space-domain inversion of the incomplete abel transform

2126 J. Opt. Soc. Am. A/Vol. 9, No. 12/December 1992

Space-domain inversion of the incomplete Abel transform

Eric W Hansen

Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, New Hampshire 03755-8000

Received March 23, 1992; revised manuscript received June 12, 1992; accepted July 7, 1992An axisymmetric object is reconstructed from its transaxial line-integral projection by the inverse Abel trans-form. An interesting variation of the Abel inversion problem is the finite-length line-spread function intro-duced by Dallas et al. [J. Opt. Soc. Am. A 4, 2039 (1987)], in which the path of integration does not extendcompletely across the object support, resulting in incomplete projections. We refer to this operation as the in-complete Abel transform and derive a space-domain inversion formula for it. It is shown that the kernel of theinverse transform consists of the usual Abel inversion kernel plus a number of correction terms that act to com-plete the projections. The space-domain inverse is shown to be equivalent to Dallas's frequency-domain inver-sion procedure. Finally, the space-domain inverse is demonstrated by numerical simulation.

INTRODUCTION

An axisymmetric function f(r) and its transaxial (line-integral) projection g(x) are related by the Abel trans-form pair'

Forward g(x) = I f(V + y)dy 7 2f(r)rdr

Inverse f(r) = - - gx (1)

Figure 1(a) illustrates the Abel transform geometry. TheAbel transform and its inverse are applied in the solutionof several practical problems in optics, and several meth-ods have been devised for performing the Abel inversion(see Ref. 2 and the references therein for a survey of appli-cations and inversion methods). For example, if f(r) isthe point-spread function (PSF) of an imaging system, theng(x) is the line-spread function (LSF) of that system, andthe Abel inversion may be used to reconstruct the PSFfrom measurements of the LSF.'

Recently Dallas et al., motivated by the practical prob-lem of characterizing a space-variant imaging system,presented a variant of the line-spread problem leading to ageneralization of the Abel inversion.4 They noted that, ifa system is only approximately space invariant and thelength of the line being imaged-the path of integrationin the forward Abel transform- exceeds the size of an iso-planatic patch, the standard Abel inversion yields an er-roneous PSF estimate. They defined a finite-length LSFbased on a path of integration (i.e., a measurement slit)that stays within an isoplanatic patch; for the ideal case ofan infinitesimal-width, uniform (unshaded) measurementslit, the finite-length LSF is defined as

(L/2(x)=J p(VX + y2)dy, (2)

-L/2

where L is the length of the projection line, p(r) is theradially symmetric PSF, and 1(x) is the LSF [Fig. 1(b)].The inversion formula, which estimates the PSF for theisoplanatic patch from the finite-length LSF of the in-

complete Abel transform, proceeds by means of an inte-gral transform:

f(y)= fJp(x)exp(i27ryx2)dx

=2Ip(Ve)exp(-i21rye)d6,

which is a square-root coordinate mapping (I = x2) fol-lowed by a Fourier transform. The result, for the idealcase, is a transfer function relationship between the inte-gral transforms of the finite-length LSF and the PSF:

(y)fv)=Y I{C(LV) + ilVy)}

1 i2iryl(y)

7 -2iV,-{C(LV/-Y) + iS(L V7 )}

where C and S are the Fresnel integrals5 :

C(z) = |cos 2 dt S(z) = Jsin 2 dt.

(3)

In a later study, Barrett6 carried out a numerical inver-sion of the transfer function in order to determine thespace-domain kernel, here denoted h(x, r; L), of the finite-length line-spread inversion. He observed that the in-verse kernel displayed discontinuities-steps or kinks-atregular intervals x2 - r2= n(L/2)2 , which were unex-plainable on the basis of the frequency-domain derivationand which raised questions about how the inverse kerneldoes its work (Fig. 2).

This paper presents a solution to the inversion problemderived directly in the space domain rather than throughthe frequency domain. Since the finite-length path inte-gration represents a kind of truncation of the ideal Abeltransform, we shall henceforth refer to it as the incom-plete Abel transform. I derive analytically a space-domain inversion of the incomplete Abel transform thatgeneralizes the complete Abel inversion formula [Eqs. (1)],Fourier transforms to Dallas's transfer function [Eq. (3)],and, most importantly, neatly explains the discontinuities

0740-3232/92/122126-12$05.00 © 1992 Optical Society of America

Eric WZ Hansen

Vol. 9, No. 12/December 1992/J. Opt. Soc. Am. A 2127

ftr)~ ndescribed by partial projections gp(x):

gp (X) = fL2 + y2 )dy = Ir 2f(r)rdr

I L/2x Vr-(4)

(Table 1 shows several transform pairs that can be statedanalytically.) We refer to the missing data as complemen-tary projections:

gc(x) = g(x) - gp(x) = I 2f(r)rdr-fv2 an1)

(5)

which depends on values of f(r) outside /x2- (L/2)2 .

