Fast Algorithm for X-ray Cone-beam Microtomography

Ang Shih,1 Ge Wang,2 and Ping-Chin Cheng1*

1Advanced Microscopy and Imaging Laboratory, Department of Electrical Engineering, State University of New York at

Buffalo, 315 Bonner Hall, Buffalo, NY 142602Department of Radiology, University of Iowa, Iowa City, IA 52242

Abstract: Cone-beam X-ray microtomography attracts increasing attention due to its applications in biomedi-

cal sciences, material engineering, and industrial nondestructive evaluation. Rapid volumetric image recon-

struction is highly desirable in all these areas for prompt visualization and analysis of complex structures of

interest. In this article, we reformulate a generalized Feldkamp cone-beam image reconstruction algorithm,

utilize curved voxels and mapping tables, improve the reconstruction efficiency by an order of magnitude

relative to a direct implementation of the standard algorithm, and demonstrate the feasibility with numerical

simulation and experiments using a prototype cone-beam X-ray microtomographic system. Our fast algorithm

reconstructs a 256-voxel cube from 100 projections within 2 min on an Intel Pentium IIt 233 MHz personal

computer, produces satisfactory image quality, and can be further accelerated using special hardware and/or

parallel processing techniques.

Key words: X-ray computed tomography, cone-beam, microtomography, image reconstruction, real-time,

nondestructive evaluation

INTRODUCTION

Due to its penetration ability and contrast mechanism,

cone-beam X-ray microtomography is a powerful tool in

studies on 3D microstructures of opaque specimens in bio-

logical, medical, material, and industrial applications (Russ,

1988; Kinney et al., 1989; Cheng et al., 1991; Pan et al., 1997;

Wang et al., 1999). With an X-ray point source and a 2D-

detector array, X-rays intersecting a spherical specimen

form a cone, giving rise to the nomenclature cone-beam.

Compared to parallel-beam or fan-beam approaches, the

cone-beam geometry is beneficial for faster data collection,

higher image resolution, better radiation utilization and

easier hardware implementation (Cheng et al., 1991; Wang

et al., 1999).

Despite impressive progress in exact cone-beam recon-

struction, approximate cone-beam algorithms remain im-

portant, which allow incomplete scanning loci and partial

detection coverage. The Feldkamp cone-beam image recon-

struction algorithm is the most popular approximate cone-

beam algorithm, but it is limited by circular scanning,

spherical specimen reconstruction, and longitudinal image

blurring (Feldkamp et al., 1984; Kak et al., 1988). We de-

veloped a prototype cone-beam X-ray microtomographic

system (Cheng et al., 1991), and generalized the Feldkamp

algorithm to allow flexible scanning loci including helical/

helical-like scanning patterns, and reconstruct spherical,Received January 11, 1999; accepted June 13, 2000.

*Corresponding author

Microsc. Microanal. 7, 13–23, 2001DOI: 10.1007/s100050010053 Microscopy AND

Microanalysis© MICROSCOPY SOCIETY OF AMERICA 2001

rod-shaped, and planar specimens (Wang et al., 1992, 1993,

1999).

A major problem with the cone-beam image recon-

struction has been lack of fast algorithms for real-time or

near real-time performance. The computational complexity

of the standard Feldkamp-type reconstruction requires ex-

pensive computing resources and/or leads to lengthy com-

putational time, which preclude many important applica-

tions, such as on-line inspection of industrial products,

massive screening of certain objects, and dynamic studies of

time-varying specimens/processes.

Substantial efforts have been made to shorten the re-

construction time by algorithmic improvement. A

STRETCH algorithm approximates bi-linearly interpolated

projection data with nearest neighbors that are efficiently

computed with a sufficiently small step (Peters, 1981). An

incremental reconstruction algorithm utilizes beam geom-

etry-based pixels, instead of Cartesian grid points (He et al.,

1993). The beam-oriented pixels are also useful in emission

computed tomography (CT) (Gullberg et al., 1996).

