An explicit inversion formula for the exponential Radon transform using data from 180 ~

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Ark. Mat., 42 (2004), 353 362

An explicit inversion formula for the exponential Radon transform using data from 180 ~

H a n s R u l l g s

A b s t r a c t . We derive a direct inversion formula for the exponential Radon transform. Our formula requires only the values of the transform over an 180 ~ range of angles. It is an explicit formula, except that it involves a holomorphic function for which an explicit expression has not been found. In practice, this function can be approximated by an easily computed polynomial of rather low degree.

1. I n t r o d u c t i o n

T h e set of' all o r i e n t e d lines in t h e p l a n e c a n b e i d e n t i f i e d w i t h t h e s p a c e S 1 x R b y a s s o c i a t i n g t h e p a i r

(O,s)ES 1

• w i t h t h e line

L(O,s)--{xCR2;x.O=s}.

A p a r a m e t e r i z a t i o n of t h i s line is t h e n given b y t h e m a p p i n g

t~--~ sO+~O •

w h e r e 0 • is o b t a i n e d b y r o t a t i n g 0 c o u n t e r c l o c k w i s e t h r o u g h a r i g h t angle.

L e t f b e a s m o o t h , c o m p a c t l y s u p p o r t e d f u n c t i o n in t h e p l a n e R 2. T h e R a d o n t r a n s f o r m of f is t h e f u n c t i o n

R f

on S 1 x R d e f i n e d so t h a t

Rf(O, s)

is t h e i n t e g r a l of f a l o n g t h e line

L(O, s).

If # is a r e a l n u m b e r , t h e e x p o n e n t i a l R a d o n t r a n s f o r m

R , f

is d e f i n e d b y t h e w e i g h t e d i n t e g r a l

F

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R,I(O, s) = f(sO+tO• "t dr.

O 0

N o t e t h a t t h e o r d i n a r y R a d o n t r a n s f o r m is o b t a i n e d as a s p e c i a l case of t h e e x p o - n e n t i a l R a d o n t r a n s f o r m w h e n # = 0 .

B o t h t h e R a d o n t r a n s f o r m a n d t h e e x p o n e n t i a l R a d o n t r a n s f o r m , as well as t h e still m o r e g e n e r a l a t t e n u a t e d R a d o n t r a n s f o r m , arise in a p p l i c a t i o n s t o m e d i c a l i m a g i n g , see [2]. It is t h e n of i n t e r e s t t o i n v e r t t h e t r a n s f o r m , t h a t is t o d e t e r m i n e t h e f u n c t i o n f f r o m m e a s u r e m e n t s of its t r a n s f o r m . F o r t h e o r d i n a r y R a d o n t r a n s f o r m , a n e x p l i c i t i n v e r s i o n f o r m u l a was f o u n d b y J. R a d o n in 1917, [6], a n d a g e n e r a l i z a t i o n

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to the exponential Radon transform was derived by O. Tretiak and C. Metz in 1980, [71. A further generalization to the attenuated Radon transform was recently discovered by R. Novikov, see [5] and [3]. An iterative algorithm for inversion of the exponential Radon transform, which only requires R , f ( O , s) to be known for 0 on half of the unit circle has been published by F. Noo and J. M. Wagner, [4].

The goal of this paper is to find an explicit inversion formula, solving the same problem as the algorithm of Noo and Wagner. More precisely, we consider the following problem.

P r o b l e m . Let D c R be a compact set, let # be a real number, and let $1=

{OESI;01>_O} be the right half of the unit circle. Find an explicit formula for computing f , given the values of R , f on $1+ • R , where f is any smooth function with supp f c D .

We will present an essentially explicit formula meeting the requirements of this problem. For the purpose of numerical computations it is not clear that this formula offers any advantages over the algorithm given in [4]. We hope that nevertheless an explicit inversion formula might be of some interest.

2. S t a t e m e n t o f r e s u l t s

The adjoint of the exponential Radon transform is known as the dual Radon transform and is denoted R~. It takes functions on S i x R to functions on R 2 and is given by the formula

(2) R~g(x) = Ji,1 g(0, x-0)e " x ' ~ dO,

where dO denotes arc length measure on the unit circle. Computation of R~g is also known as backprojection. The first step ill our inversion formula is to compute lg~,9 , where g OR~f/Os on S+ 1 x R and 0 on the other half of S 1 • R.