Now, if those values are available, e.g., as the result of ear-

g(x) -X.. x

(a)

(a)

X.; ..: K....t I A

g(x)

TL

(b)

2.0< -

-~~~~~~~~~lx |r t L)

10

.- I I I I I I X;21 - r20.0 1.0 2.0 3.0 4.0 5.0 6-0 (LJ2)2

0.0 1.0 2.0

x

(b)Fig. 1. Geometry for (a) complete and (b) incomplete Abel trans-forms. The incomplete transform matches the complete trans-form on an outer annulus (a disk object is shown here).

observed by Barrett. Numerical results are shown thatverify the space-domain inverse's correctness.

SPACE-DOMAIN INVERSION

OverviewThe mathematical form of the Abel transform [Eqs. (1)]makes it evident that the projection g(x) depends on theobject f(r) only on an annulus R 2 r 2 x (where R is theradius of the object support) and that the reconstructionof f(r) requires knowledge of g(x) only on an intervalR Ž x 2 r. With reference to Fig. 1, the incomplete andcomplete Abel transforms are identical over an interval atthe edge of the object, R 2 x 2 /RW - (L/2)2 (where L isthe length of the integration path). Hence the usual com-plete Abel inversion will successfully reconstruct the ob-ject at the edge [Fig. 3(a)].

A different approach must be followed for x <VR2

- (L/2) 2, where we have the incomplete transform

(c)

0.0 1.0 2.0

3.0 X

3.0 x

(d) 0.00Jii I I i0.0 1.0 2.0 3.0 X

Fig. 2. Kernels for inverting the incomplete Abel transform.h(x, r; L) is inverse kernel for reconstructing at radius r, given aprojection line of length L. (a) Kernel plotted against the normal-ized transformed coordinate (X2 - r2)/(L/2)2 , showing discontinu-ities at regular intervals. (b)-(d) Inverse kernels plotted againstspatial coordinates, demonstrating the shift variance of the ker-nel. Note also that the kernel approaches a constant asymptotefor large x.

.. I V....I.. "".I"""'."'.""' . ......... .........................11 ;;K

, .

Eric W Hansen

I- f.>::':..1 :d p k k :.: 3 Ala .1 p k k a: f.:t' :.1 :.:.Q: i- ..:

__J_� -


(a)

ffr) PARTIAL FULL or_t= _, RESTORED

h(x, r; L)

7s

r

iI1 x, r; L)

"I -_\ ir2 + L12

2/

Vr2 + (L/2)~

(c)

r; L)

r \

I I - I I I

I I

I I h(x, r; L)I

?r\2 + ( 2 \ L21r2 + 2(L122

(b) (d)Fig. 3. Schematics of space-domain reconstruction, showing the reconstruction kernel h(x, r; L) used at four different depths in theobject. (a) The outer annulus is reconstructed by using the zero-order (complete) Abel inversion kernel. (b) The first-order correctionkernel adds information from the reconstructed annulus to complete the projection at r. (c) The third annulus is reconstructed by usinginformation from the second annulus, filtered by the first-order correction kernel, and from the first annulus, filtered by the second-orderkernel. (d) Reconstruction proceeds until the diameter of the innermost unreconstructed circle is smaller than the projection line.

lier computation based on complete projection data, thenthe complementary projections may be computed and usedto restore the partial projections. Once this is done, theusual Abel inversion is employed to reconstruct the object.

Of course-and this is the key to the derivation thatfollows-once f (r) is reconstructed from restored projec-tions at some radius, it may be used as a source of comple-mentary projections for points even deeper in the object(closer to the center). And so, beginning with the knowncomplete data at the outside and working inward, the en-tire object may be reconstructed. In what follows I makethis argument precise and ultimately derive an inversekernel connecting the partial projections with the recon-structed object.

Derivation of the Inverse KernelThroughout this paper we shall work in a transformed co-ordinate system in which the Abel transform becomes aconvolution integral.' We define the transformed coordi-nate and p in terms of the spatial coordinates x and r, by

= x2 and p = r. Applying this coordinate transforma-

tion to the Abel transform pair [Eqs. (1)] results in

=I (A)dp [fk]()

pP '(()d-)Ap) = U1 ,

(6)

where f(4) = f (V), etc., U( ) is the unit step function,and P = R2 is the radius of the object support in thetransformed coordinates.