In this article, we report a fast reconstruction algorithm

for generalized Feldkamp image back-projection using a

table mapping strategy on a personal computer. In the next

section, both a prototype cone-beam X-ray microtomo-

graphic system and the generalized Feldkamp algorithm are

described, then the generalized Feldkamp algorithm is re-

formulated using curved voxels and mapping tables, and the

computational complexity analyzed. In the third section, a

speed performance review of our new algorithm is numeri-

cally and experimentally demonstrated using a personal

computer. In the last section, several relevant issues and

further research topics are discussed.

MATERIALS AND METHODS

Cone-beam Microtomographic System

A cone-beam X-ray microtomographic system has been

constructed at the Advanced Microscopy and Imaging

Laboratory (AMIL), State University of New York at Buffalo

(SUNY/Buffalo). The system consists of an X-ray source, a

3D translation and rotation stage, an X-ray scintillation

phosphor screen (built at SUNY/Buffalo), a high resolution

(1288 × 1004) slow-scan cooled charge-coupled device

(CCD) camera (Photometric C200, Tucson, AZ; Kodak

KAF 1400, Rochester, NY), and a personal computer (Mi-

croexpert, Cheektowaga, NY). The X-ray source can be ei-

ther a dental X-ray source (Aztech 65, Boulder, CO) or a

micro-focused electron optical column, depending on the

image resolution requirement. Figures 1 and 2 are a pho-

tograph and a diagram of this cone-beam imaging system,

respectively.

Generalized Feldkamp Algorithm

The generalized Feldkamp cone-beam image reconstruction

algorithm is expressed as follows:

g~x,y,z! =1

2 *0

2pD2

~D − s!2 *−`

`

R~p,z,b!

h~q − p!D

=D2 + p2 + z2dpdb, (1)

where g (x,y,z) is a 3D image, D the distance between the

source and the z-axis, b the source rotation angle relative to

the z axis, R(p,z,b) cone-beam projection data,

5 z =Dz

D − s

q =Dt

D − s

(2)

H t = x cos b − ysin bs = −xsin b − ycos b, (3)

as illustrated in Figure 3.

The generalized Feldkamp reconstruction can be di-

vided into the following three steps:

1. Obtain weighted projection data

R8~p,z,b! =D

=D2 + p2 + z2R~p,z,b!, (4)

2. Filter the weighted data

Q~q,z,b! = R8~q,z,b! * h~q!, (5)

3. Weight and backproject the filtered data

g~x,y,z! = *0

2p D2

2~D − s!2 Q~q,z,b!db. (6)

14 Ang Shih et al.

Our first cone-beam image reconstruction software was di-

rectly based on a discrete version of the above formulas to

test the feasibility and provide the gold standard.

Fast Reconstruction Techniques

The Feldkamp-type cone-beam reconstruction is cast in the

filtered backprojection framework. Specifically, the Feld-

kamp-type reconstruction consists of two processes: (1) fil-

tration, and (2) backprojection. In the filtration process,

horizontal projection profiles are ramp-filtered in the Fou-

rier domain after zero padding. In the backprojection pro-

cess, an image volume is reconstructed by backprojecting

filtered data along X-ray paths with appropriate weights. Let

us consider reconstruction of an N voxel cube from M

projections. Roughly speaking, a standard Feldkamp-type

algorithm requires MN2 weighting operations, 2MN2 +

10MN2 log2 2N filtration operations, and 17MN3 backpro-

jection operations. Typically, more than 80% of the total

reconstruction time is devoted to backprojection. There-

fore, the backprojection is the computational bottleneck of

filtered backprojection-based reconstruction and, in this

subsection, we will focus on acceleration of backprojection

for Feldkamp-type reconstruction.

The table mapping can be very effective for reduction

of computational time. The limiting factor is available ran-

dom memory space. Typically, a current personal computer

has a random memory space of 128 megabytes to 1 gi-

gabytes. As a representative problem, we consider recon-

struction of a 256-voxel cube from 100 cone-beam projec-

tions. Clearly, a direct table mapping in terms of x, y, and z

indexes would require a random memory space of 4 gi-

gabytes in this case (i.e., 2563 for every reconstruct voxel,

times 100 for different directions, and times 3 for separate

tables of x-axis index, y-axis index, and coefficients). How-

ever, this requirement may be significantly reduced if we

reformulate the generalized Feldkamp algorithm in the dis-

crete domain as follows:

g~x,y,z! = (l=1

M

W~s!Q@A~t,s!,A~z,s!,b1#, (7)

Figure 1. Diagram of the X-ray

cone-beam microtomographic

system of the Advanced

Microscopy and Imaging

Laboratory (AMIL), State

University of New York at

Buffalo (SUNY/Buffalo). CCD,

charge-coupled device.

