an improved cone-beam reconstruction algorithm for the circular orbit

SCANNING Vol. 18,572-581 (1996) 0 FAMS, Inc.

Received November 13, 1995 Accepted January 17, 1996

An Improved Cone-Beam Reconstruction Algorithm for the Circular Orbit HUI Hu

Applied Science Laboratory, GE Medical Systems, Milwaukee, Wisconsin, USA

Summary: By reformulating Grangeat’s algorithm for the circular orbit, it is discovered that an arbitrary function to be reconstructed,f(?), can be expressed as the sum of three terms:fl?)=f,, (?)+ f,,(?)+ fN(F‘) where f,,(F’)corresponds to the Feldkamp reconstruction,f,, (?) represents the information derivable from the circular scan but not utilized in Feldkamp’s algorithm, andfN(?) represents the information which cannot’be derived from the circular scanning geometry. Thus, a new cone-beam reconstruction algorithm for the circular orbit is proposed as follows: (1) computef,,(?) using Feldkamp’s algorithm, (2) computefMl(?) using the formula developed in this paper, and (3) estimatefN(?) using a priori knowledge such as that suggested in Grangeat’s algorithm. This study shows that by including thef,, (7) term, the new algorithm provides more accurate reconstructions than those of Feldkamp even without thef,(?) estimation.

Key words: cone beam reconstruction for the circular orbit, relationship between Grangeat’s algorithm and Feldkamp’s algorithm, improvement upon Feldkamps’s algorithm, filtered-backprojection algorithm

1. Introduction

Three-dimensional (3-D) reconstruction from a set of cone- beam projections has been the subject of many studies. Appli- cations of cone-beam reconstruction include cone-beam x-ray microtomography (Wang et al. 1993a,b) and cone-beam x-ray medical (Ning et al. 1988, Robb et al. 1980) and industrial (Rizo et aZ. 1991) computed tomography.

The theoretical development of analytical formulae for exact cone-beam reconstructions can be categorized into two groups. The first group includes Tuy’s algorithm (Tuy 1983) and Smith’s algorithm (Smith 1983,1985), which are based on the seminal work of Kirillov (1961). Their research

Address for reprints:

Hui Hu Applied Science Laboratory GE Medical Systems

Milwaukee, WI 53201, USA P. 0. BOX 414 NB - 922

revealed the sufficient condition of an exact 3-D reconstruction which requires that any plane intersecting the object intersect the scanning orbit. The scanning orbit is the curve along which the vertex of the cone-beam moves during a scan. The reconstruction algorithms in this group did not lead to practical implementations. The second group includes Grangeat’s algorithm (Grangeat 1990) which is based on the Radon inversion formula (Natterer 1986). Grangeat revealed the relationship between cone-beam projections and the first derivative of the 3-D Radon transforms. He also provided a unified treat- ment for various scanning orbits. Grangeat’s algorithm can be implemented with moderate computational requirements. Recently, Defrise and Clack (1994) and Kudo and Satio (1994) demonstrated that Grangeat’s algorithm can be reformulated in the shift-variant filtering and cone-beam backprojection form.

In practice, the circular orbit is the most commonly employed scanning orbit for cone-beam tomography (Feld- campetal. 1984,KakandSlaney 1988,Ningetal. 1988,Rizo et aZ. 1991, Robb etal. 1980, Webb et al. 1987), thanks to its inherent simplicity and rotational symmetry. Unfortunately, a circular orbit does not satisfy the sufficient condition for an exact reconstruction and therefore only approximate reconstructions exist. Feldkamp et al. (1984) proposed a reconstruction algorithm, which was deduced by extending the two-dimensional (2-D) fan beam filtered-backprojection algorithm (Rosenfeld and Kak 1982) to 3-D cone-beam geometry. As a filtered-backprojection type algorithm, Feldkamp’s algorithm can be implemented both accurately and efficiently. For this reason it has become the most popular cone-beam reconstruction algorithm for the circular orbit. The point-spread function of the Feldkamp reconstruction was formulated by the author (Hu 1989a,b), Yan and Leahy (1991) and Wang et al. (1992), and the performance of this algorithm was evalu- ated in many studies.

