Localized motion artifact reduction in projection imaging

An x-ray CT scanner acquires projection data from a series of projections during a scan of a patient's chest Movement of the patient's chest due to respiration is also sensed during the scan and this acquired motion data is employed along with an warping function which models chest motion to calculate factors which correct the acquired projection data and reduce motion artifacts in an image produced by back projecting the acquired projection data.

BACKGROUND OF THE INVENTION 
The present invention relates to medical imaging, and particularly, to the 
reduction of motion artifacts in images produced using a projection method 
of reconstruction. 
There are a number of modalities used to produce medical images. These 
include x-ray computed tomography (CT), magnetic resonance imaging (MRI), 
single photon emission computed tomography (SPECT), and positron emission 
computed tomography (PET) methods. In all cases, the data used to 
reconstruct the desired image are acquired over a period of time in a scan 
comprised of a series of projections. Each projection is a snapshot of the 
patient from a different angle, or perspective, and a scan typically 
includes tens, or hundreds of projections. In the case of x-ray CT the 
entire data set may be acquired in a few seconds, whereas an MRI scan 
typically requires a few minutes to complete. The methods used to 
reconstruct an image from such data sets presume that the patient is 
motionless during the entire scan and that the same fixed object is the 
subject of all acquired projections. To the extent this is not true, 
artifacts such as ghosts, smearing and fuzziness appear in the 
reconstructed image. 
Efforts to reduce patient motion during a scan can significantly improve 
image quality. However, artifacts caused by respiration are a significant 
problem in chest scans where suspension of breathing is not possible or 
poor instructions are provided to the patient by the scanner operator. 
Children and comatose patients are routinely scanned with no attempt to 
synchronize respiration with scanning, and it is expected in such cases 
that a number of poor quality images will be produced and will be 
discarded. 
One approach to reducing motion artifacts in medical images is to 
retrospectively correct the acquired data to offset the effects of motion. 
One such method, for example, is disclosed in U.S. Pat. No. 4,937,526 and 
is applied to acquired MRI data. The corrections that are made may be 
determined from an examination of the acquired raw data itself, or 
additional information, such as a signal from a cardiac monitor or a 
respiration monitor, may be used. The manner in which the corrections are 
made to the acquired raw data is determined by the particular 
reconstruction technique that is used. In the above patent, for example, a 
2D Fourier transformation is used to reconstruct an image from the 
acquired MRI data, and the correction methods disclosed are limited to 
that technique. 
The back projection method for image reconstruction is employed to some 
extent in all computed medical imaging modalities. It is the predominant 
method used in x-ray CT, and there is a need to correct acquired data used 
in projection imaging for the effects of patient motion. 
The related application Ser. No. 07/615,778, now U.S. Pat. No. 5,121,128, 
referred to above, provides a method of modifying the back projecting 
process to accommodate motion of the patient occurring during acquisitions 
of the projections. In particular, the patient motion is modeled as a two 
dimensional magnification and offset of the volume elements (voxels) of 
the patient such as might occur with expansion of the chest during 
breathing. 
The possibility of the above modification to the back projecting process 
was founded on an analyses of the image reconstruction process as a 
Fourier transform using the Fourier slice theorem. The results of this 
analyses were then applied to the more typically used backprojection 
process. The results suggested that the modification was appropriate 
provided the motion could be modeled as a simple, global magnification and 
offset of the voxels of the patient. Nevertheless, in general, the patient 
motion does not conform to a simple magnification and offset of the 
patient voxels but is a more complex function of time. 
SUMMARY OF THE INVENTION 
The present invention recognizes that the backprojecting process can be 
modified to correct for patient motion other than uniform magnification 
and offset providing the motion reduces locally to magnification and 
offset. The correction of the present invention occurs not on a global 
basis but point by point for each voxel of the patient, subject to the 
ability to construct a suitable modeling function of normal physiological 
motion. 
Specifically, the method produces an image of a moving patient composed of 
a plurality of voxels, each voxel having a reference coordinate defined at 
a reference time during the patient motion, and a displaced coordinate, 
which at the reference time may equal the reference coordinate but at 
other times may not equal the reference coordinate. The patient is scanned 
to acquire a plurality of projections of a projection set at different 
times and throughout a range of different projection angles (.theta.). 