Cone-beam X-ray Microtomography 15

where x,y and z are defined only at finitely many prespeci-

fied locations (a regular 3D Cartesian grid), t and s depend

on x and y,

W~s! =D2Db

2~D − s!2,

(8)

and

A~u,s! =Du

D − s(9)

are 1D and 2D, respectively.

To accumulate the terms W(s)Q[A(t,s),A(z,s),bl] effi-

ciently, we introduce curved voxels that are discretely in-

dexed by (r, u, z), as shown in Figure 4. With curved voxels,

backprojection components W(s)Q[A(t,s),A(z,s),bl] for

Figure 2. Photograph of the

X-ray cone-beam

microtomographic system of

AMIL, SUNY/Buffalo. X-ray

source (A); specimen holder (B);

scintillator (C); relay lens (D);

slow-scan cooled CCD (E);

x-axis translation stage (F);

z-axis translation stage (H); tilt

and yow stage (I).

Figure 3. Generalized Feldkamp cone-beam

reconstruction system and variables.

16 Ang Shih et al.

each source angle bl can be easily accumulated after an

appropriate translation in u. Also, backprojection based on

curved voxels naturally avoids computation for voxels out-

side the reliable reconstruction cylinder, in which voxels are

seen from all the X-ray source positions.

To rapidly compute (t,s) from (r,u), two tables are de-

signed for conversion from polar coordinates (r,u) to Car-

tesian coordinates (t,s), which are denoted as T(r,u) and

S(r,u), independent of z.

As shown in Figure 5, intervals in t and s should be

made small so that quantization errors are insignificant, but

these intervals should not be too small to result in manage-

able sizes of memory space for W(s) and A(u,s), T(r,u) and

S(r,u). For every curved voxel (r,u,z) and each source angle

bl, (t,s) are well approximated from T(r,u) and S(r,u), W(s),

A(t,s), and A(z,s) subsequently retrieved, and W(s)Q

[A(t,s),A(z,s),bl] computed. After backprojection over the

curved voxels is done, the reconstructed image volume in-

dexed by (r,u,z) should be transformed into the conven-

tional format indexed by (x,y,z) for direct use with image

visualization and analysis software. Hence, two more tables,

R(x,y) and U(x,y), are needed to find values of voxels at

(x,y,z) from values at cylindrical coordinates (r,u,z).

A refinement should be made to correct the drawback

of the curved voxels in Figure 4, in which voxels along inner

rings are substantially smaller than those along outer rings.

With loss of generality, let us assume an N-voxel cubic

volume and the unit size of cubic voxels. Suppose there are

K centered rings participating a transaxial slice, as shown in

Figure 6 while K = 4, the thickness of each ring D = N/2K,

the perimeter of the ith ring Pi = 2pD(i − 1/2), i =

1, ? ? ? , K, and we would distribute Vi = ceil(Pi/D) voxels

along the ith ring to produce a roughly squared pixel.

To loop through curved voxels rapidly, we stored Vi as

a function of the ring index i in table V(i). A trade-off must

be made in selection of K. The larger K, the higher recon-

struction accuracy will be achieved, but the larger memory

space and the longer computational time will be involved.

Practically, we may just set K to N/2. In this case, only p/4

voxels will be processed as compared to the Cartesian sys-

tem-based reconstruction. Figure 7 shows a flowchart for

the table mapping reconstruction algorithm.

Computational Complexity

We use a Pentium IIt 233MHz computer under Windows

NT 4.0 and Microsoft Visual C++ 4.2. Table 1 lists the

central processing unit (CPU) time in seconds for each of

Figure 6. Arrangement of curved voxels of similar sizes.

Figure 4. Curved voxels in a cylindrical coordinate system.

Figure 5. Small intervals in t and s for reduction of quantization

errors in the mapping process.

Cone-beam X-ray Microtomography 17

basic data operations, including addition, multiplication,

division, sine, cosine, tangent, and indexing of a 1D array.