Besides Feldkamp’s algorithm, Grangeat proposed another practical cone-beam reconstruction algorithm for the circular orbit (Grangeat 1990). From his general algorithm, Grangeat demonstrated that cone-beam projections acquired from a circular orbit only support a torus-shaped region in the 3-D Radon space. He proposed to fill by interpolation the region outside the torus, also called the shadow zone, that is not supported by the circular orbit. He and others (Defrise and Clack 1994, Kudo and Saito 1994) concluded that the Grangeat reconstruction will reduce to the Feldkamp reconstruction if the shadow zone is left unfilled.

H. Hu: Improved cone-beam recc mstruction algorithm for circular orbit 573

In this paper, we reformulate Grangeat’s algorithm for the circular orbit. This analysis leads to a new circular orbit cone- beam reconstruction algorithm, which includes, besides the Feldkamp reconstruction, another measurement space component that has been overlooked in the previous studies. The paper is organized as follows: in section 2, Feldkamp’s and Grangeat’s algorithms are briefly reviewed to prepare for our discussion; in section 3, Grangeat’s algorithm is reformulated for the circular orbit; in section 4, a new cone-beam reconstruction algorithm for the circular orbit is proposed; in section 5 , the additional component is examined and the improvement of the new algorithm over Feldkamp’s algorithm by including this term is demonstrated.

2. Brief Review of Feldkamp’s and Grangeat’s Algorithms

Let f(7)denote the function to be reconstructed, where ?is a position vector. Let’s assume that the function f(F‘) has a finite support, denoted as Q , and has continuous partial deriv- atives of all orders. Let rdenote the scanning orbit. For a circular orbit with a radius of d centered at the origin0, the vertex position of a+cone-beam, denoted as OS, can be characterized as: OS= (d cos @, d sin @, 0) . Let the unit vec- tors (i’, y, 2’) define a rot3ting coordinate system (Fig. I). whereY and 2’ are along SO and the axis of rotation, and i‘ = 2’ x y. A set of cone-beam projections can be characterized by P,,, ( Y , Z ) , where ( X Z ) are coordinates of the rotating coordinate system. Note in Figure 1 that the 2-D detecting sur- face is assumed to be flat, containing the origin 0. Actual physical detector arrangements can be converted to this form by mapping. Let P a ( X Z ) denote the weighted projection, defined as follows:

2.1 Feldkamp’s Algorithm

The algorithm proposed by Feldkamp et al. ( 1984) can be expressed as follows:

Grangeat’s Algorithm

As shown in Figure 1, a plane P in the object space can be characterized by its normal, it, and its algebraic distance, p, to the origin. The unit vector it can be further parameterized by the colatitude angle, 8, and the longitude angle, 4, as follows: if= (sin 8 cos @, sin 8 sin @, cos 8). The point (p, it) is the characteristic point of the plane P. The Radon transform of f (?), denoted as Rf(p, it), is defined as the integral off (?) over the plane P.

Let S(Q) and S ( 0 denote the sets of the characteristic points of the planes that intersect the support Q and the scanning orbit r, respectively. When the sufficient condition for an exact 3-D reconstruction (Smith 1985, Tuy 1983) holds, the sets S(Q) and S ( 0 are equivalent. Otherwise, S ( I J is equivalent to a subset of S(Q). Let S(N) denote the remaining subset of S(Q), then one has S(Q) = S(r) + S(N). The Radon transforms in the set S ( 0 are supported by the circularly scanned projection measurements, whereas those in the set S(N) are not. The set S(N) is called the shadow zone in Grangeat’s paper (Grangeat 1990).

The characteristic points in the set S(0 can also be characterized by a detector coordinate system. As shown in Figure 1, the detector coordinate system (Z, @03) [or (l, 0, @)I is chosen, where a plane is characterized by the vertex point, 03 (or @), and the line of intersection with the detector plane whose

and FPa(w,, WJ denotes thc 2-D Fourier transform of

To simplify the comparison, the convolution step in Feld- kamp’s algorithm is expressed in Eq. (2a) in the Fourier space rather than in the object space.

p a (EZ). .

FIG. 1 coordinate system, (l,O, a), and the relationship between them.

