Grangeat-Type Helical Half-Scan CT Algorithm for ... · Currently, cone-beam CT and Micro-CT scanners are under rapid development for major biomedical applications. Half-scan cone-beam

Grangeat-Type Helical Half-Scan CT Algorithm for Reconstruction of a Short Object

Seung Wook Leea, b and Ge Wanga

aCT/Micro-CT Lab.

Department of Radiology

Department of Biomedical Engineering

University of Iowa

Iowa City, IA 52242, USA

bDepartment of Nuclear and Quantum Engineering

Korea Advanced Institute of Science and Technology

Daejeon, 305-701, South Korea

Corresponding AddressSeung Wook Lee and Ge WangCT/Micro-CT LaboratoryDepartment of RadiologyUniversity of IowaIowa city, IA 52242, USA319-384-5616 (Tel)319-356-2220 (Fax)[email protected]@ct.radiology.uiowa.edu

7/3/2003 5:39 PM

ABSTRACT

Currently, cone-beam CT and Micro-CT scanners are under rapid development for major biomedical applications. Half-scan cone-

beam image reconstruction algorithms assume only part of a scanning turn, and are advantageous in terms of temporal resolution

and image artifacts. While the existing half-scan cone-beam algorithms are in the Feldkamp framework, we have published a half-

scan algorithm in the Grangeat framework for a circular trajectory 1. In this paper, we extend our previous work to a helical case

without data truncation. We modify the Grangeat's formula for utilization and estimation of Radon data. Specifically, we

categorize each characteristic point in the Radon space into singly, doubly, triply sampled, and shadow regions respectively. A

smooth weighting strategy is designed to compensate for data for redundancy and inconsistency. In the helical half-scan case, the

concepts of projected trajectories and transition points on meridian planes are introduced to guide the design of weighting

functions. Then, the shadow region is recovered via linear interpolation after smooth weighting. The Shepp-Logan phantom is

used to verify the correctness of the formulation, and demonstrate the merits of the Grangeat-type half-scan algorithm. Our

Grangeat-type helical half-scan algorithm is not only valuable for quantitative and/or dynamic biomedical applications of CT and

micro-CT, but also serve as an intermediate step towards solving the long object problem.

Key words: half-scan, Grangeat-type reconstruction, cone-beam CT, helical CT, reconstruction

Running title: Lee and Wang: Helical Half-scan Grangeat formula

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Seung Wook Lee and Ge Wang: Helical Half-Scan Grangeat 1

I. INTRODUCTION 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Sixteen-slice helical computerized tomography (CT) scanners are already commercially available. C-arm systems and

several micro-CT systems are also based on cone-beam acquisition and reconstruction 2-4. Prototypes of the cone-beam

systems were recently reported 5. With rapid increment in the number of detector rows, the concept of cone-beam CT or

volumetric CT will become more and more popular. New applications are made possible by these new fast volumetric

imaging technologies, such as for cardiac and lung examinations, CT angiography, and interventional procedures 6,7. In

those applications, high temporal resolution is one of the most important requirements.

Half-scan techniques were developed to improve the temporal resolution for axial and spiral CT images 8,9.

While half-scan cone-beam algorithms in the Feldkamp framework have relatively long history 10-13, the half-scan cone-

beam algorithms have been recently developed in the Grangeat framework for the circular scanning geometry by our

Laboratory as well by Noo and Heuscher independently 1,14. The difference between these results is substantial. That

is, our work is in the rebinning framework 15, while the formulation by Noo and Heuscher is in the filtered

backprojection format 16.

This research is a natural extension of our half-scan algorithm from circular to helical scanning geometry to

solve the short object problem, which assumes that the object is completely covered by the X-ray cone beam from any

source position. Even though the geometry for data truncation along the axial direction is the most practical, research

with the short object geometry is not only valuable on its own, such as for micro-CT imaging of spherical samples, but

also essential as an intermediate step toward solving the long object problem 17,18.