The operations time was recorded with the instruction ex-

ecution time (set up in a 224 loops) subtract the empty loop

execution time. Table 2 compares time consumed by the

standard Feldkamp-type reconstruction and our accelerated

implementation for a 256-voxel cube from 100 cone-beam

projections keyed to each of basic operations. Clearly, our

Figure 7. Flowchart for

cone-beam reconstruction using

curved voxels and mapping

tables.

18 Ang Shih et al.

fast reconstruction schemes improve the computational ef-

ficiency by more than 22 times.

RESULTS

Our earlier standard software for generalized Feldkamp re-

construction was used as the reference, and modified to

implement the fast reconstruction techniques described

above. The programs were coded in Microsoft Visual C++

4.2, and tested on an Intel Pentium IIt 233 CPU installed

on a Gigabytet 686LX2 motherboard with 128 megabyte

SDRAM under NTt 4.0. Nearest neighbor interpolation

was applied whenever interpolation was needed. Both nu-

merical simulation and physical experiments were per-

formed to refine reconstruction parameters and verify pro-

grams.

A point X-ray source, point detectors, and circular

scanning were assumed. The source-to-origin distance was

set to 6 cm. The image volume to be reconstructed was

assumed to be a 256-voxel cube. The detector plane was 2.2

by 2.2 cm with 256 by 256 detectors, and for the ease of

computation it was scaled to pass through the z-axis of the

reconstruction coordinate system. Specifically, the detector

plane was so placed that its center is at the origin of the

x–y–z system, its horizontal axis p stays in the x–y plane,

and its vertical axis z was superimposed upon the z-axis. A

3D version of Shepp and Logan’s (1974) head phantom was

used as the testing object. One hundred cone-beam projec-

tions with a 3.6° angular interval were used for the recon-

struction. Cone-beam projection data were synthesized

based on analytically derived formulas.

This phantom was reconstructed using both our direct

implementation of the generalized Feldkamp algorithm and

the table mapping-based software to evaluate effects of re-

construction parameters on image quality. As mentioned in

the proceeding section, over-sampling in t and s would

reduce the reconstruction error without compromising the

reconstruction speed, but this gain is at cost of enlarged

mapping tables. The standard deviation of reconstruction

errors against the over-sampling size were plotted in Figure

8, which helped determine the lengths of t and s intervals for

design of related tables. We also monitored the change in

edge sharpness as a function of table sizes, but would not

include those curves for sake of brevity. In this study, we

used 512 intervals of the same length for both t and s.

Compared to the ideal values of the phantom, the stan-

dard deviation of reconstruction errors with the direct

implementation and the table-mapping improvement was

0.03 and 0.05, respectively. The original image and two

reconstructed counterparts of a representative slice at z =

−0.25 are shown in Figure 9. The intensity profiles of origi-

nal and reconstructed values for the line at (y,z) = (0.2,

−0.25) were plotted in Figure 10. An analysis on a 100 by

100 homogeneous region in the slice at z = 0 indicated that

the image noise standard deviation with the direct imple-

mentation and the table-mapping improvement are 0.0125

and 0.0122, respectively. Visual inspection with different

settings of brightness and contrast of display suggested that

our fast algorithm produced comparable image quality rela-

tive to our earlier standard software.

As a real example, Figure 11 includes two rendered

views of a small electrolytic capacitor, which was scanned

with our prototype microtomographic system described in

the preceding section, and reconstructed using our table-

mapping-based software. The specimen-to-source distance

was 8.13 cm. The specimen-to-detector distance was 3.81

cm. Hundred cone-beam projections were taken. Each pro-

jection was exposed and integrated for 0.5 sec. The recon-

struction time was 2 min, much shorter than 20 min taken

by our earlier software that directly implemented the gen-

eralized Feldkamp formula. In comparison to the incre-

mental algorithm (He et al., 1993), our version of incre-

mental back-projection implementation has a reconstruc-

tion time of 5 min.

DISCUSSION AND CONCLUSIONS

The system setup we described in this article was for testing

the feasibility of our table-mapping-based generalized Feld-

kamp reconstruction. The next step is to use a micro-

focused X-ray source and a higher density CCD detector

array to obtain microtomographic image volumes in near

real-time. We are actively working along this direction.

