Systems and methods for reconstructing an image in a CT system performing a cone beam helical scan

A method for reconstructing an image in a computed tomography system performing a cone beam helical scan is described. In accordance with one embodiment, a point is selected for which image data is to be generated and a ray pair is identified wherein each ray passes through the selected point. Further, each ray in the ray pair is related according to view angle and detector angle associated with each ray. Projection data of each ray is then weighted to generate image data for the selected point.

FIELD OF THE INVENTION 
This invention relates generally to computed tomography (CT) imaging and 
more particularly, to reconstructing images from projection data obtained 
in cone beam helical scans. 
BACKGROUND OF THE INVENTION 
In at least one known CT system configuration, an x-ray source projects a 
fan-shaped beam which is collimated to lie within an X-Y plane of a 
Cartesian coordinate system, generally referred to as the "imaging plane". 
The x-ray beam passes through the object being imaged, such as a patient. 
The beam, after being attenuated by the object, impinges upon an array of 
radiation detectors. The intensity of the attenuated beam radiation 
received at the detector array is dependent upon the attenuation of the 
x-ray beam by the object. Each detector element of the array produces a 
separate electrical signal that is a measurement of the beam attenuation 
at the detector location. The attenuation measurements from all the 
detectors are acquired separately to produce a transmission profile. 
In known third generation CT systems, the x-ray source and the detector 
array are rotated with a gantry within the imaging plane and around the 
object to be imaged so that the angle at which the x-ray beam intersects 
the object constantly changes. A group of x-ray attenuation measurements, 
i.e., projection data, from the detector array at one gantry angle is 
referred to as a "view". A "scan" of the object comprises a set of views 
made at different gantry angles during one revolution of the x-ray source 
and detector. In an axial scan, the projection data is processed to 
construct an image that corresponds to a two dimensional slice taken 
through the object. 
One method for reconstructing an image from a set of projection data is 
referred to in the art as the filtered back projection technique. This 
process converts the attenuation measurements from a scan into integers 
called "CT numbers" or "Hounsfield units", which are used to control the 
brightness of a corresponding pixel on a cathode ray tube display. 
To reduce the total scan time required for multiple slices, a "helical" 
scan may be performed. To perform a "helical" scan, the patient is moved 
in the z-axis synchronously with the rotation of the gantry, while the 
data for the prescribed number of slices is acquired. Such a system 
generates a single helix from a fan beam helical scan. The helix mapped 
out by the fan beam yields projection data from which images in each 
prescribed slice may be reconstructed. In addition to reduced scanning 
time, helical scanning provides other advantages such as better control of 
contrast, improved image reconstruction at arbitrary locations, and better 
three-dimensional images. 
Cone beam helical scanning also is known. A cone beam scan is performed 
using a multi-dimensional detector array instead of a linear detector 
array as is used in a fan beam scan. In a cone beam helical scan, the 
x-ray source and the multi-dimensional detector array are rotated with a 
gantry within the imaging plane as the patient is moved in the z-axis 
synchronously with the rotation of the gantry. Such a system generates a 
multi-dimensional helix of projection data. As compared to fan beam 
helical scanning, cone beam helical scanning provides improved slice 
profiles, greater partial volume artifact reduction, and faster patient 
exam speed. 
Generally, in cone beam helical scanning, approximately one-helical-pitch 
worth of data on each side of the image slice is used to generate the 
image data. Specifically, image data on each side of the image slice, and 
360.degree. apart is interpolated to reconstruct an image slice. The 
method for generating the image using one-helical-pitch worth of data on 
each side of the slice is sometimes referred to as a "360.degree. 
interpolation" method, and is effective in reducing inconsistency 
artifacts. 
The one-helical-pitch interpolation method described above generally 
requires a total of two-helical-pitch (720.degree.)worth of data to 
reconstruct each slice, i.e. , one-helical-pitch worth of data on each 
side of the slice. The 360.degree. interpolation method, therefore, does 
not provide satisfactory results at end regions along the z-axis. 
Specifically, any slice to be reconstructed within the "first" or "last" 
helical pitch worth of data along the z-axis does not have the requisite 
two-helical-pitch worth of data. These end regions are sometimes referred 
to herein as "dead regions." Moreover, using a long range interpolation to 
reconstruct slices broadens the slice profile. 
Also, in some applications, reduced scanning time or increased volume 
coverage is required. Without changing other system parameters, the time 
reduction and coverage increase can be achieved by moving the table 
faster, which results in each slice being supported by less than 27r worth 
of cone beam data. 
