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. .IEEE TRANSACTIONS O N ACOUSTICS, SPEECH, A N D SIGNAL PROCESSING, VOL. ASSP-31, NO. 4, AUGUST 1983 979

An Algorithm for the Numerical Evaluation of theHankel and Abel Transforms

DOUGLAS R. MOOK

Abstruct-A procedure for the efficient numerical evaluation of theHankel and Abel transforms is proposed. The A bel transform is reducedto a convolution.which is evaluated in part analytically and in part withan FFT. The Hankel transform is obtain ed by following th e Abel trans-form with an FFT.

I numerous fields including the areas of astronomy, optics,physics, seismology, and tomography, it is necessary to nu-merically evaluate the Abel or Hankel ransforms. The Abeltransformprovides theprojection ofacircularlysymmetricfun ction onto its radial axis. The Hankel ransform providesthe two-dimensional Fourier transform (2-D FT) of a circularlysymmetricunction [ l ] , [8 ]. When an Abel transform isfollowed by a Fourier transform the result is a realization ofthe Hankel ransform [ I ] . In this paper we propose an effi-cient me th od , for evaluating the Abel transform which, whenfollowed by an FF T, results in an efficient meth od for evaluat-ing the Hankel transform.Because of the impo rtance of the H ankel ransform, herepresently ex ists a variety of nume ric schem es for its evaluation.These may be divided into hree general categories: slow ac-curate methods that require on the order of NM operations(where N is the number of input points and M s the numberof ransformed points required) [3 ], [ 9 ] , [ l o ] , approximatemeth ods which require on the order of N log ( N ) operations(where N is both the number of input and output points) butwhich are correct only asymptotically [ 5 ] , and fast convolu-tional methods which require on the order of N log ( N ) opera-tions, but which require samples on an exponential grid andoutp ut on a similar grid [6] , [l o ] . Hybrid methods combin-ing more han one f the above have also been uggested [4] .We propose a fastN log (N ) onvolutional transform tha t usessamples on afl rid rather than an exponential grid.

THE:ABEL AND HANKEL TR ANS F OR MSAs sta ted, the Hankel transform is related to th e 2-D FT of acircularly symmetric fun ction . Specifically, iff,(x,y) is a cir-

cularly symmetric function and if F c ( k x ,k y ) is its 2-D FT sothat

J - m J -,Manuscript received August 5, 1981; revised February 8, 1982 andMarch 18, 198 3. This work was suppo rted in part by th e Office of NavalResearch under Contra ctN00014-77-C-0196, by he Advanced ResearchProjects Agency monitored by ONR under Contract00014-81-K-0742-NR-049-506and by he NationalScience Found ation underGrantECS80-07102.The au thor s with theResearch Laboratory of Electronics, Massachu-setts Institute of Technology, Cambridge, M A 02139 and t he WoodsHole Oceanographic Institutio n, Woods Hole, MA 02543.

then the Hankel ransform of a slice of f c ( x ,y ) generates aslice of its 2-D FT [7]. In particular

F c ( k , , O ) = H T . f c ( x , O ) EC ( x , O ) J o ( k , x ) x d x . (2 )I -Because the 2-D FT of a circularly symmetric function mustalso be circularly symmetric, the Hankel transform determinesthe entire 2-D FT:Fc(kx ,k y ) = F c ( W , ). (3)

We can use the projectio n slice theorem [7] for Fourie r tr ansforms to relate the Abel and Hankel transforms. The projec-tion slice theorem shows that instead of using the Hankel trans-form we could have com puted a slice of the 2-D FT by firstforming the projection

(Abelransform) (4)followed by the one-dimensional FT

m

F ( k x ,O)= 1, ( x ) e i k X X d x .This provides us with the equality

HT * f ( r j = FT . [ A *f(r)] ( 6 )where we have written f ( r ) instead of fc (r , 0). Hereafter wewill switch entirely to a radial coordinate system and also writF ( u ) fo r Fc(v ,0).Oppenheim, Frisk, and Martinez [8] have suggested tha t th eHankel transform be evaluated nu merically by forming the pro -jection p( r ) , which is transform ed with an FFT. Direct calcu-lation of the projectionss computationally expensive, howevrequiring on the order f N 2 operations and numerous functioevaluations.

