Allowing for variable resolution and constantattenuation in SPECT

D.G.H. Tan, BScJ.X. Qu, MEK.L. Garden, ME, PhDProf. R.H.T. Bates, DSc(Eng), FEng, FRSNZ, FIEE

Indexing terms: Image processing, Biomedical applications, Biological effects, Computer applications, Computed tomography

Abstract: By representing an image as a distribu-tion of Gaussian blobs, due account is taken, for asingle photon emission computed tomography(SPECT) system, of finite detector resolution(leading to spatially varying image resolution) andconstant attenuation of the radiations on theirpassage through a body. A simple enhancementprocedure is proposed. It is argued that the effectsof variable attenuation (as must occur in practice)are unlikely to be important in general.

1 Introduction

Single-photon-emission computed tomography (SPECT)has been investigated for more than a dozen years [1, 2].From the beginning, there has been interest in compen-sating for the attenuation suffered by the radiations ontheir passage through the body. A significant difficulty isthat, even if conventional computed tomography (CT)measurements [3] are made initially, using radiation ofthe same energy as that responsible for the SPECTimage, there is no guarantee that a convergent algorithmexists to account for the attenuation [1, 4]. Although thegeneral attenuation problem for SPECT remainsunsolved (and may well be unsolvable), the constantattenuation problem is well understood now [5-7]. Inthis latter problem, one assumes that the attenuationcoefficient is invariant throughout the body. Thisassumption only leads to useful practical results if thesurface of the body is known from prior observation;using calipers or by echo location, for instance [4]. Somesuccess has been achieved with post-processing based onassuming different values of the attenuation coefficient indifferent parts of the body [5].

Received signal levels are usually high enough in con-ventional CT that one can use collimated detectors pos-sessing narrow beams. It thus makes good physical senseto think of the radiation received by an individual CTdetector as being confined to a thin pencil of rays.

Paper 5064A (S9, E4), first received 7th January and in revised form 9thApril 1986The authors are with the Electrical & Electronic Engineering Depart-ment, University of Canterbury, Christchurch 1, New ZealandJ.X. Qu is on leave from the Radio & Electronics Department, Uni-versity of Science & Technology of China, Hofei, Anhwei, People'sRepublic of China

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SPECT signals, on the other hand, tend to be photon-limited, so that it is usually necessary to employ detectorshaving appreciable beamwidths. Surprisingly, the effect ofa finite SPECT beamwidth, which leads to variableresolution within the body [9], seems to have receivedcomparatively little attention [10, 11].

In this paper we examine both constant attenuationand variable resolution. We also discuss the kind ofimage processing which can ameliorate their deleteriousconsequences. Of special significance is that we allowsimultaneously for both effects without having to resortto iterative procedures. Despite the greater generality ofour analysis, we find ourselves able to propose a verysimple enhancement strategy which, nevertheless, appearsto be potentially useful. The results of this processingcould probably be slightly improved by any of the com-monly practised post-processing image-restorationschemes [4—13].

The general SPECT problem is formulated in Section2, while Section 3 lists the simplifications which we findare necessary to obtain quantitative image-processing cri-teria. Section 4 presents some general analysis fromwhich specific results are obtained in Sections 5 and 6.The general SPECT problem is re-examined in Section 7where we conclude that it may not be worth worryingtoo much about trying to take due account of variableattenuation, because little advantage is likely to accrueeven if a practicable means of accomplishing it could bedevised. We assess, in Section 8, the practical implica-tions of our approach.

2 General SPECT problem

The closed curve C in Fig. 1 represents the perimeter of aparticular cross-section of a 3-dimensional body contain-ing a distribution of radiating sources. The distribution isdenoted by/(x, y, z), where the Cartesian co-ordinates x,y and z are taken to be fixed in the cross-section. The <!;',Y\ Cartesian co-ordinates are rotated in the plane of thecross-section by an (arbitrary) angle (/> with respect to thex, y co-ordinates. The Cartesian and polar co-ordinatesof the arbitrary point P in the body are (x, y, z) and (r; 9,z), respectively. Note that a semicolon always precedes anangular variable in this paper, so that we can write

f(x,y,z)=f(r;0,z) (1)

without being in any way ambiguous concerning thetypes of co-ordinate implied in each case.