Next, when one defines A = (L/2)2 , the incomplete Abeltransform takes the form

- ( +A f(p)dp

k () =U(-6)U(6 + A) ()VC

lI .|

Eric W Hansen

1- _� (P),

Ir


and, similarly, the complementary projections in trans-formed coordinates are given by

g (f = J f(p)dp= f O ])f [f *kJ(6),

kd~g) = U(-6 - A) (8)

The Abel transform's half-order integral property-applying the Abel transform twice in succession to a func-tion integrates it-will be useful in what follows. It is alsoexpressed by a convolution in transformed coordinates':

T * T* g'= -7ig. (9)

Now, assuming a finite (circular) support for the objectf(p), the partial projection is formed from the annulusp E (, f + A) and the complementary projection from theremaining outer annulus ( + A, P). We begin the recon-struction at the edge of the object, computing f(p) fromcomplete projections on the outer annulus (P - A, P) bythe usual Abel inversion [Eqs. (6)] [Fig. 3(a)]. Then, usingEqs. (6) and (8) in Eq. (5), we calculate the complementaryprojections and use them to restore the partial projections:

where in the last step we used the convolution identity(a * b)' = a' * b = a * b' to shift the derivative from theprojection to the complementary projection kernel. Theconvolution k *k 'c is spatially limited by the step functionU(- - A) in kc, so the convolution ( *Xk') * j is takenover the limits > t > p + A, at the edge of the objectsupport where g is identical to gp. Hence Eq. (10) may berewritten as

= + I (11)

and we may apply the Abel inversion formula [Eqs. (6)]to reconstruct A(p). Substituting Eq. (11) into Eqs.(6) re-sults in

=~+-- *[ + ( r c

k *gp +(-k * k * )*gp,

but, by the half-order integral property [Eq. (9)], k * k *T' = -rkc, and

g = + g = gp + kc *f= gp +

=g + - * **

7= -I(k + kc) * 'p .ir

(12)

(10) That is, the reconstruction kernel is the sum of the ordi-nary Abel inversion and a correction term that is identical

Table 1. Examples of Incomplete Abel Transforms'

Object, f(r) Transform, gp(x)

Diskcirc(r/a)

Hemisphere(a2 - r2)1/2 circ(r/a)

2(a2 - X2)1/2 a2 x- a2 (L/2)2L x2 < a2 _ (L/2)2

(a2 _ x2)2

[a -(L/2) -x]" + (a' - arcsin L/

a' 2 X2 2' a2

_ (L/2)2

X2 < a2 _ (L/2)2

Paraboloid(a2 - r2)circ(r/a)

1

a2 + +r

Gaussianexp(-r2/2o'2)

r2 exp(-r /2o-2)

4 (a2 _ x2)3123

a2 2 X2 2 a2 _ (L/2)2

L(a 2 - -(L/2)2 -x2) x2 < a2 _ (L/2)2

2 arctan L/2Xi' x' \ 2 + x'

V_- L --> 00

2a + x'2

V7ro- erf[L2/2ao)exp(- X2 /2a-2)

12ro exp(-X2/2o-2), L -> co

{N/'o- erf[L/(2N/2ov)](x2 + o-2) - Lcr exp[-(L/2)2/2u-2]}exp(-x2/2o-2)

,7ro(X2 + o2 )exp(-x /2-2, L X

a sinc(ax), L -X -Jinc

Jc(irar)

2ar

aFor objects with finite support, substituting (L/2)' = a' - x2 converts the partial projection into the complete projection. For objects with infinite sup-port, let L - .

bAn incomplete projection for the jinc function is not available in closed form.

Eric WZ Hansen

1 - Ik - k *_9IT


to the complementary projection kernel. In full,

P)+ U( - P- A)]-'7rf = I6 dj'p({) By (' ]

fP g p ()d + fP g ()dg

P _ up p+ up

The first integral, in spatial coordinates, is taken over theinterval x E (r, R); this is the usual Abel inversion at ra-dius r, which uses all the data outside r. It is incorrect forr < N/R' - (L/2)2, where the projections become partial,but the second integral supplies the needed correction,pulling in data from the interval x E (VR2 - (L/2)2, R)and contributing it to the reconstruction of the partialprojections. The point /R' - (L/2)2 is precisely the lo-cation of the first discontinuity in the inverse kernel,which is now seen to be a jump that is due to the correc-tion term.

This first-order correction [Eq. (12)] holds for an annularzone immediately inside the complete projection annulus[Fig. 3(b)]. As projections successively closer to the cen-ter of the object are considered, a point is reached wherethe correction requires more information than just theouter annulus [Fig. 3(c)]. At this point one repeats theprocess described above, using the reconstruction ob-tained in Eq. (12), to generate the complementary projec-tions for the next round. Substituting Eqs. (12) and (8)into Eq. (10), we obtain

= + k * -- (k + kc) * ,

whence the Abel inversion [Eqs. (6)] yields

_ _ 1 - - -Surf = k * ' + -* -kc * (k + V'm) jT'p

'r

= [k + T + * (- T~c* c)] * gPc (13)

The kernel now contains a second correction, derivedagain from the Abel inversion kernel and the complemen-tary kernel. This term,

h2 =k -kc * ') U(-p) * 2U(-p - 2,A)W N/-V - pV r p-~X

= 1 1-2 arcsin -A U(-p-2k),,_.A,1--rc P +AIUP-2)

Proceeding for annular zones successively closer to thecenter of the object ultimately gives a general expressionfor the incomplete Abel inversion:

