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A three-dimensional reconstruction algorithm for an inverse-geometry

volumetric CT system

Taly Gilat Schmidt,a?Rebecca Fahrig, and Norbert J. Pelcb?

Department of Radiology, Stanford University, Stanford, California, 94305

?Received 11 December 2004; revised 14 August 2005; accepted for publication 19 August 2005;

published 14 October 2005?

An inverse-geometry volumetric computed tomography ?IGCT? system has been proposed capable

of rapidly acquiring sufficient data to reconstruct a thick volume in one circular scan. The system

uses a large-area scanned source opposite a smaller detector. The source and detector have the same

extent in the axial, or slice, direction, thus providing sufficient volumetric sampling and avoiding

cone-beam artifacts. This paper describes a reconstruction algorithm for the IGCT system. The

algorithm first rebins the acquired data into two-dimensional ?2D? parallel-ray projections at mul-

tiple tilt and azimuthal angles, followed by a 3D filtered backprojection. The rebinning step is

performed by gridding the data onto a Cartesian grid in a 4D projection space. We present a new

method for correcting the gridding error caused by the finite and asymmetric sampling in the

neighborhood of each output grid point in the projection space. The reconstruction algorithm was

implemented and tested on simulated IGCT data. Results show that the gridding correction reduces

the gridding errors to below one Hounsfield unit. With this correction, the reconstruction algorithm

does not introduce significant artifacts or blurring when compared to images reconstructed from

simulated 2D parallel-ray projections. We also present an investigation of the noise behavior of the

method which verifies that the proposed reconstruction algorithm utilizes cross-plane rays as effi-

ciently as in-plane rays and can provide noise comparable to an in-plane parallel-ray geometry for

the same number of photons. Simulations of a resolution test pattern and the modulation transfer

function demonstrate that the IGCT system, using the proposed algorithm, is capable of 0.4 mm

isotropic resolution. The successful implementation of the reconstruction algorithm is an important

step in establishing feasibility of the IGCT system. © 2005 American Association of Physicists in

Medicine. ?DOI: 10.1118/1.2064827?

I. INTRODUCTION

Conventional computed tomography ?CT? systems are rap-

idly evolving to acquire increasingly thicker volumes per

circular rotation using multirow detectors or flat panel digital

detector technology. These volume CT approaches provide

several advantages over single slice acquisition, including

faster scan times, thinner slices, and reduced motion arti-

facts. The ability to scan an entire organ in one rotation could

have important clinical impact, for example, in perfusion

studies and other dynamic applications.

The increased volume thickness comes at the expense of

larger cone-beam angles. Because of the diverging x-ray

beam in the axial, or slice, direction, a circular scan cone-

beam acquisition does not acquire sufficient volumetric

data.1Although approximate reconstruction algorithms are

commonly used,2the resulting artifacts can be significant for

large cone-angles. While exact reconstruction is possible for

helical cone-beam scanning for certain pitch values,3–6this

paper focuses on sufficient volumetric acquisition in one cir-

cular scan.

We have previously proposed a volumetric CT system that

can sufficiently sample a thick ?on the order of several cen-

timeters? volume in one fast circular scan.7This inverse-

geometry volumetric CT system ?IGCT? uses a large-area

scanned source and an area detector with a smaller extent in

the transverse direction. The sampling is fanlike in the trans-

verse direction, and in the axial direction the source and

detector have the same extent, providing sufficient volumet-

ric coverage and avoiding cone-beam artifacts. In addition,

the smaller detector area may provide significant advantages

over conventional cone-beam systems with respect to cost

and detected scatter radiation.

Previous work studied the feasibility of the IGCT system

with respect to sampling and photon flux and found it pos-

sible to sample a 30-cm wide field of view ?FOV? with

15-cm volume thickness in less than half of a second.7In

fact, the source scanning is sufficiently fast so that the scan

time is limited by gantry speed rather than sampling. Another

important feasibility question is whether the acquired IGCT

data can be reconstructed accurately ?from an artifact per-

spective? and efficiently ?from a noise perspective?. The pur-

pose of this paper is to present a reconstruction algorithm for

the IGCT system.

The data acquired by the IGCT geometry are very similar

to that from a multiring positron emission tomography ?PET?

geometry. Therefore a PET reconstruction algorithm can be

used. As in a three-dimensional ?3D? PET system, the IGCT

data consists of in-plane rays which connect each source row

to the opposed detector row, and cross-plane rays which con-

nect each source row to other detector rows. It is the in-plane

rays that ensure a sufficient dataset for accurate volumetric

reconstruction, while the cross-plane rays improve the

signal-to-noise ratio ?SNR?.

32343234Med. Phys. 32 „11…, November 20050094-2405/2005/32„11…/3234/12/$22.50© 2005 Am. Assoc. Phys. Med.

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Numerous algorithms have been proposed for 3D PET.

One class of algorithms uses 3D filtered backprojection.8–10

The data are rebinned into 2D parallel-ray projections at

multiple tilt and view angles, and the central slice theorem is

used to derive appropriate filters in frequency space. The

filtered projections are then backprojected into the volume.

The IGCT reconstruction algorithm proposed in this paper

follows this 3D filtered backprojection approach. Although

this type of algorithm has been thoroughly studied for PET

imaging, the application to a CT system merits additional

research. CT produces images of higher spatial resolution

and lower noise than PET and therefore demands more ac-

curate reconstruction. Further, the process by which IGCT

data are converted for use by this type of algorithm has not

been explored.