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We postulate a linear array of identical detectors withtheir faces (i.e. collecting apertures) in a plane which isparallel to the £', z-plane. The £-axis, which lies in the x,y-plane at a distance R from the < '̂-axis (to which it isparallel), passes through the centres of all detectors in thearray. The centre of a typical detector lies at Q. We

Fig. 1 Basic geometry and co-ordinate systems for SPECTC is the perimeter of a particular cross-section through a 3-dimensional body

denote its total response to the complete distribution ofsources within the body by ps(£; <$). In the absence of anyattenuation of the radiations from these sources, ps(£,; </>)can be expressed quite generally as

;' cos (<f>)

— r\ sin ((/>), % sin (0) + rj cos ((/>), z)

x D(£ -£,rj + R, z) d% dr\ dz (2)

where the infinite limits here (and in similar instanceslater in this paper) merely imply that all of the body isbeing considered, and where £>(£ — £', rj + R, z) is thedetector's response to a point source at P. Note that, inthe arguments of/(x, y, z) in eqn. 2, the co-ordinates xand y have been expressed in terms of £' and r\.

We imagine a continuous array of detectors, whichcan be effectively realised either by scanning a singledetector along the £-axis (this is of course impracticablein the real medical world because it would take too long)or with a discrete array whose spacing is close enough tosatisfy whatever resolution constraints appertain in anyparticular application (this is practical) [14]. We alsopostulate that responses are recorded for all (p, which isanother idealisation which can be effectively realised, inpractice, if responses are recorded for discrete values of (f>,again spaced sufficiently closely to satisfy the prevailingsampling constraints [15].

The general SPECT problem is to recover /(x, y, z)from ps(£; 4>), as defined by eqn. 2, recorded in all planesz for — oo < £ < oo and 0 ^ </> < 2n, on the understand-ing that the attenuation coefficient 2n n{x, y, z), corre-sponding to each point P in the body, is unknown apriori. As we have no idea at present how to attack thisproblem, and because it probably does not possess aunique solution, it makes sense to pose the restrictedSPECT problem, i.e. recover/(x, y, z) when n(x, y, z) hasbeen reconstructed previously from a set of conventional

CT measurements. It is worth emphasising that the con-ditions under which even this restricted problem canhave a unique solution are as yet unclear. This is why welower our sights and only give detailed consideration tothe simplified problem introduced in Section 3.

Like everyone before us, we have to introduce approx-imations before we can even begin to analyse theproblem. In particular, we note that it is far from clearhow to modify the integral in eqn. 2 so as to take accountof attenuation. It cannot be done with complete gener-ality by merely inserting a factor such as

h(P, Q) = exp -2n f H(x, y, z) dl (3)

into the integrand. This factor represents the attenuationalong the straight line path from an arbitrary point in thebody to the centre of a typical detector. We are not sug-gesting that the radiations do not travel in straight lines,just emphasising that the face of a real-world detector hasa finite area. Consequently, the general SPECT problemis even more challenging than is customarily intimated[4,8,10,11].

3 Simplified problem

Because the detectors used in SPECT systems have mod-erate beamwidths (between 5° and 10° say), and as Rmust be roughly twice the radius of the circle whichcircumscribes C (to accommodate patients of variousshapes and sizes), and owing to the comparatively lowresolutions and poor signal-to-noise ratios achieved inpractice, we feel it is reasonable to adopt the simplifica-tions introduced here.

Each detector (specifically the one with its centre at Q)is characterised by a simple beam emanating from thedetector's centre with its axis parallel to the ty-axis. Whilethe width of the beam increases with distance from Q, theshape of the beam is circularly symmetric in any planeparallel to the z, £-plane. This means that we now can, infact, account for attenuation by inserting into the inte-grand of eqn. 2 the factor h(P, Q) defined by eqn. 3. Notethat dl in eqn. 3 is the element of length along thestraight line path from P to Q.

