Along with a review of some of the mathematical foundations of computed tomography, the article containsnew results on derivation of reconstruction formulas in a general setting encompassing all standard formulas;discussion and examples of the role of the point spread function with recipes for producing suitable ones;formulas for, and examples of, the reconstruction of certain functions of the attenuation coefficient, e.g.,sharpened versions of it, some of them with the property that reconstruction at a point requires only theattenuation along rays meeting a small neighborhood of the point.
1. Introduction
The divergent beam radiograph of the function ffrom the source point a in the direction 0 is defined by
ZOaf(O) = J f(a + t)dt. (1.
Physically f is the x-ray attenuation coefficient of theobject x rayed, a is the x-ray source, or tube focal point,and 0 is an x-ray detector. af() represents the atten-uation in the x-ray beam along the ray with origin a anddirection . The attenuation coefficient f is assumedto be square integrable on Rn and to vanish outside afixed bounded open set Q, i.e., to lie inLO2(Q). The roleof computed tomography is to reconstruct this func-tion from a number of x rays.
In a strict sense, the word tomography refers to thereconstruction of cross sections of f, i.e., to the 2-Dproblem, and most of the activity in the field hascentered on two dimensions, but the 3-D problem isbeginning to attract more attention. To accommo-date both, the present discussion takes place in Rn.
A point in Rn is an n-tuple of real numbers usuallydenoted by a single letter: x = (x1, .. . ,xn). The innerproduct and absolute value are defined by
n
(x,y) = xjyj and Ixx = (xx).
A direction is a point of absolute value 1, usually denot-ed by 0. The unit sphere Sn-i is the set of all direc-
The authors are with Oregon State University, Mathematics De-partment, Corvallis, Oregon 97331.
tions. For any point x, x is the subspace perpendicu-lar to x, i.e., the set of points y satisfying (x,y) = 0. Indimension 2 the coordinates are often written out, with(x,y) in place of ( 1,x2) and with some confusion al-lowed between the direction 0 and angle Ap so that 0 =(cosA sinV9).
Historically, computed tomography began with par-allel beam x rays in which the photons travel alonglines with a fixed direction 0 rather than along raysemanating from a fixed source a. The parallel beamradiograph is defined by
'P of(x) = f(x + tO)dt, x -'. (1.2)
Parallel x-ray beams are difficult to produce physical-ly, but the parallel beam transform can be producedmathematically by the process called rebinning shownin Eq. (1.5), and sometimes is approximated by using asizable distance from x-ray source to object.
The Radon transform, using planar integrals insteadof line integrals, is defined by
Rof(t) = JI,) = f(x)dx. (1.3)
The very lively field of nuclear magnetic resonanceinvolves the reconstruction of f from its Radon trans-form.
In addition to the x-ray and Radon transforms, twoothers will play a role:
LCfly = Jf[x + tylyj)]dt,
!LJ(y) = I (yx) jy I -V-f(Y) (1.4)
The following relations are clear:
.L;(O) = Z(0) + DJ(-0) = Pof(Eox), (1.5)
where E is the orthogonal projection on 01.These mathematical transforms represent idealized
approximations to the complex relationships between
the object and measured data. Discrepancies result-ing from numerical approximation, the positive diame-ter of the x-ray sources and detectors, the mixed ener-gies of the x-ray beam, etc. are not discussed here.
From a mathematical point of view two of the mostinteresting aspects of computed tomography are thequestions of uniqueness and stability. These ques-tions are discussed, and the results are summarized inRef. 1. Uniqueness and nonuniqueness theorems areproved in Refs. 2-4. The most complete theorems onstability are given in Refs. 5-7; earlier special casesoccur in Refs 8-12.
II. Inversion Formulas
In this article the Fourier transform is given by
/() = (2rYn/2J exp(-i(x,())f(x)dx.
g(x) = (1/2)C(n,1)A J Pog(Eox)dO
= (1/2)C(n,1) f AP 0g(Eox)d0. (2.10)
When the actual x-ray sources lie on the sphere A (ofradius R surrounding Q), the divergent beam inversionformula is obtained by making a change of variable inEq. (2.9):
= (a-x)la-x , dO = (11R) (x -a,a) ix -a I -nda.