Each projection measures a physical characteristic of the voxels at their 
displaced coordinates. Also acquired with each projection is a motion 
parameter indicating the movement of the patient as the projection is 
acquired. 
The acquired projections are reconstructed by back projecting using a back 
projection formula which is modified by a predetermined warping function 
(G,H) that relates the displaced coordinates of the voxels to their 
reference coordinates as a function of the motion parameter. Artifacts, 
produced in the image by movement of the patient between projections are 
thus reduced. 
In one embodiment, the back projecting employes the warping functions G and 
H to determine, along with the motion parameter, the displaced coordinates 
of each voxel (identified by its reference coordinate) at the time of the 
acquisition of the projection and backprojects the voxel at the reference 
coordinate but employing the displaced coordinate to identify the relevant 
information from the projection. 
It is thus one object of the invention to provide an improved correction of 
motion in a scanned patient, such motion which may cause image artifacts, 
wherein the correction is responsive to a general warping function. 
In a first embodiment, a method of calculating a Jacobian, employed prior 
to the modification of the backprojecting, is also provided, however, it 
has also been determined that in certain applications the Jacobian need 
not be employed, thus simplifying the calculational burden. 
The foregoing and other objects and advantages of the invention will appear 
from the following description. In the description, reference is made to 
the accompanying drawings which form a part hereof, and in which there is 
shown by way of illustration a preferred embodiment of the invention. Such 
embodiment does not necessarily represent the full scope of the invention, 
however, and reference is made therefore to the claims herein for 
interpreting the scope of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
CT Hardware 
While the present invention may be applied to many different imaging 
systems that employ back projection image reconstruction methods, the 
preferred embodiment is employed in an x-ray CT scanner such as that 
illustrated in FIG. 1. 
As shown in FIG. 1, a CT scanner used to produce images of the human 
anatomy includes a patient table 10 which can be positioned within the 
aperture 11 of a gantry 12. A source of highly collimated x-rays 13 is 
mounted within the gantry 12 to one side of its aperture 11, and one or 
more detectors 14 are mounted to the other side of the aperture. The x-ray 
source 13 and detectors 14 are revolved about the aperture 11 during a 
scan of the patient to obtain x-ray attenuation measurements from many 
different angles. 
A complete scan of the patient is comprised of a set of x-ray attenuation 
measurements which are made at discrete angular orientations of the x-ray 
source 13 and detector 14. Each such set of measurements is referred to in 
the art as a "projection" and the results of each such set of measurements 
is a projection set. As shown in FIG. 2A, the set of measurements in each 
projection may be obtained by simultaneously translating the x-ray source 
13 and detector 14 across the acquisition field of view, as indicated by 
arrows 15. As the devices 13 and 14 are translated, a series of x-ray 
attenuation measurements are made through the patient and the resulting 
set of data provides a transmission profile at one angular orientation 
(.theta.). The angular orientation of the devices 13 and 14 is then 
changed (for example, 1.degree.) and another projection is acquired. These 
are known in the art as parallel beam projections. An alternative 
structure for acquiring each transmission profile is shown in FIG. 2B. In 
this construction, the x-ray source 13 produces a fan-shaped beam which 
passes through the patient and impinges on an array of detectors 14. The 
detectors 14 can be curved as shown in FIG. 2B, or they can be aligned in 
a straight line (not shown in the drawings). Each detector 14 in this 
array produces a separate attenuation signal and the signals from all the 
detectors 14 are separately acquired to produce the transmission profile 
for the indicated angular orientation. As in the first structure, the 
x-ray source 13 and detector array 14 are then rotated to a different 
angular orientation and the next transmission profile is acquired. 
As the data are acquired for each transmission profile, the signals are 
filtered, corrected and digitized for storage in a computer memory. These 
steps are referred to in the art collectively as "preprocessing" and they 
can be performed in real time as the data is being acquired. The acquired 
transmission profiles are then used to reconstruct an image which 
indicates the x-ray attenuation coefficient of each voxel in the 
reconstruction field of view. These attenuation coefficients are converted 
to integers called "CT numbers", which are used to control the brightness 
of a corresponding pixel on a CRT display. An image that reveals the 
anatomical structures in a slice taken through the patient is thus 
produced. 