The most important of the physical processes affectingthe quality of SPECT imagery is Compton scattering. Itgives rise to the major part of the attenuation suffered bythe radiations on their passage through the body. It alsoaffects the effective beamwidths and beam shapes of thedetectors owing to multiple scattering. A ray proceedingfrom a particular source towards a particular detectormay be scattered out of its initial path and then be re-scattered into another detector. This increases the effec-tive width of each detector's beam [12]. As thedistribution of Compton-scattering material, which isapproximately proportional to n(x, y, z), varies through-out the body, the beamwidth and beam shape of eachdetector must, in general, be different, and must alter as <\>changes. By the central limit theorem, however, theaverage beam shape can be expected to be Gaussian. Asthe beam shape in the absence of multiple scattering isclose to Gaussian [16], it is reasonable to assume theoverall average beam shape to be Gaussian, which iswhat we do in the following.

The distance from any point within the beam to Q isapproximated by R plus the rj co-ordinate of the point.This means that the beam can be expressed in the generalform <D([(f' - 02 + z2]/(ri + R)202), where 0 is the effec-

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987 137

tive beamwidth. Radiation emanating from a pointsource spreads out spherically, weakening according tothe inverse square of distance from the source, so that thedetector's response to a point source is given by

Dtf -?,ri + R, z) = (l/4*fo + *)2)O([(<r - £)2

+ z2V(rj + R)262) (4)

It is also assumed that the straight-line path implied bythe integral in eqn. 3 can be taken parallel to the rj-axis,so that dl is replaced by dr\, i.e.

(5)h(P, Q) = exp -2TT H(X, y, z) drj-R

The distribution f(x, y, z) is assumed independent of zover the beam, so that

/ (x , y, z) =f{x, y) (6)

implying that the distribution of sources throughout a3-dimensional body can be built up from 2-dimensionaldistributions existing in stacked parallel cross-sections,each of which can be reconstructed from a separate set ofmeasurements, as in conventional CT [3].

The attenuation is taken to be zero outside the bodyand constant inside (as already intimated in Section 1,this simplification has to be made in order for us to makeany headway). The constant attenuation coefficient is2nji. The curve C is understood to be determined byprior measurement, implying that the position of thepoint Q' (see Fig. 1) is known. The upshot is that ( — R)and fj,(x, y, z) are replaced in eqn. 5 by r\{^\ <£) and Ji,respectively, where fj(£; 4>) is the rj co-ordinate of Q'. Fur-thermore, h(P, Q) can be normalised by multiplying it byexp { — 2nJiY\{^\ (j))), thereby giving

h(P, 0 = exp ( - (7)

because rj is, by definition, the rj co-ordinate of P.We have now reached the same position with regard

to attenuation as those [5, 7, 17, 18] who have gonebefore us, but we are simultaneously taking account ofthe finite beamwidth of the detectors. It is impracticableto attempt to recover the form of/(x, y) from ps(£; 4>)until a functional form for O ( ) is specified. For thereasons given in the third paragraph of this section, andbecause we are approximating /x(x, y, z) by ji, we feel it isappropriate to take each detector to be an average detec-tor, which is characterised by a Gaussian beam shapehaving an effective beamwidth denoted here by 0. On thisunderstanding, ps(£; </>) is seen to play the part, familiar inconventional CT [3], of the projection at angle </>.Accordingly, we call it the SPECT projection which, byincorporating eqns. 4 to 6 into eqn. 2, can be written as

cos (0) — t] sin ((j>), £' sin ($)

+ rj cos (<f>)) exp {-2njir] - [(£' - £)2

+ R)262} d? dr] dz/4n(rj + R)2 (8)

The simplified SPECT problem is to recover/(x, y) fromPs(€; <t>\ a s defined by eqn. 8, given for — oo < £ < oo and0 ^ 0 ^ 27i, for particular values of JL, R and 0.

Whenever we refer in the rest of the paper to theactual reconstruction of images, we find it convenient (forease of exposition) to do so in terms of direct Fourierinversion [15]. We emphasise, however, that the moreefficient modified back-projection approach remains

applicable, in any of the manifestations which are nowstandard for conventional CT [1, 9].