Table 1. Central Processing Unit (CPU) Time (in seconds) for

Basic Data Processing Operations, Each of Which Was Executed

224 Times in a Loop

+ × / Sin Cos Index

Short

integer 0.778 1.798 2.937 NA NA 1.313

Float 2.564 2.314 7.272 37.067 42.203 0.904

Double 3.447 4.713 4.480 NA NA 1.059

NA, not available.

Cone-beam X-ray Microtomography 19

Table 2. Relative Time Consumed by the Direct Implementation of the Generalized Feldkamp Reconstruction and the Accelerated

Implementation for a 256-Voxel Cube from 100 Cone-beam Projections, Keyed to Each of Basic Operations

Basic operation Direct implementation Mapping implementation

Operation Relative time per instruction Instruction no. Relative time Instruction no. Relative time

+ 2.564 4 10.256 1 2.564

× 2.314 9 20.826 1 2.314

/ 7.272 3 21.816 0 0

Sin 37.067 2 74.134 0 0

Cos 42.203 2 84.406 0 0

Index 0.904 1 0.904 5 4.520

Total relative time 209.778 9.398

Figure 8. Plots of the standard deviation

of reconstruction errors against the

over-sampling size.

Figure 9. Comparison of

reconstructed image quality for

slice z = −0.25. a: The image

reconstructed using our direct

implementation of the

generalized Feldkamp

reconstruction; b: the

counterpart using our

table-mapping algorithm.

20 Ang Shih et al.

Further refinements of our table-mapping and curved

voxel techniques are possible. For example, utilization of

mapping tables can effectively shorten the backprojection

process. However, a paradox is that large mapping tables are

not practical in many cases, while small mapping tables will

not yield satisfactory image quality. A balance must be ap-

plication-dependent. The general relationships between im-

age quality indexes and table design parameters may be

analytically quantified (Kriete, 1994).

Although our emphasis was put on backprojection,

similar fast processing techniques can be applied to improve

the speed of data preprocessing and filtration. Some rela-

tively time-consuming parts could be even implemented in

assembly language for minimum computational overhead.

Also, all the software work could surely be done by specially

designed hardware. Furthermore, Feldkamp-type recon-

struction is highly parallel in its computational structure.

With rapid development of computing technologies, real-

time cone-beam tomography should be a reality in near

future.

Our fast reconstruction techniques may be adapted for

iterative cone-beam reconstruction as well (Wang et al.,

1996, 1999). Currently available noniterative cone-beam al-

gorithms require that projections should not be truncated

along at least one direction. Therefore, satisfactory cone-

beam reconstruction with these algorithms is impossible in

Figure 10. Examination of a

representative profile at y = 0.2,

z = −0.25. a: The original

profile and the reconstructed

profile using our direct

implementation of the

generalized Feldkamp

reconstruction; b: the

counterparts using our

table-mapping algorithm.

Examination of a representative

profile at y = −0.6, z = −0.25.

c: The original profile and the

reconstructed profile using our

direct implementation of the

generalized Feldkamp

reconstruction; d: the

counterparts using our

table-mapping algorithm.

Cone-beam X-ray Microtomography 21

cases where objects contain X-ray opaque components and/

or are larger than the cone-beam aperture defined by effec-

tive detection area and X-ray source position. Recently, it

was demonstrated that both EM- and ART-type algorithms

are effective for metal artifact reduction and local region

reconstruction. These iterative algorithms share a fully par-

allel reprojection and backprojection format, and can be

efficiently implemented using the techniques we developed

in this article.

The fast table-mapping reconstruction algorithm has

simplified the back-projection process to noncalculation in-

tensive steps. Therefore, the reconstruction procedure

would not require high-speed floating point calculation

anymore; in fact, in our experiment, even though our work-

station has three times the GFLOP performance. Our gen-

eralized personal computer outperforms our 10 times ex-

pensive workstation. Furthermore, the noncalculation in-

tensive algorithm is a parallel process and pipe-line

architecture friendly algorithm for load distribution imple-

mentation. On the other hand, the table-mapping algo-

rithm has less advantage on an high-performance worksta-

tion, due to the algorithm not depending on fast processors.