Referring particularly to FIG. 3, the operation of the CT system is 
controlled by a programmed data processing system 25 which includes a 
computer processor 26 and a disk memory 27. The disk memory 27 stores the 
programs the computer processor 26 uses in patient scanning and in image 
reconstruction and display. It also stores on a short-term basis the 
acquired data and the reconstructed image data. The computer processor 
includes a general purpose minicomputer with input and output ports 
suitable for connection to the other system elements as shown It also 
includes an array processor such as that disclosed in U.S. Pat. No. 
4,494,141 which is incorporated herein by reference. 
An output port on the computer processor 26 connects to an x-ray control 
circuit 28, which in turn controls the x-ray source 13. The high voltage 
on the x-ray source 13 is controlled and its cathode current is controlled 
to provide the correct dosage. The high voltage and cathode current are 
selected by an operator who enters the desired values through an operator 
console 30 and the computer processor 26 directs the production of the 
x-rays in accordance with its scan program. 
The x-rays are dispersed in a fan-shape as described above and received by 
the array of detectors 14 mounted on the opposite side of the gantry 
aperture 11. Each individual cell, or detector element, examines a single 
ray originating from the x-ray source 13 and traversing a straight line 
path through a patient located in the aperture 11. The currents formed in 
each detector element are collected as an analog electrical signal and 
converted into a digital number by A/D converters in a data acquisition 
system 31. The digitized measurements from all the detectors is a complete 
projection. U.S. Pat. Nos. 4,112,303 and 4,115,695 disclose details of the 
gantry construction, U.S. Pat. No. 4,707,607 discloses the details of the 
detector array 14, and the data acquisition system is disclosed in U.S. 
Pat. No. 4,53,240. All of these patents are incorporated herein by 
reference. The digitized signals are input to the computer processor 26. 
The digitized attenuation measurements from the data acquisition system 31 
are preprocessed in a well-known manner to compensate for "dark currents", 
for uneven detector cell sensitivities and gains, and for variations in 
x-ray beam intensity throughout the scan. This is followed by beam 
hardening corrections and conversion of the data to logarithmic form so 
that each measured value represents a line integral of the x-ray beam 
attenuation. This preprocessing is performed in real time as the scan is 
being conducted, and as shown in FIG. 4, each projection is comprised of a 
set of attenuation values 32 which define a transmission profile, or 
projection, indicated by dashed line 34. 
In addition to the transmission profile data 34, two other pieces of 
information are input during the acquisition of each projection. The first 
is the angle (.theta.) which indicates the angular orientation of the 
x-ray source 13 and detectors 14 with respect to the vertical reference 
axis. Typically, for example, the projections are acquired at 1.degree. 
increments over a range of 180.degree.. 
Patient Motion 
The second piece of information acquired with each projection is a distance 
value (D) that is indicative of the position of the patient's chest cavity 
and which is a parameter employed in a geometric model of the chest cavity 
during respiration. Referring particularly to FIG. 5, this geometric model 
is illustrated by a schematic cross-section taken in a transverse plane 
through the patient's chest as indicated at 40. As the patient breathes 
the posterior chest wall 41 which rests on the supporting table 10 does 
not move any significant amount, whereas the anterior chest wall 42 moves 
vertically as indicated by the dashed line 43. As will be described below, 
the size and shape of the patient's chest cavity at any point in the 
respiratory cycle can be approximated by monitoring the vertical position 
of the anterior wall 42. Accordingly, an ultrasonic range finder 44 is 
mounted to the gantry 12 and is positioned to measure the vertical 
distance (D) to the patient's chest. This measured (D) is input to the 
computer processor 26 (FIG. 3) along with each projection of acquired 
data. 
a. The Magnification-Shift Model of Patient Motion 
Referring still to FIG. 5, as the patient breathes and the anterior chest 
wall 42 moves up and down, the contents of the chest cavity may be modeled 
as magnifying and shrinking along the vertical axis (y). This 
magnification does not occur about the center near y=0, but instead, at 
the posterior wall 41 located at y=-y.sub.p. There is virtually no 
magnification along the horizontal (x), but to the extent that there is, 
it occurs about the center of the chest cavity at x=0 (if the patient is 
centered on the table 10). Using this model and the measured parameter D, 
a first embodiment of the present invention corrects the acquired 
projection data and employs the corrected projection data in the back 
projection reconstruction process such that the chest cavity appears 
stationary in a reference position during the entire scan. As a result, 
motion artifacts are significantly reduced or eliminated. 