4 Analytical considerations

Note that the z-integration in eqn. 8 can be done imme-diately. To proceed further we introduce, in the spirit ofthe projection theorem of conventional CT [3, 15], the1-dimensional Fourier transform, with respect to £,, ofPs(£; ((>)• We denote this transform by Fs(p; <f>). Adoptingexp {jlnpQ for the Fourier kernel leads to

1' cos ((/>) — 77 sin (0), <!;' sin^s(p; 0) = (02/2)

+ r\ cos (</>)) exp (2n[jp^ - Jir\

r, (9)

It is now instructive to state the result of a particularrearrangement of the integral in eqn. 9. We require the2-dimensional Fourier transform F(p; <j>) of/(r; 6) =f(x,y). After lengthy, but nevertheless straightforward, manip-ulations it transpires that

Fs(p; <t>) = (n

x exp ( - 2n2R262p2) exp (-12/262)

x F([l + T2]1/2p; 4> - dt (10)

where x = t — jji/p — j2nR62p. In situations where it isallowable to neglect the detector's beamwidth (i.e. 6 a 0)we see that the Gaussian function in the integrand hasthe character of a delta function, implying that eqn. 10reduces to

= F((P2 - lWp)) (11)which emphasises Clough and Barrett's [8] elegantinsight that useful information concerning the conven-tional spectrum F(p; $) is given only by the part of theSPECT spectrum Fs(p; (f>) lying outside the circle ofradius ji centred on the origin of Fourier space. It isunclear how to make practical computational use ingeneral of values of Fs(p; (j>) for p <Ji, because thesevalues define F(p; </>) for imaginary values of its radialco-ordinate. The fact that Fs(p; <j>) defines F(p; <f>) explic-itly only for complex values of <f> does not mean that F(p;<f>) cannot be readily evaluated for real values of (p.Inspection of eqn. 11 reveals that, when Fs(p; <p) and F(p;4>) are both expressed as trigonometrical Fourier series in</>, their corresponding Fourier coefficients are directlyproportional to each other, with the quantity tanh"1

(ji/p) appearing in the proportionality factors.We are interested in situations in which the factor exp

( — t2/262), in eqn. 10, does not have the character of adelta function. The form of eqn. 10 is then decidedly dis-couraging, and we have found it more profitable toreturn to eqn. 9.

Various practical difficulties cause the resolutionachievable with SPECT systems to be quite limited [14,19]. It is, consequently, adequate to represent/(x, y) asan array of 'Gaussian blobs'. The spacing of the blobsand the effective radius, a say, of each blob characterisethe resolution which one hopes a particular SPECTsystem will achieve.

Because/(x, y) is linearly related to either ps(£; <f>) orFs(p; (f>), the performance of a SPECT system can be

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

assessed by examining a single blob, centred at the point

= exp{-l(x-bcos(p))2

+ (y-b sin (0))2]/2<r2} (12)

When eqn. 12 is substituted into eqn. 9, both integrationscan be performed immediately (but care must be takenwhen manipulating the integrands never to allow r\ tobecome implicitly less than (— R), as is clear from inspec-tion of Fig. 1), giving

Fs(p; 4>) = H(p; </>, Ji, a, R, b; 0; 9)F(p; <j>, b; P) (13)

where F(p; </>, b; /?) is the spectrum of the single blob, i.e.

F(p; (f>,b;P) = 2na2 exp ( - 2n[no2p2

-jbp cos ( 0 - / 0 ] ) (14)

and H(p; <$>, Ji, a, R, b; /?; 9) represents the spatial fre-quency distortion due to both (constant) attenuation ofthe radiations and (variable) finite resolution of thesources of these radiations, i.e.

H(p; (p, Ji, a,R,b\ p; 9) = (92/2co) exp (-Q) (15)

where

co = (1 + 47 tV0V) 1 / 2 (16)

and

Q = 27r{7i[l - 4nJia2/R-]R292p2 - nji2a2

- Jib sin (<f> - P) - 2nbR02p2 sin (0 - j?)

