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Fourier rebinning algorithm for inverse geometry CT

Samuel R. Mazina?

Department of Radiology, Stanford University, Stanford, California 94305

Norbert J. Pelc

Department of Radiology, Stanford University, Stanford, California 94305

and Department of Bioengineering, Stanford University, Stanford, California 94305

?Received 1 June 2008; revised 26 August 2008; accepted for publication 27 August 2008;

published 13 October 2008?

Inverse geometry computed tomography ?IGCT? is a new type of volumetric CT geometry that

employs a large array of x-ray sources opposite a smaller detector array. Volumetric coverage and

high isotropic resolution produce very large data sets and therefore require a computationally

efficient three-dimensional reconstruction algorithm. The purpose of this work was to adapt and

evaluate a fast algorithm based on Defrise’s Fourier rebinning ?FORE?, originally developed for

positron emission tomography. The results were compared with the average of FDK reconstructions

from each source row. The FORE algorithm is an order of magnitude faster than the FDK-type

method for the case of 11 source rows. In the center of the field-of-view both algorithms exhibited

the same resolution and noise performance. FORE exhibited some resolution loss ?and less noise?

in the periphery of the field-of-view. FORE appears to be a fast and reasonably accurate recon-

struction method for IGCT. © 2008 American Association of Physicists in Medicine.

?DOI: 10.1118/1.2986155?

Key words: volumetric CT, inverse geometry CT, reconstruction algorithm, Fourier rebinning

I. INTRODUCTION

Inverse geometry computed tomography ?IGCT? is a new

volumetricCT geometry

investigation.1–5Instead of using a single x-ray focal spot to

acquire a projection, IGCT employs a large array of focal

spots opposite a detector array that has the same axial extent

as the source array but is much smaller in the transverse

direction ?Fig. 1?.

It is important to find a reconstruction algorithm for this

geometry that is accurate, fast, and signal-to-noise ratio

?SNR? efficient. Computational efficiency is important due to

the need to process a very large data set that is inherent to

IGCT due to the presence of rays at multiple angles with

respect to the axis-of-rotation.

Previous work focused on a three-dimensional filtered-

backprojection ?3DFBP? approach.2Although 3DFBP is ac-

curate and SNR efficient, it is computationally expensive.

Positron emission tomography ?PET? has a similar sam-

pling distribution to IGCT ?after a rotation?. The 3D re-

projection algorithm6?3DRP? is a PET algorithm that per-

forms a two-pass reconstruction. The 3DRP algorithm is

accurate, however, it is also slow due to a costly reprojection

step.

The purpose of this work is to evaluate a much faster

algorithm based on a Fourier rebinning ?FORE? algorithm

developed by Defrise et al. for PET,7and also applied to

multiple orbit SPECT.8

thatis currentlyunder

II. METHODS

II.A. FORE algorithm description

FORE was designed for a PET geometry and we will

attempt to use the same notation as described in Ref. 7. Each

line-of-response ?LOR? can be characterized by four param-

eters: two in-plane parameters ?s,?? and two longitudinal

parameters ?z,?? ?Fig. 2?. s and ? are the radial distance and

angle, respectively, of the LOR projected onto a transverse

plane. z is the midpoint of the LOR and ? is the tangent of

the angle the LOR makes with a transverse plane.

Any LOR can then be described as the projection

p?s,?,z,?? =?

−?

?

f?s cos ? − l sin ?,s sin ? + l cos ?,z

+ l??dl.

?1?

For each z and ??0, we call the two-dimensional ?2D? data

as a function of s and ? an oblique sinogram. The ?=0

subset data are the in-plane sinograms. One can take a 2D

Fourier transform of p with respect to s and ? to get

P??,k,z,??, where ? and k are the frequency variables as-

sociated with s and ?, respectively. One of the main results

of Ref. 7 is the approximation

P??,k,z,0? ? P??,k,z + ?k/???,??,

?2?

which suggests that one can rebin all of the oblique sino-

grams into in-plane sinograms. After the rebinning process,

the data is in the form of a stack of in-plane 2D sinograms

that can then be reconstructed using any 2D algorithm. The

data set is reduced in size, resulting in a very computation-

ally efficient algorithm. The basic steps of the FORE algo-

rithm are as described in Ref. 7:

?1? Initialize a stack of 2D Fourier-transformed in-plane si-

nograms P??,k,z,0? to zero.