The corrections to the acquired data and the manner in which the corrected 
data is employed in the back projection process has been determined for a 
general case in which patient motion produces magnification along two axes 
(x and y) and the point about which magnification occurs is shifted, or 
offset, from the origin (x=0, y=0). The correction factors for this 
generalized case have been determined and will now be described. 
Let f(x,y) be the cross-section of the patient which is to be 
reconstructed. A magnified and shifted version of this same cross-section 
during various stages of the patient motion cycle is given by: 
EQU f'(x,y)=f(.alpha..sub.x +.beta..sub.x x, .alpha..sub.y +.beta..sub.y y)[1] 
where .beta..sub.x and .beta..sub.y are magnification factors along the 
respective x and y axes and .alpha..sub.x and .alpha..sub.y are shift 
factors and where .beta. and .alpha. are functions of .theta.. For 
parallel beam projection data the formula for projection of this magnified 
image at gantry rotational angle .theta. is given by: 
##EQU1## 
where .delta.(t) is the Dirac-delta function known to those skilled in 
this art. The Fourier transform of this projection can be found: 
##EQU2## 
Now make the following changes of variables 
EQU x'=.alpha..sub.x +.beta..sub.x x 
EQU y'=.alpha..sub.y +.beta..sub.y y [4] 
When [4] is used in [3], the following is obtained: 
##EQU3## 
Let F(u,v) be the two-dimensional Fourier transform of f(x,y). Then it is 
seen that: 
##EQU4## 
This equation is a version of the Fourier Slice Theorem in the case of 
projections acquired from a magnified and shifted object function. It says 
that the Fourier transform of the projection at gantry position .theta. is 
a spoke of the two-dimensional Fourier transform of the object function at 
angle 
##EQU5## 
after a phase term and a scaling factor have been removed. 
Equations [6] and [7] could form the basis of a reconstruction algorithm 
which employs a two-dimensional Fourier transform of the data which 
results from the mapping of the one-dimensional Fourier transform of the 
projection data into the Fourier transform of the patient. This is not the 
preferred method of reconstruction. Instead, a filtered back projection 
method has been developed and will now be described in detail. 
A filtered back projection reconstruction formula will now be derived for 
reconstructing f(x,y) using p'(.theta.,t). The inverse Fourier transform 
of F(u,v), is given by 
##EQU6## 
Consider the following change of variables 
##EQU7## 
The components of the Jacobian in this change of variables are given by 
##EQU8## 
where .beta.'.sub.x and .beta.'.sub.y are the derivatives of .beta..sub.x 
and .beta..sub.y with respect to .theta.. The Jacobian, J(u,v,.omega., 
.theta.), can be determined using [10] resulting in 
EQU J(u,v,.omega., .theta.)=.vertline..omega..uparw.g(.theta.) [11] 
where 
##EQU9## 
The value g(.theta.) is a weighting factor which is applied to the 
projection data at gantry position .theta.. In the parallel beam 
acquisition, this weighting factor is a constant value which is applied to 
each attenuation value 32 in the projection profile 34 at the position 
.theta. (FIG. 4). In some situations where the density of the object being 
imaged decreases as it is magnified, this weighting factor is modified to 
more closely approximate the geometric model of motion in equation [1]. 
More specifically, in such cases the weighting factor g(.theta.) should be 
multiplied by the value .beta..sub.y .beta..sub.x. The derivatives 
.beta.'.sub.x and .beta.'.sub.y can be calculated with numerical 
differences using the adjacent values of .beta..sub.x and .beta..sub.y. 
Using [9] and [11], [8] becomes 
##EQU10## 
When [6] is used, [13] reduces to 
##EQU11## 
Equation [14] represents a filtered back projection formula for 
reconstruction of parallel projections that are acquired from the 
magnified and shifted object. The formula is valid for reconstructing any 
point in the x-y plane. 