+ nb2d2p2 sin2 (0 - j3)}/«2 (17)

Eqns. 12 to 17 suggest a simple image-processing schemefor SPECT, as we explain in the following two Sections.Before embarking on this, we point out that Fs(p; 0) isnot, as it stands, a suitable basis for image reconstructionbecause it is not conjugate symmetric, i.e. Fs(p; </> + n) #F*{p; cf>), where the asterisk denotes complex conjuga-tion, implying that the image is not real. It is, therefore,appropriate to introduce what we call the oppositelyaveraged SPECT spectrum Fs{p; </>) defined by

which is conjugate symmetric. Note that using Fs(p; 4>) asa basis for image reconstruction is equivalent to averag-ing diametrically opposed projections, which is a stan-dard SPECT procedure [5].

5 Resolution considerations

To generate useful SPECT imagery, the resolution mustbe much less than the average diameter of the cross-section. So, we must insist that a <€ R. The term in Q, asdefined by eqn. 17, which most limits the spatial fre-quency content of the SPECT spectrum is 2n2R262p2. Itfollows that co, as defined by eqn. 16, differs little fromunity throughout the range of p for which exp( — 2n2R292p2) has appreciable value. Consequently, cocan be replaced by unity in eqn. 17 without introducingany significant error.

If Ji is large enough that the attenuation is appreciableover a distance equal to the effective radius of the blob,then either this radius is set too large or the magnitude ofthe attenuation is so severe that the SPECT system isunlikely to be of much practical use. Consequently, thefactor 2njio must be appreciably less than unity, say nogreater than 1/3. Therefore, (4n2Jio2/R) can be no greater

than 2a/3R, which is itself considerably less than unity(see the second sentence of this Section). Similarly,2nji2o2 cannot exceed 1/18. We then see that eqns. 15 to17 reduce effectively to

H(p; </>, Ji, a, R, b; 0; 9)

= (92/2) exp (-2n2R292p2 - A) (19)

where

A = 27r2{sin2 {$-$)- [2R/b + Jt/nb92p2~]

x sin (0 - p)}b292p2 (20)

from which, it appears that attenuation only contributesappreciably to image degradation at low spatial fre-quencies. We must not, of course, forget the normal-isation involved in the definition of h(P, Q) in eqn. 7. Thisnormalisation glosses over the very significant adverseeffect of the attenuation on the detected signal-to-noiseratio. Because the maximum value of b can hardly begreater than 2.R/3, say, in practice, and is likely to becloser to R/2, we see from eqns. 13, 14, 18, 19 and 20 thatany blob's oppositely averaged SPECT spectrum isdominated by the factor

M(p) = (no292) exp [-27r2(<72 + R292)p2~\ (21)

which determines the resolution which one shouldattempt to achieve in any particular instance. The pointis that | Fs{p; </>) | is only appreciable in that part ofFourier space wherein M(p) exceeds a threshold set bythe prevailing noise level.

The total flux of radiation received from a blob is pro-portional to (a0)2, with the constant of proportionalitydepending on the various system parameters such as thespecies and densities of the sources of radiation, the timedevoted to measuring each SPECT projection and thenature of the detectors. Because M(0) is directly pro-portional to this flux, as is l/R2, the ratio of M{p) tonk2/R2 (where the constant k depends on the systemparameters) characterises the signal-to-noise ratio for thecomponent of the spectrum at the spatial frequency p. Asthe effective radius of each blob is a, we require M{p) toexceed the lowest spectral level that can be determined toa useful accuracy for all p up to I/a. Taking nk2/R2 to bethis level, we see that

= nk2/R

Combining eqns. 21 and 22 gives

a2 = n2l{q + loge a)

(22)

(23)

where a = (<r/R9) and q = loge (R292/k) - n2.

In any particular instance, the system parameters areknown and so eqn. 23 can be solved for a. This deter-mines the system resolution because it fixes the choice ofa.

One can attempt to remove some, small, part of theblurring inherent in H(p; <$>, Ji, a, R, b; /?; 9) by invokingan image restoration strategy of the kind referenced inSection 1. As such strategies are covered adequately inthe quoted References, we content ourselves with pro-pounding a simple enhancement scheme which is sug-gested by the analytical approach introduced in thispaper.