THE ABEL TRANSFORMWe have developed an algorithm for generating the projec-tionsquicklyby fficiently valuating heAbel ransformshown below:

0096-3518/83/0800-0979$01.00 0 1983 IEEE


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980EE E TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-31, NO. 4 , AUGUST 1983

As suggested by Bracewell [ I ] , [ 2 ] , we wrote this transformas a convolution by defining

(8)

andF(r) p(+) r > o (9)F(r)-f0 (10)

which leads to the convolution formulaF(r)=.+Ih(r). (1 1)

P o 3 = p"(r21. (1 2 )p ( r ) is determined by

Bracewell (21 proposedevaluating 11)by hiftsand ums;however, because th e Fourier transform of h(r) is the knownanalytic functioni + i 1H(u ) =--+ f i for all u ,F(r) can be determined in principle by means of the Fouriertransform

where&u) is the FT off@).Unfo rtunately the singularity at u = 0 makes this functiondifficult to repres ent n he com puter. We havesuccessfullyremoved the factor l / f i from the numerical part of the trans-form (14) in such a way that the remaining function behavesas well as @ ( u ) within the numerical portion of the transform.Th e singlular p art of t he transform is performed analyticallyand added in.To this end the transform (14)s written

where b is a parameter chosen as described later.The integrands n the first and third integrals of1 5b) do nothave singularities at u = 0. Because both he num erator anddenom inator of these ntegrands approach zero as u approaches

0, they can be evaluated by 1'Hopital's ru le to show that as uapproaches zero they approach@(O) .Upon definingP u = oL(u)E (@ ( u ) - F ( 0 ) e-blul ( i - f i))/f i otherwise(16)

and performing analytically the two integrals that do not de-pend on @( u ) , we have

L(u) was defined in (16) such tha t L ( 0 )=F(0) and L(u)@ ( u ) for large u . Theparameter b was chosen so that L(u)moves smoothlybetw een ts limits. It effectively onvertsthe Fourier transforms in (15b) that do not depend on @(u)into Laplace ransforms.Without b , L(u) would have a dcterm h at would ransform to an mpulse. Theoretically hiswould be canceled by the singularities in the por tion of thetransformperformed nalytically [see (17)] ,but omputa-tional error s wo uld certainly cause problem s. If b were infinite,L(u) would suffer from the l/G ingularity at the origin. b ischosen to smooth out the singularity somewhat between theselimi ts. We havebeenusingvaluesof b such hat e-bv hasdecayed to e-' after roughly six samples ofF(u).

EXAMPLESWe pre sent hreeexamples of functions Abel-transformedwith the algorithm described above.

Example 1 a)The first example s the transform of the pillbox fun ction

(18)

10 24 samples of this functio n were generated on afi rid,with T = &, and transformed. The result is shown in Fig. 1 asdots superimposed over a solid line,which is the transformc om p ute d a naly tically ( 2 v ' P ) . T h eoutputmatches heanalytic solution well.Example 2(a)

For the second example we transformed the function

where w(y) is a Hanning window. 2048 simples on fi ridwith T = were input (Fig. 2). Fig. 3 shows the ou tput (dots)


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MOOK: EVALUATION OF HANKEL AND ABEL TRANSFORMS 9812

1

Y

0

I I I

\\

I I II 2 4 6

XFig. 1. Superposition of the numerically generated Abel transform of apillbox (dots) an d the correct transform..6

.4

Y .2

(

_ I

I I I

(0 20 30X

Fig. 2. T h e input (J1( f l ) /m)( n T ) w i t hT = 1/2 an d n = 0, 1 ,2 ,. ,2047 where w ( n T ) is a Hanning window.superimposed over the correct transform (solid line). The cor- The output is coincident with the correct solution.rect Abel transform was computed by evaluating the Fourier-Bessel series [113 to obtain a slow but accurate Hankel rans- ExffmPleform of th ewindowed nput .TheHankel ransform was For he hirdexample weagain transfomed 2048 pointsofthen inverse Fourier transformed to generate the Abel trans-form.nhe bsence fhewindowhe resultwould ave J ~ ( r )been sin ( r ) / r .