The paper begins with a brief description of the IGCT

system, followed by an overview of the theoretical founda-

tion of the reconstruction algorithm. The key difference be-

tween the IGCT and 3D PET geometries is the ray sampling,

which is accounted for during rebinning. Once the data are

organized into 2D parallel-ray projections, the geometry is

equivalent to that of 3D PET after rebinning and the already

established filters can be used. Therefore, we focus much of

our investigation on the rebinning algorithm and only briefly

review the filter design. Gridding is used to rebin the data.

We show that errors can arise due to the location of acquired

data samples relative to the output grid point, and we present

a new method for reducing this gridding error. The paper

then investigates the image artifact, resolution, and noise per-

formance of the algorithm through simulations. Finally, al-

ternative reconstruction methods are briefly discussed.

II. SYSTEM DESCRIPTION

The basic system geometry is illustrated in Fig. 1. The

IGCT system consists of a large-area scanned x-ray source

mounted on a CT gantry opposite a smaller array of fast

photon-counting detectors. During an acquisition, the elec-

tron beam is electromagnetically steered over a transmission

target, dwelling behind each of an array of collimator holes

which limit the resulting x rays to those that illuminate the

detector area. For each source position, the entire detector

array is read out, creating a 2D divergent projection of a

fraction of the field of view. The scanning of the source

positions is fast relative to the gantry rotation.

III. RECONSTRUCTION ALGORITHM

A. Rebinning

The goal of the rebinning algorithm is to estimate, from

the rays in the IGCT geometry, a full set of 2D parallel-ray

projections. The parallel-ray geometry is illustrated in Fig. 2.

We define the axis of rotation to be along the z axis, and axial

planes to be perpendicular to the axis of rotation. We assume

that a parallel-ray projection is formed by the set of rays

normal to a virtual planar detector. The rotation of the pro-

jection about the axis of rotation ?i.e., view angle?, is defined

as ?, while the rotation from the axis of rotation ?i.e., colati-

tude or tilt angle? is defined as ?. Parameters u and ? repre-

sent the local coordinates within each projection ?i.e., where

a ray falls on the detector?. For all projections, the u axis lies

within an axial plane.

These four parameters, ?, ?, u, and ?, can be calculated

for each ray in the IGCT geometry. We define ? to be the

azimuthal angle of a ray, ?i.e., the angle about the z axis in

the absence of gantry rotation?. The parameters are illus-

trated in the context of the IGCT geometry in Fig. 3. A ray

with ? equal to zero and ? equal to ?/2 is parallel to the x

axis, and a ray with ? equal to zero is parallel to the z axis.

FIG. 1. Proposed IGCT geometry shown with the x-ray beam at one position

in the source array.

FIG. 2. 2D parallel-ray geometry to which the IGCT data is rebinned is

illustrated using a virtual detector. ? is the projection view angle, ? is the

colatitude angle, and u and ? are the coordinates within the projection. For

comparison, two virtual detectors are shown, one with ? equal to ?/2 and

one with a smaller value of ?.

FIG. 3. Four geometry parameters, ?, ?, u, and ?, shown for a ray in the

IGCT geometry where ? is the azimuthal angle.

3235 Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3235

Medical Physics, Vol. 32, No. 11, November 2005

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The parameters depend on the 3D locations of the source

and detector element that define the ray and can be calculated

using the following equations. The coordinates ?sx,sy,sz? de-

fine the location of the source spot before gantry rotation,

where −sxis the source-to-isocenter distance ?SID?. Simi-

larly, each detector has coordinates ?dx,dy,dz? before gantry

rotation, where dxis the detector-to-isocenter distance ?DID?.

Parameters ?, ?, u, and ? are independent of the gantry ro-

tation and are calculated using the coordinates of the unro-

tated source and detector. Parameters ? and u can be calcu-

lated by considering the projection of the ray onto the x-y

plane.

? = arctan?sy− dy

dx− sx?,

?1?

u = dy· cos??? + dx· sin???.

?2?

The total view angle ? depends both on ? and the gantry

rotation angle ?gantry.

? = ? + ?gantry.

?3?

The parameters ? and ? can be calculated by considering the

plane defined by the ray and the source column from which

the ray originates.

2− arctan?

? =?

sz− dz

??sx− dx?2+ ?sy− dy?2?,

?4?

? = dz· sin??? + ?dx· cos??? − dy· sin????cos???.

?5?

In this formulation, the distance of the ray to isocenter is

parametrized by the two perpendicular components u and ?,

which are equivalent to the parallel-ray detector coordinates

shown in Fig. 2.

The four parameters, ?, ?, u, and ?, are sufficient for

reorganizing the IGCT data into 2D parallel-ray projections.

However, for a discrete implementation with regularly

sampled output 2D projections that are equally spaced in the

two angles, some form of interpolation must be used.

In order to better understand the rebinning algorithm, it is

helpful to visualize the data in projection space. For a 2D

reconstruction from 1D projections, such as those acquired

by conventional single slice CT systems, each ray is de-

scribed by two parameters, the rotation angle ? and the ra-

dial distance to isocenter ?. For these single slice CT sys-

tems, projection space is two dimensional with coordinate

axes ? and ?. Each ray in a 1D projection samples one point

in the two-dimensional projection space, and a 1D parallel-

ray projection, comprised of data at one ? value and a range

of ? values spanning the field of view, samples a horizontal

line in projection space.