This paper is organized as follows. In Section II.A the Grangeat algorithm is briefly reviewed for

completeness. In II.B, the rebinning equations for a helical trajectory are introduced. In II.C, the helical half-scan

Grangeat formula is described. In II.D, the weighting functions are derived. In II.E, the interpolation method used in

this study is explained. In II.F, the implementation procedure is summarized. In Section III, the results are presented

and discussed. Finally, In Section VI, the paper is concluded.

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II. MATERIALS AND METHODS

A. GRANGEAT FRAMEWORK

1

2

3

4

5

6

7

8

9

A.1. GRANGEAT'S DERIVATION

For cone-beam CT, it is instrumental to connect cone-beam data to 3D Radon data. Smith, Tuy and Grangeat

independently established such connections 15,19,20. Grangeat’s formulation is geometrically attractive and becomes

popular. Here, we review Grangeat's derivation process. It largely follows the notations defined in 21.

In cone-beam geometry as in Fig. 1, the 3D Radon transform at the characteristic point is defined byC

2/

2/ 0

),,()( rdrdrnfnRf , (1)

which means the plane integration of the gray triangle with one of the vertices being the source point . The plane is

normal to the vector

Sn . Any point on the plane is represented with the polar coordinate ),(r on itself.10

11

12

13

A cone-beam projection is the line integration from to any point on the detector plane and is

mathematically represented by

S A

0

),,()( drrnfnXf . (2)

As you can notice, there is r in Eq. (1), which prevents us from using the measured cone-beam projection. To

eliminate this, the first derivative with respect to

14

is applied to Eq. (1) and we have15

2/

2/ 0

),,()( rdrdrnfnRf . (3)16

To first order, we have the following relation at any point on the plane, ),,( rn :17

drd )cos( , (4)18

where is the angle between and .SO SC19

20 By substituting

cos1

rdd

dd

, (5)21

22 we have

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2/

2/ 0 cos1),,()( ddrrnfRf . (6)1

Here, r is cancelled out and the cone beam projection can be directly utilized as 2

2/

2/ cos1),()( dnXfRf (7)3

The two angular variables, and can be changed to the variables on the detector plane, s and t , with

the following geometrical relationships.

4

5

tanSOs (8)6

tanDSCt (9)7

8 Differentiating the two yields

dSOds 2cos (10)9

dSAdtcos

(11)10

11 Finally, we have

dttnsXfSASO

snRf )),((

cos1)( 2 (12)12

13

14

15

16

A.2. CONE-BEAM DATA TO DERIVATIVE DATA IN THE RADON DOMAIN

Fig. 1 is transformed to Fig. 2 to show the geometrical relationship between the detector plane and the meridian plane

in the Radon domain. Accordingly, Eq. (12) can be rewritten as

,)(,),(cos

1

)(,),(cos

1

)()(

2

2

dtntnsfXs

dtntnsXfSA

SOs

nfRnRf

w

D (13)17

where )(,),( ntnsXf is the detector value, which is defined by the distance s from the detector

center O along the line t perpendicular to on the detector plane extending an angle

18

D DDCO D from the197/3/2003 5:39 PM


x -axis, the distance between the source and the detector center, the distance between the source and an

arbitrary point

DSO SA1

A along t , and the angle between and . Given a characteristic point on a

meridian plane in the Radon domain, the plane orthogonal to the vector

DSO SC C2

M n is determined. Then, the

intersection point(s) of the plane with the source trajectory

3

)(a can be found, and the detector plane(s)

specified, on which the line integration ought to be performed. Let denote the intersection of the detector plane

with the ray that comes from and goes through C . The position can be described by a vector

D4

5 DC

D S DC Dns .