- r )r (20)


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9 8 2 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, A N D S I G N A L P R O C E S S I N G , VOL. ASSP-31, NO. 4, AUGUST 1982

1

Y

0

- 1

I

0 10 20 30 40X

Fig. 3. Th e Abel transform of Fig. 2. The dots a re the output of t h eproposed algorithm. The th in line s the correct output..6

.4

Y .2

C

_ c I I I I20 40 60

X80

Fig. 4. The input (J~(m)/mi;)(nT) with T =4 an d n = 0,1, 2,. . ., 2047 where w ( n T ) s a Hanning window.onhe,grid m, ut now T was chosen to be 4. ThisnputHEANKELRANSFORMis shown n Fig. 4. Increasing the sampling nterval educedthe effect of windowing on the inp ut because a greater range To complete the Hankel ransform t is necessary to Fourieof the function was represen ted, but it also increased the Sam- transform the projection obtained from the Abel transformerpiing interval on the outp ut. Fig. 5 shows the output (dots) Unfo rtunately this is available on afi rid and not the evesuperimposed over the correct ransform . Again, there is no grid requir ed by he FFT . T o generatep(r) on an evengriddiscerniblerror. is necessary to interpolate.fimplenterpolationcheme


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MOOK:EVALUATIONOF HANKEL AND ABEL TRANSFORMS 983

3

2

Y 1

0

-1 0 20 4 0 60 80 100X

Fig. 5 . The Abel transform of Fig. 4. The dots are the output of theproposed algorithm. The thin line s the correct output.

0 20 40 60 80X

Fig. 6 . Th e Hankel transform of a pillbox computed by using an FF Ton the linearly interpolated output f the Abel transformer presentedin Fig. 1.

100

used, ike sample and hold or inear nterpo lation, he result better he results. We com plete he H ankel ransform of thewill be generated rapidly but ma y suffer some degradation. If exam ples presented above using linear interpolation to generatea more sophisticated interpolator is used, bett er results can be the uniform grid.expec ted, b ut a t the expense of greater comp utation time. Be-cause the interpo lation is from an uneven grid to an even grid(and no t the reverse), it is difficult to characterize the error Fig. 6 shows the result of using an FFT on the linearly inter-theoretically beyon d the fact that the finer the initial grid, the polated projec tion generated in Example l(a). The dots are the


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984 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-31, NO. 4, AUGUST 1981.5

1

Y .5

0

-.5

I I I

-7

I

I

0 2 4 6 8X

Fig. 7. The Hankel transform of (J~(m)/m)( n T )computed byusing an FF T on th e linearly interpolated output of the Abel trans-former presented in Fig. 3 , T = 1/ 2 case.calculated output and are superimposed over the analytic solu-tion (solid line). The agreement is excellent. T he time require dto perform the total Hankel transform (1024 input points to1024 output points), including the required linear nterpola-tion, was less than 31 s on a PDP 11-55 with a floating pointprocessor.Example 2fb)

Fig. 7 shows the result of Fourier transforming th e linearlyinterpolated output o f Exa mple 2(a). Again the dots rep resentthe output of theAbel-Fourier scheme and th e solid line is theHankel ransformascomputedby th e Fourier-Bessel series[1 I ] . The agreem ent is excellent.Example 3fb)

Fig. 8 compa res the result of Fourier transforming he linearlyinterpolated output of Example 3(a) (dots) with the correcttransf orm (solid). S ignificant d istortion is appa rent in th is tr ans-form. Since the output of the Abel transform in E xample 3(a)essentially equals the output in Example 2(a) (the correct pro-jection) except for the sampling interval, we can associate thisdistortion with the linear interpolation performed before theF FT .

DISCUSSIONWe have fo und , as indicate d by the examples above, th at the

Abel transforme r works well. When its ou tpu t can be success-fully interpolated and s followed by an FF T, the esult is a fast,accurate Hankel transform as illustrated by Examples 1 and 2.As the spacing betw een output samples of the Abel transformeris increased, the suitability of a simple n terpolation schem ebecomes suspect. Example 3 was chosen to il lustrate the effectof inapp ropriate interpo lation on the resulting Hankel trans-

form . At this point we would suggest that the cau tious usedetermine t he validity of a Hankel transform performed usinthis algorithm by comparing the output for inputsof differengrid spacings, just as w ould be done if an FF T were used tcalculate the Fourier transform of a function with unknownbandwidth.

SUMMARYTheprocedure orperforming heHankel ransform HT1) Generate fu(r) =f( fi ).2) Fourier transform t o obtain F ( u ) .3 ) Generate L(u ) @(u ) - F(0)e-bll(l - fi)/fi.4)Perfo rm he inverse Fourier ransforman dadd n th

f(r) = F H ( u ) s summarized below.

analytic terms

5) Interpolate to an even grid p(r ) = F ( r 2 ) .6) Fourier transform t o obtain the Hankel transformFH(u )=FT p(r).