On recalling that the resolution is essentially deter-mined by M(p), we see, from eqns. 13 to 20, that the onlyfactor in F(p; <fi) that depends, effectively, on Ji is

J((f>, Ji,b;P) = cosh \_2nbji sin (</> - (24)

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the practical computational significance of which is con-veniently assessed by examining the multi-blob imageshown in Fig. 2. This is an idealised/(x, y) consisting of adiscrete array of identical blobs. Fig. 3 shows the recon-struction, directly from Fs(p; <f)). The attenuation coeffi-cient is typical of that occurring in practice. Although the

Fig. 2 Multi-blob image, consisting of 13 identical blobs (a = 0.08)The co-ordinates of the centres of the blobs are (0.6; n), (0.3; n), (0.0; 0), (0.3; 0),(0.6; 0.0), (0.4; rt/4), (0.8; n/4), (0.4; 3n/4), (0.8; 3TT/4), (0.5; 4n/3), (0.5; 5n/3), (0.3;3TT/2),(0.7;3JI/2)

Fig. 3 Magnitude of uncompensated reconstruction of multi-blobimage, assuming R = 1,9 = 0.05 and Ji = 0.5

shapes of the reconstructed blobs are somewhat dis-torted, their most serious defect is the large variation intheir amplitudes. The enhancement strategy described inthe following Section compensates for much of this.

6 Simple image enhancement strategy

The variation in the amplitudes of the blobs apparent inFig. 3 is mainly due to the factor J((f>, ji, b; ft) defined byeqn. 24. The enhancement strategy introduced here isbased on the following interpretation of this factor. Wefirst note, from the definition of the Fourier transform[15], that

/(0, 0)Jo Jo

F(p; 4>)p d(f> dp (25)

which suggests that the aforementioned variation in blobamplitudes may be given to a useful degree of approx-imation by the integral of J(0, Ji, b; ft) over the p, em-

plane. As J{<j>, n,b;P) is independent of p, we need onlyconsider the factor

,jJL)= \ J { < t > , H , b ;Jo

A(b,jJL)= (26)

By expanding J((p, Ji, b; /?) as a series of modified Besselfunctions [20], we obtain

A(b, fi) = Aipb) = lo{2njib) (27)

the form of which is displayed in Fig. 4, for the interval0 ^ Jib ^ 0.5.

Fig. 4 Radial amplitude variation factor A{Jib)

The enhancement procedure is to take the imageobtained directly from Fs(p; 4>) and, at each point, dividethe value of the image by A(p.r). Fig. 5 reveals the resultof enhancing the image shown in Fig. 3. While this

Fig. 5 Magnitude of enhanced reconstruction of multi-blob image

Same parameters as for Fig. 3

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

enhancement procedure is certainly approximate, we findit to be effective over a wide range of values of ji. Blobs atdifferent radii are reconstructed to somewhat differentresolutions, but the variations are likely to be acceptablein practice. The amplitude variations with radius arereduced markedly. We demonstrate this quantitativelywith the aid of the 5-blob image, shown in Fig. 6, which

Fig. 6 5-blob image, consisting of identical blobs (a = 0.08) all centredon x-axis at points defined by x = 0, +0.3, ±0.6

consists of a linear discrete array of blobs each identicalto each of those shown in Fig. 2. Fig. 7 shows the recon-struction of the 5-blob image, directly from Fs(p; (j>),without any kind of restoration being attempted. The

Fig. 7 Magnitude of uncompensated reconstruction of 5-blob image,assuming R = 1,6 = 0.05 and Ji = 0.5

attenuation coefficient is the same as that invoked forFig. 3. When the image shown in Fig. 7 is enhanced, it isquite similar to Fig. 6, although there is a significant lossof resolution. Fig. 8 shows line profiles along the x-axisthrough the original image and also through the uncom-pensated and enhanced images. The main differencesbetween the profiles through the original and enhancedimages are that the latter profile exhibits slightly broaderpeaks of somewhat unequal amplitudes and its nulls areless than half as deep.

Our suggested image-processing strategy is thus torecord a set of measured projections within the angularrange 0 ^ (j> < 2n, normalise each projection by h(P, Q)

as defined by eqn. 7, thereby generating a set of what wecall (in Section 3) SPECT projections, average theSPECT projections at angles differing by n, thereby gen-erating a set of 'averaged' projections within the angularrange 0 ^ <j) < n, filter each of these averaged projectionssuch that their spatial frequency content corresponds tothe resolution estimated by the procedure characterisedby eqns. 22 and 23, apply a conventional CT imagereconstruction algorithm (refer to final paragraph ofSection 3), and, finally, divide each pixel value in thereconstructed image (whose radial co-ordinate is r) byAQir), where 2nji is the available estimate of the averageattenuation coefficient. It is the combination of the filter-ing and the division by A(Jir) which represents theoutcome of the theoretical development introduced inthis paper.