When the homogeneity and symmetry
.Lag(ty) = Lag(y), Cxg(a - x) = Lag(x - a)
are used, the result is
(2.1)
For use in potential theory Riesz13 introduced thefunctions Ra defined by
R.(Q) = (27r)n/2 I - < a < n. (2.2)
From the homogeneity and invariance under orthogo-nal transformations it follows that
R,(x) = C(n,a) Ix I an, with C(n,a) = 2[(n - a)/2] (2.3)Ifthoprator isdefinedby 2n/(a/2)
If the operator Ac, is defined by
(2.4)(WAf)^) = 1 I '7Q)
then A' inverts convolution by Ra, i.e.,
Aa(Ra*f) = f.
The operator A is expressed directly by the formula
(2.5)
g(x) = [C(n,1)/2R]A JA lag(x - a)da. (2.11)
The corresponding Radon inversion formula is
g(x) = (1/2)(27r)' An' J YSn g((x,0))d0. (2.12)
Because of the singularity of A, it is expedient inpractice to seek a reconstruction of an approximationto f rather than a reconstruction of f itself. Normallythe approximation has the form e*f, where e is anapproximate function called the point spread func-tion.
In view of the formula
'PO(e*f) = Poe*Pof,
replacement of g by e*f in Eq. (2.10) gives
(2.13)
e*f(x) = f k*Psf(Eox)dO, withk = (1/2)C(n,1)APoe. (2.14)
The corresponding formula for the divergent beam,n
Af = - (R 1/xj) * f/ax,1=1
(2.6)
the convolution being a Cauchy principal value.The role of the Riesz potential in x-ray theory comes
from the following:
THEOREM (2.7). If g > 0, then for every point x,
[1/C(n,1)]R1*g(x) = I x 1Jg(O)d0 = (2) I 'Pg(Ex)dO, (2.8)f~~n-1 ~ 2 J
provided +PROOF.
tion y = t,
- is allowed as a value.In the integral below make the substitu-
dy = tn-ldtd0 = I y I n-ldtd0 to get
fSn-5 Jxg(0)dO = Sn- g(x + tO)dtd0 = J g(x + ) I I -ndy
and then use Eq. (1.5).Application of A to both sides of Eq. (2.8) yields the
J 'P f(y)K(y)dy = (1/2R) JA .J(0)K(E 0 a)da (2.16)
with K(y) = k(Eox - y). Equation (2.16) is obtainedby making the change of variable y = E0a, dy =(a,0)/RI dO on the upper and lower hemispheres of A.
Equation (2.12) is the original formula of Radon,14
and Eq. (2.10) probably has been known equally long.Equation (2.11) and the simple proofs above appearedfirst in Ref. 3. The general form of Eq. (2.14), valid fordimension n and an arbitrary point spread function,appeared first in Ref. 3. In the case of dimension 2 anda particular point spread function, Eq. (2.14) is theoriginal convolution formula of Ram and Lak,15 andEq. (2.15) is the divergent beam formula of Lak, 6 as ismade clear in the derivation of Scudder. 17
Some common point spread functions and kernelsare described in Refs. 3 and 18.
The above formal derivations of the inversion for-mulas are not difficult to justify when both the attenu-ation coefficientf and the point spread function e have
bounded support. Among the common point spreadfunctions, the logarithmic one18 is the only one withbounded support. (Some are not integrable.) Thefollowing sections contain derivations in a general set-ting that includes all standard examples.
Ill. Riesz Potentials and Singular Integrals
The pointwise derivative
aR/Oxj = (1 n)C(n,l)xj/lx n,, (3.1)
is not locally integrable, so it cannot be used as aconvolution operator in the absolutely convergentsense. The formula,
The use of distributions in x-ray theory allows appli-cation of Fourier transforms to functions too big at for the classical Fourier transform and also has somelocal conveniences. is the Schwartz space of rapidlydecreasing CO functions, and its dual S' is the space oftempered distributions.1 9 Distribution derivativesare denoted by Dj and pointwise derivatives by /Oxj.The following formulas are not hard to prove:
DjRj = vpd9R1/Oxj, (DjR1)^ = (27r)-n/2ijJ t I -'. (3.4)
The Calder6n-Zygmund theory of singular inte-grals2 0 gives the following:
THEOREM (3.5). If g e L2, then the limit in Eq.(3.3) exists almost everywhere, and in the L2 sense, thefunction (DjRD)*g = (vpaR 1 /axj)*g is in L2 , and[(DjRj)*g]^ = idl -g. If R1*g exists a.e. as an abso-lutely convergent integral, then (/Ox j)(Rj*g) existsa.e. and is equal toDj(R1*g) = (DjR,)*g; if A is given byEq. (2.6), then A(Rl*g) = g a.e.