?2? For each ?z,?? sinogram:

48574857 Med. Phys. 35 „11…, November 20080094-2405/2008/35„11…/4857/6/$23.00 © 2008 Am. Assoc. Phys. Med.

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?a?

Take the 2D Fourier transform in s and ? to form

P??,k,z,??.

For each ??,k?, rebin P??,k,z,?? to P??,k,z

−?k/???,0? using interpolation onto the discrete

samples in z.

?b?

?3? Normalize the rebinned stack to account for variable

density of contributions by applying the same procedure

to unit data ?P=1?.

?4? Take the inverse 2D Fourier transform to get the stack of

in-plane sinograms p?s,?,z,0?.

?5? Perform a series of 2D reconstructions, slice by slice,

using any method.

II.B. Adaptation to IGCT

The IGCT sampling geometry is similar to that of PET.

After a 2? rotation, the IGCT projection data can be repre-

sented as a set of in-plane and oblique sinograms. We call a

line of focal spots at the same axial coordinate a source row

and similarly we call a line of detector elements with the

same axial coordinate a detector row. Using the same nota-

tion as in Sec. II A, we can treat each ray measurement from

source to detector like a LOR and thus represent the data set

in the same form p?s,?,z,??. There are five deviations from

the PET case:

?1? In IGCT, if the source and detector arrays are both flat

and parallel to each other, all ray measurements from a

fixed source row to a fixed detector row have the same

?. In PET, the ? values of LORs between two detector

rings depend on s. In Ref. 7, it is assumed that s is

always much smaller than the detector ring radius so that

? is approximately uniform. This approximation does

not need to be made in IGCT.

?2? PET data are sampled uniformly in z. IGCT data sam-

pling in z is nonuniform due to the different sample

spacings within the source and detector arrays in z. Be-

cause Ref. 7 used a gridding scheme for axial interpola-

tion of the rebinned data, no changes for the IGCT sam-

pling scheme were necessary.

?3? If the magnification factor for IGCT is not 2, we need to

redefine z from its original definition as the midpoint of

the LOR. Let zAand zBdenote the endpoints of a ray

measurement from source spot A to detector element B

in IGCT. Let SDD and SID be the IGCT source-to-

detector and source-to-isocenter distances. We redefine

z=??SDD-SID?/SDD?zA+?SID/SDD?zB. This is consis-

tent with the definition for the PET geometry where the

effective magnification is 2 ?i.e., SDD=2SID?.

?4? In Ref. 7, it was noted that PET data are sampled over a

range ???0,??. Therefore, to properly form the Fourier

(a) (b)

(c)(d)

FIG. 3. Reconstructions of the slice at z=−3 mm windowed to 400 HU

around 0. ?a? FDK and ?b? FORE reconstructions of the torso phantom. ?c?

FDK and ?d? FORE reconstructions of the truncated torso phantom. The

boxes in ?a? depict the regions where noise was estimated.

FIG. 4. ?a? FDK and ?b? FORE reconstructions of the Shepp–Logan

phantom.

FIG. 1. Inverse geometry CT.

s

ϕ

A

B

(a)

A

B

θ

z

δ = tanθ

(b)

FIG. 2. ?a? Transverse view and ?b? longitudinal view of a PET system. A

LOR is characterized by four parameters. s and ? are in-plane parameters, z

and ? are longitudinal parameters.

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Medical Physics, Vol. 35, No. 11, November 2008

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transformed oblique sinogram, one has to combine the

data from the two half-sinograms ?z,?? and ?z,−?? using

the symmetry relation p?s,?+?,z,??=p?−s,?,z,−??.

This step is not necessary if one views the data as being

sampled over the full range ???0,2??.

?5? For low frequencies ??????limand ?k??klim?, Ref. 7

used the approximation P??,k,z,0??P??,k,z,??, with

?limand klimset to small nonzero values. We did not

apply this approximation because we did not observe

any benefit from its use ?i.e., we used ?lim=0 and klim

=0?.

The FORE algorithm requires that each projection in a

sinogram be composed of parallel rays that are equally

spaced in s. As in PET, rebinning is required for IGCT data.