In the IGCT geometry, each ray is described by two

angles and two distances and is represented in a 4D projec-

tion space. Each ray samples one point in the 4D projection

space, but the sample points from all acquired rays are not

uniformly distributed. Rebinning the data to 2D parallel-ray

projections is equivalent to interpolating the nonuniform

samples onto a 4D Cartesian grid in projection space. The

problem of resampling nonuniform data onto a uniform grid

arises in many different fields and has been the subject of

much work. We are using a gridding approach11in which

each acquired data point contributes to all output grid points

within some neighborhood. In this implementation, a bin

width is selected for each of the four projection space param-

eters, defining the 4D neighborhood of measured data points

used to estimate each grid point. Each data point in this bin is

weighted based on its 4D location with respect to the grid

point and a chosen 4D kernel shape. The interpolated value

at the grid point is the sum of the weighted data points,

normalized by the sum of weights for that point.

The important design parameters for the rebinning algo-

rithm are the bin widths, kernel shape, and output grid sam-

pling density. For application in magnetic resonance ?MR?

reconstruction, the effect of each of these parameters on the

gridded data has been described in detail.12Although most

medical imaging applications, including MR, apply gridding

in frequency space, the analysis in Ref. 12 is based on gen-

eral signal processing theory and is relevant for other appli-

cations. When gridding in projection space, special care must

be taken to properly combine rays that are physically close

yet separated in angle. For example, rays near ?=2? must

be considered when gridding data at ?=0.

B. Rebinning error correction

One important step in the gridding algorithm is compen-

sation for the nonuniform and/or asymmetric location of the

acquired data points. That is, the estimated grid point value

should not be biased by the number or the distribution of

measured data points used in the estimation. Errors can occur

if the sampling is not accounted for properly.

The simplest method for performing this correction is

post-compensation, where the value at the output grid point

is normalized by the total sum of the deposited weights. Af-

ter this normalization, and considering gridding of a 1D

function f?x?, the gridded value at a point xois

fˆ?xo? =?

i

kif?xi?,

?6?

where f?xi? is the ith input sample and kiis the normalized

kernel value for that sample. This method corrects for the

number of data points that contribute to a grid point and

gives an unbiased estimate if the data are locally constant.

That is, if f?xi?=f?xo? for all i, Eq. ?6? gives the correct

answer since the sum of the kiis one. However, consider the

particular but relatively simple case where the input function

is linear with slope G.

f?x? = f?xo? + G?x − xo?.

?7?

Straight-forward gridding yields

fˆ?xo? =?

i

ki?f?xo? + G?xi− xo??,

?8?

which reduces to

3236Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3236

Medical Physics, Vol. 32, No. 11, November 2005

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fˆ?xo? = f?xo? + G?

i

ki?xi− xo?.

?9?

Since the desired value is f?xo?, the second term on the right-

hand side of Eq. ?9? is the gridding error ?.

? = G?

i

ki?xi− xo?.

?10?

If the kernel is even and the samples are symmetric about xo,

the error is zero. In general, though, there is an error propor-

tional to the slope of the input function. In our implementa-

tion, we are gridding the projection measurement data.

Therefore, it is the gradient of the projection of the object

that determines the amount of error in the gridded value.

In addition, we have found that the error caused by the

linear term and the asymmetric sampling can be coherent in

adjacent gridded projection angles, causing an artifact to ac-

cumulate in the image. This can be understood by consider-

ing the distribution of data points about a particular grid

point. If the data points are asymmetrically distributed in the

radial direction, the interpolated value at the grid point will

be biased in the direction with more samples. For example, if

the projection measurements are higher on the side with

more samples, the gridding output may overestimate the cor-

rect value. The asymmetric sampling will likely bias a grid

point at a nearby radial location in the opposite direction

?note that the gain of the gridding process is unity?. In our

system, each view samples data from a range of azimuthal

and radial positions. The radial sampling varies slowly with

azimuthal angle within each IGCT view, and repeats for each

gantry position. Since the overall trends of projections also

vary slowly with view angle, rebinned projections at nearby

azimuthal angles will contain similar errors. In other words,

the gridding error will vary rapidly in the radial direction and

slowly in the azimuthal direction, which is the type of error

to which CT is particularly sensitive.

A more sophisticated gridding approach preweights the

data by the inverse sampling density of the measurements.

That is, data from highly sampled regions are deemphasized

while data from sparsely sampled regions are emphasized by

the preweighting factors. For certain sampling patterns, such

as spiral sampling in MR, these density weights can be cal-

culated analytically.13Several other approaches, including

computational and iterative methods, have been proposed to

determine the weights for arbitrary sampling patterns.14–16

While preweighting should reduce errors, we note that Eq.

?10? predicts residual errors even with uniform sampling

density.

