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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.

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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

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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

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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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