaps the few lowestvalues of n. On taking the RMS value (over a) of 77,, asfound from eqn. 26, we see that the expected value of dan

depends only on | bZx/Zx \ and is therefore independent ofn. It therefore makes sense to write

Sccn ~ da (28)

On substituting eqns. 4 and 6 into eqn. 1, and invoking theform of Bessel's equation [5], we find that

A =

where

(29)

X Jm(<*l,m,nPM

so that

I (5 A / A I =

(30)

(31)

The effect of errors in alm „ on \J/{ is again (as arguedabove for i//,) only significant for those terms for whichm = I. Since at is, by definition, the smallest value of asatisfying eqn. 23, it is necessarily of the order of | /1 . Con-sequently, for all values of /, most an are large comparedwith unity. It then follows from eqn. 30 that

~ (p/r) exp <x2nBn J\{an p/r) dan (32)

Inspection of eqns. 27 and 32 reveals that the first terminside the magnitude signs on the right-hand side of eqn.31 can be expected to dominate the second throughoutmost of the interval 0 < p < r. The amplitude of i£, is, ofcourse, likely to fluctuate markedly inside the measure-ment circle. However, it must have a definite value at anyparticular point, P, say. When estimating 3A/A, we find itconvenient to introduce the normalisation

\j}l = (p/r) exp (jlcf)) at P

It then follows that

I (5 A/A I = \5a t^nJ\(anp/r)\

(33)

(34)n = l

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 8, NOVEMBER 1984 619

The values of a that satisfy eqn. 23 belong to a sequence ofincreasing real numbers, successive members of whichdiffer by roughly 2K, as the form of eqn. 14 makes clear. Inactual fact [5], it transpires that

«„= \l\ (35)

where | /?„ | is a small fraction of | / j (except for low valuesof | / | ) and /?„ approaches some constant value, /? say, as nincreases. To well within an order of magnitude, then, /?can be neglected, so that

<*„ ^ + Inn (36)

It is explained in Section 7 of Reference 1 that data mustbe gathered for many values of both / and k if we are tohave any hope of reconstructing an arbitrary A(p; </>). Theformulas developed in this Section therefore apply to allvalues of /, i.e. to all the incident waves used to probe anyparticular body.

The resolution of a measurement scheme depends uponthe number of independent parameters that are abstractedfrom the data. In order to resolve detail of finite size (Asay) in A(p; (j>), we need only account for a finite number ofFourier coefficients in eqn. 4. The summation thus runsfrom — M to M, where the integer M is large enough forthe summation to be capable of reproducing detail of therequired size at the circumference of the cross-section, i.e.

M ~ nr/A (37)

Similarly, only a finite number of terms (Nm say) need beconsidered in the summation in eqn. 6. Each basis function(Jm(a, m „ p/r) exp {jm<t>)) c a n be usefully thought of as atwo-dimensional spatial frequency component. The spatialwavelength in the p-direction is (27ir/a, m „). We then seefrom eqns. 36 and 37 that detail of size A can be resolved if

Nm = r/2A - \l\/2n (38)

from which it is clear that an order of magnitude estimateof the total number of spatial frequency components, towhose complex amplitudes the measurement scheme mustbe sensitive, is

MNm ~ nr2/2A: (39)

5 Resolution limits on imaging with conservativefields

Suppose that the medium enclosed by the measurementcircle is circularly symmetric and piecewise constant:

a = <r_ for p < r0

= a. for r0 < p < r (40)

a situation which is envisaged in Fig. 6a. By invoking theconformal transformations summarised in Fig. 7, thegeometry of Fig. 6a can be transformed into either of thegeometries of Figs. 6b and c, or vice versa. The latter twogeometries have important applications in engineeringscience, as is to be discussed in a later paper.

It is unnecessary to consider more than one excitationhere. This follows from the circular symmetry of themedium depicted in Fig. 6a. Furthermore, since k = 0, thenotation introduced in eqns. 4 and 5 can be convenientlycompressed:

= \J/m(p)

Fig. 6 Simple piecewise constant cross-sections for which a equals cr_and a + , respectively, inside and outside each of the small circles

Note that r is normalised to unity. Each geometry is associated with a particularcomplex plane. The connections between the three planes are described by the con-formal transformations listed in Fig. 7, which also indicates the relations existingbetween r0 ,-f, f, c, d and ha Symmetric geometry (z-plane)b Offset geometry (w-plane)c Flat geometry (t-plane)The straight broken line in a transforms into the curved broken line in b

nsymmetric geometry, z-planeL—,

w=(d*z)/(zd-1)

f =ro(l-d2)/(l-rjd2)

c = d(1-r(2

))/(i-r2

)d2)

d =[l •c2-r2-/(Uc2-r2)2-/,c'!]/2c

z=(d-w)/(wd-1)

ro=[i-c2»F2-y(l-c2rr2)2-A72l/27

d =[uc2-r-2-/(Uc2-r2)2-4c2]/2c

offset geometry.w-plane

Ht= j(w-l)/(w»l)

f" = 2r/[(Uc)2-r2]

h=[Ur2-c2]/[(Uc)2-r2]

w = -(Uj)/(t- j)

r = 2r/[(Uh)2_?2]