To compute the derivative of Radon value at , the line integration is performed along , which is orthogonal to the

vector

6

7 C t

Dns .

f

d8

12

2/

/ 2

2

nn)xf ( 2

2

),y z ,'(y

h

SO ), h,cosSO()(a

n

,,( )

)( )

),,

n x

8

9

10

Once the first derivative of Radon is calculated, the original function can be reconstructed with the

following 3D Radon inversion formula: 22,23

dxRf sin][()0

. (14)11

12

13

14

B. REBINNING EQUATIONS FOR A HELICAL TRAJECTORY

The geometry for a helical half-scan is shown in Fig. 3, where is the reference coordinate system,

denotes a meridian plane at

,(x )'z

from the x-axis, and the source vertex, , ranges from 0 to m2 . If the helical

pitch is denoted by , the source trajectory can be parameterized as

15

16

sin 0z . (15)17

To compute the derivative of the Radon value at , we need first calculate the line integration point

on a detector plane, through which a line integration is performed along the line normal to OC . There is a

geometrical relationship between the characteristic point

DC18

19 D

) and the line integration point ,,(s . In other

words, we can find ,,s from ( ,, .

20

21

The 3D Radon value at a characteristic point ( is the integration of an object on a plane satisfying the

following equation:

22

23

. (16)24

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The intersection of the plane with the source trajectory is found by replacing x with )(a to solve1

)(an (17)2

where )cos,sinsin,cos(sinn denotes the characteristic point, and ),sin,cos()( 0zhRRa

the source position. The equation can be written as:

3

4

5

6

cos)()sinsincos(cossin0cos)(sinsinsincossincos

0

0

zhRzhRR

(18)

While we can calculate the line integration point analytically in a circular trajectory case, we can only do it numerically

in a helical trajectory case. Once we acquire , we have line integration point ( , s ) on the detector plane ,

where

D7

2. The equations are expressed as 17:8

))sin(

cot(tan 1 , (19)9

cossin)cos)(( 0zh

s . (20)10

Therefore, the Radon value at the characteristic point defined by ),,( can be calculated by the integration along

the line represented by

11

),,( s according to Eqs. 18-20.12

13

14

15

16

17

C. GRANGEAT-TYPE HELICAL HALF-SCAN FORMULA

With a circular half-scan, there are three types of regions: shadow, singly and doubly sampled regions, respectively 1.

With a helical half-scan, in addition to those three types of regions, there may be triply sampled regions as well. They

are schematically illustrated in Fig. 4. Hence, the Grangeat-type helical half-scan formula must be in the following

format:

)()(3

1

nfRnRfg

gg , (21)18

where g denote the weighting functions, g is a group identifier to be explained in detail later, and19

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dttsXfSASO

snfRg ),,(

cos1)(2 . (22)1

2

3

4

The form is basically the same as that with a circular half-scan trajectory but three weighting functions and

corresponding Radon values are needed for each characteristic point, while we only need two weighting functions in a

circular half-scan case.

In Eq. (22), the value of the group identifier g is determined according to the following criteria5

g = 1, 10 t , (23)6

2, 21 tt ,7

3, mt 22 .8

where 1t and 2t are tangential vertices and will be explained in detail in Sec D.2. This means that the calculated

Radon data must belong to one of the three groups depending on the

9

. Regarding the weighting functions, they must

satisfy

10

11

12

13

14

15

16

17

18

19

3

1

1g

g

. (24)

If we assume that there is no motion or data inconsistency during the scan, we can simply average Radon data

from different vertices. In this case, the weighting functions become discontinuous. However, in real situations, motion

effects and data inconsistency should be taken into account . Then, the discontinuous weighting functions could

cause artifacts. Therefore, the smooth weighting functions are needed for satisfactory image quality. The next section

will be devoted for this purpose.

D. WEIGHTING FUNCTIONS

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1

2

3

The concepts of projected trajectory and transition points are needed before our weighting functions are designed.

Hence, we first introduce them in D.1 and D.2. Then, we design the weighting functions in D.3.