The spherical coordinate system, (p,O& and the detector

574 Scanning Vol. 18,8 (1996)

algebraic distance to the origin 0 is 1 and whose normal inclines at an angle 0 with respect to the 2 axis.

Grangeat (1990) revealed the following fundamental relationship between the weighted cone-beam projections P#Z) and the Radon transforms, Rf(p, d), in the set S(T):

where Zo3 (LO) represents the integration of the weighted

projection, Pd(YZ), along the line of intersection, (Z, O),

between the plane P, characterized by ( p = 6 S . n',Z ), and the detector plane, characterized by 6S :

For the circular orbit, Grangeat's (1990) algorithm can be summarized as follows:

JRf JP

JRf

1. Calculate - over the set S ( 0 (including its boundary)

using Eqs. (3) and (4).

2. Estimate 7 over the set S(N) by interpolation, based on

JRf JP

JRf JP

those - on the boundary of the set S ( r ) .

3. Once - are computed over the entire set S(Q , compute

f (7) using the Radon inversion formula(Grangeat 1990,

Marr et al. 1980):

where the integration is over the unit sphere.

Reformulating Grangeat's Algorithm for the Circular Orbit:

First, using Fourier Slice Theorem (Bracewell 1956, Rosen- feld and Kak 1982) Eq. (4) can be expressed as follows :

where FPo3 (w, 0) is the 2-D Fourier transform of PCs ( Y , Z)

expressed in the polar coordinate system.

Second, in the set S ( 0 , the Radon inversion formula (Eq. 5) can be transformed from the spherical coordinate system (p, 8, Cp) to the detector coordinate system (Z, 0, a).

Recalling the properties of the delta function, the Radon inversion formula can be rewritten as follows:

where the integration is over the entire set S(Q) .

f,(y', and fN(r'), one has: Breaking the above equation into two terms, denoted as

(8)

(9) f , (7) = -- -(p,n) 6 (7 . n' -p)dpdii 8r2 qj-) J la2Rf ap2 -

1 a'Rf f N ( 7 ) = -- jj -(p,ii) 6 (7. ii -p)dpdii 8x2 S(N) aP2

where the integration is over the sets S(T) and S(N), respectively. From an image recovery point of view [Hanson 1987, Hu 1989a1, f,@) and fN(r') represent the measurement space component and the null space component off (?I, respectively.

For the circular orbit of a radius of d , the set S ( 0 is defined by: Ipl I d sin 8 [Grangeat 1990.1. Thus, Eq. (9) can be expressed in the spherical coordinate system (p, 8,Cp) as

( 7 . ii - p)sin 8 dp d8 dCp

Integrating by parts, Eq. (1 1) can be expressed as the sum of the following two terms:

f,(3 = f , o m + f , , (7)

J6 -(7. ii - p)sin 8dpdOdCp JP

H. Hu: Improved cone-beam reconstruction algorithm for circular orbit 575

Note that the contribution tofMo(?) comes from the entire set S(T), while the contribution tofM,(?) comes from those points on the boundary of the set S(T) only.f,,(r') is referred to as the boundary term.

In what follows we will prove thatf,,(?) is equivalent to the Feldkamp reconstruction:

In the set S(T), (p, 8, @) and ( I , 0, @) have the following relationship (see Appendix Al):

@=@+arccos - ( d i n 8 1

Equations (15) describe how the point (p, 8, @) in 3-D space is projected along the projection angle @ to the point (L 0) on the detector plane.

Therefore, the Jacobean for the transformation from (p, 8, @) to (LO, @) is (see Appendix A2):

s in0 d" sin8 (d' + 1 2 ) '

J = -

(16)

From Eqs. (3) and (6) and from A13 in Appendix Al, one has:

d' + I 2 I FPo3 ( w , 0) j2nw ellrrd d w dRf - -(OS .ii,ii) = - JP d2

Furthermore, the derivative of the delta function in Eq. (13) can be rewritten as follows (refer to Appendix A3):

d6 - - d2 + I 2 d2 dP d2 (d+? .? ) '

d6 -(Yo s in0 + Z,, cos0 - I )

- (r .n - p ) = -

dl (18)

where Yo and Z, are defined in Eq. (2b).

ten in the detector coordinate system (LO, @) as: With Eqs. (16), (17), and (18), Eq. (13) can be rewrit-

Recalling the properties of the delta function, Eq. (19a) can be reduced to:

It is concluded by comparing Eq. (19b) to Eq. (2a) that fMo(?) is equivalent to the Feldkamp reconstruction.