The image f(x,y) can be reconstructed, therefore, from a set of parallel 
beam projections acquired over a range of 180.degree. gantry positions by 
modifying the conventional data acquisition and reconstruction method in 
the following manner. First, for each projection of the patient, not only 
is the attenuation profile data acquired, but also, the parameters 
.alpha..sub.x, .beta..sub.x, .alpha..sub.y and .beta..sub.y are measured. 
In the preferred embodiment, .alpha..sub.x is set to zero and .beta..sub.x 
is set to one since the patient is usually centered on the table 10 and 
there is very little magnification of the chest cavity along the x axis 
during respiration. Only .beta..sub.y and .alpha..sub.y are required, 
therefore, to significantly reduce motion artifacts in the chest cavity 
and these are measured indirectly. As shown in FIG. 5, the distance 
between y=0 and the point about which magnification occurs is fixed at 
-y.sub.p. Also, the distance between the range finder 44 and this same 
point is fixed. As a result, the values for .beta..sub.y and .alpha..sub.y 
can be calculated from these fixed values and the measurement (D) produced 
by the range finder 44 as follows: 
EQU .beta..sub.y =1/(1+(D-D.sub.0)/(y.sub.0 +y.sub.p)) [16] 
EQU .alpha..sub.y =-y.sub.p (1-.beta..sub.y) [17] 
where y.sub.o is a reference position for the anterior chest wall which is 
selectable by the operator and which determines the shape and size of the 
reconstructed image, and D.sub.0 is the range finder measurement at this 
reference position. Consequently, the distance measurement (D) is acquired 
along with each projection profile and this measured parameter is 
sufficient to indicate the shape and size of the patient's chest cavity at 
the moment the projection was acquired. 
b. A Generalized Warping Function Model Of Patient Motion 
In a second embodiment, it has been recognized that this technique of 
correction may be expanded to two-dimensional temporally and spatially 
varying motion as may be described by a general function dependent on 
space and time f'(x,y)=f(x',y') and x' and y' are determined from 
generalized warp functions: 
EQU x'=G(x,y,.theta.) [18] 
EQU y'=H(x,y,.theta.) [19] 
where (x', y') are the Cartesian coordinates of some voxel displaced from a 
reference position (x,y) and .theta. is gantry position and proportional 
to time and G and H are warping functions. Employing a similar analysis as 
that provided above, projections acquired from the body undergoing the 
generalized motion of equations [18] and [19] is given by the following 
Radon transform: 
##EQU12## 
where P'(.theta.,t) is the projection value at t and .theta. and .delta. 
is the Dirac delta function. After the projections P'(.theta.,t) are 
filtered, the filtered projections q(.theta., t) are backprojected as 
follows. 
##EQU13## 
where G.sup.-1 and H.sup.-1 are the solutions of equations [18] and [19] 
for x and y respectively. 
Equation [21] follows from equation [14] and the assumption that equation 
[4] is satisfied by the warping functions G and H over a small area. In 
other words, the warping functions must be capable of approximation 
locally as magnification and offset, e.g. the warping functions must be 
non-rotational. The validity of this assumption has been established 
through computer simulation. 
As can be seen from equation [21], the voxels (x,y) are reconstructed 
according to their reference coordinates, however, the data of the 
filtered projections employed in the backprojecting is determined from the 
displaced coordinates (x',y') of those voxels as determined from the 
warping functions G and H, not from the reference coordinates (x,y). 
Because warping functions G and H are not restrained to be uniform 
magnification and offset, in general no .beta. values exist to compute the 
Jacobian or the value of g(.theta.) as computed in equation [11] above. 
Nevertheless, a Jacobian can be computed by considering an arbitrarily 
small volume of the patient in which the generalized warp functions G and 
H devolve to a simple magnification and shifting as described above. 
When we consider more complex warping functions such as those in [18] and 
[19], the derivatives with respect to x in the .alpha. and .beta. terms 
simply become partial derivatives. To compute g(.theta.), we need .beta.x 
and .beta.y which are given by: 
##EQU14## 
Using these .beta. values, it is now possible to compute the Jacobian 
g(.theta.) and recover f(x,y) from projections of f(x',y'). It has been 
determined that the Jacobian values are approximately unity for actual 
patients where the density of the patient decreases with increase in 
patient volume and hence the Jacobians may be ignored in many practical 
situations. 