N-1

f = -- E 1: n p )'r nO

h2 = * (--kc * 'c

4,1 = * -- kc * ') (16)

where the superscript *v indicates repeated self-convolu-tion (a*' = , a*' = a, a*2 = a * a, etc.). (Closed-form ex-pressions for the inversion kernel terms have not beenobtained beyond the third correction.) Each convolutionshifts the result by A along the axis, owing to the pres-ence of the step function U(-{ - A) in kc. Hence thenth kernel term is nonzero on the interval (-mc, - nA), andat every unit of A along the g axis a new kernel term ap-pears [Fig. 2(a)]. In spatial coordinates, new kernelterms appear at intervals x2 - r2 = n(L/2)2, preciselywhere the discontinuities have previously been observed[Figs. 2(b)-(d)].

The number of correction terms, N required dependson the radius of the object support, R, relative to thelength of the projection line, L. The iterative restoration-reconstruction procedure terminates when the projectionline covers a disk of radius -L/2 at the center of the object[Fig. 3(d)]. The object radius, R, lies between \/N(L/2)and VN+ I (L/2), and we have N = 1 + int[(2R/L)2].

Connection with Transform-Domain InversionThe inverse kernel, as given by Eqs. (16), may be Fouriertransformed to check it against Dallas's transfer function[Eq. (3)] and to strengthen the connection betwen the twomethods. The upper limit of the summation is extendedto infinity (for objects of arbitrary and potentially infinitesupport), and under the convolution theorem, the repeatedconvolutions become powers. At length, one obtains

(y) = 1 _ 2 2 [ K(y) + KCy)]G(y) IT V-

1 - _ vX -K(y)i2,7yKJ(Y)

Xr(14)

is zero for p > -2A, so it produces a second discontinuityin the inverse kernel, corresponding to the point where thefirst correction fails to complete the projection data, i.e.,

> p + 2A, or, in spatial coordinates, r < VR' - 2(L/2)2[Fig. 3(c)]. Continuing still deeper into the object, thethird order of correction leads to the kernel term

=kc* (i--N* U(-p - A) 2(-p - 2A)T,3 = Up* ITjk *k') -7 pV p A

= A 16--arcsin + j)U(-p - 3A),V __ IT p +k

(15)

which contributes for r < N/R' - 3(L/2)' [Fig. 3(d)]. [Theconvolutions in Eqs. (14) and (15) were performed with theaid of a tabulated integral.']

=-2iy[K(y) + Kc(y)] E [-2ivKC'(v)Yv'-0

= -2iy K(y) + Kc('y)1 + 2iyK2(y)

provided that -2ivK'I < 1 so that the geometric seriesconverges. The Fourier transform Kc is found to be

_(A exp(-i27ryg)dg= TI: = J

%1 -t 2e dtV[2 fr(Ly)1- rL V- 2 Y 2

Eric W. Hansen


where fr( ) is the complex Fresnel function,

fr(z) = C(z) + iS(z).

The convergence condition 1-2iyK,21 < 1 reduces, aftersome algebra, to

C(LV/jy) + S(Lj I >1C2(V\/IY) + S2(L/ji/ 1)

Both Fresnel integrals C(z) and S(z) approach zero linearlyas z -,8 so the ratio is approximately 1/z for small z;furthermore, their maximum values are less than unityfor positive arguments. Hence the series converges for ally, and the transfer function directly follows:

Fly) _G('y)

1 + [1 - \/C-2 fr(LV-)] - VY1 - - V--ifr(L - )]2 fr(LN/-)

which is identical to Dallas's result [Eq. (3)].

Limiting CasesTwo special extreme cases of the inverse kernel maybe considered. For L Ž 2R (complete projections every-where), the location of the first discontinuity, x2 -r2 =

(L/2)2, is past the edge of the object (moving to infinity asL -> ), and the kernel reduces to k, the complete Abelresult. In the transfer function, fr(L/'y) -> (1 + i)/2, andthe transfer function becomes i2wTy/V^Ty = i2ITryK, thetransfer function connecting a complete projection g(e)with its Abel inversion f(p).

At the other extreme, L << R, the partial projection is,approximately,

_ r+ f(p)dp f f+A dp

= 2(g)\/V I6+A =

that is, within a scale factor L the incomplete Abel trans-form is identical to the object as the projection line ap-proaches a point. In this case, the inverse transformshould be the identity operator, scaled by 1/L. For smallL, the Fresnel function fr(LVy) is approximated by theleading term of its series expansion, which is simply L/y,8and so the transfer function reduces to 1/L, as expected.Demonstrating this in the space domain hinges on show-ing that the infinite sum in the inverse kernel becomes,for small L,

--*

whence

-- (k + kc) * i -- k * k' c 2 *IT - 2L

and, by the half-order integral property [Eq. (9)], com-bining this limiting form of the kernel with the projectiongp Lf yields

L ITk*k L

The summation is readily verified in the frequency do-main, i.e.,

[-2K ()]n=O

1 11 + 2iyKC2(y) 1 + 2iy1 - /

1 1 _ K(y)1 -[1l- V/=2 fr(L- )]2 > 2L y 2L

However, direct proof in the space domain seems to beconsiderably more difficult and has been elusive to date.