T h e o r e m 1. / f f e 6 ~ ( R 2 ) , then

(3) , ~ OR~fos (0, x.O)e -"x'~177 dO = 2 o~ f ( x x + t , x2)-" cosht #t dr,

where the singularity at t = 0 in the integral on the right-hand side is treated as a principal value.

Let ch, denote the distribution

cosh #t

ch.(t) t

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An explicit inversion formula for the exponential R a d o n t r a n s f o r m using d a t a from 180 ~ 355

with a principal value at the origin. (Later, we will also use ch~ to denote the meromorphic function defined by the same expression.) T h e next step is to find a compactly s u p p o r t e d distribution u such t h a t the convolution u*chu restricted to a given compact set is a point mass 50 at the origin. This is t r a n s f o r m e d into a problem about functions of one complex variable by means of the following definitions.

Let p be a function, holomorphic in the whole complex plane except on some subset of the real line. Define B + ~ and B _ ~ to be the b o u n d a r y values of F on the real line from above and from below

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provided t h a t these limits exist as distributions, and let B2qo=B+qo+B ~ and B A ~ = B + c R - B qo.

Furthermore, if ~ and ~b are holomorphie outside some compact set in the complex plane, define an entire iunction [p, ~b] by the formula

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[~,r

~ ~ ( 0 r

where F is a closed curve, depending on z, so large t h a t the integrand is holomorphic in the unbounded component of C \ F .

T h e o r e m 2. Let r < R be positive numbers, and let F be a ]:unction holomor- phic in C \ I - R , R]. I f

(6) a'rtd

1

(7) [ch,, F] O,

the,~ ~=B~F s~ti4/es 2(~r fo~* Itl<,.

Finally, we show t h a t it is possible to find functions F satisfying the hypothesis of T h e o r e m 2. See Section 4 for some remarks on the c o m p u t a t i o n of F.

T h e o r e m 3. Let w((~) be a positive ]:unction on the interval [0, 1], and let

G(~) be dej~ned fo~" ~ C \ [ - R , R] by

(s) G(~) = / J f d~,

J0 r

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where for any positive real number c, we have chosen the branch of 1/ c~ZT-z 2 which is holomorphic outside the interval [ - , / ~ , , / c ] and satisfies z / c,/7~z2-z 2 -+i, as z--+oc. For generic choice of r and R there exist a number a and an odd, entire function h such that

satisfies the hypothesis (6) and (7) in Theorem 2.

Remark. It is likely that the conclusion of the theorem holds for all r, R and w, but we do not have a proof.

Combining these results, we obtain an inversion formula for the exponential Radon transform.

C o r o l l a r y 1. Let D c R 2 be a compact set, and let r > s u p { l x l yll;x, y E D } . Choose a number R > r and a positive function w on the interval [0, 1], let F be the function constructed in Theorem 3 which satisfies the hypothesis of Theorem 2, and

let u = B ~ F. I f f is any function of class C 2 with supp f c D , then

(10) f ( x ) = / _ ~ u ( t ) L " OR#'f (o, Ol(xl+t)+O2x2)e "(~176 Os

for all z E D .

3. P r o o f s

Proof of Theorem 1. Let 00 denote the directional derivative in direction 0.

Then it follows from the definition of the exponential Radon transform, that OR, f (0, s) = Oo f(sO+tO• ~t dt

O S o o "

and hence

OR~f (O,x.O)e-#x~177 L I ~

Os ~

=L,L

=/2L,

Oof((x.O)O+tO• ~t-~x~177 dt dO

0 o f ( x + rO• )e ~ dT dO

Oo f ( x + rO• )e ~ dO dr,

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An expiicit inversion formula for the exponential R a d o n t r a n s f o r m using d a t a

from

180 ~ 357

where we have made the change of variables

T=t--x.O •

For fixed x and % 0 is the tangent vector to the semicircle

{x+rO • ;OES1},

so that

Oof(x+rO ) dO f ( x l +%

x 2 ) - f ( x l

--r,

X2)

r

Combining these computations, it follows that

~ OR'f(O,x.O)e ~x~177 / ' ~ (f(x~+r'X2)--f(z~--r'Z2)e~~dr

Os ~ r

=lira [ e"~+e-"~f(~+~,~)d~

e-+o Jl~_l> e 7-

[]

T h e o r e m 2 is a direct consequence of the following identity.

L e m m a 1.