11IU11/}

1

1)ii

ii>\

-1.2 0 1.2

Fig. 8 Line profiles along x-axis for the 5-blob imageFig. 6Fig. 7magnitude of enhanced image

7 Effects of variable resolution

When, as is always true in practice, the attenuation isvariable, the SPECT spectrum of a blob is given(approximately, but nevertheless to reasonable accuracy)by Fs(p; <f>), as developed in Sections 4 and 5, multipliedby a factor Z(p; (j), b; ft) which of course depends, ingeneral, on where the blob is located in the cross-section.As this factor represents only the variation of the attenu-ation about its average value, it can usually be expectedto merely affect the resolution to which the blob isreconstructed. We remark that this factor has an identi-cal form to that which arises when the amplitude of theimage f(x, y) varies from projection to projection. Wehave previously studied such 'temporal variations'(because projections are usually measured sequentially,the aforesaid variations are usefully thought of asoccurring in time) and have shown that they have sur-prisingly little effect on image detail [21, 22]. Boundariesbetween regions of different 'density' tend to be clearlyrecognisable, even when the variations are large [21].

The main purpose of a SPECT system is to identifylocations of appreciable concentrations of sources ofradiation. One seldom attempts to recover fine detail(which is just as well, considering the relatively poor

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 2, FEBRUARY 1987 141

resolutions and signal-to-noise ratios of SPECT systems).We therefore feel confident in asserting, on the basis ofour previous experience [21, 22], that only rarely will beeffects of variable resolution be actually noticeable inimages reconstructed from SPECT projections.

We are not, of course, inferring that attenuation hasno important effects. It seriously degrades the signal-to-noise ratio, as we have already emphasised. Com-pensation for the average value of the attenuationcoefficient is mandatory, unless one is prepared to put upwith image degradation of the kind illustrated in Figs. 3and 7. We have, however, shown in Section 6 how toameliorate the effects of average attenuation. What weare arguing in this Section is that variable attenuation isusually comparatively unimportant.

8 Conclusions

Our combined analysis of variable resolution and con-stant attenuation in SPECT has persuaded us that:

(a) variable resolution is conveniently handled by rep-resenting the image as a distribution of blobs whose effec-tive diameters are chosen according to the simpleprescription introduced in Section 5

(b) the only really serious image degradation due toconstant attenuation (apart from its adverse effect on thesignal-to-noise ratio) is its tendency to cause differentparts of the image to be reconstructed with differentamplitudes (this tendency can be markedly inhibited bythe simple enhancement procedure introduced in Section6)

(c) it is probably a waste of time to attempt to solvethe variable attenuation problem.

We are planning experiments to evaluate (in a clini-cally meaningful manner) the usefulness of the image-processing strategy suggested in Section 6 and tocompare it with more sophisticated processing, such asthat recently reported by Faber et al. [23] and Webb[24].

9 Acknowledgments

D.G.H. Tan acknowledges the award of a PostgraduateResearch Scholarship from the New Zealand UniversityGrants Committee, while J.X. Qu thanks the Universityof Science and Technology of China, for providing finan-cial support, and the Ministry of Foreign Affairs of NewZealand, for making the necessary arrangements for hisvisit. K.L. Garden and R.H.T. Bates thank TechnicareCorporation of Cleveland, Ohio, for financial assistancefor our research. We are especially grateful for technicalcorrespondence with George Kambic and Steve Gotts-chalk. R.H.T. Bates thanks Anne Clough for helpful dis-cussions during an early stage of the work reported here.