In addition, recalling the properties of the delta function, Eq. (14) can be simplified as follows (see Appendix A4):

and H,(@,r,y/,z)=J[rcos(@- ~ ) - ? d ] ' + z 2 ( 2 h )

Transforming Eqs. (20) from the spherical coordinate system (p, 8, @) to the detector coordinate system, (Z, 0, @) one has (see Appendix A5):

where

= j dZ 'h l (Z- Z')o,(Z')

576 Scanning Vol. 18,s ( 1996)

FoJq) is the Fourier transform of o,(Z). Notice that thef,, (7) reconstruction, as formulated in

Eqs. (21), is in the desired filtered-backprojection form. Furthermore, the filtration, h l(Z), is along the z direction.

4. A New Cone-Beam Reconstruction Algorithm for the Circular Orbit

Combining Eqs. (8) and (12), it is discovered that for the circular orbit an arbitrary function to be reconstructed,f(r'), can be expressed as the sum of the following three parts:

wherefMo(7) corresponds to the Feldkamp reconstruction, fMI(T) is the remaining part of the measurement space component that is also supported by the circular orbit but cannot be reconstructed by Feldkamp's algorithm, and fN (?') represents the null space component that is not supported by the circular orbit.

Equation (22) leads to a new cone-beam reconstruction algorithm for the circular orbit. This algorithm computes three terms in Eq. (22) in following three separate steps: Stepl: Computef,,(?') using Eqs. (2), i.e., Feldkamp's

algorithm, whose implementation is described in Feldkamp et al. (1984).

Step 2: Computef,,(?') using Eqs. (21). Its implementation can be summarized as follows: 1. Multiply each cone-beam projection p,(XZ)

by a weighting factor, as shown in Eq. (I) , to get the weighted projection P,(XZ).

2. Sum the (2-D) weighted projection P,?(XZ) along the row (the Y) direction, as shown in Eq. (21d), to get the (l-D) row sum o,(Z).

3. Filter the row sum oJZ) by a 1-D filterjw-, as shown in Eq. (21c), to get the filtered row sum p,(Z). This step can also be done by convolution or by directly differentiating the row sum o,(Z), as shown in Eq. (21c).

4. The filtered row sump,(Z) from each projection is weighted by a position-dependent factor and then backprojected, as shown in Eq. (21a), to form the f, I (r') reconstruction.

Step 3: EstimatefJ?) based on a priori knowledge or cer- tain assumptions. For example, thef,(r') estimation method proposed by Grangeat (1 990) can be adopted as follows:

3Rf JP

1. Estimate - over the set S(N) by interpola-

tion, based on those aRf on the boundary of the JP set S(T).

2. Compute f,,(r') using Eq. ( 5 ) , assuming that

F% is zero outside the set S(N). dP

The implementation of Step 3 is similar to Grangeat's algorithm(Grangeat 1990),

Finally, combinef,,(r'),f,,(?) andfd?') to form thef(7) reconstruction.

It is concluded thatf,,(;) and.f,,(r') combined consti- tutes the entire measurement space component, represent- ing the best reconstruction from the circularly scanned cone-beam projections.

Furthermore, bothf,,(r') andfMI(?') reconstructions, as formulated in Eqs. (2) and (211, respectively, are in the desired filtered-backprojection form, and therefore can be implemented accurately and computationally efficiently.