If 9~ and ~ are holomorphie outside a compact subset of the real line, then

1 Proof.

Let z E R and suppose that qo(() and ~ b ( z - ( ) are holomorphic for ( outside the interval [a, b]. Then

[g), g)] (z) = liin 1 (jfab

~.o ~ ~(t-ic)r162

fb a 1

27ci 1 47ri

- 4~ri ((Be~BLxr*

1

(BA~Ber*

\

~(t + i e ) r (t + ie) ) dr)

~b

- - - (B ~ ( t ) B + ~ r B+~(t)B ~ r

--/b(B2~(t)B:,V,(~-t)-BA~(e)B2~(~ ~)) dt

Proof of Theorem

2. To prove the theorem, take 9~=ch, and "~b--F. Then B 2 9 ~ = 2 c h , and

Ba9~=-2rci6o.

From the assumption that [ c h , , F ] = 0 it follows that

2 ch~ * u = B z c h ~ . BA F = BA ch~ */32 F = - 2rci50 * B~ F.

The conclusion follows from the assumption on

BzF. []

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Proof of Theorem 3. We must show that there exist a number a and an entire function h such that

a

satisfies (6) and (7). Note that for p and ~ holomorphic outside the reai line, (12)

provided that the products on the right make sense. Since B 2 G = t 3 A h = O in ( - r , r), it follows that in the interval ( - r , r),

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x/r2 +(R 2 r2)a

Hence, the condition (6) will be satisfied precisely if

( 1 4 ) a =

fO

4~2 w(cd

x/r2 + ( t~2- r2)~

^da

It remains to determine h so that (7) is satisfied. To prove the existence of h it is useful to reformulate the condition by means of the following lemma.

L e m m a 2. Let h and ~ be entire holomorphic functions, and let ~ be holomor- phic outside a compact set, with a zero of order k > 0 at infinity. Then [z -1, h~] = O if and only if h= [z -1, r + P for some polynomial P of degree at most k - 1.

Proof. Note first t h a t for any ~ holomorphie outside a compact set, [z -1, Pl is the unique entire function with the property that p - [ z -1, ~ ] - O ( [ z [ ~), as z ~ o c . From this it follows that

is ^apolynomial of degree at most k - 1 . [B Rewrite the condition (7) as

(15) [l cl= ]

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An explicit inversion formula for the exponential Radon transform using data from 180 ~ 359

Since G has a zero of order 1 at co, it follows from L e m m a 2 t h a t this condition will be satisfied if

Write t h e r i g h t - h a n d side of (16) as

-q)(h)-H,

where H is an entire function a n d (I) is a linear o p e r a t o r on t h e space of entire functions. Let K be a c o m p a c t set c o n t a i n i n g t h e interval I - R , R] in its interior, a n d let

A(K)

be t h e B a n a c h space of functions continuous in K a n d h o l o m o r p h i c in t h e interior of K . Since ( c o s h ( # z ) -

1)/z

is an entire function, t h e c o n t o u r of integration in the definition of

cosh( z ) - l , h a

=2~i ( h ( z - ( ) G ( z - ( ) d (

can be chosen so t h a t t h e a r g u m e n t of h always is in t h e interior of K . F r o m this it is clear t h a t 9 can be e x t e n d e d to a b o u n d e d o p e r a t o r from

A(K)

to

A(K')

for any c o m p a c t set K / c C . If K is c o n t a i n e d in t h e interior of K / , t h e restriction

A(K')--+A(K)

is c o m p a c t , and it follows t h a t (I) is a c o m p a c t o p e r a t o r on

A(K).

So unless 1 h a p p e n s t o be an eigenvalue of q), t h e e q u a t i o n

h+~2(h) H

has a unique solution

hEA(B2).

Since b o t h H a n d (I)(h) are entire f\mctions, it follows t h a t h is also entire. Since r takes o d d tunctions to o d d functions, even functions to even functions a n d H is odd, it follows t h a t h is odd.

Finally, note t h a t q) d e p e n d s analytically on r a n d R, a n d the n o r m of converges t o 0, as r a n d _R a p p r o a c h 0. Hence, - 1 is n o t an eigenvalue of (I) for generic choices of r a n d / ~ . In fact, numerical experiments seem to suggest t h a t t h e eigenvalues of (I) are always positive. []

4. N u m e r i c a l t e s t s

To use the inversion formul~ on numerical d a t a , it is necessary to choose a weight w, c o m p u t e a p p r o x i m a t i o n s for t h e c o r r e s p o n d i n g a a n d h to find a function

F=(a/z+h)G

satisfying t h e h y p o t h e s i s of T h e o r e m 2, a n d t h e n c o m p u t e a list of values of

U=BAF.

Choice of weight.