A 2D gridding algorithm9was employed to resample the rays

from each source/detector row pair into parallel ray oblique

sinograms with uniform sample spacing in s and ? using a

2D Kaiser–Bessel kernel with parameter ?=9.1375,10fol-

lowed by postdensity compensation using unit data as the

input.

II.C. FDK-like algorithm for comparison

To compare FORE to another reconstruction algorithm

that similarly does not require a reprojection step, we devel-

oped an algorithm based on FDK.11

In IGCT, the data from each source row is very analogous

to the data from a circular source trajectory. We can therefore

treat the entire data set as multiple circular source trajecto-

ries, one for each source row. As in FORE, we first grid the

data from each source row and detector row pair into parallel

rays forming a 2D oblique sinogram. All sinograms corre-

sponding to a single source row are then one-dimensional

?1D? filtered and backprojected into the 3D volume. To ac-

count for a variable number of contributions to each voxel,

unit data were backprojected using the same process and the

result is used to normalize the final reconstruction.

II.D. Computational analysis

Let N and M denote the number of detector rows and

source rows, respectively. Let P denote the number of radial

samples per oblique sinogram, and assume that the number

of views is also P. There are therefore NM oblique sino-

grams, each with P2samples. Further, we assume that the

reconstructed volume will have on the order of N slices and

P2pixels per slice.

The FDK-like algorithm consists of 2NMP 1D Fourier

transforms of length 2P to filter the sinogram data. The fac-

tor of two in 2NMP comes from the need to perform both

forward and inverse transforms. The factor of 2 in 2P comes

from the need to zero pad the data in s prior to filtering. This

is followed by 3D backprojection for all source rows requir-

ing MP operations for each of the NP2pixels. Therefore, the

total computational effort of the FDK-like algorithm is on

the order of 4NMP2log?2P?+NMP3, where we have as-

sumed that a 2P-length FFT requires 2P log?2P? operations.

If we divide by the number of reconstructed pixels, NP2,

then the per-pixel effort is 4M log?2P?+MP. In the

backprojection-limited case where P?M, the FDK-like al-

gorithm requires on the order of MP operations per recon-

structed pixel.

The FORE algorithm requires NM forward and N inverse

2D transforms. After zero padding in s, each sinogram has

2P?P samples. Each 2P?P 2D Fourier transform can be

decomposed into P 1D transforms of length 2P ?s direction?

and 2P 1D transforms of length P ?? direction?. As in the

FDK analysis, the s direction transforms require a total of

2NMP2log?2P? operations for the forward direction and

2NP2log?2P? for the inverse direction. Similarly, the ? di-

rection transforms require an additional 2NMP2log P

+2NP2log P operations. The Fourier rebinning step requires

2NMP2operations ?one for each data point?. Finally, N 2D

reconstructions require NP3backprojection operations. Thus,

FORE requires2N?M+1?P2log?2P?+2N?M+1?P2log P

+2NMP2+NP3operations. If we again divide by the number

ofreconstructedpixels,

NP2,

+1?log?2P?+2?M+1?log P+2M+P operations per pixel. In

the backprojection-limited case where P?M, FORE re-

quires on the order of P operations per pixel.

The FDK-like algorithm therefore requires roughly M

times as many operations per reconstructed pixel as the

FORE algorithm.

FORE requires2?M

TABLE I. Numerical torso phantom simulation parameters.

SDD ?Source-to-detector distance?

SID ?Source-to-isocenter distance?

Detector axial coverage

Source axial coverage

Radial coverage ?s?

View coverage ???

Number of rebinned slices

950 mm

550 mm

1.0 mm?50 rows

5.0 mm?11 rows

1.0 mm?512 samples

2? ?1024 samples?

50

FIG. 5. Profile plots along the central vertical axis of the Shepp–Logan

reconstructions.

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II.E. Simulations

Projection data of a numerical torso phantom ?uniformly

sampled in s and ?, but sampled under the IGCT geometry

in z and ?? were simulated, modeling finite source and de-

tector blurring, under the geometry specified in Table I.

Noisy data were generated using a Poisson model. The

entrance intensity, I0was set to 20 000 photons per ray.

A second simulation was performed with the torso phan-

tom truncated in z at the central slice for a more stringent test

of the 3D performance of the algorithms. We call this phan-

tom the truncated torso phantom.

To test low contrast performance, a 3D Shepp–Logan

phantom was simulated.