In conclusion, we have significantly accelerated the

generalized Feldkamp algorithm using curved voxels and

mapping tables, and improved the reconstruction efficiency

by an order of magnitude relative to a direct implementa-

tion, in comparison with an incremental implementation of

the standard algorithm which is, to our best knowledge, the

fastest method (He et al., 1993). The table-mapping algo-

rithm improved the reconstruction efficient more than a

factor of two. The table-mapping algorithm accelerates the

generalized computer to provide a low-cost solution for

computer tomography reconstruction. Using our tech-

niques, a 256-voxel cube from 100 projections of 256 by 256

detectors can be reconstructed within 2 min on an Intel

Pentium IIt 233MHz personal computer. With our

method, computationally intensive backprojection has been

turned into simple mapping, which allows even faster hard-

ware implementation. With ever-improving computing re-

sources, we believe that real-time or near real-time cone-

beam tomography will play a substantial role in various

applications.

ACKNOWLEDGMENTS

The authors thank Mr. W. Schulze for excellent machining

work, Mr. C. Shen for technical assistance, and Dr. Carl

Crawford for helpful discussion. This project was supported

in part by the Academic Development Fund of the State

University of New York.

REFERENCES

Cheng PC, Lin TH, Shinozaki DM, Newberry SP (1991) Projection

microscopy and microtomography using X-rays. J Scanning Mi-

crosc 13(I):10–11

Feldkamp LA, Davis LC, Kress JW (1984) Practical cone-beam

algorithm. J Opt Soc Am 1:612–619

Gullberg GT, Hsieh YL, Zeng LG (1996) An SVD reconstruction

algorithm using a natural pixel representation of the attenuated

Radon transform. IEEE Trans Nucl Sci 43:295–303

He YJ, Chi A, Sun JA (1993) Increment backprojection algorithm:

modification of the searching flow scheme and utilization of the

Figure 11. a,b: Two rendered views of

a small electrolytic capacitor, which

was scanned with our prototype

microtomographic system and

reconstructed using our table-mapping

algorithm.

22 Ang Shih et al.

relationship among projection views. IEEE Trans Med Imag 12:

555–559

Kak AC, Slaney M (1988) Principle of computerized tomographic

imaging. New York: IEEE Press, pp 99

Kinney JH, Johnson QC, Nichols MC, Bonse U, Saroyan RA,

Nusshardt R, Pahl R (1989) X-ray microtomography on beamline

X at SSRL. Rev Sci Instrum 60:2471–2474

Kriete A (1994) Image quality considerations in computerized 2-D

and 3-D microscopy. In: Multidimensional Microscopy, Cheng PC,

Lin TH, Wu WL, Wu JL (eds). New York: Springer-Verlag, pp

209–230

Pan SJ, Shih A, Liou WS, Park MS, Wang G, Bohne B, Newberry

SP, Kim H, Cheng PC (1997) Cone-beam X-ray microtomography

of human inner ear. J Anal Morphol 4:216–217

Peters TM (1981) Algorithms for fast back- and re-projection in

computed tomography. IEEE Trans Nucl Sci NS28:3641–3647

Russ JC (1988) Differential absorption three-dimensional mi-

crotomography. Trans Am Nucl Soc 56:14

Shepp LA, Logan BF (1974) The Fourier reconstruction of a head

section. IEEE Trans Nucl Sci NS21:21–43

Wang G, Lin TH, Cheng PC, Shinozaki DM (1992) Cone-beam

X-ray reconstruction of plate-like specimens. J Scanning Microsc

14:350–354

Wang G, Lin TH, Cheng PC, Shinozaki DM (1993) A general

cone-beam reconstruction algorithm. IEEE Trans Med Imag 12:

486–496

Wang G, Snyder DL, O’Sullivan JA, Vannier MW (1996) Iterative

deblurring for metal artifact reduction. IEEE Trans Med Imag 15:

657–664

Wang G, Zhao SY, Cheng PC (1999) Exact and approximate cone-

beam X-ray microtomography. In: Focus on Multidimensional Mi-

croscopy, Vol. 1, Cheng PC, Huang PP, Wu JL, Wang G, Kim HG

(eds). Singapore: World Scientific, pp 233–261

Cone-beam X-ray Microtomography 23