304 J. Opt. Soc. Am./Vol. 71, No. 3/March 1981

Theory of circular harmonic image reconstruction

Eric W. Hansen*

Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford,California 94305

Received March 31, 1980

A method is described for image reconstruction that processes the data in polar rather than rectangular coordinatesand does not require backprojection. It is based on the decomposition of the object and its shadow (set of projec-tions) into circular harmonics, or radial modulators of angular Fourier components. Inverse filters are derived thatenable the radial modulators of the object to be reconstructed from those of the shadow. An optical system for re-construction using the method is proposed.

INTRODUCTION

In computed tomography and similar imaging modalities, itis desirable to reconstruct a cross-sectional image of an objectfrom projections taken through the object. The reconstruc-tion problem may be visualized with the aid of Fig. 1. Theobject of interest is represented by the two-dimensionalfunction f(r,O). Line integrals are taken along a path denotedL(R,0), where R is the perpendicular distance from the originto the path and 0 is the angle of the normal to the path relativeto the 0 = 0 axis. The set of line integrals at all values of R fora particular angle 0 is called a projection. If projections arecollected for a large number of angles, the set of projectionsmay be regarded as a two-dimensional function g(R, 0). Thisfunction is called the shadow, represented mathematicallyby the well-known Radon transform

g(R, b) = SL tf(rO)dl

= J 2 'f f(r,0)5[rcos(0-0)-R]rdrd0, (1)

where 6[ I is the Dirac delta function.Various methods for reconstructing an image from the

shadow have been developed and reviewed in the literature.1-3

Of these, the filtered backprojection algorithms have enjoyedthe greatest commercial success. However, the processingcapability required by future applications, such as real-timecross-sectional imaging of the beating heart,4 motivates thesearch for faster reconstruction procedures. The methoddescribed in this paper circumvents the two chief disadvan-tages of backprojection-based techniques. It processes thedata in the polar coordinates in which they are taken, ratherthan interpolating them onto a rectangular grid, as is done fordigital processing. It also completely avoids the use ofbackprojection, which is often the slowest part of a recon-struction algorithm. The method appears to be particularlywell suited for coherent optical processing, which has the ca-pability for high-speed, high-space bandwidth operation.

Aspects of the circular harmonic method presented herehave been previously investigated by Cormack, 5 Ein-Gal,6

Perry,7 and Hansen and Goodman.8 This paper represents

the method's fullest theoretical treatment to date and offersa unifying framework for the earlier results. An alternativeapproach leading to similar results has recently been offeredby Verly.9 In Section 1 of this paper we present the theoret-ical foundation of the method. In Sections 2 and 3 specificreconstruction filters are derived; they are compared in Sec-tion 4. Practical implementation is discussed in Section 5.Experimental verification of the theory has been performedwith both digital and optical processors' 0'1' and will be thesubject of a future paper.

1. CIRCULAR HARMONIC TRANSFORM

The object and shadow are assumed to be periodic in anglewith period 27r. Hence they may be written as Fourier series,and the Radon transform [Eq. (1)] becomes

g. (R) = r fn (r)cn(R/r)dr, (2)

where= 2R,

g. (R) =- 1 (R</>e-jn1d0,,27r f

(3a)

Path of Integration,IL

Fig. 1. Projection geometry.

0030-3941/81/030304-05$00.50 © 1981 Optical Society of America

Eric W. Hansen

Vol. 71, No. 3/March 1981/J. Opt. Soc. Am. 305

g(R,O) = E g.(R)einsl';n=--

fn(R) =- f f(r,O)e-inOdO,27r fo

f(ro) = E fn(r)ejno;

2wrCn(r) = fr (cos k-r)e-Jnkdo

u(1 -r) (27 [r 0 - 0

+ 5(0 + cos-1r)]e-jnkd = 2Tn(r)u(1 - r)

(3b)and

B(x,y) = r(x)r(y)/r(x + y)

(4a) is the Beta function.'4 Because the CHT is reduced by theMellin transform to a product of Mellin transforms (analogous

(4b) to the relation between convolution integrals and Fourier orLaplace transforms), we say that the CHT has the form of aMellin-type convolution.