ExampleFor a simple analytical example, we consider a disk object(Table 1); in transformed coordinates, the partial projec-tion and its derivative are

_ 2 V a_- > > agP{ L a -A > > 0

g 'P~= # U(-e + a)U[(6 - (a -A)],

where a = a2 . Using the (complete) Abel inversion kernelwith no corrections for incomplete projections, we have

) _ gP(

a 2p2 a-A

p < a-A

Using a tabulated integral,7 we find that

(P) = 1 1 . p-A)I-- -arcsin l 2 7T ~p-aJ

a p a-A

p<a-A

It is apparent that the inversion is correct only in theouter annulus (a - A, a). Inside p = a - A, we must ap-ply the first correction [Eq. (12)], so that

(P) = arcsin(2 IT \p a1 {+ ' ()deI +A P

1 . -A 1fa _ _d_- -- arcsin __2 iT p-aa 7TJP+A-/ve{

1 1 . pA 1 a dg{ -i-arcsinP 1Ja-AVg/U1 IT__a__ __/a-__ _ ~____

a-A> p

a - A > p Ž a - 2A

p < a - 2A

.

Eric W Hansen

- 1fa de

IT,-Va - eye - p- 1fa de

Ir _,k - Va_ _-e V�e- p�


and, using the same tabulated integral, we arrive at theresult that

1- - arcsin(1 + 2A)Vr p- a

a Ž p 2 a - 2A

p < a - 2A

The correction term has done its job. If L is sufficientlylarge that a - 2A < 0 (L 2 aVd), then the first correctionis sufficient. If not, further corrections must be applied.

NUMERICAL IMPLEMENTATION

Formulation of the AlgorithmThe simplest numerical realization of the inverse incom-plete Abel transform is to calculate samples of the inversekernel [Eqs. (16)] and employ trapezoid integration for theconvolution. The observation that the inverse kernel ap-proaches a constant asymptote (Fig. 2) suggests an ap-proximate kernel that uses the closed-form expressionsfor orders 0-3 and the constant asymptote for > 3A,thereby avoiding repeated convolutions for higher-orderkernel terms. This procedure works and may be suffi-cient for some purposes, but more-accurate results are ob-tained by using the algorithm described below.

We begin by rewriting the inverse transform formula[Eqs. (16)]:

1 N-1

IT n-O

=Tk*g' +C * g +*Tk*' *k '

+ t *t* *gp + t*k ' + *-

= -- k * k'¢.7r

Defining the sequence of partial sums [v and correctionterms fiv, where the usual (zero-order) Abel inversion is f°and the fully corrected reconstruction is fNl, we proceedinductively to a general iterative form:

Ir

f3 fl + t- - l + ?C3I

fv+2

= f + g*+2, V = 1, 3,...,2 int[N/2] - 1,

fc-+2 = * f ?l = f,

= 1- -, 2\ U(-g-2A)I = - * -c=IT IT o /

(17)

where N = 1 + int[P/A] = 1 + int[(2R/L) 2 ] is the numberof kernel correction terms. Each step in the iterationadds two terms in the summation [Eqs. (16)], and the itera-tion stops when v + 2 is the smallest odd integer ON - 1.If N - 1 turns out to be even, the step functions in theconvolutions automatically delete the last (odd) term inthe series.

Three convolutions must be performed, using the ker-nels k, kc, and . In each of these the signal, denoted s(either g'p or f), will be approximated piecewise linear,i.e., on the interval between sample points (n, en+l):

Sn+l Sn

Each convolution then becomes the sum of integrals overintervals (nn el+). The value of k * s and p = pm is

(k *)=- s () d M- 1mŽ0irPm e-P

1 1 (S - fin) + /e d

IT nrm n \/f71M-1 f I+-P. - A6 .

IT=n _ m Isn /(n Pm)

IT nrn J~nPm L X/~+ IuV'1de',

where M is the number of sample points. Assumingequally spaced samples (n = nA, Pm = mA, etc.), we ob-tain, after performing the integrations,

M-1

(k * )m = E (bn mn + bm Sn+i)

n-mrM -m20,

bn = 4<-[n12 (n + 3/2) - (n + 1)3/2],3 +-

b '= 43IT [(n + 1)1/2(n - 1/2) - n32 (18)

Similarly, the convolution kc * S is found by

(kc* rs)m deI +d,X7 pm+A P.