C =[1 «r2-h2]/[(Uh)2-f 2]

flat geometry.t-plane

Fig. 7 Standard conformal transformations [<5] connecting the threegeometries depicted in Fig. 6

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

and

ZUk)=l/Cm (41)

where \jj remains a function of both p and 0. The thirdrelation is introduced to agree with the notation of a laterpaper. Two parameters needed in later expressions are

p = aja+ and q = (ro/r)2 (42)

A consequence of eqn. 40 is that eqns. 1 and 3 reduce to

V2\j/ = 0 (43)

for both p < r0 and r0 < p < r. Since i// and (a dip/dp) mustboth be continuous across p = r0, it follows that [6]:

as follows from inspection of eqn. 45. Fig. 9 shows plots ofsx and s2 against In (p) for several different values of q.

= AJpm + (1 - p)r2omp-m/(l + p)) (44)

for r0 < p < r, where Am is an expansion coefficient. Com-parison of eqns. 5, 41 and 42 with eqn. 44 further showsthat

Cm = (r/mXl + p + (1 - p)qmW + P - (1 - p)qm) (45)

In practice, we measure \j/ and dxjj/dp on the measurementcircle, compute their Fourier coefficients and from thelatter calculate as many of the (m as we think we require, inorder to be able to estimate the values of p and q in thepresence of measurement noise. It is therefore vital to beable to detect the alteration in (m that occurs when pchanges from unity to any particular value different fromunity. We denote by

Co, m = r/m

the value that £m reduces to when p = 1.The quantities vm, defined by

Vm = (Cm - Co, m)/((m + Co, J

(46)

(47)

are normalised measures of the differences between the (m

and the Co, m- We call them 'visibilities' by analogy withsimilarly defined quantities in interferometry [7]. Substitu-ting eqns. 45 and 46 into eqn. 47 shows that

vm = (l- P t e 7 ( l + P) (48)

Fig. 8 shows plots of vl and v2 against In (p) for several

1.0

-A - 2 0 2 A - A - 2 0 2 A

Fig. 8 Visibility vm for m = 1 and 2, with q as parameter

different values of q. Note that, for any given value of p,the difference between a + and a _ is more visible the largerr0 is. The smallest value of r0 for which this difference isvisible (i.e. for which vm is nonzero to the accuracy of themeasurements) is the limit to the size with which detail canbe resolved.

It is also important to be able to detect a prescribedchange in p, given any particular value of q. A normalisedsensitivity of (m to changes in p is conveniently defined by

= 4p<r/[(l + P)2 - (1 - p ) V m ] (49)

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

0.00- A - 2

Fig. 9 Sensitivity sm for m = 1 and 2, with q as parameter

Note that, for any given value of q, the sensitivity falls as| In (p) | increases. The smallest change in p for which sm isnonzero to the accuracy of the measurements is the limitto the accuracy with which p can be found.

6 Discussion

In most important practical applications involving remoteprobing of bodies with wave motion, the wavelength isvery small compared with the linear dimensions of thebodies. This implies that one is unlikely to be able to fixthe positions of sensors, relative to a reference point withinthe body, to better than the wavelength. Consequently, forany /, the quantity et defined in eqn. 17 can be expected tolie with roughly equal probability anywhere between thelimits set down in eqns. 21. Since, as is explained in Section3, | Rt | is usually appreciably less than unity, the expectedvalue of e, is 1/2 \Ri\.

The expected error in each of the Rt can usually begauged from the experimental conditions and the recordedsignal/noise ratio. Inspection of eqn. 17 then shows that| <5Z,/Z, | is roughly | SRl/2Rl \. Since eqn. 26 indicates thatthe expected value of n, is (l/23/2a), it is seen from eqn. 24that Sa, which is largely independent of n (as argued inSection 4) but does depend in general on / and k, is givenapproximately by

da~ \5Rll25l2Rl\ (50)

which relates the expected errors in the eigenvalues a, , „to the errors in the measurement of the reflection coeffi-cients Rt.

When probing a body of radius r, and hoping to resolvedetail of size A, one must attempt to reconstruct roughly(nr2/2A2) two-dimensional spatial frequency componentsof the field existing inside the measurement circle. This isthe result embodied in eqn. 39. It is only meaningful, ofcourse, if the wavelength is less than A.