The uniform resampling algorithm ?URS?, which is opti-

mal in the minimum norm least square sense, and the block

uniform resampling algorithm ?BURS?, a computationally

feasible locally optimal gridding algorithm, have also been

proposed.17These algorithms indirectly incorporate the sam-

pling pattern when estimating the grid points by formulating

the gridding problem as a linear set of equations and using

least-squares methods to solve for the values at the grid

points. These methods are sometimes ill-conditioned and

may be sensitive to noise or measurement errors. A regular-

ized version has also been proposed which provides stability

at the expense of accuracy.18

Most of the methods listed above were developed for

gridding in frequency space and are largely applied to MR

imaging. Gridding in projection space has slightly different

challenges.19Due to the ramp filter in CT reconstruction,

errors that are high in frequency in the radial direction are

greatly amplified. Also, the dynamic range ?the range of re-

constructed values divided by the noise level? of CT de-

mands a higher signal-to-artifact level compared to MR or

PET. For example, CT is sensitive to errors on the order of a

few Hounsfield units ?HU?, where one HU is a change in

signal that is one tenth of one percent of the attenuation of

water, while the range of values may be 400% of the density

of water.

Therefore, we propose a new gridding correction that is

motivated by Eq. ?10?. We note that if the sum in the error

term was zero, the grid point value would be correct ?for this

case? regardless of the slope. We modify each kernel value ki

by an amount which depends on the distance between the

data point and grid point. We define the new kernel values,

k_newias

k _ newi= ki+ ??xi− xo?,

?11?

and solve for the value of ? such that the sum in Eq. ?10?,

and therefore the error, equals zero.

0 =?

i

?ki+ ??xi− xo???xi− xo??12?

? =

−?iki?xi− xo?

?i?xi− xo?2

.

?13?

By using the kernel values defined in Eqs. ?11? and ?13?,

the zero and first-order terms of the projection data are esti-

mated correctly at the grid points. This local kernel correc-

tion strategy can be generalized to ensure that higher-order

terms are correctly estimated, but since we only use the data

in a small neighborhood about each grid point, the higher-

order terms should be small. In addition, the higher-order

terms are less likely to be similar in neighboring projections

and should not lead to the coherent errors.

Although the proposed correction does not explicitly

compute the measurement sampling density, the modified

kernel values in Eq. ?11? can be thought of as compensating

for this as well as resymmetrizing the kernel based on the

distribution of data points. A post-compensation step to en-

sure that the total sum of weights at each grid point is one is

still required. The gridding correction can produce negative

kernel values which may cause the sum of the kernel values

at the grid point to be very small. This occurs when the

measured data points are clustered close together on one side

of the grid point. When the kernel value sum is very small,

the post-compensation step amplifies the contribution of

some data points and the noise. Therefore, a threshold is set

3237 Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3237

Medical Physics, Vol. 32, No. 11, November 2005

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on the sum of the corrected kernel values. If the sum is

below the threshold, the original kernel values are used.

This method can be easily extended to multiple dimen-

sions. In the case of 2D gridding, the locally linear function

is

f?x,y? = f?xo,yo? +?f

?xf?x − xo? +?f

?yf?y − yo?.

?14?

The grid point value at ?xo,yo? estimated from data points at

?xi,yi? is

fˆ?xo,yo? =?

i

kif?xi,yi?,

?15?

and the adjusted kernel values are defined by

k _ newi= ki+ ?x?xi− xo? + ?y?yi− yo?,

?16?

where ?xand ?yare determined by solving the following

equations:

?x?

ii

= −?ki?xi− xo?

??y?

ii

= −?ki?yi− yo?.

The solution in Eq. ?17? is not well-defined when the

system of equations is ill-conditioned. This could be the case

in sparsely sampled regions where there is an insufficient

distribution of data points surrounding a grid point. This will

cause the calculated ? values to be very large, which may

lead to unstable performance. A threshold on the allowed

size of ? can be set, and for grid points for which this thresh-

old is exceeded, either the original kernel values can be used,

or the region size used to estimate the grid point can be

expanded.

For our geometry, the gridding correction is applied in

four dimensions, which requires solving a system of four

equations to ensure that the linear term is correctly esti-

mated.

?xi− xo?2+ ?y?

?xi− xo??yi− yo?

?yi− yo?2+ ?x?

?xi− xo??yi− yo?

?17?

C. Filtered backprojection

Once the data are organized into 2D parallel-ray projec-

tions, the central slice theorem can be used to design the

appropriate reconstruction filter. The theorem states that a 2D

parallel-ray projection of a 3D object samples the 3D Fourier

transform of the object along the plane that is perpendicular

to the projection direction and that passes through the origin.

Therefore the ensemble of parallel-ray projections sample

the Fourier transform of the object, with some areas of fre-

quency space sampled more than others.

The role of the reconstruction filters is to weight the fre-

quency content of each projection so that, when they are all

superimposed during backprojection, the 3D Fourier trans-

form of the object is properly reconstructed. One solution is

to define the filter applied to each projection to be the inverse

of the density of measurements in frequency space on the

plane sampled by that projection.

An analytical expression for this filter, known as the

“Colsher” filter, has been previously derived8,10and is stated

without proof below. The derivation assumes 2D parallel-ray

projections continuously and uniformly distributed between

? equal to zero and 2? and colatitude angle between ?min

and ?/2, where ?minis the colatitude angle of the most ob-

lique projection. These assumptions are reasonable if the dis-

tance between adjacent projections is small in both angular

directions. The density of measurements, stated without

proof, is

M arcsin?cos????

?k cos??min?

D??k,?? =

sin????

,

?18?

k =?ku

? = arcos?k?sin ?

?? = max??min,?

2+ k?

2,

?19?

k?,

2− ??,

?20?

?21?

where kuand k?are the coordinates of the 2D Fourier trans-

form of the projection and M is the total number of projec-

tions.

