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Image Reconstruction for Fan Beam X-Ray

By Anna Johnson,2015-03-15 06:37
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Image Reconstruction for Fan Beam X-Ray

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    Image Reconstruction for Fan Beam X-Ray

    hy Using General Hankel Tomograp

    Transform Pair (2)

    Zhao Shuang-Ren

    Doubletask, Toronto, Canada

    Abstract: A new transform pair has been introduced for the

    fan-beam image reconstruction[10]. This new integral transform

    plays the same role as the Hankel ( Fourier Bessel) transform in the

    parallel beam case and includes the Hankel transform as a special

    case. In this paper we develop this method for the equal-angle fan

    beam geometry. We call this new pair the General Hankel transform

which gives transform and inverse transform from an object

function f(r,?? to fan beam projections R?(?). One member of the

    pair can be used to calculate projection data for simulation; the

    other can be used as an algorithm for fan beam reconstruction

without interpolation in polar coordinates. Compared with other

    algorithms for fan beam reconstruction the general Hankel

transform method is especially useful when f(r,??! the image to be

reconstructed, has only low frequencies in the;? angle direction and

only few projections are available.

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    1. Introduction

    The problem of reconstructing an object from a set of its

    projections arises in among other fields, computer-aided tomography

    CAT, radio astronomy, electron microscopy, and spotlight-mode

    synthetic aperture radar. Traditional algorithms of reconstruction for

    parallel or fan beam projections is the convolution backprojection

    method[1-6], the Fourier method[7], the Hankel transform method [8].

    The convolution-backprojection method is fast and widely used. The

    Fourier method is faster than the convolution-backprojection method, but

    the reconstruction quality is poor because interpolation in the Fourier

    domain introduces additional artifacts[9]. For a special case where the

object to be reconstructed has no high frequencies along the;? angle, the

    Hankel transform method can be used and good quality of reconstruction

    can be achieved by few projections. However, the Hankel method is only

    used for parallel beam projections. Ref.[10] has introduced a new integral

    transform pair which extended the Hankel method for the fan beam

equal-space case. This new integral transform pair called General

    Hankel transform pair plays exactly the same role in the fan-beam case as

    the Hankel transform pair in the parallel-beam case. In this paper we

    developed this method for the fan beam equal-angle geometry.

    2. Two integral formulas and General Hankel

    Transform pair for fan beam tomography

    The spirit of the Hankel transform method for parallel tomography

is first to obtain the integral transform from the object function f(r,?) to

the projection P?(u) and the inverse transform from the projection P?(u)

    to the object function f(r,?), then expands these into Fourier series. We

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follow this spirit to obtain the general Hankel Transform pair which can

be applied to fan beam geometry.

From Ref. [11] the method of back projection for the case of equi-

angle detectors is available

2; 1;?m (1) ? ? f(r,??( ? L2 ?cos(?)R????g(?'-?) Dd?d?; 0 -?m

where

2 1 ?;(2) ) h(?) g(?)=2 ( sin(?)

;1 1

|?|exp(j??)d?;h(?)= 2 2; ?? D -; ; 1 |?|exp(j?D?)d?;= (3) 2; ??

-;;

2

L= (4) 1+? +2;? sin(?;?)

? cos(?;??;?'=tan-1( ) (5) +? sin??;??;

and f(r,?) is the image function to be reconstructed in polar coordinates

(r,??, R?(?) denotes a fan beam projection as shown in Fig.1 where;? is

the fan beam span angle corresponding to the detector bank and;? is the

rotation angle; R'?(?) is a modified fan beam projection, D is the distance

r

from the source point;? to the origin O.;?;(D;! If we define

(6) R?(?)=0 if;? is not in [-?m,;?m]

the integral limit in Eq.(1) [-?m,;?m] can be extended to (-;,;).

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    ?;

     ?;object fuction.

    t

     ?; D u (r,??; ?;?;O

    Fig 1. Fan beam geometry.

Define a modified fan beam projection

     (7-a) W??t)=cos(?) R???)|?=t/D

    so

     1

    W??t??t(D?;R?(?)|?(7-b) cos(?)

where t is the arc which has the radius of D

    Considering the Fourier transform

     ;;;;;;;;;;;;;;;;; 1 ?exp(j?t)d?(8) ? t= ??exp(-j?t)dt; t 1?(2; ? -; -;;

     Define the one dimensional Fourier transform of W??t) as

     G????(? t(W??t)) ;;

     = ??W??t)exp(-j?t)dt -;;

     ;;

    (9) = ??cos(?) R???)exp(-j?D?)Dd?;

    -;;

So

    W??t)

    =cos(?) R???)

    4

     http://www.paper.edu.cn ; 1 ?G(?)exp(j?D?)d?= (10) 2; ?

    -;;

where t=D?, We also have

    t

    g^(?)=? tg(D??;

    ;;

    ? t

    ( g(D)exp(-j?t)dt ?

    -;;

    ;;

     g(?) exp(-j?D?) D d?; ???,,?;( -;; and g(?) =t;??g^(?))|t=D;?; ; 1 (2; ??g^(?) exp(j?t)d? -;;

     ; 1 ?g^(?) exp(j?D?)d??,:?;(;2; ?

    -;;

    Combining (1) - (12) and assuming that all functions are

integralable, so that the order of integration can be changed, we have

    2;;

    ?m ? ;1 1 ? ? f(r,??(;L2 cos(?)R??2; ?g^(?)exp(j?D(?'-?))d?Dd?d?