The next step is to determine when Rl*g exists a.e.LEMMA (3.6). There is a constant C, that if is
integrable and (1 + Ix I )no is bounded, then
I Rl*o(x) I < C(1 + x ) n[IIllL + 11(l + x I )no 1 L4- (3.7)
PROOF. Assume that 0 > 0 and let M = 1(1 +Ix )no FLU, so that 0(x) < M(1 +I x I )n. Separate inte-gration over Ix - y 1 and Ix - y I > 1 gives
Rl*k(x) CMJ izI-ndz + C||IIIL,.
The range of integration for Rl*o(x) is split intothree parts. In the range I I Ix 1/2, Ix -y I > Ix 1/2,so the integral over this range is bounded byC x I 1njj 01b~. In the rangely > 21, I - I> x 1and the integral is bounded similarly. In the rangex 1/2 < ly I < 2 x 1, write x = Ix 10 and y = x Iz. The
integral becomes
IX 1l-n A1- Q1° -Z 1-~n((X Iz)IX n dz.
Sinceo ( xIZ)X n= O(X Z) ( X IZ )nlZ-n,<2%this isbounded by
CMIx1n JI-w IW I'1-dw.
THEOREM (3.8). If (1 + Ix )l-ng L1, thenRl*g isdefined a.e. by an absolutely convergent integral and islocally integrable. Moreover,
I (R,*g ,)1 = I R 1 *g(x)(x)dx I
- CII(1 + Ix I ) ng"Ll1110U + 11(1 + Ix p)no IL.
If (1 + Ix )l-ng L', and g > 0, then Rl*g = +everywhere.
PROOF. The first part is immediate from the lastlemma, and the last part is easy to check.
According to Theorems (3.8) and (2.7), a natural L 2-type domain for the Riesz potentials and the x-raytransforms is
Dx, {g a L2 : (1 + Ix )lflg L'I,
with the norm
llglIDxr = lOglIL2 + 11(1 + Ix )lngIL,.
(3.9)
(3.10)
LEMMA (3.11). Ifg e Dxr and (1 + Ix I)n-1 e eL1,the e*g is in Dx,, and
IIe*gIID., < CII(I + IX )n-le iLlg"Dl (3.12)
PROOF. The classical inequality of W. H. Youngshows that e*gl]L2 < leflLljlgJJ2. The elementary in-equality 1 + (a + b)2 (1 + a2)(1 + b)2 gives (1 +Ix 12)(1-n)/2 + IX - y 2)(1-n)/2(l + ly I)n1, which inturn gives
S lel*lgI(1 +IX12 )(1-n)/2 dx
< ff (1 + I -y 1 2,(1-0/ lg(x - (1 + I )n-11 e(y) I dydx
= le,-gi 11Ll - Ilelld~llglllLl with
e = (1 +IXI)n-leI andg, = (1 +x1 2 )(1-n)/2jg.
LEMMA (3.13). C is dense in Dx,.PROOF. It is plain that the functions with bounded
support are dense in Dxr, so it is enough to approximateone of these, sayg. It is a standard result that if e e Lohas integral 1, and e(x) = r ne(x/r), then for any g L2, e*g converges to g in L2 as r - 0. Since allsupports are contained in a fixed bounded set, thisimplies that er*g - g in Dxr.
LEMMA (3.14). Fix e Lo-, and set e(x) =r-ne(x/r).
(a) If t e L1 + L2, then (u,er) 0 as r oa(b) If g e Dxr, then (Rl*g,er) 0 as r -PROOF of (a). If u L2, then
I (ued)l < IlullL2lIerIIL2 = rn/211uIL21leIlL2.
If a L, then(u,er) = (2x0-n/2 f exp(i(x,))(Q)er(x)dtdx
This goes to 0 because a is bounded and e(rt) - 0 foreach t 5d 0.
PROOF of (b). It is enough to prove (b) when e isthe characteristic function of the unit ball, which willbe assumed from now on.
Suppose first that g = Q outside B(0,ro). For Ix I >2ro, IR 1*g(x) I < C lx I -n gIID.,r, and
I (Rl*ge,)l < Cr-n J IR,*g(x) dx
+ Cr-n 11D J2roz I r I I-ndx.