The 2D filter for a parallel-ray projection at a colatitude

angle ? is then given by

G??ku,k?? =

W?k?

D??k,??,

?22?

where W?k? is a window function used to control the impulse

response. Substituting the expression for D?the resulting 2D

filter is

G??ku,k?? =

?k cos??min?

M arcsin?cos????

sin????

W?k?.

?23?

As can be seen in Eq. ?23?, the filter depends on the colati-

tude angle ? but is the same for all view angles at that ?.

The window function W?k? can be designed to recover

some of the resolution lost during the rebinning step. The

gridding algorithm convolves the input data with a 4D kernel

causing some apodization in frequency space. During the

filtering step, the Fourier transform is performed in two spa-

tial dimensions, u and ?. Therefore, in these two dimensions,

the blurring due to gridding can be undone by incorporating

into the filter window the inverse of the Fourier transform of

the gridding kernel. The blurring in the two angular dimen-

sions cannot be reduced during the normal filtering step but

could be deapodized in a separate step prior to backprojec-

tion.

The filter in Eq. ?23? is defined as a continuous function in

frequency space. Implementing the filter discretely can intro-

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Medical Physics, Vol. 32, No. 11, November 2005

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duce low frequency artifacts in the reconstructed image due

to aliasing of the filter in image space.20To reduce these

artifacts, the discrete filters are first oversampled in fre-

quency space, windowed in image space, and then trans-

formed back to frequency space.21In this way the filter for

each colatitude angle is calculated as part of the preparation

for image reconstruction. During reconstruction, each 2D

parallel-ray projection is filtered with the 2D filter for the

appropriate ? and backprojected into the 3D volume. We

used a pixel-driven backprojection with linear interpolation.

D. Projection truncation

Due to the finite longitudinal extent of the source and

detector, oblique rays do not encompass the entire field of

view, as illustrated in Fig. 4. Depending on the size of the

object, the rebinned 2D parallel-ray projections at these ob-

lique colatitude angles will be truncated and cannot be di-

rectly incorporated into the filtered backprojection algorithm.

One known method for dealing with these truncated projec-

tions performs an initial reconstruction from the complete

projections at ?=?/2 and then uses reprojection to estimate

the missing rays.22The completed set of projections can then

be used to reconstruct the 3D volume using the method de-

scribed above.

For the preliminary investigation of this algorithm, the

reprojection algorithm was not implemented. Instead, the

longitudinal FOV was reduced somewhat and only projec-

tions at colatitude angles that contain the entire object were

used to reconstruct the volume. Although this simplification

inefficiently uses the collected data and would suffer a large

SNR penalty in a real system, it is acceptable for the prelimi-

nary investigation of the integrity of the algorithm in the

absence of noise. When studying the noise performance of

the algorithm, projections from all colatitude angles were

used as will be described in Sec. IV D.

E. Noise considerations

The IGCT geometry contains cross-plane ray measure-

ments, and in order to provide suitable noise performance,

the reconstruction algorithm must use these rays efficiently,

ideally as efficiently as the in-plane rays. To study this re-

quirement we use the metric of photon utilization efficiency.

We distribute the same total number of photons to the IGCT

system and to a parallel-ray geometry that uses only in-plane

rays, where both systems have comparable resolution and

field of view, and compare the noise in the resulting images.

The noise in a reconstructed CT voxel depends on the

spatial resolution and the number of photons that passed

through the voxel and were detected as expressed in the fol-

lowing equation:23

?2= A ·?

j=1

m

1

Nj

,

?24?

where Njis the mean detected photon density that has passed

through the voxel in the jth projection, m is the number of

projections, and A is the integral of the reconstruction filter

squared. For parallel-ray reconstruction using only in-plane

rays, A can be expressed as

m2?

−?

−?

A =?2

??

?

ku

2?W?ku,k???2dkudk?,

?25?

where kuand k?are the coordinates of the 2D Fourier trans-

form of the projection, and W is the window function, in our

case a radial Hanning window with frequency cutoff kc,

2?1 + cos???ku

W?ku,k?? =1

2+ k?

2

kc?? ???ku

2+ k?

2

2kc?.

?26?

By combining Eqs. ?24?, ?25?, and ?26?, and assuming that

the photon density N is the same for all projections, we can

calculate the photon density required to achieve a specified

noise variance.

?2?

−?

−?

m?2

N =

??

?

ku

2?W??ku

2+ k?

2??2dkudk?

.

?27?

From the photon density, which is defined as the number

of photons per unit area, the total number of photons in the

parallel-ray acquisition is

Ptotal= N · pixu· pix?· aread· m,

?28?

where pixuand pix?are the detector dimensions in pixels,

and areadis the area of a detector element.

By distributing Ptotalphotons to the IGCT geometry and

measuring the resulting noise, we can compare the IGCT

photon utilization efficiency to that of the in-plane parallel-

ray geometry. That is, we can examine whether the proposed

reconstruction algorithm uses the cross-plane rays as effec-

tively as the in-plane rays.

IV. SIMULATIONS

To test the reconstruction algorithm, projection data were

simulated for the IGCT geometry. The specifications of the

simulated system are summarized in Table I and are based on

hardware components developed by NexRay Inc. ?Los Ga-

tos, CA?,24but with SID and DID typical of a CT geometry

?but reversed?. At each gantry position data are collected

FIG. 4. Profile of the source and detector illustrates that while the in-plane

rays span the entire longitudinal field of view, cross-plane rays at oblique

colatitude angles do not cover the entire object, leading to truncated parallel-

ray projections.