The ability to employ a generalized warping function allows different 
models of particular patient motion to be employed, models that may more 
accurately reflect the actual motion. For example, one such model, 
differing from that described above with respect to equations [1], is 
radial expansion about a point (x.sub.0, y.sub.0) as given by the 
following equations: 
##EQU15## 
where m(.theta.) is the expansion factor and linearly related to D the 
motion parameter and d is the distance between the center of expansion and 
the point that is moving and 
##EQU16## 
Equations [24] and [25] are an empirical description of actual respiration 
which has been shown by computer simulation to satisfy the assumptions of 
equation [21]. 
CT Software 
The preferred embodiment of the invention will now be described with 
reference to the flow chart of FIG. 6. While most of the steps are carried 
out in dedicated hardware so that the processing can be carried out in 
"real time", the process itself is controlled by a program executed by the 
computer 26 which performs the scan. 
This control program is entered at 75 and the CT system is initialized at 
process block 76 to acquire the data for the first projection. This 
includes receiving input data from the operator such as the reference 
chest position y.sub.o and reference range finder distance D.sub.0, and 
orientation of the gantry to the desired starting position of 
.theta.=-90.degree.. A loop is then entered in which the profile data for 
the first projection is acquired and preprocessed as indicated at block 
77. The distance measurement (D) from the range finder 44 (FIG. 5) is 
acquired at process block 78 and motion factors are calculated at process 
block 79. The motion factors may be the values for .beta..sub.y and 
.alpha..sub.y calculated using the equations [16] and [17] or may be 
related to the more complex warping functions of equations [18] and [19]. 
In the latter case, distance measure D and the reference value D.sub.0 are 
used to deduce the parameters of the warping functions and to synchronize 
the warping function with the actual motion of the patient. 
The acquired projection data is filtered in the usual fashion at process 
block 80 and then it is weighted at process block 81 by multiplying each 
value in the profile data set by the weighting factor g(.theta.) 
calculated in accordance with equation [12] or with respect to warping 
function G and H, as indicated generally at equations [18] and [19]. 
The corrected projection data is employed in reconstructing an image using 
the back projection technique as indicated at process block 82, however, 
as indicated by equation [14] and [21], this process is modified to 
account for motion. 
Referring particularly to FIG. 7, a 512 by 512 pixel image 85 is created by 
determining which of the values in the corrected and filtered projection 
data set 86 contribute to the brightness value of the pixel located at 
reference coordinates (x,y). In a parallel beam acquisition, the 
conventional back projection formula for determining which value (t) to 
use is as follows: 
EQU t=x cos.theta.+y sin.theta. [26] 
where (x,y) is the location of the pixel, .theta. is the projection angle 
for the projection, and t is the location in the projection data set from 
which an attenuation value 87 is read. This conventional back projection 
is shown in FIG. 7 by the dashed line 88. Typically, t is located between 
two samples in the acquired data set and interpolation is used to 
determine a more accurate value to be added to the CT number for pixel 
(x,y). For each projection, all of the pixels in the image 85 are 
processed in this fashion to determine the contribution to their 
accumulated CT numbers. 
To practice the present invention this back projection technique is changed 
to select a different value (t') from the corrected projection data set 
86. This selection is made as follows: 
EQU t'=x'cos.theta.+y'sin.theta. [27] 
where in the first embodiment: 
EQU x'=(x-.alpha..sub.x)/.beta..sub.x [ 28] 
EQU y'=(y-.alpha..sub.y)/.beta..sub.y [ 29] 
and in the second generalized embodiment: 
EQU x'=G.sup.-1 (x,y,.theta.) [30] 
EQU y'=H.sup.-1 (x,y,.theta.) [31] 
Note that x' and y' are used differently than previously defined. 