Theorem (3.8) (with o the characteristic function ofthe ball with radius 2ro) gives
I (Rl*g,er)l < Crl'n(l + ro)n j1g11D, for r 1. (3.15)
In general, write g = g, + g 2, with g = g for I x I < 1and g1 = 0 for Ix I > 1. Formula (3.15) shows that
I (Rl*gl,er)l < Crl- 11911Dx for r > 1.
The Frostman mean value theorem21 gives
I (Rl*g2 e,)l < CR1*1g2 () = C J I I-g(y) I dy
< CIlgIW.r
The last two formulas show that the linear forms r(g)= (Rl*g,er) are uniformly bounded on Dxr, and formu-la (3.15) shows that Ir(g) - 0 on the dense set of g withbounded support. Therefore, r(g) - 0 for every g eDxr.
THEOREM (3.16). If g & Dxr, and q5 e 9 with 0(0)= 0, then
( (R1*g) ,P)= f I S I -I'gd. (3.17)
If I t I -'g is locally integrable (always true if n > 2 or if gE LP, p < 2), then (Rl*g)^ = Ik1 -kg.
PROOF. By Theorem (3.8), Rl*g is a tempereddistribution, and by Theorem (3.5) the Fourier trans-form satisfies
It is easy to check that bj E S. Equation (3.18) givesn
((Rl*g) ,¢) (= E(R*g) ,tjPj)1=1
n
= (j(R1*9) ,0j)
1=1n
f=l
-JkI -do.
If g l is locally integrable, then it is a distribu-tion, and the above gives ((R1 *g) ,>) = (I I :'k¢) for 0E with 0(0) = 0, which implies that (Rl*g) = I t I -Ig +cb. I -I is the sum of an LI function and an L2function, so its inverse Fourier transform u satisfies Cte L1 + L2 and Rl*g = u + c. If e has integral 1, thisgives (Rl*g,e,) = (u,e,) + c, and Lemma (3.14) showsthat the first two terms go to 0. Therefore, c = 0.
IV. Fourier Transforms
Theorems (2.7) and (3.8) give the following:-THEOREM (4.1). If g e Dxr, then, for almost all x,
Oxg is defined a.e. on Sn-i by an absolutely convergentintegral and is integrable on Sni1. If g > 0, and (1 +lx )l-n g L1, then for no x is Oxg integrable on Sn-1.
THEOREM (4.2). If g e Dxr, then, for almost all 0,P0g is defined a.e. on 91 by an absolutely convergentintegral. If k e Li and (1 + Ix )n 0 E LX, then
fJ~ (P~ogPo)dO= (Rl*g,4),
Sn-1 (Pog,0P)d0 I CIlglD [IIkllLl + 11(1 + Ix pno 'L']'
(4.3)
(4.4)
LEMMA (4.5). If g e L', then, for every 0, Pog eL'(0 ) and (Pog)^(Q) = (2ir)1/2g(Q) on 0-.
This result, sometimes called the "projection slicetheorem," is obvious from the definition of the Fouriertransform.
LEMMA (4.6). If h is nonnegative and measure-able, then
I~ I~ lflhQ)dtdO = IS.-21 JRh)d#.
PROOF. Consider first functions h on Sn-1.these, the formula becomes
Is-1 Jn-lnol h(O)dkdo = ISn"21 I h(o)dO,
(4.7)
For
which is true because both sides determine rotationinvariant measures on the sphere, and they agree for h= 1. Equation (4.7) results from this and the polarcoordinate expressions of the integrals.
THEOREM (4.8). If g e Di., then, for almost every0, (a) I 1
1/2i e L 2(0 I-). (b) P0g is a tempered distribu-
tion on 0A. (c) If 0 E S '(0) with q5(0) = 0,
((Pig) 0) = (241)1/2 I' jQ)¢()d#.
(d) If I I -g is locally integrable, then i is locallyintegrable on 0A, and (og) = (27r)1/2i on 0'.
PROOF of (a). By Lemma (4.6)
(4.9)Js-LIlfll=(t)1 dtd = Cllgll 2_C11911D 2,
and (a) holds for any 0 where the inner integral is finite.PROOF of (b). Theorem (4.2) with (x) = (1 +
for almost all 0. For any such 0, (b) holds.PROOFof(c). Letg - ginD.,,,g, & L1. By(4.10),
sn-l JLI I Pog(x) - ¶P0gn(x) (1 + x I )-dxdO - 0.