3239Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3239

Medical Physics, Vol. 32, No. 11, November 2005

Page 7

from each of the 2000 source spots. We assumed the acqui-

sition of these data was instantaneous, i.e., no rotation during

source scanning. Sixty-three gantry positions over 2? were

determined necessary for sufficient sampling ?note that each

view contributes to many 2D projections?.7The source and

detector apertures were simulated by averaging data from

discrete subsources and subdetectors spanning the finite ap-

erture sizes.

The data were rebinned into a 2D parallel-ray geometry

using the parameters specified in Table II. While a thorough

optimization was not performed, the gridding parameters

were selected experimentally to provide acceptable perfor-

mance.The projectionsampling

0.125 mm?0.125 mm, as using an oversampled grid im-

proves the gridding performance.12For computational con-

venience, the number of rebinned views was 1008, which is

16 times the number of acquired views. By using this num-

ber of rebinned views, the distribution of measurements in

the 4D projection space repeats for every 16 grid points in

the ? direction. The precorrection kernel weights were based

on a separable 4D Hanning window. The implemented kernel

widths provided a reasonable tradeoff between the resulting

blur and having sufficient measurements to estimate each

grid point. Further, the selected kernel widths were large

enough to ensure that all data points were used. The thresh-

old on the sum of the kernel values was initially set to zero

and was then increased until the related artifacts were quali-

tatively absent from the rebinned projections. The maximum

? value was chosen by examining a histogram of all calcu-

lated ? values and choosing a reasonable threshold.

waschosento be

The reconstruction filter was apodized with a Hanning

window with a cutoff of 15 lp/cm unless otherwise noted.

For the rebinned IGCT data, the projections were further

windowed with the inverse of the Fourier transform of the

gridding kernel, as discussed in Sec. III C, unless otherwise

stated.

A. Water sphere

To measure the artifact level in the reconstructed volume,

data from the IGCT system and a comparable 2D parallel-ray

geometry were simulated through a water sphere located at

?1 cm, 1 cm, 0 cm? with a radius of 2 cm.

B. MTF

A 0.006 25 cm radius sphere at isocenter was simulated in

order to investigate the resolution effects of the reconstruc-

tion algorithm. The IGCT source focal spot and detector ap-

erture were modeled as part of the simulation. The modula-

tion transfer function

?MTF?

reconstructing a volume containing the sphere, projecting the

volume perpendicularly to the axis of rotation, and comput-

ing the 2D Fourier transform of the result. The horizontal

radial line through the transform gives the MTF in the in-

plane direction, while the vertical radial line gives the MTF

in the slice direction. The small sphere was also simulated

centered at ?4, 4, 1 cm? to study the resolution away from the

center of the field of view.

For comparison, 2D parallel-ray projections of the geom-

etry in Table II were simulated for the 0.006 25 cm sphere at

isocenter. The parallel-ray data was simulated with focal spot

and detector aperture blurring equivalent to the IGCT geom-

etry so that any discrepancies in the MTF would be due to

the IGCT reconstruction algorithm. For all MTF studies, the

reconstruction filter was apodized with a Hanning window

with a 40 lp/cm cutoff.

wascalculatedby

C. Resolution phantom

To further investigate the algorithm performance for high

resolution objects, a phantom was simulated with 32 spheres

arranged into four resolution patterns with 0.7, 0.6, 0.5, and

0.4 mm spheres, respectively. For example, the 0.5 mm pat-

tern contained eight 0.5 mm diameter spheres centered at the

vertices of a 1 mm cube. Coronal and axial planes through

the patterns were reconstructed. The cutoff of the reconstruc-

tion filter Hanning window was 40 lp/cm.

D. Noise

The photon utilization efficiency of the IGCT reconstruc-

tion was examined by using Eq. ?27? to calculate the photon

density required for an in-plane 2D parallel-ray geometry

with 15 lp/cm bandwidth to achieve a noise standard devia-

tionof 10 HU. Theresulting

?106photons/cm2, yields a total of 2.7?1011photons for

the projection sampling described in Table II. Noisy IGCT

data and in-plane parallel-ray data were simulated using this

number of photons detected through air, and the central axial

photondensity,5

TABLE I. Simulated IGCT geometry.

Source dimensions ?transverse?axial?

Number of source locations ?transverse?axial?

Detector dimensions ?transverse?axial?

Number of detector elements ?transverse?axial?

SID

DID

FOV ?transverse?axial?

Number of views over 2?

Source focal spot width

Detector aperture width

25?5 cm

100?20

5.4?5.4 cm

48?48

41 cm

54 cm

12?5 cm

63

0.06 cm

0.11 cm

TABLE II. Rebinning algorithm parameters.

2D projection sampling

2D projection dimensions

Number of views over 2?

Maximum colatitude angle

Minimum colatitude angle

Colatitude angle spacing

Radial kernel width ?u,??

Angular kernel width ??,??