The basis functions in which wave-like emanations areexpanded, in Sections 3 and 4, are the Jt(ap/r), where a isan abbreviation for a, , „. These Bessel functions are onlyof appreciable amplitude when (ccp/r) is comparable to orgreater than | /1. Throughout most of this range of p, thefunctions J,(ap/r) and 3\{ap/r) behave like damped sinu-soids in quadrature with each other. Their RMS valuesare, in fact, essentially the same within intervals of extentno less than 2nr/a. This allows us to estimate the magni-tude of the errors in the spatial frequency components ofSA/A. The /, nth such component is (i/(an p/r) exp (//<£))> o r

equivalently (J\{an p/r) exp {jl<t>)). Inspection of eqns. 6, 32and 34 then reveals that the expected error in the /, nthcomponent is (a^a), where Sec is given by eqn. 50. This

621

reduces to (l2Sct) for low values of n, from eqn. 36. It growsrapidly, however, with increasing n, when the latter islarge.

The above argument shows that knowledge of measure-ment error magnitudes (determined mainly by signal/noiseratios, which can usually be assessed in practice to usefulprecision) permits expected errors in the reconstruction ofthe spatial frequency components of A to be quantitativelyassessed. The latter errors are irreducible, in the sense thatthey are the smallest possible ones. They are likely to belarger in practice because, as intimated in Section 1, sweep-ing approximations are needed to obtain solutions toinverse problems in most situations of scientific and techni-cal interest.

While we think it is worth reiterating that the exacttheory presented in Section 5 of Reference 1 is probablytoo complicated ever to be useful in practice, we emphasisethat it provides what seems to be a much deeper insightthan previous analyses into the dependence upon spatialfrequency of the errors in ideal inversions of measureddata.

The inherent resolution of conservative fields is, ofcourse, much less than that of wave-like emanations. It isnevertheless possible to devise measurement schemes fordetecting where the internal constitutive parameters of abody change significantly. Such changes can only besensed with certainty if the measurement errors are lessthan the differences to be expected in the values of idealmeasurements made on a homogeneous body and a bodyexhibiting the said changes. Such differences, appropriatelynormalised, are quantitatively expressed by the visibilitiesand sensitivities introduced in Section 5. These quantitat-ive results can be immediately applied to offset and flatgeometries (see Fig. 6) with the aid of the algebraic rela-tions listed in Fig. 7. It is important to recognise that the

visibility and sensitivity are always greatest for m = 1, sothat measurement schemes should always be designed sothat v^ and sx can be evaluated.

While the abovementioned visibilities and sensitivitiesrelate to physical situations of less generality than thoseexamined in Sections 3 and 4, it is worth noting that theyare derived without recourse to approximations. Theirapplication to the assessment of schemes for imaging withlow-frequency electric currents will be demonstrated in alater paper.

7 Acknowledgments

A.D. Seagar acknowledges the awards of a postgraduatescholarship and a postdoctoral fellowship from the NewZealand University Grants Committee and MedicalResearch Council, respectively. T.S. Yeo acknowledges theaward of a postgraduate scholarship from the NewZealand Government Bilateral Aid Programme.

8 References

1 BATES, R.H.T.: 'Full-wave computed tomography Pt. 1: Fundamentaltheory', IEE Proc. A, 1984,131, pp. 610-615

2 BORN, M, and WOLF, E.: 'Principles of optics' (Pergamon Press,Oxford, 1975, 5th edn.), Chap. 7

3 PRICE, L.R.: 'Electrical impedance computed tomography: a newimaging technique', IEEE Trans., 1979, NS-26, pp. 2736-2739

4 BATES, R.H.T, McKINNON, G.C., and SEAGAR, A.D.: 'A limi-tation on systems for imaging electrical conductivity distributions',ibid., 1980, BME-27, pp. 418-420.

5 WATSON, G.N.: 'Theory of Bessel functions' (Cambridge UniversityPress, 1958, 2nd edn.), Chaps. 3 & 7

6 MORSE, P.M., and FESHBACH, H.: 'Methods of theoretical physics'(McGraw-Hill, New York, 1953), Chaps. 4 & 10

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

Errata'Consultancy', Special issue of IEE Proceedings Part A,1984,131, (6), p. 341:

It is regretted that in the penultimate paragraph of theEditorial on p. 341 of the above special issue, the amountof fees earned overseas by UK consultants was erroneouslystated as £50 million per year. The correct figure is inexcess of £500 million.

3503A

WIAK, S.: 'Method of calculation of the transient electro-magnetic field in ferromagnetic materials for the knownvector of induction B on the surface', IEE Proc. A, 1984,131, (5), pp. 293-300

In Fig. 1 'differential' should read as 'difference'

In eqn. 4 A + B) should read as A+ B1}

In eqn. 5 A — B" should read as A_ B"

Eqn. 10 should read asC^ — 1 II D"ll 2 -" C *" ISKl \\t> II ^ on ^ A.

Eqn. 17 should read as

3 2Ax2[>d3]?

In the caption to Fig. 2 B(0, #t) = B(d, # t) should read asB(0, t) = B(d, t) and (1 - c~tlT) should read as (1 - e~tlT)

In the captions to Figs. 5-12 y should be replaced by o

In the caption to Fig. 7 T = 10 ~3 m should read asT = 10"3 s

In the caption to Fig. 11 d = 5 x 10~3s should read asd = 5 x 10" 3 m

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