For some subsequence (still called g),
I¶'Pog(x) - ¶Peg(x)j(1 +Ix I)-dx - 0 (4.12)
for almost all 0. If (4.11) and (4.12) hold, then Pog,converges to P0 g in 8'(), so that
(24)1/2 j ) = ((P 0 gn) At -> ((qP'g)F,). (4.13)
By Eq. (4.9),
J I 1 . ()dO I 0.
For some subsequence (still called g),
lIl 11/2(i -id "L'(0-1) - 0 (4.14)for almost all 0. For such a 0, and e & Y(0),
10ltm I jQ) j,,) 0Q)d( 4 1 I0 11L2111 41 '/(i - id L2(01 -0
If (0) = 0, then I 0)l < cl 1, and
I01jj< IgQ) - j.Q)II -0Q) d,
f J Ifd4) 1Ijf1/2(j -in) ['L2(0) O .
Therefore, (c) holds for any 0 where Eqs (4.11)-(4.14)hold.
PROOF of (d). By Lemma (4.6)
Jsn-' ft,50 al Ig()Idtdo = c JI 1I -1 g(Q)l d#.
so i is locally integrable on 0' for almost all 0. Fix 0 sothat this holds and so that (b) and (c) hold. Theseconditions imply that (Pog) = (27r)1/2i + c on 0'.Hence, if u is the inverse Fourier transform of on 0',
.that P0 g = u + c.Lemma (3.14) shows that
I~n ~ <P Ig 1,Poe > dO = (R1 *lg ,e,) - as r a,
therefore, that for some sequence rn,(PogPoern) 0-for almost all 0. Since (oe), = Per, Lemma (3.14) (inRn-1) shows that (u,Poer) - 0, which implies that c =0.
V. Reconstruction Formulas
The Sobolev space Ns is defined by
WS ge L2:I|IsgE L2}, IIgl2 =llgL 22 + |l |gIL2.
If s is an integer, then
Ws= {ge L 2 : Dge L2 forIj I As.
The local space Yjocs consists of the locally integrablefunctions g such that each point has a neighborhood onwhichg is equal to some function in ]S. Bloc1 consistsof the locally integrable g with Djg in L2 locally. If g E]fioc', then g can be redefined on a set of measure 0 sothat the pointwise derivatives exist a.e. and coincidewith the distribution derivatives Djg (see Ref. 22).
THEOREM (5.1). If e Dx,. n W1 12 and l-l islocally integrable, then for almost all 0
(a) 'Poe W10c(0')
(b) With any coordinates in 0', Dj P oe E L 2(0 1'), and
(DjfPe)^ = (2,r)4/2itjg on 0--. (5.2)
(c) APoe = - E DjRj*DjP0 e exists a.e. on 0', andj=i
which gives I I E &L 2 for almost all 0. Fix 0 so that thisholds along with the conclusions of Theorem (4.6).Part (d) of Theorem (4.8) gives part (b) of the presenttheorem, and part (b) implies part (a), since Poe islocally integrable. Part (c) follows from Theorem(3.3). If 0 L0
2, then Poo E L02, and part (c) gives
(APoe,'P 0 ) = (27)1/2 j I IQ)pO)d#.
Lemma (4.6) gives
fSn (A'PePOO)dO = (27r)"/1Sn-2 (e,-P).
Equation (5.4) now follows by taking 0(y) = f(x - y).(The fact that the constants match is left to the read-er.)
The corresponding divergent beam formula isproved as before [Eq. (2.15)]:
e*f(x) = (1/2R) f I .(laf)k(Eox - Eoa)dOda. (5.5)
As shown in examples below, it can be useful toreconstruct Af, either in place of f, in addition to f, or toform some combination (1 + E )f + E Af. Ordinarily fis not in Y1i so Af is not defined, but A(e*f) is obtainedsimply by replacing e by Ae in Eqs. (5.4) and (5.5).
THEOREM (5.6). If e E W3/2, Ae E Dx,, and f E L02,
PROOF. Apply Eq. (5.4) with e replaced by Ae,noting that Ae E 11/2 with I -'(Ae) = e & L2 locallyintegrable.