Minimum sum of corrected kernel values

Maximum ? value

0.0125?0.0125 cm

352?960 pixels

1008

?/2 rad

?/2 - 0.03 rad

0.003 rad

0.07 cm

0.0126 rad

0.6

1000

3240Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT 3240

Medical Physics, Vol. 32, No. 11, November 2005

Page 8

slice was reconstructed. The simulations were repeated five

times. The full range of colatitude angles, +/−0.03 rad, was

used in the IGCT reconstruction. Because the reprojection

algorithm was not implemented, some of the projections

were truncated. While the missing values were set to zero,

the truncation did not significantly affect the image noise

because the projections contained only noise and because

only the central axial slice was reconstructed. The recon-

structed noise standard deviation did not change significantly

when the missing projection data was replaced by values

with standard deviation equivalent to the measured rays.

V. RESULTS

A. Water sphere

The simulated IGCT water sphere data were first rebinned

into the 2D parallel-ray geometry specified in Table II. Fig-

ure 5 compares a profile through an ideal parallel-ray projec-

tion and an IGCT projection rebinned with and without the

gridding correction. In other words, this graph plots one line

in the 4D projection space. The region highlighted in Fig.

5?a? is expanded in Fig. 5?b? to more clearly show the grid-

ding errors. Figure 6 displays the difference between the

parallel-ray projection and the IGCT rebinned projection

with and without correction. The relationship between the

gridding error before correction and the gradient of the pro-

jection is demonstrated in Fig. 6, as the error is highest

where the curve in Fig. 5?a? is steep. The error at the edge of

the object, still present in the corrected projection, is due to

the blurring incurred during rebinning.

The rebinned projections were then filtered and back-

projected to reconstruct the central axial plane of the sphere.

Figure 7?a? shows the reconstructed plane without the grid-

ding error correction. The image is displayed with a window

centered 0 HU, and a width of +/−1 HU, that is pixels val-

ues below −1 HU are black, and pixel values above 1 HU

are white. Figure 7?d? plots the central horizontal profile

through the image. The high frequency gridding artifacts

seen in the sphere are reduced to well below 1 HU by the

gridding correction, as shown in Figs. 7?b? and 7?e?. This

demonstrates that the artifacts in Fig. 7?a? are caused by the

asymmetric sampling around each output grid point, com-

bined with the locally non-constant projection values which

are relatively similar in adjacent views. With the correction

up to linear terms, the IGCT reconstruction is comparable to

that from direct parallel-ray data, shown in Figs. 7?c? and

7?f?. Although the gridding errors displayed in Fig. 7 are

relatively small, on the order of 1–2 HU, the gridding error

is proportional to the gradient of the projection, and the ob-

ject simulated in this experiment is by no means the worst

case. Without the gridding correction, significant artifacts

will result for more challenging objects.

Figure 8 compares the central coronal plane of the sphere

simulated by both the IGCT and parallel-ray geometries. The

gridding correction has been applied in the IGCT reconstruc-

FIG. 5. Profile of a water sphere projection for the ideal parallel-ray geom-

etry and the IGCT geometry rebinned with and without the gridding correc-

tion. The highlighted region of ?a? is expanded in ?b?.

FIG. 6. Difference between the parallel-ray projection and ?a? the IGCT

rebinned projection without gridding correction and ?b? IGCT rebinned pro-

jection with correction.

3241Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3241

Medical Physics, Vol. 32, No. 11, November 2005

Page 9

tion. The images are displayed at two different windows to

show artifacts in both air and water, and the central horizon-

tal profile is plotted for each reconstruction. The IGCT image

contains more prominent view aliasing artifacts. However,

these artifacts are small enough to be acceptable. Other than

this difference, the two reconstructions have comparable im-

age quality.

In the IGCT reconstruction, only rebinned parallel-ray

projections containing the complete object were used. That

is, only seven of the possible 21 colatitude angles were used,

while the ideal parallel-ray projections were simulated at all

21 colatitude angles. Therefore the results of the IGCT simu-

lations verify the integrity of the rebinning algorithm and the

integrity of the filtered backprojection for small colatitude

FIG. 7. Reconstructed central axial plane of the off-center water sphere for ?a? the IGCT simulation without gridding correction, ?b? the IGCT simulation with

gridding correction, and ?c? the parallel-ray simulation. All images are windowed to a level of 0 HU +/−1 HU. The central horizontal profile through the

image is plotted for ?d? the IGCT simulation without gridding correction, ?e? the IGCT simulation with gridding correction, and ?f? the parallel-ray simulation.

3242Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT 3242

Medical Physics, Vol. 32, No. 11, November 2005

Page 10

angles, while the results of the parallel-ray simulations verify

that more oblique projections are handled properly by the

filtered backprojection and do not introduce artifacts.

For efficient noise and dose performance in a real IGCT

system, the reprojection algorithm would be implemented to

utilize data from all colatitude angles.

B. MTF

The MTF was calculated for the IGCT and parallel-ray

geometries, both reconstructed with a Hanning windowed

reconstruction filter. IGCT reconstructions were made with

and without the deapodization described in Sec. III C and

with and without the gridding correction. Since the object is

small, 21 colatitude angles were used in the IGCT simula-

tion. Only in-plane rays were used in the direct parallel-ray

simulation. Figure 9 compares the resulting MTF curves.

Both the in-plane and slice MTFs are displayed for the IGCT

geometry. As can be seen, while the rebinning algorithm

does introduce some blurring, the resolution can be largely

recovered by using the deapodization window during filtered

backprojection. The slight difference between the in-plane

and slice MTF is likely due to discretization errors. The 10%

value of the MTF is at 16 lp/cm. Even higher spatial reso-

lution is possible if the Hanning window is omitted or re-

placed with a function with higher amplitude at high spatial

frequencies. Without the Hanning window, the modulation at

16 lp/cm is expected to have been 15%.