By letting x = e-t in Eq. (7), it is found that

(5)

where Tn(x) = cos(n cosI x) is the Chebyshev polynomial ofthe first kind, order n,' 2 and

x >0

x =0

x <0

is the Heaviside unit step function. Equations (3) and (4)express the shadow and object in terms of circular harmonics,or radial modulators of angular Fourier components. Thisdecomposition is called the circular harmonic expansion(CHE). Equation (2), connecting the radial modulators ofobject and shadow, is called the circular harmonic transform(CHT).

When the object is circularly symmetric, hence withoutangular variation, the CHT assumes the simple form

go(R) = SR- 2fo(r)rdr,

which is the well-known Abel transform.' 3

F(s) = f(e-t)e-st dt, (10)

which is the two-sided Laplace. transform of the functionf( ) subjected to a coordinate distortion. Letting r = e-7 , R= e-t in Eq. (2) gives

gn(e-t) = 3 fn(e-T)cn[e-(t-T)]e-7dr, (11)

which is a standard convolution with respect to the pseudo-time variables t and T. Laplace transforming Eq. (11) willagain yield the transfer-function relationships Eqs. (8) and(9).

Of course, what is ultimately desired is the inverse of theCHT, which enables the radial modulators of the object to becalculated from those of the shadow. In the frequency do-main, Eq. (8) is solved for Fn(s), giving

where

Hence

(6)

2. FREQUENCY-DOMAIN ANALYSIS

The circular harmonic transform [Eq. (2)], although linear,is space variant. Thus the usual Fourier methods cannot beused to reduce it to a frequency-domain representation.However, it can be transformed to a transfer-function rela-tionship by way of the Mellin transform, which is definedas' 3

F(s) = Atlf(x)} = |f(x)xs-1dx, (7)

Fn(s) = -(s - 1)Gn(s -)Hn(S)

(S- 1)Hn(s)Cn(s - 1) = 1.

Hn(s) = -- Bs + n2 N)

= 2

2s2 )r(s n)Pr (s)

The Gamma function F(z) has simple poles for z = 0, -1, -2,.... As a result, Ha(s) has poles at s = +n - 2i (i = 0, 1, 2,... ) and zeros at s = 0, -1, -2,.... Many of the poles andzeros cancel, leaving a well-arranged set of simple poles andsimple zeros on the real axis. These are considered in moredetail in the next section.

Taking the inverse Mellin transform of Eq. (12) gives theinverse CHT:

where

f(x) = M-V'F(s)} = F(s)x-sds27rj -,-

and s = a + jw is a complex variable. Applying the Mellintransform to Eqs. (2) and (5) yields

fn(r) = R 9'n(R)hn(r1R) R . (15)

It remains to calculate the point-spread function hn ( ).

3. POINT-SPREAD FUNCTIONS FOR THEINVERSE CIRCULAR HARMONIC TRANSFORM

Gn(s) = Fn(s + 1)Cn(s),

where

Cn (S) = 21-s Res > 0s + 1 + n s + 1 -n

sB (

(8) The point-spread function hn (r) is obtained as the inverseMellin transform of Hn (s); alternatively, the inverse Laplacetransform of Hn (s) yields the pseudo-time-domain func-

(9) tion

hn(e-t ) = 1 j- Hn (s)e t ds.2 7ri -j-

(12)

(13)

(14)

Eric W. Hansen

(16)

I1

U W = � 1/2

01

306 J. Opt. Soc. Am./Vol. 71, No. 3/March 1981

This integral is evaluated using Cauchy's integral theorem.[We assume that n > 0, since Hn (s) is an even function of n.The results are easily extended by substituting In for nthroughout.]

The choice of the parameter c locates the vertical path ofintegration in the complex s plane. Because the two-sidedLaplace transform is used rather than the one-sided formfamiliar from circuit theory, the path may lie in any singu-larity-free region.15 The poles of Hn(s) that lie to the left ofthe path contribute to hn (e -t) for t > 0 (the causal part). Thepoles to the right of the path contribute to hn(e-t) for t < 0(the anticausal part). Thus Hn(s) may be additively de-composed as