VI. Point Spread Function
The usual way to obtain a point spread function eand reconstruction kernel k is to fix a function el withe1 bounded and continuous at 0, and Pl(0) =(27r)-n/2(integral e = 1 if e is integrable), and to take e = er and k= k, with
e,(x) = r-ne1 (x/r), k,(x) = r-nk1 (x/r), (6.1)
where k1 is the reconstruction kernel for el. Depend-ing on additional properties of el, er*f - f in varioussenses, as r - 0. If r is small, the reconstructed func-tion e*f is an approximation tof. Its value at point x isa weighted average of the values of f, the weight beingthe function e shifted to the center x. In currentpractice el is radial, so e and k are radial, and k isindependent of 0.
A good choice of the number r in Eq. (6.1) is based onnumerical considerations and on the effective pointspread radius, the smallest number R, such that
J e(x)dx < E (6.2)R'< I xR'
whenever R R' < R", with some fixed small e. (Theauthors usually use 0.01.) The reconstructedvalue e*f(x) is effectively the average of the values fwith the wieght e on the ball with center x and radius R= R(e). Since R(er) = rR(el), Eq. (6.1) can be used toproduce e with R(e) prescribed.
A good choice of R(e) depends on the required reso-lution, the noise, the presence or absence of small highcontrast features, etc.-the resolution being the num-ber p so that the smallest features to be resolved have adiameter of -2p. Too large a radius entails a loss ofresolution, and too small a radius amplifies noise andproduces streak artifacts. In the authors' experi-ments, R(e) 1.5p has proved a good compromise, butother kinds of problems might call for something else.From now on it is assumed that a radial el is fixed witha radius that is satisfactory in respect to the abovephysical constraints.
There are also numerical constraints. A typical ker-nel (see Ref. 3) has sharp curvature at 0 and at thepoint where it is minimum and is rather flat elsewhere.To obtain good numerical values for the convolutions,the data should be known at these points, i.e., thereshould be x-ray detectors situated at 0 and at thekernel minimum or very close by.
Suppose (2-D parallel beam case) that the actualdetectors are situated at the points nh,n = 0,±1,+2,..., and that the minimum of the initial el
(with a satisfactory radius) occurs at z1. Then theminimum of e, occurs at rz1 , and r should be chosen sothat rz, is a multiple of h. To disturb the initial goodradius as little as possible, r should be close to 1. Thusthe point spread function e is determined as follows.
(1) Choose an initial el with a radius that is satifac-tory in respect to the physical characteristics of theproblem.
(2) Take e = e, with r = mh/zi or r = (m + 1)h/z,where el is minimum at zl, and m is the integer part ofzil/h.
VII. Discussion
A. Mathematical Derivations
The formal derivations in Sec. II (also in Refs. 1 and18) are easy to justify when the attenuation coefficientf and the point spread function e both have boundedsupport. Usually the authors take
(x 2 (1 - Ix 2)(u2)/2, Ix I <1,to, IxI>1,
with , = 15, but most of the common point spreadfunctions do not have bounded support. For example,the Ram-Lak point spread e(x) = (1/21x )Jl(x I).which behaves like Ix 1-3/2 at -, is not integrable. TheShepp-Logan point spread function is given by
el(x) = 4 Jxk(y)(IxJ2_y2)1/2dy,; ~~~~~~~~~o
with k(j) = 1/7r2(1 - 4j2), and k linear between integers.This el is integrable but Ix 13 el is not, and el is notdifferentiable at 0.
In these special cases, special derivations can beconcocted, but the aim of the article is to providegeneral derivations under conditions close to the bestpossible, i.e., to the conditions necessary for the exis-tence of the x-ray transforms themselves.
B. Reconstructions with A
Contrary to the statement at the beginning of thearticle, the role of computed tomography is not thereconstruction of the x-ray attenuation coefficient f.It is the reconstruction, from the x-ray data, of func-tions providing significant information about thephysical object. In this light, Af and (1 - )f + e Afshould not be viewed as approximations to f but asfunctions with supplementary and complementary in-formation about the physical object.
In regard to the interpretation of this information,the major disadvantages are:
(1) In the presence of noise (including the mathe-matical noise introduced by discretization), Af is noisi-er than f.
(2) Even without noise, Af has a large magnitude,both positive and negative, near high contrast inter-faces. The rapid changes hide low contrast detailsnear high contrast interfaces. In the combinations (1- e )f + e Af, Af can be truncated so as to avoid most ofthe loss.