5. The Boundary Termf',(r') and Its Impact in Reconstruction

Equation (22) reveals that besides the Feldkamp reconstruction there exists another measurement space component,f,,(?'), that is also supported by the circularly scanned cone-beam projections. This appears to contradict the conventional wisdom (Defrise and Clack 1994, Grangeat 1990, Kudo and Saito 1994) that for the circular orbit the Grangeat reconstruction and the Feldkamp reconstruction coincide when is ignored. This discrepancy results from the discontinuity of the redundancy function in the set S(T) (the torus region). The redundancy function is defined as the number of times the orbit intersects the plane to be con- sidered. The conventional wisdom is derived assuming that the redundancy function equals 2 for every point in the set S(T). This assumption is correct in general except for those points on the boundary of the set S(T) where the redundancy function equals 1. Consequently, Feldkamp's algorithm correctly represents the contribution from the points inside the set S(T), but misrepresents the contribution from the boundary of the set S(T) (the torus region). Note that the error so induced occurs in the measurement space. Recalling that the boundary temfMI(r') represents the contribution from the boundary, and thatf,,(?) andf,,(i') combined constitute the entire measurement space component, it is concluded that thefM,(?) term, previously overlooked, corrects the error in the measurement space induced by the Feldkamp reconstruction.

The backprojection weighting factors in Eqs. (2) and (21) indicate that as d approaches infinity, f,,(r') is in the

order of ( f ) ' whilef,,(r') is in the order of (iy. Thus,

f,,(?') is a high order correction to,f,,(;). This correction becomes increasingly significant as the cone angle


[reflected in the weighting factor ( d + r , ., i , ) 2 in Eq. (21a)l

increases and/or as the function f ( ? ) changes increasingly

along the z direction [reflected in the ___ term in Eq. dods ( Z ) dZ

(21c)l. It correctly reduces to zero at the midplane or for a z uniform f(.'), when the Feldkamp reconstruction is known to be exact.

The effect of the boundary term fMl(?) was evaluat- ed by a computer simulation. Two mathematical phantoms were used: one consists of a uniform ball with a radius of 20 cm centered at the location (0, 0, 3) cm, the other consists of five identical uniform ellipsoids with the axes of x: 15 cm, y: 15 cm, and z: 3 cm and centered at the axis of rotation with z= -20, -10, 0, 10, 20 cm, respectively. The radius of the circular orbit was assumed to be 100 cm, resulting a maximum cone angle of rough-

Three sets of volume data were computed. They are (1) the Feldkamp reconstruction, fMo(?); (2) the new reconstruction ignoring fN(f') term, fM(?) =fMo(.') + fMo(?); and (3) the true phantom distribution,f(?). Recall from Eq. (22) that the difference between the second and the first data sets corresponds to the boundary term fMl(?) and the difference between the third and the second data sets represents the term fN(?). Studying these three sets of data allows us to decompose the three terms in Eq. (22) and assess the contribution of each term. The corresponding images on the sagittal plane of y=O are displayed in the same order in Figures 2 and 3 for the two phantoms, respectively. The corresponding z profiles at x= 0, 12 cm are shown in Figures 4 and 5.

Figures 2-5 demonstrate that the Feldkamp reconstructions suffer a density drop-off as the cone angle increases. The ball phantom study (Fig. 4a,b) shows that the new reconstruction will reduce this density drop-off by more than 60%. Consequently, the image of the ball is more uniform in Figure 2(b) than in Figure 2(a). It is worth pointing out that in this case more than 60% of the drop-off presented in the Feldkamp reconstruction arises from the term fMl(?), as opposed to 40% from the fN(?) term. The ellipsoid phantom study (Fig. 5a,b) shows that including the term f M l (?) does not reduce the overall (i.e., from ellipsoid to ellipsoid) density drop-off. Therefore, the overall density drop-off arises mainly from the term fN(?). However, Figure 5(a,b) shows that including the term f M l (?) reduces the density variation within each ellipsoid. Consequently, its z profile is closer to the rectangular shape than that of the Feldkamp reconstruction. In both studies, the correction term, fMl(?) , could account for more than 5% of the total measurement space component.

This simulation study demonstrates that the contribution of the additional measurement space component, fMl(?), though small compared with the Feldkamp reconstruction, can be rather significant.

ly *lo".

(c) FIG. 2 Display of three sagittal images of the uniform ball phantom. (a) Feldkamp reconstruction,f (7); (b) reconstruction from

Ma+ the new algorithm ignoring fN(7 ) , fM(r) = ,fMa(7) + fMI(7); and (c) true phantom distributionf(7). To study the density uniformity, the display window was chosen to be 1k0.075, around the true density of one. The ball in (b) is more uniform than that in (a).