Each of steps 2)-6) requires no more than the orderof .N log( N ) operations. Therefore, the total transform can be accomplished on th e orde r of N log ( N ) operations. Direct compu-tation f rojectionsrom the two-dimensional ircularlysymm etric function would have required a t least N functioevaluations and N sums for each of N points before the finaFF T, which leads to an algorithm requiring on theo rd er o f Noperations. Therefore, for sufficiently large N this method o


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MOOK: EVALUATION OF HANKEL AND ABEL TRANSFORMS 9851.5

1

Y . 5

0

- I

...-...--..-

0 2 4 6 8X

Fig. 8. Th e Hankel transform of (J1 (J.?;)/J.?;) w ( n T ) computed byusing an F FT on the linearly interpolated output of the A bel trans-former presented in Fig. 5, T = 4 case.

calculating th e p rojections an provide a considerable advantagein speed.Steps 1) an d 5) above indicate that in two places interpola-tions may be required. In man y cases, howev er, the functionto be transformed can be generated initially on a grid evenlyspaced in + . F u r t he r , f ( m ), as required by th is a lgori thm,has the desirable feature that equal reas of the underlying two-dimensional function f ( r , 0) are represented between samples,If stationary white Gaussian noise (SWGN) corrupts the mea-surement of f(m),= 0, l , . * , hen the Hankel transformon a fi rid will be corrup ted by SWGN (corru ption of equalareasof heunderlying wo-dimensional unctionproducescorruption of equal areas of the und erlying 2-D FT). This isnot t rue f f ( nT) s corrupted by SWGN.To. implemen t a Hankel transform using this metho d, it isnecessary to perform he nterpolations of step 5). Becauseof the speed of the Abel transform portion of this algorithm,we have foun d it sufficient in many cases to sim ply generatean oversampled version of p(r)and to use linear interpolationto obta inp(r) .

CONCLUSIONFor many applications, this method of calculating the Abel

transform appears to perm it he efficient calculation of heHankel transform for large data files. A detailed study of theproperties of this transform technique has not yet been carriedout. Issues thatmustbeexplored urther ncludeadequaterepresentation off@) and p ( r ) by samples onfi nd r 2 grids,respectively.Becauseof theequal area propertydescribedearlier fo r f ( f i ) and because the speed of this algorithm per-mits oversampling in p ( r 2 ) , t is not expected that these ssueswill pose serious problems.

REFERENCESR. Bracewell, Strip ntegration n radio astronomy, Aust. J.-, The Fourier Transform and its Applications. New York:McGraw-Hill, 1965.J. Bruno1 and P. Chavel, Fourier transformation of rotationallyinvariant two-variable functions: Com puter mple men tationofHankel transform, e o c . ZEEE, pp. 1089-1090, Ju ly 1977.S . M. Candel, Dual algorithms o r fast calculation of the Fourier-Bessel transform, IEEE Trans.Aco ust. , Speech, SignalProcessing,vol. ASSP-29, no. 5, pp. 963-972, Oct. 1981.F . R.DiNapoli and W. B. Deavenport, Theoretical andnumericalGreens function field solution in a plane multilayered system,J. Acoust. SOC.Amer., vol. 67, pp. 92-105,198 0.H. K. Johansen and K. Sorensen, Fast Hankel transforms, Geo-phys. Prosp.,vol. 27,pp. 876-901, Dec. 1979.R. Mersereau and A.V. Oppenheim, Digital reconstruction ofmulti-dimensional signals from theirprojections, Proc. IEEE,A. V. Oppenheim, G. V. Frisk, and D. Martinez, Computationof th e Hankel transformusing projections, J.Acoust. SOC. mer., .vol. 6 8 , no. 2 ,pp. 523-529, 1 980.A. Papoulis, Syste ms and Transforms with Applications in Optics.New York: McGraw-Hill, 1968 .

Phys., vel. 9, pp. 198-217,1956.

v01.62, pp. 1391-1338, 1974.

A. E. Seigman, Quasi fast Hankel transform, Opt . Let t . ,V O ~ . ,pp. 13-15, July 197 4.G N. Watson, Theory of Bessel Functions, 2nd ed. New York:Cambridge University Press, 1966.Douglas R. Mook was born in Englewood, NJ,on July 17, 1952 . He received the S.B., S.M.,and E.E. degrees in electrical engineering fromthe Massachusetts Institute of Technology,Cambridge, in1978,and he Sc.D. degree inelectrical and oceanographic engineering fromthe M.I.T.m.H.O.1. joi nt program in Jan uary1983.He iscurrentlya Postdo ctoral Associate a tM.I.T. and a V isiting Investigator at th e WoodsHole Oceanographic Institution, Woods Hole,

MA. His inte rest s include digital signal processing of acoustic signals.

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