M - ml - 1 m 0

+f S (e) doe

IT Pm fnA ~(

nm+ml+1 Jfn

where ml = int[A/A]. The first panel of the numericalintegration, over the interval (Pm + Agmi+i), is handledspecially because the lower limit of integration, Pm + A,will not, in general, coincide with a sample point. Usingthe same piecewise linear representation for the signal,we obtain

(kc * )m =--{ (ml - m + 1)3/2

2Vr23+ (A) [A _(ml- m + 1)]}sm,

+ 2E (ml - m + 1) 112(ml - m + 1/2)

(A 12[ A+ A)3 (mim)Jf Smi1

Af-1

+ 2 (bn_mSn + b-m-n+l),n-m+ml+(

M -ml --12m 0, (19)

Eric W Hansen


where the filter coefficients {b'} and {b} are as defined inEqs. (18).

Finally, we consider the convolution t * s, which also istaken in two pieces:

17T p2A ( P.)Vg - P

M - M2- 12 m 2 0

_ 2\/ fJCr2

+i -~ t)d

IT pm+2 ( Pm)Vg - Pm - A

1 M- frn+, de

n=m+m2+1 C. (e -P-We /-PM-A

where m2 = int[2A/A]. When a tabulated integral is used,9

this works out to

a,,, = [(ict n U 2 - M)A/A - 11 - 1

IT 4 [(M 2 - m)A/A - 1]1/2 + 1j

X [( 2 - m + 1)s2 - ( 2 -M)sm2+

4 A+ - {[(m2 - m)A/A - 1]1/2 -1}(sm2+1 - Sm2)

ITA

N-1

+ 2 {[(n - m + 1)dno-m -dnm]Snn-m+m2+1

-[(n -m)d-m - di-m]-n +},

M - M2-1 m 0, M2 = int[2A/A], (20a)

where

4 f [(n + 1)A/A - 1]/2 - [nA/A + 1]1/2 l

n 1+ [(n + 1)A/A - 1]1 2(nA/A - 1)1/2

dn = - {[(n + 1)A/A - 1]1/2 - (nA/A - 1)1/2}. (20b)IT A

The filter coefficients {dn} and {dn'} may be computedonce, stored, and reused for successive convolutions. Foreach successive convolution t * fC, the upper limit M isreplaced by M - M2 - 1, and fewer samples are involvedeach time.

These formulas [Eqs. (17)-(20)] are the core of the in-version algorithm. If greater accuracy is required, thesignals may be modeled piecewise as polynomials of orderp-the integrals can still be performed analytically, pro-ducing filters with coefficient sets {bn} through {be} and{d.} through {dn}. Also, the procedure may be generalizedto nonuniform sampling grids, viz., for reconstructionfrom data taken on an undistorted grid (r, x) instead of(p, 6). The convolutional form of the summations will belost, however.

Simulations with Test FunctionsThe numerical algorithm was tested with several knownincomplete Abel transform pairs (Table 1). Objects withfinite support were taken to be confined to the unit disk(a = 1). For the others, scale factors were chosen suchthat the samples were nearly zero at r = 1. Derivatives of

the incomplete transforms were calculated in closed form,except for the jinc function, whose incomplete transformwas computed [by a variation on Eqs. (18) and (19)] anddifferentiated numerically by a three-point formula."Naturally, the derivatives would be estimated from thedata in a real application, with appropriate data smoothingor other precautions to minimize noise (see Ref. 2 for sev-eral references to papers discussing noise issues in Abelinversion). Programming was done in Symantec THINKPascal on an Apple Macintosh Plus computer, and execu-tion times ranged from 3 s for the simplest cases to 336 sfor the longest runs (101 points with L = 0.25, requiring a64th-order reconstruction).

When the derivative of the partial projection is taken tobe an impulse, ' = (e), then the reconstructed object isthe inverse transform kernel, which asymptotically ap-proaches a constant value beyond p = 3A. Observing theapproach to the asymptote with successive corrections, Ifound that the asymptotic value was effectively reachedwhen just over half of the theoretically required number ofcorrections was applied, which means that the reconstruc-tion algorithm can be made faster by truncating the cor-rection sequence. The following empirical relationshipwas found to apply:

Neff = 1 + it[1 + int[N/2]1

where N = 1 + int[(2R/L)2 ] is the theoretically requirednumber of correction terms {1 + int[N/2] is the number ofconvolutions, since one convolution performs two correc-tions [see Eqs. (17)]} and Neff is the effective numberof convolutions actually needed to come within 0.1% ofthe asymptote. This simplification was used in mysimulations.

Results for three test functions are shown in Figs. 4-6.Parts (a) and (b) of each figure consist of three graphs:the differentiated partial and complete projections (upperleft); the reconstruction of the partial projection by usingthe zero-order (uncorrected complete Abel inversion)kernel, the reconstruction of the partial projection byusing the full-order (corrected) kernel, and the true object(right); and the error between the full-order reconstruc-tion and the true object (lower left). The reconstructionerror graph also displays the ratio of the rms error to therms signal strength, as a single relative measure of thequality of the reconstruction. In all cases, the number ofsample points was chosen to keep this rms error ratio at orbelow 0.01.