It would be desirable to provide, in a cone beam helical image 
reconstruction, a manner for reducing the extent of the dead regions and 
slice profile broadening. It also would be desirable to enable the table 
speed to be increased in a cone beam system. 
SUMMARY OF THE INVENTION 
These and other objects may be attained in a cone beam helical scanning 
image reconstruction system which, in one embodiment, reduces the extent 
of the dead regions and slice profile broadening. Specifically, for a 
slice to be reconstructed, the system identifies ray pairs relating to the 
slice and uses projection data of the ray pairs to reconstruct the slice. 
Particularly, by dividing one rotation worth of cone beam data into two 
groups, and by combining the corresponding rays from each group to 
compensate for errors caused by the table movement, rays can be identified 
and utilized to reconstruct an image. A two-dimensional helical 
extrapolative algorithm may be extended, for example, to three dimensions 
and used for reconstructing the slice. 
Reconstructing an image slice using the above described method requires 
180.degree. worth of data on each side of the slice. The method is 
sometimes referred to herein as a "180.degree. interpolation" method. 
While still maintaining the effectiveness in reducing inconsistency 
artifacts, the method also reduces the extent of the dead region on both 
ends to one half-helical-pitch and facilitates reducing slice profile 
broadening. 
In another aspect, the present invention enable increasing table speed, 
i.e., a faster scan. More particularly, and with respect to the slice to 
be reconstructed, the cone beam projection data that have contributions to 
reconstruction of the slice are identified. The redundant data is weighted 
so that redundant ray weights have a sum of unity, or one. By limiting the 
contribution of redundant rays using the above described algorithm, the 
amount of data required for cone beam helical image reconstruction can be 
reduced. Reconstructing an image slice using the above described method 
requires less than 360.degree. worth of data and enables table speed to 
increase while maintaining image quality.

DETAILED DESCRIPTION OF THE DRAWINGS 
Referring to FIGS. 1 and 2, a computed tomograph (CT) imaging system 10 is 
shown as including a gantry 12 representative of a "third generation" CT 
scanner. Gantry 12 has an x-ray source 14, or focal spot, that projects a 
beam of x-rays 16 toward a detector array 18 on the opposite side of 
gantry 12. Detector array 18 is formed by detector elements 10 which 
together sense the projected x-rays that pass through a medical patient 
22. Each detector element 20 produces an electrical signal that represents 
the intensity of an impinging x-ray beam and hence the attenuation of the 
beam as it passes through patient 22. During a scan to acquire x-ray 
projection data, gantry 12 and the components mounted thereon rotate about 
a center of rotation 24. 
Rotation of gantry 12 and the operation of x-ray source 14 are governed by 
a control mechanism 26 of CT system 10. Control mechanism 26 includes an 
x-ray controller 28 that provides power and timing signals to x-ray source 
14 and a gantry motor controller 30 that controls the rotational speed and 
position of gantry 12. A data acquisition system (DAS) 32 in control 
mechanism 26 samples analog data from detector elements 20 and converts 
the data to digital signals for subsequent processing. An image 
reconstructor 34 receives sampled and digitized x-ray data from DAS 32 and 
performs high speed image reconstruction. The reconstructed image is 
applied as an input to a computer 36 which stores the image in a mass 
storage device 38. Preferably, the reconstructed image is stored as a data 
array. The methods described herein may be performed by image 
reconstructor 34, computer 36, an external computer (not shown), or a 
combination of such apparatus. 
Computer 36 also receives commands and scanning parameters from an operator 
via console 40 that has a keyboard. An associated cathode ray tube display 
42 allows the operator to observe the reconstructed image and other data 
from computer 36. The operator supplied commands and parameters are used 
by computer 36 to provide control signals and information to DAS 32, x-ray 
controller 28 and gantry motor controller 30. In addition, computer 36 
operates a table motor Controller 44 which controls a motorized table 46 
to position patient 22 in gantry 12. Particularly, table 46 moves potions 
of patient 22 through gantry opening 48. 
Referring to FIG. 3, cone beam three dimensional rays are used to 
reconstruct an image with two-helical-pitches (720.degree.) worth of data. 
In helical scanning, gantry 12 translates one helical pitch, .DELTA.Z, in 
each gantry rotation. The trajectory of x-ray source 14 is shown as line 
50. During a scan, gantry 12 translates several (N) helical pitches. As 
shown in FIG. 7, there are dead regions 52 for which it is difficult to 
provide a satisfactory image reconstruction. As shown, each dead region 52 
is typically one helical pitch, .DELTA.Z, in length. 