D.1. PROJECTED TRAJECTORY ON A MERIDIAN PLANE AT4

The projected trajectory on a meridian plane at is useful for design of the weighting functions. Several projected

trajectories on different meridian planes are shown in Fig. 5. The source trajectory of Eq. (15) is first transformed

from to coordinate systems, as shown in Fig. 3. Therefore, for a given meridian plane at

5

6

),,( zyx )',','( zyx , the

projected trajectory on this plane is described as

7

8

0

))2

(sin(

zhz

SOy . (25)9

10

11

12

13

Dashed lines in Fig. 6(a) represent the planes normal to the meridian plane. The integration result(s) on the plane(s)

corresponding to each line must be equal to the Radon value at . In the Grangeat framework, the derivative

of Radon data is calculated from the detector planes as denoted by the colored lines associated with the intersected

vertex points. Geometrically, there can be maximally three intersection points. For a given , the number of

intersecting points determines the degree of redundancy and depends on

),,(

),(

.14

15

16

17

D.2. TRANSITION POINTS ON THE PROJECTED TRAJECTORY

Every projected trajectory has two end points expressed by

)0)),2

(0sin((),( 01 zSOzye , (26)18

))2()),2

()2sin(((),( 02 zhSOzye mm . (27)19

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For a given on a meridian plane , there may be planes normal to and tangent to the projected trajectory. The

tangential points can be analytically specified by

20

21


))),2

(sin((),( 0111 zhSOzyt tt , (28)1

))),2

(sin((),( 0222 zhSOzyt tt , (29)2

where 1t , 2t are calculated in the following:3

4 A projected trajectory on a meridian plane is determined as

))2

(sin(SOy , (30)5

0zhz . (31)6

To express as a function of z , we obtain7

hzz 0

, (32)8

9 substitute Eq. (32) into Eq. (30) , and have

))2

(sin( 0

hzzSOy . (33)10

11 Then, we compute the derivative and set it to the slope of the colored line:

cot))2

(cos(1 0

hzzSO

hdzdy . (34)12

13 In other words,

01 ))

2()cot((cos z

SOhhz . (35)14

15 Then, we have

))2

()cot((cos 1

SOh

. (36)16

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The solution of this equation is meaningful only when m20 , resulting in up to two solutions: 1t and 1

2t . It is assumed that 2t is greater than 1t if it exists. Recall that these 1t and 2t are used for grouping

vertices in Subsection C.

2

3

In terms of the above end points and tangential points, we can find the transition points as follows:4

11 ee5

22 ee6

11 tt7

22 tt , (37)8

where )cos,(sin),( zy . As shown in Fig. 6(a), we can identify the type of a region according to the

following criterion:

9

2e2t , indicating a doubly sampled region; 12 ee , a singly sampled region;10

1t1e , a doubly sampled region.11

12

13 D.3. SMOOTH WEIGHTING FUNCTIONS

Weighting functions are designed in reference to 1e , 2e , 1t , 2t , which are functions of and .

Mathematically speaking, there are up to two possibilities when we have neither

14

1t nor 2t , which are 2e1e

and

15

21 ee . Similarly, there are up to six cases when we have only either 1t or 2t . Furthermore, there are up to

twenty four cases when we have both

16

1t and 2t . However, all the statements are not meaningful after our case-by-

case inspection. It is found that there exist only eleven cases actually. For example, in absence of

17

1t and 2t ,

we always have the case of

18

21 ee , and never have the case of 2e1e . Similarly, it is impossible to have the

cases of

19

21 et1e and 1e1t2e in absence of 2t . Those eleven cases are: i) 21 ee in absence of20

1t and 2t ; ii) 2e1e1t in absence 2t ; iii) 1t12e e in absence of 2t ; iv) 1e2e1t in 21

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absence of 2t ; v) 121 tee in absence of 2t ; vi) 1122 teet ; vii) 2211 teet ; viii)1

221 ete 1t ; ix) 12 te12 et ; x) 21 et1e2t ; xi) 12 te2t1e .