It is apparent that, in these idealized cases, the proce-dure works well and can recover the object even with ashort projection line. To graphical accuracy, the recon-structions are indistinguishable from the original func-tions. Figure 4 displays the least challenging object,a paraboloid with unit disk support. With L = 1.00, afourth-order reconstruction is needed (four correctionterms), but only 21 points were required for an rms errorratio of 0.007 to be achieved. With L = 0.33 (36 correc-tion terms), 51 points gave an rms error ratio of 0.006.

Figure 5 shows a more challenging object, a Gaussianweighted by a parabola. For an object not having a finitesupport, it was found necessary to begin sampling the pro-

Eric W Hansen


0.00

-0.50

-1.00

-1.50

-2.000.

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-4.0e-3

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0.00

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-1.00

-1.50

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O.Oe+O

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-2.0e-3

-3.0e-3

-4.0e-3

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dProjection

_.......... ....................

.......... . . .. : .. . .

I00 0.25 0.50 0.75 1.

RE

x

-construction error: 4.26e-3 / 5.85e-1

Jo

00 0.25 0.50 0.75 1.00r

dProiection

x

r

(a)

1.00

0.75

0.501econstruction error: 3.67e-3 / 5.80e-1

0.25

...............................

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I I .l

0.50 0.75 1.00r

0.25

n nn

Reconstruction

._O

. f

. I

0.25 0.50 0.75 1.00

(b)Fig. 4. Reconstructions of a paraboloid object with unit disk support from incomplete projections. (a), (b) Upper left: Differentiatedpartial and complete projections g', and g'. Right: True object f, full reconstruction from partial projection fN-l, and zero-order Abelinversion of partial projection P0. Lower left: Reconstruction error f - fN-1. (a) Projection line length L = 1.00 (half-diameter of theunit disk); 21 points used. The ratio of rms error to rms (true) signal was 0.007. Reconstruction required 3 s. (b) Projection linelength L = 0.33; 51 points used. The rms error/signal ratio was 0.006. Reconstruction time was 64 s.

jection well out on the tail of the object, to ensure that theprojection is nearly zero at the edge. In simulations, thiswas achieved by choosing the Gaussian parameter or =0.225 to confine the object effectively to the unit disk.With L = 1.00 and 51 points, the rms error ratio was 0.008,and with L = 0.25 (64 correction terms) and 101 points,an rms error ratio of 0.004 was achieved. In the lattercase particularly, the partial and complete projections (g'pand g') differed strongly, and the zero-order reconstruc-tion of the partial projection is seriously inaccurate.

Finally, Figure 6 shows the most challenging object, ajinc function; the parameter a was chosen to be 42335,which places the fourth zero of the jinc at r = 1. Becausethe partial projection was not available in closed form, theprojections were computed and differentiated numerically.

In this case, it was found to be quite important to makesure that the main lobe was adequately sampled. On thequadratic grid used in the simulations, this meant that,even with L = 1.00, 101 points were required for goodaccuracy (rms error ratio 0.013) to be obtained. How-ever, once this sampling was done, the inversion couldbe performed from L = 0.25 with the same rms errorratio (0.014).

CONCLUSIONS

The incomplete Abel transform results when an axi-symmetric object is projected with a line of integrationsmaller than the object support. In this paper I havederived a space-domain inversion formula for the incom-

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Eric W Hansen

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plete Abel transform. The resulting kernel is equivalentto the transfer function inversion formula developed byDallas et al., although that method is more general in thatit more readily accommodates shaded projection lines, asmight be encountered in a line-spread application. Thekinks in the inverse kernel, first observed by Barrett,6 arenow seen to result from successive corrections to the usual(complete) Abel inversion kernel.

To demonstrate the inversion process, a numerical algo-rithm was designed and used to reconstruct several testfunctions with known incomplete transforms. The re-

0.40

0.20

0.00

-0.20

A A

0.Oe+0

-2.0e-4

-4.0e-4

-6.0e-4

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dProjection

f :

< ~ ~ . . ...

_ ....... ....................... . ......... :..........

I I

0.00 0.25 0.50x

0.75 1.

construction error: 1.27e-4 / 1.59e-2

sults were good, but I emphasize that these were idealcases: the test functions were sampled on a quadraticgrid so as to exploit the convolutional form of the recon-struction, and the data were noise free. Both sampling ge-ometry and signal conditioning (particularly the need todifferentiate the data without ruining the signal-to-noiseratio) are significant practical issues for Abel inversion,which have been discussed at length in the literature. It isto be expected that these issues will be no less importantin practical applications of the incomplete Abel transform.It is also important to possess correct values of the pa-

JO

Reconstruction

(a)

0.40

0.20

0.00

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Al

0.

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-5.0e-5

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dProjection

R ;

_ ....... _

.. . .:.. . .:_ ; ~~~~~.....

I I I

Jo 0.25 0.50x

0.75 1.