10 References

1 PETERS, T.M.: 'Image reconstruction from projections'. Ph.D.thesis, Engineering Library, University of Canterbury, Christchurch,New Zealand, 1973

2 GULLBERG, G.T., and BUDINGER, T.F.: 'The use of filteringmethods to compensate for constant attenuation in single-photonemission computed tomography', IEEE Trans., 1981, BME-28, pp.142-157

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

• 4 GULLBERG, G.T.: 'The attenuated Radon transform: theory andapplication in medicine and biology'. Ph.D. thesis (Tech. Rep. LBL-7486), Lawrence Berkeley Lab, University of California, 1979

5 GREER, K.L., JASZCZAK, R.J, and COLEMAN, RE.: 'An over-view of a camera-based SPECT system', Med. Phys., 1982, 9, pp.455-463

6 GULLBERG, G.T.: 'The attenuated Radon transform: applicationto single photon emission computed tomography in the presence ofa variable attenuating medium'. Tech. Rep. LBL-10276, LawrenceBerkeley Lab, University of California, 1980

7 TRETIAK, O.J, and METZ, C : The exponential Radon trans-form', SI AM J. Appl. Math., 1980, 39, pp. 321-335

8 CLOUGH, A.V, and BARRETT, H.H.: 'Attenuated Radon andAbel transforms', J. Opt. Soc. Am., 1983, 73, pp. 1590-1595

9 GARDEN, K.L.: 'An overview of computed tomography'. Ph.D.thesis, Engineering Library, University of Canterbury, Christchurch,New Zealand, 1984

10 HSIEH, R.C, and WEE, W.G.: 'On methods of three-dimensionalreconstruction from a set of radioisotope scintigrams', IEEE Trans.,1976, SMC-6, pp. 854-862

11 ANSARI, A, and WEE, W.G.: 'Reconstruction from projections inthe presence of distortion'. IEEE Conf. Proc. Decision & Control(New Orleans), 1977,1, pp. 361-366

12 FLOYD, C.E, JASZCZAK, R.J, GREER, K.L, and COLEMAN,R.E.: 'Deconvolution of Compton scatter in SPECT', J. Nucl. Med.,1985,26, pp. 403-408.

13 MADSEN, M.T, and PARK, C.H.: 'Enhancement of SPECTimages by Fourier filtering the projection image set', J. Nucl. Med.,1985, 26, pp. 395-402

14 LIM, J.S.: 'Performance analysis of three camera configurations forsingle photon emission computed tomography', IEEE Trans., 1980,NS-2, pp. 559-568

15 LEWITT, R.M.: 'Reconstruction algorithm: transform methods',Proc. IEEE, 1983,71, pp. 390-^08

16 GOTTSCHALK, S.C., SALEM, D , LIM, C.B., and WAKE, R.H.:'SPECT resolution and uniformity improvements by noncircularorbit', J. Nucl. Med., 1983, 24, pp. 822-828

17 CHANG, L.T.: 'Attenuation correction and incomplete projectionin single photon emission computed tomography', IEEE Trans.,1979, NS-26, pp. 2780-2789

18 DEUTSCH, M.: 'New inverses of the attenuated Abel integral equa-tion', J. Phys. A, 1984,17, pp. L939-L944

19 JASZCZAK, R.J, COLEMAN, R.E, and WHITEHEAD, F.R.:'Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT)',IEEE Trans., 1981, NS-28, pp. 69-80

20 ABRAMOWITZ, M, and STEGUN, LA. (Eds.): 'Handbook ofmathematical functions' (Dover, New York, 1965), p. 376

21 GARDEN, K.L, BATES, R.H.T, WON, M.C., and CHIK-WANDA, H.: 'Computerized tomographic imaging is insensitive todensity variation during scanning', Image & Vision Comput., 1984, 2,pp. 76-84

22 GARDEN, K.L, and BATES, R.H.T.: 'Image reconstruction fromprojections VII: interactive reconstruction of piecewise constantimages from few projections', Optik, 1984, 68, pp. 161-173

23 FABER, T.L., LEWIS, M.H, CORBETT, J.R, and STOKELY,E.M.: 'Attenuation correction for SPECT: An evaluation of hybridapproaches', IEEE Trans., 1984, MI-3, pp. 101-107

24 WEBB, S.: 'Comparison of data-processing techniques for theimprovement of contrast in SPECT liver tomograms', Phys. Med. &BioL, 1985, 30, pp. 1077-1086

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