C h o o s i n g w to be a piecewise linear flmction makes t h e com- p u t a t i o n of G straightforward. In order to make G fairly s m o o t h , it is advisable to make w(0) w(1) 0.

Computation of a and h.

T h e c o n s t a n t a is found directly from (14). T h e function h is c o m p u t e d directly from t h e e q u a t i o n (7) r a t h e r t h a n (16). More

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precisely, h is a p p r o x i m a t e d by an odd polynomial. Choose a positive integer n, and let c h , and G be the Laurent series expansions of eh~ and G up to terms of some finite degree, s~v 4n. Use the relations

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I (--1)J@l( ~I__j)Z j§

EzJ,zk] = ( _ l ) k ( _ L k ) zJ+k+l ' 0,

if j < 0 and j + k + l _ > 0 , if/c < 0 and j + k + l _ > 0, otherwise,

and the bilinearity of [ . , . ] to c o m p u t e an odd polynomial hn of degree 2 n + 1 such t h a t the Taylor series of

vanishes up to t e r m s of degree 2n. Note t h a t this expression is an even function, so we have n + l linear equations in the n + l unknown coefficients of h~. T h e n it is easy to show" t h a t if a solution h of (16) exists, h~ converges to h, as n - + o c .

Computation of ~t. The distribution u is readily c o m p u t e d by the formula

(18) //

_a• ,r2)/(R2-,r~)} ~/r2 + ( R 2 - r 2 ) a t 2 da.

Here 'u is t r e a t e d as a function rather t h a n a distribution. W h e n computing the integral in (10) numerically, it is necessary to deal with the singularity of u at the origin. One simple-minded approach is to use the trapezoid rule on a set of nodes s y m m e t r i c with respect to the origin to a p p r o x i m a t e the principal value integral. More accurate results can be obtained by using the m e t h o d s described in [2, C h a p t e r III].

A reconstruction was made with the values r 1, R - 1 . 5 and ten nonzero terms in the polynomial h,r,, see Figures 1 and 2. T h e test object consists of circular discs, and the R a d o n t r a n s f o r m was sampled at 200 values of 0 equally spaced over $1+, and 101 values of s equally spaced between 0.5 and 0.5. T h e width and height of the image are 1 and the attenuation # = 3 .

Acknowledgements. I am grateful to m a n y people who have given valuable c o m m e n t s on this work. In particular, I would like to t h a n k J a n B o m a n for patiently reading preliminary versions of the present p a p e r and providing valuable suggestions for improvements and simplifications.

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An explicit inversion formula for the exponential Radon transform using data from 180 ~ 361

Figure 1. Exact image and reconstruction obtained using the inversion formula.

0 5 ~ 0 4 0 3 0 2 - 0 1 O 0.1 0 2 0 3 0.4 0 5

J

0 5 0 4

i i i i i i i i

-0.3 - o 2 o l e o.1 o z 0 3 o 4 o.~

Figure 2. Cross section of exact image (dotted) and reconstruction (solid) along the horizontal (left) and vertical (right) axes through the center of the image.

R e f e r e n c e s

1. KUCHMENT~ P. a n d SHNEIBERG, [.~ Some inversion formulae in the single p h o t o n enfission c o m p u t e d t o m o g r a p h y , Appl. Anal. 53 (1994), 221-231.

2. NATTERER, F., The Mathematics of Computerized Tomography, Wiley, Chichester, 1986.

3. NATTERER, F., Inversion of t h e a t t e n u a t e d R a d o n transform, Inverse Problems 17 (2001), 113-119.

4. NOO, F. and WAGNER, J . - M . , I m a g e r e c o n s t r u c t i o n in 2D S P E C T w i t h 180 ~ acqui- sition, Inverse Problems 17 (2001), 1357~1371.

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5. NOVIKOV, R. G., A n inversion formula for the a t t e n u a t e d X-ray transformation, Ark.

Mat. 40 (2002), 145 167.

6. RADON, J., 15ber die Bestimmung yon Funktionen durch ihre Integralwerte 1/ings gewisser MannigfMtigkeiten, Her. Ver. Siichs. Akad. 69 (1917), 262 277.

7. TRETIAK, O. and METZ, C., The exponential Radon transform, S I A M o r. Appl. Math.

39 (1980), 341 354.

Received December 13, 2002 in revised f o r m August 28, 2003

Hans Rullggrd

D e p a r t m e n t of Mathematics Stockholm University SE-10691 Stockholm Sweden

emaih [email protected]

An explicit inversion formula for the exponential Radon transform using data from 180 ~