Reconstruction error: 6.79e-5 / 1 .60e-2

0.25 0.50 0.75 1.00

(b)Fig. 5. Reconstructions of a parabola-weighted Gaussian. The Gaussian had o- = 0.225 to ensure that the function was small at r = 1.(a), (b) Upper left: Differentiated partial and complete projections, g, and g'. Right: True object f full reconstruction from partialprojection fN-1, and zero-order Abel inversion of partial projection fP. Lower left: Reconstruction error f - fN-1. (a) With the pro-jection line length L = 1.00 and 51 points, the rms error/signal ratio was 0.008, and reconstruction required 12 s. Although the partialand complete projections appear to be nearly identical, substantial error results from using only the zero-order kernel on the partial pro-jection. (b) With projection line length L = 0.25, there is substantial deviation of the partial projection from the complete projection,but, with 101 points, the full-order reconstruction tracks the true curve, with an rms error/signal ratio 0.004. Reconstruction required336 s.

-v.su-

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Eric W Hansen

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0.25 0.50x

0.75 1.'

ReconstructionA AA

2.001

0.001

Reconstruction error: 6.92e-3 / 5.43e-1

00 0.25 0.50 0.75 1.Cr

dProjection

_ ....... . ...- .....

. . . .. . . .

I I.00 0.25 0.50 0.75 1.

x

Reconstruction error: 7.76e-3 / 5.43e-

_.. . . . . .. .. .. . .If-f

I II

0.25 0.50 0.75r

0.00 1.

-2.0

A A

)0 0.

(a)

4.00

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00

)0 0.25 0.50r

Reconstruction

r(b)

Fig. 6. Reconstructions of the jinc function. Parameter a = 4.2335 was chosen to place the fourth zero of the jinc at r = 1. In this case,the projections were computed and differentiated numerically. (a), (b) Upper left: Differentiated partial and complete projections,g'p and g'. Right: True object f full reconstruction from partial projection f N and zero-order Abel inversion of partial projection f 0.Lower left: Reconstruction error f - jN-1. (a) With projection line length L = 1.00, there is little obvious discrepancy between recon-structions with zero-order and full-order kernels, but holding the rms error/signal ratio to 0.013 requires 101 points. Reconstruction re-quired 40 s. (b) For L = 0.25, the (differentiated) partial and complete projections clearly differ, but the full-order reconstruction using101 points tracks the true curve with an rms error/signal ratio 0.014. Reconstruction required 336 s.

rameters L and R, so as to be able to construct the appro-priate reconstruction kernel. In line-spread applications,in which the projection line is an illuminated slit used as aprobe of the imaging system, L would certainly be knowna priori. The radius R of the object support would also bespecified in the data collection setup.

ACKNOWLEDGMENTS

The author thanks H. H. Barrett and the Optical SciencesCenter, University of Arizona, for their hospitality duringthe author's 1987-1988 sabbatical, during which this studywas initiated. This study was presented in part at the1988 Annual Meeting of the Optical Society of America."1

Readers interested in experimenting with the numericalimplementation may obtain the Pascal source code by writ-ing to the author by conventional or electronic mail([email protected]).

REFERENCES

1. R. N. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill, New York, 1978), pp. 262-266.

2. E. W Hansen and P.-L. Law, "Recursive methods for comput-ing the Abel transform and its inverse," J. Opt. Soc. Am. A 2,510-520 (1985).

3. J. C. Dainty and R. Shaw, Image Science (Academic, NewYork, 1974), pp. 210-215.

4. W J. Dallas, H. H. Barrett, R. E. Wagner, H. Roehrig, and C.

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40.00

20.00

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-20.00

-40.00

0.

0.040

0.020

0.000

-0.020

-0.040

0.

40.00

20.00

0.00

-20.00

-40.00

0.

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0.020

0.000

-0.020

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0.75 1.1

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Eric W Hansen

I

I

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-1

Eric W Hansen Vol. 9, No. 12/December 1992/J. Opt. Soc. Am. A 2137

N. West, "Finite-length line-spread function," J. Opt. Soc.Am. A 4, 2039-2044 (1987).

5. M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions (Dover, New York, 1970), §7.3.1, 7.3.2, p. 300.

6. H. H. Barrett, Optical Sciences Center, University of Ari-zona, Tucson, Ariz. 85721 (personal communication, 1988).

7. I. S. Gradshteyn and I. M. Ryzhik, Table of Integals, Series,and Products, 4th ed. (Academic, New York, 1965), §2.261,p. 81.

8. Ref. 5, §7.3.11, 7.3.13, p. 301.9. Ref. 7, §2.246, p. 78.

10. Ref. 5, §25.3.4, p. 883.11. E. W Hansen, W J. Dallas, and H. H. Barrett, "Space-domain

inversion of the finite-length line spread function," inAnnual Meeting, Vol. 11 of 1988 OSA Technical Digest Series(Optical Society of America, Washington, D.C., 1988),paper FOL.

space-domain inversion of the incomplete abel transform

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