Hn (s) = Hn(s) + H- (s),

where H+ is the transform of the causal part of hn and H- isthe transform of the anticausal part. An examination ofHn(s) reveals that it has poles in both left and right half-planes. If any of the poles of H+ lies in the right half-plane,then its inverse h+ will be unstable. Likewise, if any of thepoles of HnI lies in the left half-plane, then h- will be unstable.Although there are many possible choices for the location ofthe path, only two are of interest here. Placing it to the rightof all poles results in a completely causal, but unstable, inverse.Placing it along the jw axis results in a noncausal inverse,which is stable. In the space domain (r = e-t), the causalinverse is confined to r < 1, whereas the noncausal inverseextends over all r:

The right-most singularity of Hn (s) is at s = n so that, forthe causal inverse, the contour is placed to the right of n. Thecausal inverse is tabulated' 6 :

hn(r) = - 1 Tn (1 /r)u(1-r7r / P

(17)

where again Tn ( ) is the Chebyshev polynomial of the firstkind, order n.

In calculating the noncausal inverse, the contour is placedalong the jw axis, and Hn (s) is additively decomposed intoright and left half-plane parts in a partial fraction expan-sion:

Hn (S) = i+ F B (18)i s -X\ j s -Pj

where {A, I are the residues at the left half-plane poles Xi andJBj} are the residues at the right half-plane poles pj. Then thepoint-spread function is obtained as

hn(r) = i A~r-iu(l - r) + E (-Bj)r-Piu(r - 1)i i

where= h+(r) + hn(r),

Ai = lim(s -i)Hn(s),s bXi

Bj = lim(s -pj)H (s),s Bpj.

since the poles are all first order.The left half-plane poles are

Xia=c-nd - 2i id=O ,l ,..f..

and the corresponding residues are found to be

(19)

(20)

A, 2- [(ni + l)12)14(n + 2)/2]i (21)(n +1I)i!

where (x) A r(x + i)/r(x) is Pochhammer's symbol, usedin defining Gauss's hypergeometric function

i(a)i(b)iziF(a,b;c;z) -AE ( .i i!

i=o (c)i i!

The causal part, then, is17

h+(r) = - - FF 1 ' 2 ;n + 1;r2)u(1 -r)

(22)- V 1 1 u(1-r).7 1 +1

The right half-plane poles are

pj=n-2j j=0,1,...,[(n-1)/2],

where [ I] denotes the greatest integer less than or equal to.The corresponding residues are

Bj = - (-l)i2n-2j-1 (n-1-j7 (-(n -1- 2j)!j!

so that the anticausal part is

1 [(n-l)/21(r)=- FZ

irr ,=~o

(n - 1 - j)! (2|n-2j-1

(-l)i (n - 1 - 2j)!j! r) I

=- Un-, (l/r) u(r - 1),7rr

(23)

u(r- 1)

(24)

where

U l(X) =sin(ncos'1 x)sin(cos'1 x)

is the Chebyshev polynomial of the second kind, order n- 1.12,18

The general noncausal point-spread function, then, is

I 1 1 r n

hn(r) = 1 +A

I Un-1(1/r)r

1 >r>O,n > 0

r > 1, n >0.

(25)

We point out that the noncausal form is stable, i.e., boundedas r - 0, whereas the causal form is unstable, growing withoutbound as r - 0. This is in keeping with the respective polelocations relative to the choices of contour for inversion of theMellin transform.

The causal form of the inverse CHT [Eq. (17)] was originallyderived in the space domain by Cormack. 5 It was later foundby Ein-Gal 6 by using the Mellin transform. Perry7 first ob-tained the noncausal form [Eq. (25)] by a space-domainmethod. The present work has derived both forms througha unifying frequency-domain analysis and has classified themas causal-unstable and noncausal-stable.

4. CAUSAL AND NONCAUSALRECONSTRUCTIONIt has been shown that the problem of inverting the CHTadmits of two solutions-one causal and the other noncausal.

Eric W. Hansen

Vol. 71, No. 3/March 1981/J. Opt. Soc. Am. 307

The implications of this for reconstruction are illustrated byFig. 2. The CHT is a causal operator, since the point-spreadfunction c, (r) is zero for r > 1. This means that the shadowat a radius R is a function only of the object outside the diskof radius R (Ref. 6) [Fig. 2(a)]. Similarly, reconstruction withthe causal inverse CHT uses only a portion of the shadow; apoint in the image at a radius r receives contributions onlyfrom the shadow for R > r [Fig. 2(b)]. A noncausal recon-struction, however, uses the entire shadow at all image points[Fig. 2(c)].