     ? -;;

    ? -?m

    0

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2;;

? ?;;;;?m 1 1 ? (L2g^(?) ?cos(?R??exp(-j?D?)Dd?exp(j?D?')d?d?2;; ? -?m

? -;

0

2;;;; 1 ? 1 ( ?,;?;?g^(?)G??exp(j?D?')d?d?2; ?L2 ? -;0

Eq.(7), (9) and (13) can be used to calculate f(r,?) from R???). In

order to obtain the general Hankel transform pair, we need another

equation by which R???) can be calculated from f(r,?).

The parallel beam projection P??u) is[12]

:;;a

? f(r,????u-r cos(?;???rdrd?;?,~?;? P?(u)= ??

?(; r=0

where u is the distance from the origin to the line of x-ray, see Fig 1, a is

limited such that if r>a, f(r,??(;,;

Considering the fan beam geometry Fig. 1

u=D sin?;

?(?,?;?,??;

as in ref. [10] we have

1 ??u-r cos(?;???(;???-?') (16) cos(???D+r sin(?;???

where;?' are defined in equation (5). Considering Ref.[11] and Eq.(14) -

(16) the fan beam projection R???) is

R???)=P?(u)|u=D sin(?);?(?,?;

:; a 1 = ? ? f(r,??;???-?')rdrd?;? ? (17) cos(??[D+r sin(?;???;

?(; r=0

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    Using equations (17) and assuming that all function are

    integralable, so that the order of integration can be changed , equation (9)

becomes

    ;;

    G????( ??cos(?) R???)exp(-j?D?)Dd?(;

    -;;

;;

    ?;;;;:; a 1

    cos(? ?? ?? f(r,?cos(?[D+r sin(?;?????-?') rdrd?exp);j?D?]Dd??;;;;?(; r=0

    -;;

     ;;;;;;;;;;;;;;;a;D ? ( ?? ? ? [D+r sin(?;???f(r,? ???-?') exp);j?D?]d? rdrd?

    ?(; r=0 -;;

    :;;

     ? a1 ? ( ?,??; f(r,??exp[-j?D?']rdrd?;

     ?U ? r=0

     ?(;; where

    U=1+? sin(?;?) (19)

    Equation (13) and (18) are two important integral formulas. (13)

    can be used for image reconstruction. (18) can be used to calculate the

    projection data from the image function. These data are needed for

    simulation. These two integral formulas can be expanded using Fourier

series,

     ;;

    ?:;;a?;f(r,??( ?fm(r)exp(j m;??;

    m=-;;

     2; 1 (20-b) ?f(r,?)exp(-j m?)d? fm(r,?)= 2; ? 0

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and

     ;;

    (21) ?Wm(?)exp(j m;??;W??D?)=

    m=-;;

     2; 1 ?W;?D?)exp(-j m?d?(22) Wm(?)= 2; ?;?;

    0 ;;

    ?:;?;G?(??( ?Gm(??exp(j m;??;

    m=-;;

combining definition (9) and equations (21) ,(22) and (23) gives

    ;;

    (24) Gm(??( ??Wm(?)exp(-j D?;??D d?; -;;

     These leads to the new integral transformation pair ; ;(25-a) fm(r)=exp(j m2) ??Gm(?Hm(r,?g^?d?

    ?(;;;

where

    :;;

     1 ? 1 exp[j(m;?,D;?';??? d?;?:?;b) Hm(r,??( 2; ?L ?(;;

and

     a ; (26-a) Gm(?)=2; exp(-j m;???fm(r)Im(r,?rdr :; 0 where

     :;;

     1 ? 1 26;b) Im(r,??( exp[-j(m?D?'???d?2; ?U

    ;;

    In these two formulas the substitution;?;?(?,;(: has been used and

    U=1+?cos(??;

     2 L= 1+? +2;? cos(?)

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     -? sin???;

    ?'=tan-1( ) (27) +? cos???;

    Equations

    (7)--(22)--(24)--(25)--(20-a)

    can be used as an algorithm for the fan beam image reconstruction.

Equations

    (20-b)--(26)--(23)--(10)--(7-b)

can be used to produce fan beam simulation data R?(?) from a known

    object function f(r,?).

    3. The connection between the general Hankel transform and the Hankel transform

    In this section the connection between the new integral transform

and the Hankel transform is derived.

    One of important algorithms for the parallel-beam image

    reconstruction is the Hankel transform method [8]

     ; ; (28) fm(r)= exp(j m;???Fm(?Jm(r??d? :; 0 a ; (29) Fm(?)=2; exp(-j m;???fm(r)Jm(r?rdr :; 0 where Fm(?? is defined by (30) F?(?)=? u{P?(u)} ;; (31) F????( ?Fm(??exp(j m;??; m=-;; Jm is the first class Bessel function ; 1 ?;:?;?exp[+j(m?;r? sin(????d?Jm(r;??(; 2; ? ?(;;

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     r

    we note that if;?(D;;;;>0 then;?;;;>?!;

    Hm(r,??;;;;> Jm(r;??;

    Im(r,??;;;;> Jm(r;??;

    Eq.(25);;;>Eq.(28)

    Eq.(26);;;>Eq.(29)

    Gm(?);;;>Fm(?) (33) ?his means that the Hankel transform is a special case ( where;?;;;;>0)

of the general Hankel transform.

    4. Implementation of this algorithm

    1) The implementation of (7) calculates the weighted

projections:

    (34) W(?i , tk)=cos(?) R(?i ,;?)|?(tk/D

    where --data of fan-beam projections, R(?i,?) ?i=i

    ;?!;;?(:;(?,p+1), (Np+1) --number of projections, i in [0,Np]; tk=k;t,

    ;t=(D1/(D1+D2));v, D1--distance from the source;? to the centre O, D2-- distance from the centre to detectors,;;v detector distance, k in [- Ns2,Ns2], Ns2= Ns div 2, div --integer division, Ns--detector number.

    See fig 2.

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