(c) FIG. 3 Display of three sagittal images of the uniform ellipsoid phantom. (a) Feldkamp reconstruction, f (7); (b) reconstruction from the new algorithm ignoringfN(;),fM(r) = f,,(;) + fM,(7); and (c) true phantom distributionf(7). To study the density uniformity, the display window was chosen to be 0.70k0.35. The density variation from ellipsoid to ellipsoid is comparable between (a) and (b). However, the density variation in each ellipsoid is smaller in (b) than in (a).

YO

578 ScanningVol. 18.8 (1996)

1.2' I % . I - ?he truegrofre ........ Feldkamp reconstruction rrection term 1.0 -

0.8 - -

3 0.6 - - .-

- cn C (u 0 0.4 -

0.2 -

0.0 -

-

-

-0.2 ' " 1 . " I " ' I " '

6. Discussion

- - l~:~...:-~,7,:7:.z ....... .............. 7.:: I

i I i i

! 1 - i

- ! !

i i !

- !

- ! -

- -

i

i

- I

- - The true profile - -

........ Feldkamp reconstruction Feldkamp reconstruction +the correction term .

I . . . .. ..................................... . . . . . . .

To highlight the contribution of thefM,(?) term, thefJ?) estimation is not included in this simulation study. It is demonstrated that by including the boundary term,fMl(?), the new algorithm provides more accurate reconstructions than Feldkamp's reconstructions even without thef,(r') estimation.

Ignoring thefN(7') term results in errors, as demonstrated in this simulation. These errors can be reduced, to some extent, by estimating thefN(7') term using a priori knowledge. As to thefN(?) estimation, the new algorithm is similar to Grangeat's algorithm, and therefore a comparable performance is expected. The study presented in this paper represents a first step toward the development of similar algorithms for some nonplanary scanning orbits that are capable of collecting a complete set of data.

1.0

0.8

a 0.6 .- cn c 8 0.4

0.2

0.0

1 .o

0.8

3 0.6 .- cn C 0 0 0.4

0.2

0.0

-0.2

;

-

-

-

-

1.0

0.8

2 0.6

8 0.4

.- cn C

0.2

0.0

-30 -20 -10 0 10 20 30

(4 z (cm)

-

-

-

1

-

-

1.2 L' " ' " ' " I' " " " " I " " " ' " I " " " " ' I " " " " ' I ' " " " ' ' 4

/.:::j I

p7:.:.: ...... ...........

- Thetrue profile

Feldkamp reconstruction + the correction term -0.2 I.. . . . . . ..I . . . . . . . . . I . . . . . . . . . . . . . . . . . . I,. . . . . . . . I . . . . . . . .

-30 -20 -10 0 10 20 30

(b) z (cm)

FIG. 4 Corresponding z profiles of the three ball images displayed in Figure 2(a-c) plotted with (a) 0 cm and (b) 12 cm from the axis of rotation. Inclusion of the correction temfM,(;) helps reduce the density drop-off that is presented in the Feldkamp reconstruction.

Appendix A

Al . Relationship between (p, Q, @) and (Z,O,(P)

Consider projecting a point (p, 6, +) in 3-D space along the projection angle @ to the point (Z, 0) on the detector plane. From the geometrical relationship shown in Figure 1 , it follows that

........ Feldkamp recon! Feldkamp recon!

7 - -

.r'

f-"

4 iction iction +the correction term -

- 0 . 2 1 . " I " " ' ' " " -40 -20 0 20 40

(b) (cm)

FIG. 5 Corresponding z profiles of the three ellipsoid images displayed in Figure 3(a-c), plotted with (a) 0 cm and (b) 12 cm from the axis of rotation. Inclusion of the term fM, (7) does not reduce the overall (from ellipsoid to ellipsoid) density drop-off however, it does reduce the density variation within each ellipsoid and makes its profile closer to the rectangular shape than the Feldkamp reconstruction.