Figure 10 compares the in-plane MTF of the IGCT system

with and without the gridding error correction, verifying that

the correction does not degrade the resolution.

Figure 11 shows the in-plane and slice MTF toward the

edge of the in-plane field of view, compared to the in-plane

MTF at isocenter. The slight degradation seen in the in-plane

MTF is most likely due to the azimuthal blurring introduced

during rebinning. If the cross-plane rays included in this re-

construction were approximated as in-plane rays, as is done

in single-slice rebinning,25the response at the edge of the

12-cm FOV would span more than 3 mm in the slice direc-

tion. In the IGCT reconstruction, the slice MTF is preserved

as the impulse moves off center, indicating that the gridding

algorithm properly incorporates the oblique rays.

C. Resolution phantom

Figure 12 displays an axial and coronal plane through the

resolution patterns, and in both images the 0.4 mm spheres

can be resolved. Projections at all 21 colatitude angles were

used in this reconstruction. The reduced modulation of the

FIG. 8. Reconstructed central coronal plane of the off-center water sphere for ?a?, ?b? the IGCT simulation, and ?d?,?e? the parallel-ray simulation. Images ?a?

and ?d? are windowed to 0 HU +/−1 HU, and images ?b? and ?e? are windowed to −1000 HU +/−1 HU. The central horizontal profile through the image is

shown ?c? for the IGCT reconstruction and ?f? for the parallel-ray reconstruction.

FIG. 9. Plot comparing the MTF curves for the IGCT and parallel-ray simu-

lations. The IGCT data was reconstructed both with and without the grid-

ding kernel deapodization window. For the IGCT data with deapodization

window, both in-plane and slice MTF curves are displayed.

3243Schmidt, Fahrig, and Pelc: 3D reconstruction for inverse-geometry volumetric CT3243

Medical Physics, Vol. 32, No. 11, November 2005

Page 11

smaller sphere patterns is consistent with the MTF curve.

Again, a different filter window function could be used to

preserve more uniform response across all frequencies.

D. Noise

Compared to the predicted 10 HU standard deviation, the

noise standard deviation of the in-plane parallel-ray recon-

structions, averaged across the five simulations, was 9.98

+/−0.06 HU, while the noise in the IGCT reconstruction

was 7.69+/−0.05 HU before deapodization, and 9.09+/

−0.05 HU after deapodization. The slightly lower noise in

the IGCT images, even after apodization, may be caused by

residual gridding blur. When only in-plane rays are used in

the IGCT reconstruction, the resulting noise after deapodiza-

tion was 19.54+/−0.02 HU, thereby demonstrating the ad-

vantage of incorporating the cross-plane rays. The results of

the noise investigation show that the proposed reconstruction

algorithm efficiently uses the cross-plane rays and can pro-

vide noise performance similar to a comparable in-plane

parallel-ray geometry when given the same number of pho-

tons.

VI. DISCUSSION AND CONCLUSIONS

The results presented in this paper demonstrate acceptable

performance of the 3D filtered backprojection algorithm pro-

posed for the IGCT geometry. This reconstruction method is

one of several possible approaches for the IGCT data. One

alternative is Fourier rebinning ?FORE?.26,27These algo-

rithms estimate the in-plane data from the cross-plane data,

thereby requiring only a 2D filtered backprojection. Another

advantage of these methods is that they avoid the data trun-

cation problem and do not require the time consuming re-

projection step. The FORE algorithm has been applied to

PET data with much success.26–31An exact Fourier rebinning

algorithm, FORE-J, which also avoids the reprojection step

has been proposed.27These Fourier rebinning algorithms are

very promising and their implementation for the IGCT ge-

ometry would be interesting future work, as would the inves-

tigation of iterative reconstruction algorithms.

Our work shows that the use of a gridding method can

introduce errors when the sampling pattern surrounding each

output point is asymmetric and that these errors are signifi-

cant in rebinning for CT reconstruction. We developed a cor-

rection method that effectively reduces the errors associated

with gridding. We believe this method is new and that it may

be useful in other applications such as non-Cartesian MR

imaging. With this correction, we demonstrated that high

quality images, relatively free of artifacts and additional

blurring, can be produced from IGCT data. The noise inves-

tigation demonstrates that the proposed reconstruction algo-

rithm uses the cross-plane rays as efficiently as the in-plane

rays and provides noise comparable to an in-plane parallel-

ray geometry when both systems use the same number of

photons. The simulations presented in this paper further pre-

dict that an isotropic resolution of 0.4 mm can be achieved

using realizable source and detector components and the pro-

posed algorithm. Although much work remains to fully in-

vestigate the feasibility of the IGCT system, the encouraging

performance of the reconstruction algorithm further supports

the potential for high quality volumetric scanning free from

cone-beam artifacts using the IGCT geometry.

ACKNOWLEDGMENTS

This work is supported by GE Healthcare. The authors

would like to thank Ed Solomon and Josh Star-Lack of

NexRay, Inc. and Robert Bennett of Stanford University for

helpful discussions.

a?Also at Department of Electrical Engineering, Stanford University.

b?Also at Department of Bioengineering, Stanford University.

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