The noise in the reconstructed volumes was evaluated by

calculating standard deviations within 32?32 pixel regions

both near and far from the axis-of-rotation across all slices.

Additionally, standard deviations were calculated from dif-

ference images between the FORE and FDK reconstructions.

To compare the resolution performance of the algorithms,

oversampled images of two 200 ?m spheres at 10 and

100 mm away from the axis-of-rotation were simulated.

To evaluate cone-beam effects from high spatial frequen-

cies in z, a numerical “Defrise” phantom was simulated

which consisted of 13 ellipsoids stacked along the z axis. All

ellipsoids had x, y, and z radii of 50, 50, and 1 mm, respec-

tively. Their centers were separated by 4 mm along z.

II.F. Physical experiments

The experimental setup for the table-top experiments is

the same as in Ref. 4. We used a C-arm angiography system

?NovaRay, Palo Alto, CA? that employs a large-area scanned

anode x-ray source opposite a small-area fast photon-

counting detector array.12Data from 100?100 focal spots

and 48?48 CdZnTe detectors were used in the experiment.

An anthropomorphic torso phantom was scanned in a step-

and-shoot mode.

The data for each source row and detector row pair were

regridded in s and ? into parallel data corresponding to si-

nograms as explained in Sec. II B.

Dose measurements were not conducted. However, one

way to gauge the dose level for the experiment is to calculate

the mAs product of a conventional CT scanner that would

result in an equivalent total photon flux through isocenter.

This analysis resulted in an effective mA s of 14.

III. RESULTS

III.A. Simulations

Fifty slices ?each 512?512 pixels? were reconstructed

with the FORE algorithm at a computation time of

13.6 s/slice while FDK ran at 194 s/slice. Figure 3 shows

the reconstructed images of both the noisy torso phantom

and truncated torso phantom at slice location z=−3 mm

FIG. 6. Theoretical and simulated resolution from the FDK and FORE algorithms. ?a? In-plane MTF at 10 mm and ?b? 100 mm away from the z axis. ?c?

Fourier transform along the z axis of the point spread function at 10 and ?d? 100 mm away from the axis.

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?3 mm from the truncation point?, all displayed with a win-

dow width of 400 HU. The truncated torso phantom exhib-

ited cupping artifacts in both the FORE and FDK reconstruc-

tions. The discrete nature of the cupping in the FDK

reconstruction is due to the transition when a source row

suddenly “sees” a pixel. The pixels closer to the axis-of-

rotation are seen by sources that have larger cone angles and

therefore produce more cupping error. Compared to FDK,

the FORE cupping artifact is more gradual, but more severe

at the edges. Figures 4 and 5 depict the Shepp–Logan phan-

tom reconstructions and profiles along the central vertical

axis, respectively, demonstrating similar low contrast perfor-

mance. The axial resolution degradation in FORE ?see be-

low? is evident in the periphery.

Figure 6 shows the spatial resolution at 10 and 100 mm

away from the axis-of-rotation. At locations away from the

axis-of-rotation, FORE exhibits some loss of resolution in

the z direction.

Figure 7 shows the standard deviation as a function of z in

the regions depicted in Fig. 3?a?. Near the axis-of-rotation,

FDK and FORE reconstructions yield similar noise as can be

seen from their standard deviation and also from the low

standard deviation of the difference between the FDK and

FORE images. At 100 mm from the axis-of-rotation FORE

exhibited approximately 10% less noise across all slices.

This may be related to the loss of axial spatial resolution in

this region.

Figure 8 shows FDK and FORE reconstructions of the

numerical Defrise phantom. Although both suffer from cone-

beam effects in the center of the field-of-view, FORE does

not have the long range streaking artifacts that are prominent

in the FDK reconstruction.

FIG. 7. Standard deviation from two 32?32 pixel regions across the volume. ?a? Near the axis-of-rotation. ?b? 100 mm away from the axis.

FIG. 8. FDK ?top? and FORE ?bottom? reconstructions of the numerical

Defrise phantom. In this case the vertical direction is the longitudinal ?z?

direction.

FIG. 9. ?a? A 2D FBP reconstruction of the central slice of the anthropomor-

phic torso phantom using rays from the central source row and detector row

only. ?b? FDK and ?c? FORE reconstructions.

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