The inverse formula is Similarly, one has d sin 0

Jm sin(@ - 0) sin 8 = --

(Al l )

Using Eqs. (A3) and (A1 l) , the inverse formula of Eq. (1 5c) can be expressed as The plane on which the Radon transform is performed

can be denoted as

o%ii=p ctun(@ - @) = -- 1

(A 12) d sin@

Given that n’ = (sin8 cos@, sin8 sin@, cost)) and 03= (d cos@, d sin@, O), Eq. (A2) can be expressed as Furthermore, from Eqs. (Al) and (A3) one has

bsl‘ d2+12 dsinOcos(@-@)=p (A31

It then follows that

A2. The Jacobean for Thnsform from (p, 4 4) to (4 0, @)

Because of the redundancy in the (p, $,@) coordinate system, the points (p,8,@) and (-p, a - 8, a+@) denote the same location. Thus, we can consider only + sign in Eq. (15c) without losing any solution.

Furthermore, from Figure 1, one has

From Eqs. (Al), (AIO), and (A12), one has

i’=(-cos@,-sin@,O) (A41

i’ = (0,0,1) (-45) j’ = i’ x i‘ = (sin @,-cos @,O) (A@

The projection of n’ onto the detector plane can be described as follows:

Therefore, the Jacobean for transform from (p, 8, 4) to (LO, @) is

ii.j’=-sin$ sin(@-@) (A7)

i i . i’=cos8 (A81 Recalling Eqs. (A3), (A7) and (A8), it then follows that

A3. ’kansforming the Derivative of the Delta Function fl-om (p, 4 $4 to (40, @)

In the cylindrical coordinate system, one has

r’= (r cos w, r sin w, z ) W 8 ) Recalling Eqs. (Al), (AIO), and (A1 l), the argument in the derivative of the delta function in Eq. (13) can be rewritten as follows:

0 = arccos or

With Eq. (Al), the inverse formula of Eq. (A9) or (15b) can be rewritten as 7 . n’ - p = rcos(q - y)sin 0 + zcos 8 -p

d cos 8 = -coso ,.m

5 80 Scanning Vol. IS, 8 (1996)

Note that

Eq. (A12) is used in the last step of Eq. (A20). Substituting Eq. (A20) to Eq. (A19), one has

d d+F. i ’ (Y,, sin@ + Z(, cos0 - I ) -

dF.jj’ d z where Yo =- Z“ = ~

d+7.2’ d+?. i ’

Using the properties of the delta function, one has

a6 d2+12 d 2 -(? .Z-p) =- aP d 2 (d+F. i ’ )2

a6 ai - ( Y , sin@ + Z , cos 0 - I )

A4. Simplifying Equation (14)

Equation (14) can be rewritten as follows:

f,lm=f,l(r, y,z) = --j’” 1

dQ(L+(r ,y , z ,Q)-L- (r , y,z,Q)) 8n2 0

(204 where

(A22a)

and

g,( 8,Q. r , ty ,z) = (F .ii - fdsin 0) = [rcos(Q - y) - fdlsin 8 + zcose

(A22b)

Using the properties of the delta function, Eq. (A22a) can be reduced to

(A23a)

i.e., tan e,, = 2

f d - rcos( Q - y) (A23c)

I4 H ,

sin 8,,* = - (A23d)

It then follows that

z 3 (+ z, arctan fd-rcos(Q- y)’m)

H,( Q,r, y,z) = d[ rcos(Q - ty) - fd] ’ + 2’

(20c)

AS. Thnsforming Equations (20) from (p, 4 Q) to (1; 0, @)

Given that p=+d sine, Eq. (1%) can be reduced to

@ = Q +ak where @+ = O,@- = zf (A24)

Therefore, Eq. (20c) and Eqs. (A23c and e) can be simplified as

Hi(@, r, y, z ) = H ( @ , r, y, z )

= d [ d - rcos(@ - y)]’ + z2 (A25a)

H. Hu: Improved cone-beam reconstruction algorithm for circular orbit 58 1

.. tanOo,, =

L

+d-rcos(@+@?- W ) z =+

d - rcos(@ - w )

Furthermore, Eq. (15b) can be reduced to

0, = , , c o s ( ~ ) = ~ c c o s ( + ~ )

Equation (15a) can be reduced to

(A25b)

(A25c)

(A26a)

(A26b)

(A26c)

(-427)

Based on these equations, Eqs. (20) can be rewritten as

The Fo,(o-) is the Fourier transform of o,(Z).

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