,

m

1e 2e

2t

1t t

),( 11 2 3

1e 1t15

1t 2e 1

1t 1e

1e 2e 3

1e 1t 1t 2t20

et 2 1 1e 2t

1t 1e

2t 1t 1t 2e

2

3 In addition to our geometric analysis on projected trajectories on the meridian plane, we did exclusive

numerical simulation to confirm that there are only the above eleven meaningful cases. For all the ( ) , we

calculated the transition points, sorted the values, and decided which one of the mathematically possible thirty two

cases it belongs to. Once a specific case was found, we set the flag for that case on. After this kind of numerical

verification, we eliminated the cases that never happened. Also, we repeated the simulation with respect to

representative combinations of imaging parameters, including the source-to-origin distance, helical pitch and cone

angle ( 2 ). Finally, it was confirmed that there are indeed only the above eleven cases that are feasible.

4

5

6

7

8

9

For each of those meaningful cases, smooth weighing functions are designed in terms of , , 1t ,

. The two general requirements are that (1) the sum of the weighs for each characteristic point be one, and (2) the

weight profile along the direction be continuous. To further understand the designing processing, some

representative illustrations are considered helpful. Fig. 6(b) represents the case where there is neither nor 2 for

given , and only group 1 exists. Therefore, , and and are set to zero. Fig. 6(c) corresponds to

the case where there is one tangential point, and two groups are available. The trajectory from to belongs to

group 1, while that from to belongs to group 2. The weighting function for group1, , is designed to change

gradually from 1 to 0 along the projected trajectory from to , while the weighting function for group 2, ,

increases from 0 to 1 in the same interval, and stays constant between

2

and . is set to zero since there is

no group 3. Of course, the sum of the weighting functions should be made one. Fig. 6(d) illustrates the case where there

are two tangential points, and we have group 1 between and , group 2 between and , and group 3

between and e . The weighting function for the group 1, , should be one between to , and smoothly

change from 1 to 0 along the projected trajectory from to . The weighting function for the group 3, 3 ,

should smoothly change from 0 to 1 along the trajectory from to , and be 0 between and . The23

10

11

12

13

14

16

17

18

19

21

22

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weighting function for the group 2, , is designed to smoothly increase from 0 at 2 2t until it reaches the middle

point between

1

2t and 1t , and decrease to 0 at 1t .

2t

2e

)1

1

t2 e

)12 e

1t 2

)

2

3

4

Some representative distributions of the weighting functions are included in Fig. 7. The weighting functions

are formulated as follows, keyed to each of the feasible cases:

i) 21 ee in absence of 1t and :5

6 ,11

,027

.038

ii) 11 et in absence 2t :9

1 (cos1

2 t , 11 et10

0 , 21 ee11

2 (sin1

12

t

t , 11 et12

1 , 21 ee13

3 0 14

iii) 12 ee in absence of t :15

1 0 , 12 ee16

2(sin

11

12

et

e , 11 te17

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2 1 , 21 ee1

)2

(cos11

12

et

e , 12 te2

3 0 3

iv) 121 eet in absence of 2t :4

1 )2

(sin12

12

te

t , 21 et5

1 , 12 ee6

2 )2

(cos12

12

te

t , 21 et7

0 , 12 ee8

3 0 9

v) 121 tee in absence of 2t :10

1 1 , 21 ee11

)2

(cos21

22

et

e , 12 te12

2 0 , 21 ee13

)2

(sin21

22

et

e , 12 te14

3 0 15

vi) 1122 teet16

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1 0 , 22 et1

0 , 12 ee2

)2

(sin11

12

et

e , 11 te3

2 )2

(sin22

22

te

t , 22 et4

1 , 12 ee5

)2

(cos11

12

et

e , 11 te6

3 )2

(cos22

22

te

t , 22 et7

0 , 12 ee8

0 , 11 te9

10

vii) 2211 teet11

1 )2

(cos11

12

te

t , 11 et12

0 , 21 ee13

0 , 22 te14

2 )2

(sin11

12

te

t , 11 et15

1 , 21 ee16

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)2

(cos22

22

et

e , 22 te1

3 0 , 11 et2

0 , 21 ee3

)2

(sin22

22

et

e , 22 te4

viii) 2121 ette5

1 1 , 21 te6

)(cos21

22

tt

t , 2/)( 2122 tttt7

0 , 1212 2/)( tttt8

0 , 21 et9

2 0 , 21 te10

)(sin21

22

tt

t , 2/)( 2122 tttt11

)(sin21

22

tt

t , 1212 2/)( tttt12

0 , 21 et13

3 0 , 21 te14

0 , 2/)( 2122 tttt15

)(cos21

22

tt

t , 1212 2/)( tttt16

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1 , 21 et1

ix) 1212 teet2

1 0 , 12 et3

)2

(sin21

12

12

ee

e , 21 ee4

21

, 12 te5

2 21

, 12 et6

21

, 21 ee7

21

, 12 te8

2 21

, 12 et9

)2

(cos21

12

12

ee

e , 21 ee10

0 , 12 te11

x) 2112 etet12

1 0 , 12 et13

)(sin11

12

et

e , 2/)( 1111 etee14

)(sin11

12

et

e , 1111 2/)( tete15

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0 , 21 et1

2 21

, 12 et2

)(cos21

11

12

et

e , 2/)( 1111 etee3

0 , 1111 2/)( tete4

0 , 21 et5

3 21

, 12 et6

)(cos21

11

12

et

e , 2/)( 1111 etee7

)(cos11

12

et

e , 1111 2/)( tete8

1 , 21 et9

xi) 1221 tete10

1 1 , 21 te11

)(cos22

22

te

t , 2/)( 2222 tett12

)(cos21

22

22

te

t , 2222 2/)( etet13

21

, 12 te14

2 0 , 21 te15

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0 , 2/)( 2222 tett1

)(cos21

22

22

te

t , 2222 2/)( etet2

21

, 12 te3

3 0 , 21 te4

)(sin22

22

te

t , 2/)( 2222 tett5

)(sin22

22

te

t , 2222 2/)( etet6

0 , 12 te7

8

9

10

11

E. INTERPOLATION

As we did in the circular scanning case1,24, in this study we continue using the linear interpolation strategy to estimate

missing data. The boundaries of the shadow zone were numerically determined. Then, the shadow zones were

linearly interpolated along the direction using the measured boundary values. To demonstrate this interpolation idea

graphically, Fig. 8 shows the derivatives of Radon data with no interpolation and with linear interpolation, respectively.

12

13

14

15

16

F. IMPLEMENTATION PROCEDURE

To summarize, the Grangeat-type half-scan algorithm can be implemented in the following steps:

(1) Specify a characteristic point ),,( where the derivative of Radon data can be calculated;17

(2) Calculate 1t and 2t ;18

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(3) Calculate 2121 ,,, eett ;1

(4) Calculate smooth weighting functions ),,(1 , ),,(2 , and ),,(3 ;2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

(5) Determine line integration points for the given characteristic point according to the rebinning Eqs. (18)-(20);

(6) Calculate the derivatives of Radon data using Eqs. (13), and store them according to their group membership as

determined by Eq. (23);

(7) Apply the weighting functions to the derivative of Radon data using Eq. (21);

(8) Repeat Steps (1)-(7) until all the measurable characteristic points are done;

(9) Estimate the Radon data in the shadow zone (using the linear interpolation method in this study);

(10) Use the two stage parallel-beam backprojection algorithm as defined by Eq. (14) to reconstruct an image volume.

Note that results from steps (2)-(4) can be pre-calculated and stored for computational efficiency. Similarly,

the rebinning coefficients from Step (5) can be found beforehand.

III. RESULTS AND DISCUSSIONS

We developed a software simulator in the IDL Language (Research Systems Inc., Boulder, Colorado) for Grangeat-

type image reconstruction. In the implementation of the Grangeat-type formula, the numerical differentiation was

performed with a built-in function based on 3-point Lagrangian interpolation.