In time-domain systems, causality is a real physical con-straint, since response cannot precede stimulus. In thespace-domain operations considered here, causality is moreof a notational convenience than a physical constraint (unlessone were to consider means for reconstructing the image online as the projection data came from the scanner). Thestability of the inverse is a constraint in both space and time,however, and for this reason the stable noncausal filter ispreferable.

5. PRACTICAL IMPLEMENTATIONImage reconstruction that uses circular harmonics is basedon the inverse CHT [Eq. (15)]:

f (r) = , g'n (R)h In I (r/R) dR

The absolute value of n has now been included in thepoint-spread function, making explicit the fact that Hn (s) andhn(r) are even functions of n. In the pseudo-time domain,

fn(e-t) = f [g'n(R)]R=e-i hlnl(e-(t-T))dT (26a)

S - [gn(e-))]hlnl(e-(t-r))(-eT)d-. (26b)

fnjr)

to g,(R)

(a)

s to %W(r)

(b)

s to f%(r)

(c)

g%(R)

Fig. 2. Causality of the CHT: (a) projection, (b) causal recon-struction, and (c) noncausal reconstruction.

HOLOGRAPHICTRANSFER FUNCTION

n1 Hgqv)

DETECTORARRAY

,(R,O) -

ELECTRONICS

r-6-z DISPLAY

f(r,e)

Fig. 3. Possible configuration of optical system for image recon-struction.

The noncausal inverse filter is used. Recall that, in derivingit, the path of integration was placed along thejw axis. Thisreduces the Laplace transform to a Fourier transform. HenceEqs. (26) may be written in terms of Fourier transforms. Bycombining with the circular harmonic expansion, we have

f(e-1,O) = 2 (-12 faR g R=et} (7 )

- 2y-1{2y f-et g(e } H.(/c)} ,

(27a)

(27b)

where 2f I} denotes two-dimensional Fourier transformation.Image reconstruction may thus be implemented by using thetwo-dimensional Fourier-transform capability of an opticalprocessor.

A practical optical reconstruction system might have theform shown in Fig. 3. According to Eq. (27a), the shadow isfirst differentiated with respect to radius, then undergoes thecoordinate transformation R = e-t. Differentiation incomputer simulation has been done with a four-point digitalfilter19; in a real system, such a filter could be implementedby using preprocessing electronics. The R =et coordinatetransformation can be implemented by using a real-timespatial light modulator driven with a logarithmic sweep.2 0

The shadow is input as shown, with projections from succes-sive angles stacked vertically. The first Fourier-transformlens performs both the circular harmonic expansion and thetime transform. The holographic transfer function, whichis essentially a vertical stack of discrete one-dimensionalfunctions, filters all the radial modulators simultaneously.The second Fourier-transform lens performs the inverse timetransform and the circular harmonic synthesis, producing thestretched image f(e-t,O) at the output plane. The detectorarray and postprocessing electronics perform an inversecoordinate distortion (e.g., by clocking the detector nonlin-early), and the resulting image is output on an r-O-z dis-play.

6. CONCLUSIONS

In this study we have developed the theory of a method forimage reconstruction from projections that operates on thedata directly in its polar geometry and without the use of backprojection. The basis of the method is the decomposition ofobject and shadow into radial modulators of angular Fourier

Eric W. Hansen

PREPROCESSINGELECTRONICS

308 J. Opt. Soc. Am./Vol. 71, No. 3/March 1981 Eric W. Hansen

components and the linear transform that connects them.Analysis based on the Mellin transform of the circular har-monic transform yielded two inverse filters, one causal andunstable and the other noncausal and stable. The analysispresented here unifies the results of earlier workers. An op-tical system for reconstruction using this method was pro-posed. Experimental verification of the theory has beencarried out and will be the subject of a future article.

The author wishes to thank Jacques Verly for fruitful dis-cussions and Jon Mandeville for assistance with some of theresidue calculations. The support of National ScienceFoundation grant no. ENG-75-21275 is gratefully acknowl-edged.

* Present address: Thayer School of Engineering, Dart-mouth College, Hanover, New Hampshire 03755.

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