In both cases the back projection process is modified by a displacement 
factor. This change is illustrated in FIG. 7 where (x,y) is the pixel in 
the reference image being reconstructed, (x',y') is the location of the 
same point in the patient, at the time the projection data was actually 
acquired, and the attenuation value 89 is the value selected by equation 
[19]. In other words, the geometric model and the motion parameter D 
indicate that the attenuation value to be used at the pixel (x,y) from the 
projection data 86 is the value 89 at t' rather than the attenuation value 
87. After the contribution to each pixel in the image has been computed, 
the system loops at decision block 90 to advance the gantry and acquire 
and process that data for the next projection. When 180.degree. of data 
have been acquired and processed in this manner, the scan is complete and 
the image data 85 is displayed at process block 91. The CT numbers in the 
image data array 85 are scaled and processed in the normal fashion to 
produce an image of the desired brightness level and range. 
The teaching of the present invention is also applicable to fan-beam CT 
scanners which employ the back projection technique of image 
reconstruction. As in the first and second embodiments described above, 
the acquired projection data is corrected by a weighting factor including 
a Jacobian, and the back projection process is modified by a displacement 
factor. The calculation of the weighting factors and displacement factors 
depend on the model of motion employed and the geometry of the scanner. 
Two examples, using the magnification and shifting model of equation [12] 
and a fan beam of x-rays received by a flat and curved detector are now 
provided. 
Fan Beam with Flat Detector 
For a flat detector fan-beam reconstruction the weighting factor used in 
process block 81 in FIG. 6 is as follows: 
##EQU17## 
where .alpha. is the rotational position of the gantry, s is the position 
of the x-ray detector which is being weighted with respect to the center 
of the detector array, R is the distance between the x-ray source and the 
center of the detector array, and 
##EQU18## 
In contrast to the parallel beam acquisition, this weighting factor not 
only varies as a function of gantry position .alpha., but also as a 
function of the location of the detector in the flat array. During the 
back projection process of block 82 a different formula than equation [27] 
is used for selecting the proper attenuation value for each pixel (x,y). 
Many back projection formulas are known in the art such as that described 
in U.S. Pat. No. 4,812,983 entitled "Method and Means of Correcting For a 
Shift in the Center of Rotation of a Rotating Fan-Beam CT System" which is 
incorporated herein by reference. Regardless of the formula used, the 
displacement factor of the present invention is applied by substituting 
the values of x' and y' given above in equations [28] and [29] for the 
values of x and y respectively in the particular back projection formula 
used. 
Fan Beam with Curved Detector 
For a curved detector fan-beam reconstruction the weighting factor used in 
process block 81 in FIG. 6 is as follows: 
##EQU19## 
where .alpha. is the rotational position of the gantry, R is the distance 
between the x-ray source and the central axis of rotation of the gantry, 
and .gamma. is the angle as measured at the x-ray source between the 
central array detector and the detector whose signal is being weighted. 
During the back projection process of block 82, the values of x' and y' 
given above in equations [28] and [29] are substituted for the values of x 
and y respectively in the formula used for back projection. 
While the theory indicates that the weighting factors must be applied to 
correct the projection data before it is used to reconstruct an image 
according to the present invention, experimental results have shown that 
this is not always required. In many cases, a substantial reduction in 
motion artifacts can be achieved without applying the weighting factor and 
only applying the displacement factor to the back projection process. 
It should be apparent to those skilled in the art that the present 
invention is applicable to many different back projection reconstruction 
techniques. Regardless of the back projection technique used, a weighting 
factor can be calculated for each acquired attenuation value in the data 
set and the back projection process can be modified by substituting the 
displacement factors of equations [18], [19], [28] and [29] into the back 
projection formula. This is true regardless of the modality used to 
acquire the projection data. Thus, for example, projection data acquired 
with PET, MRI or SPECT scanners can be corrected for patient motion 
according to the teachings of the present invention. 
It is well-known that in x-ray CT fan-beam reconstruction certain factors 
can be applied to projection data to diminish the effects of patient 
motion. While the mathematics suggests that the present invention will not 
work with such prior methods, experimental results have demonstrated that 
some improvement is in fact obtained when the present invention is used in 
combination with such techniques. 
The above description has been that of a preferred embodiment of the 
present invention. It will occur to those that practice the art that many 
modifications may be made without departing from the spirit and scope of 
the invention. In order to apprise the public of the various embodiments 
that may fall within the scope of the invention, the following claims are 
made.