The source-to-origin distance was set to 5. The number of detectors per cone-beam projection was 256 by

256. The size of the 2D detector plane was 3.3 by 3.3. The helical pitch was 2. The full-cone angle was about 30

degree. The scan range was from 0 to m2 . The number of projections was 210. The number of meridian planes

was 180. The numbers of radial and angular samples were 256 and 360, respectively. Each reconstructed image

volume had dimensions of 4.1 by 4.1 by 4.1, and contained 256 by 256 by 256 voxels.

19

20

21

22

23

247/3/2003 5:39 PM

One might use the same number of projections above and below each and every slice so that all slices have the

same image quality. Our preferred approach uses an asymmetric number of slices above and below the slice except for

the z=0 slice. If we use the symmetric half-scan helical Feldkamp11, which has symmetric projections for any z


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

location, all slices would have similar image quality. However, it means that each slice is reconstructed in a different

time window. The primary purpose of our work is to develop half-scan algorithms in the Grangeat framework with

superior temporal characteristics (temporal resolution and temporal consistency; the latter requires that a whole volume

of interest is reconstructed with projections in the same time window) and less image artifacts (which is achieved by

appropriate data handling in the shadow zone). The reason that we use the asymmetric projections is to maintain this

temporal consistency. We can also use half-scan helical Feldkamp methods in the same way.

The 3D Shepp-Logan phantom was used in the numerical simulation as shown in Table 1. Fig. 9 shows the

derivatives of Radon data of the Shepp-Logan phantom, the weighting functions for each group, and the combined data.

Fig. 10 presents typical reconstructed slices of the Shepp-Logan phantom. With the helical half-scan Feldkamp method

(Fig. 10(a)) and the helical half-scan Grangeat and zero-padding method (Fig. 10 (b)), the low intensity drop was

serious away from the center plane . However, this type of artifacts was essentially eliminated with the helical half-scan

Grangeat and linear interpolation method (Fig. 10(c)).

IV. DISCUSSIONS AND CONCLUSION

In the writing period of our first draft, it came to our attention that Noo and Heuscher recently published a half-scan

cone-beam reconstruction paper in an SPIE conference 25. The similarity between our work and their paper is that

both the groups considered the half-scan cone-beam reconstruction using the Grangeat method. However, their work

is in the filtered backprojection framework 16, while ours is in the rebinning framework 15. We believe that both half-

scan algorithms are complementary. It seems that data filling mechanism is more flexible in our rebinning framework.

Noo and Heuscher suggested that the parallel-beam approximation of cone-beam projection data be used to estimate

missing data, which is done in the spatial domain. This kind of spatial domain processing is also allowed in our

framework. In addition to the spatial domain approximation, the Radon domain estimation, such as linear interpolation,

spline interpolation and knowledge-based interpolation, can be done in our framework as well. However, in the filtered

backprojection framework, each frame of cone-beam projection data can be processed as soon as it is acquired, a

desirable property for practical implementation. Clearly, a systematic comparison of the two algorithms is worth of

further investigation.

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1

2

3

4

5

67

8

9

10

11

12

13

14

15

The helical scanning geometry studied in this paper is to solve the short object problem, which assumes that

the object is completely covered by the X-ray cone beam from any source position. Even though research with the short

object geometry is valuable on its own, such as for micro-CT imaging of spherical samples, the geometry for data

truncation along the axial direction is the most practical. Therefore, the extension of our Grangeat-type helical half-scan

CT work to the long object case is an important future topic.

In conclusion, we have formulated a Grangeat-type half-scan algorithm in the helical scanning case to solve

the short object problem. The smooth half-scan weighting functions have been designed to compensate for data

redundancy and inconsistence. Numerical simulation results have verified the correctness of our formulation, and

demonstrated the merits of our algorithm. We believe that the Grangeat-type half-scan algorithm is promising for

quantitative and dynamic biomedical applications of CT and micro-CT.

ACKNOWLEDGEMENT

We would like to thank Dr. Dominic J. Heuscher and Dr. Frederic Noo for mailing us their SPIE paper 14.