(3) In regions where f is constant, Af is cup-shaped.This is a mathematical property of Af, not an artifact of
Fig. 1. Reconstructions with A. Top left: mathematical phan-tom. Top right: reconstruction of the attenuation coefficient f.Bottom: central portion of Af, reconstructed with the x-ray beamconed to that portion alone. With the interior background at 100 (toexhibit percentages), the attenuation coefficients are: central darkellipses, 99; dark ellipse at 11 o'clock, 99.5; light circle at 10 o'clock,101; three ellipses at 2 o'clock, 101, 102, 100.5; two tiny cirlces aboveand below the center, 100.25 and 99.75; skull, 200. The reconstruc-tions use 128 x-ray directions and 256 detectors and are displayed ona 128 X 128 matrix. With 128 as the diameter of the circle recon-structed, the tiny circles have a radius of 0.96 (i.e., p = 0.96), and R147 - 1.5p. All features in the phantom are elliptical, even thoughsome do not look elliptical in the original-an effect of the discrete
representations on the 128 X 128 matrix.
the reconstruction, but nevertheless it is disconcerting.For the numerical reconstruction of Af, the discretekernel k should be chosen so that the sum of its valuesis 0. The cup effect is minimized by choosing k so thatthe sum is small but not 0. If h is obtained by samplingthe analytic kernel k at the detectors, this is achievedby changing the point spread radius of k a little.
Advantages of Af and (1 - )f + Af are:(1) They provide a sharp accurate delineation of
edges and provide a sharpening of small low contrastdetails (see Fig. 1).
(2) The reconstruction of Af is a local procedure:the value at a point x depends only on the x-ray attenu-ation along rays meeting a small neighborhood of x.When the point spread function e has bounded sup-port, it usually has very small support-contained in aball of radius slightly larger than the point spreadradius R -1.5p. The reconstruction of A on a region
Fig. 2. Effects of the point spread radius. The object is an explod-ed sodium-sulfur battery. After the explosion the core was anuneven mixture of sodium and less dense materials. The rings(inside to out) were steel, ceramic, an uneven mixture of sulfur andsodium polysulfides, and steel. The explosion was caused by the V-shaped crack at 2 o'clock in the ceramic ring. The four reconstruc-tions put the kernel minimum at the first four detectors, giving pointspread radii of 0.42, 0.85, 1.27, and 1.70 (1.5 would be preferredaccording to the criteria above). They use 128 x-ray directions and
452 detectors of 0.127-mm diameter.
go c Q then requires attenuation measurements onlyalong rays that pass within distance about 1.7p of go.For example, in a reconstruction of the spine with aresolution of 0.5 mm, the x-ray beam can be coned tothe spine itself plus 0.9 mm on either side. Suchconing would reduce the x-ray dose by a large factor,allow high resolution scans (with a smaller number ofsmaller and more tightly packed detectors, as in thescan of Fig. 2), allow very fast scans (with several x-raysources, each coned to its own small bank of detectors),reduce the amount of data required for high resolution3-D reconstructions; etc. As yet unanswered is thequestion of which problems can be solved by the infor-mation in A/ alone. So far the only examples are theones in Figs. 1 and 3.
C. The Point Spread Function
At present there does not seem to be much evidencedemonstrating the superiority of any particular initialpoint spread function el, although, of course, the localreconstruction of Af is possible only with one of smallsupport. The battery reconstructions in Fig. 2 dodemonstrate the importance of the point spread radi-us. There is evidence that some computed tomogra-
Fig. 3. Reconstructions with A. Top: scanner reconstruction of aphantom made of various low contrast materials with a monkeyspine near the center. Bottom: reconstruction of the central thirdof Af with the beam coned to the central third. Resolution is limitedby the small number (42) of detectors within the coned beam. Thedata and scanner reconstruction were supplied by B. Rutt, Dept.Radiology, UC-San Francisco. The scanner is a low dose machineproviding about 0.01 R/scan, and consequently noisy data. It issurprising that Af is not exceedingly noisy. (No noise elimination
was used.)
phy algorithms automatically place the kernel mini-mum at the first or second detector, which can producetoo small a point spread radius. Indeed, the batteryexample was chosen because a prior reconstructionwith a standard scanner algorithm was very much likethe first reconstruction in Fig. 2 (R = 0.42).
This research has been supported by the National'Science Foundation grant MCS-8101586 by equip-ment grants from the Cromemco and Tektronix com-panies and by the General Electric Visiting ResearchFellowship program.
This paper was presented at the symposium on In-dustrial Applications of Computed Tomography andNMR Imaging, 13-14 Aug. 1984 at Hecla Island, Mani-toba, Canada.
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