This work was supported in part by the NIH grant R01 DC03590 and EB001685.

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Lee and Wang: Half-scan Grangeat formula 1

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Lee and Wang: Half-scan Grangeat formula 1

TABLE CAPTIONS

Table 1. Parameters of the phantoms used in our numerical simulation. (a, b, c) denote the semi-axes of ellipsoids, (x, y,

z) the center coordinates of each ellipsoid, and the same as in Fig. 1. The actual density at a position is

determined by summing the densities of the ellipsoids covering that point.

FIGURE CAPTIONS

Figure 1. Grangeat's cone beam geometry. The cone beam projection and the first derivative of Radon transform are

linked.

Figure 2. Meridian and detector planes. (a) Relationship between meridian and detector planes, (b) meridian plane, and

(c) detector plane.

Figure 3. Helical Half-scan geometry.

Figure 4. Classification of the Radon space. (a) Shadow region, (b) singly sampled region, (c) doubly sampled region,

and (d) triply sampled region.

Figure 5. Projected trajectories on meridian planes of (a) 90°, (b) 110°, (c) 130°, (d) 150°, (e) 170°, (f) 190°, (g)

210°, (h) 230°, and (i) 250°.

Figure 6. Transition points for design of the weighting functions on meridian planes. (a) Parameters of the projected

trajectory on a meridian plane, (b) case i, (c) case ii, and (d) case viii.

Figure 7. (a)-(d) Region map and weighting functions for meridian plane at , (e)-(h) , and (i)-(l)

. (a)(e)(i) Region map: the brightest is for triply sampled zones and the darkest shadow zone. (b)(f)(j) The

first weighting distribution, (c)(g)(k) the second weighting distribution, and (d)(h)(l) the third weighting distribution.

90 170

250

Figure 8. First derivative Radon data of the 3D Shepp-Logan phantom after data filling. (a) zero padding, (b) linear

interpolation.

Figure 9. First derivative Radon data of the 3D Shepp-Logan phantom at for (a) group 1, (b) group 2, and (c)

group3. (d)(e)(f) Weighting functions for each group and (g) combined data.

90

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Figure 10. Reconstructed images of the 3D Shepp-Logan phantom. (a) Helical half-scan Feldkamp, (b) Grangeat-type

helical half-scan reconstruction with zero-padding, (b) Grangeat-type helical half-scan reconstruction with linear

interpolation. The same projection range has been used to reconstruct the image volume in the same time window. First

row: vertical slice at y=0.242; second row: vertical slice at x=-0.0369; and third row: transverse slice at z=0.137. The

contrast range is [1.005, 1.05].

7/3/2003 5:39 PM

Table 1.

Phantom a b c x y z θ ϕ Density

0.69 0.9 0.92 0.0 0.0 0.0 0.0 0.0 2.00.6624 0.88 0.874 0.0 0.0 -0.0184 0.0 0.0 -0.98

0.41 0.21 0.16 -0.22 -0.25 0.0 0.0 72.0 -0.020.31 0.22 0.11 0.22 -0.25 0.0 0.0 -72.0 -0.020.21 0.35 0.25 0.0 -0.25 0.35 0.0 0.0 0.01

0.046 0.046 0.046 0.0 -0.25 0.1 0.0 0.0 0.010.046 0.02 0.023 -0.08 -0.25 -0.605 0.0 0.0 0.010.046 0.02 0.023 0.06 -0.25 -0.605 0.0 90.0 0.010.056 0.1 0.04 0.06 0.625 -0.105 0.0 0.0 0.020.056 0.1 0.056 0.0 0.0625 0.1 0.0 0.0 -0.020.046 0.046 0.046 0.0 -0.25 -0.1 0.0 0.0 0.01

Shepp-Logan

0.023 0.023 0.023 0.0 -0.25 -0.605 0.0 0.0 0.01

(b)

(d)

(a)

(e)

(g) (h)

y

z

(c)

(f)

(i)

Figure 5

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