In one aspect, the present invention is directed to a method for reducing 
the extent of such dead regions. Specifically, to reduce the extent of 
such dead regions, and in accordance with one embodiment of the present 
invention, ray pairs (.beta..sub.1, .gamma..sub.1, z.sub.1) and 
(.beta..sub.2, .gamma..sub.2, z.sub.2) are identified with respect to a 
point I to be reconstructed. The projection data corresponding to the ray 
pair is then weighted to reconstruct the image at point I. 
To further understand the problem overcome by the present method, FIGS. 
4(a), 4(b) and 4(c) indicate data inconsistency in cone beam helical 
scanning. Specifically, and referring to FIG. 4(a), plane P represents the 
image to be reconstructed from a helical scan. As shown, plane P is 
centered in one helical pitch .DELTA.Z (360.degree.) worth of data. If a 
distance from x-ray source 14 to plane P is designated as H.sub.0, the 
relationship between H.sub.0 and view angle .beta. (or projection angle), 
as shown in FIG. 4(b), is: 
EQU .beta.=(H.sub.0 .DELTA.Z+0.5).pi.. (1) 
While the source to plane distance, H.sub.0, would ideally be zero for 
every view angle, .beta., distance H.sub.0 is generally non-zero in cone 
beam scanning. Furthermore, due to the table advancement in helical 
scanning, the value of H discontinues (from -0.5.DELTA.Z to 0.5.DELTA.Z) 
in the direction where the first view (.beta.=0) meets the last view 
(.beta.=2 .pi.) as shown in FIG. 4(c). This discontinuity is known to 
cause inconsistency artifacts in reconstruction. The larger the value of 
H, the greater the inconsistency artifacts. 
In known cone beam systems, and to reduce inconsistency artifacts, two 
rays, (.beta..sub.1,.gamma..sub.1,z.sub.1) and 
(.beta..sub.2,.gamma..sub.2,z.sub.2), from opposite sides of the plane P 
passing through a point I are used to reconstruct point I in a plane P. 
The projection data of corresponding rays are 
P(.beta..sub.1,.gamma..sub.1,z.sub.1) and 
P(.beta..sub.2,.gamma..sub.2,z.sub.2). 
For example, and referring specifically to FIG. 5, to reconstruct a point I 
on the plane P, the data from a view V.sub.1 is combined with the 
corresponding data from a view V.sub.2 which is on the other side of the 
plane P, and one helical pitch, .DELTA.Z (360.degree.), away from view 
V.sub.1. Using (.beta..sub.1,.gamma..sub.1,z.sub.1) and 
(.beta..sub.2,.gamma..sub.2,z.sub.2) to denote two rays coming from 
V.sub.1 and V.sub.2, respectively, and passing through the point I, the 
contribution of these two rays to the point I, denoted as .DELTA.R, is: 
EQU .DELTA.R=w.sub.1 P(.beta..sub.1 .gamma..sub.1,z.sub.1)+w.sub.2 
P(.beta..sub.2,.gamma..sub.2,z.sub.2), (2) 
where: 
EQU .beta..sub.2 =.beta..sub.1 -2 .pi., and (3a) 
EQU .gamma..sub.2 =.gamma..sub.1, (3b) 
where 2 .pi. represents that V.sub.1 and V.sub.2 are 360.degree. apart, 
i.e., one helical pitch apart, and w.sub.1 and w.sub.2 are weighting 
functions. 
However, as shown in FIG. 5, reconstructing a slice using this method 
requires one-helical-pitch (360.degree.) worth of data on each side of the 
slice. This method does not provide satisfactory reconstructions in 
regions of one helical pitch length on both ends of the z extent covered 
by a helical scanning, i.e., dead regions. Moreover, this long range 
interpolation broadens the slice profiles. 
In fan beam helical scanning, it is known to use a helical extrapolation 
weighting function, w.sub.hs, to permit reconstruction with 360.degree., 
or one rotation, worth of data. However, cone beam projection data space, 
because of its three-dimensional characteristics, is more complicated than 
fan beam projection space. Therefore, a helical extrapolation weighting 
function, w.sub.hs, is not immediately applicable to cone beam projection 
data. 
In accordance with one embodiment of the present invention, a cone beam 
helical extrapolation weighting function, w.sub.cs, is applied to the cone 
beam projection data. The cone beam helical extrapolation weighting 
function, w.sub.cs, in one embodiment, is related to w.sub.hs, in 
accordance with the following equation: 
EQU w.sub.cs (.beta.,.gamma.,z)=w.sub.hs (.beta.-.beta..sub.0 +.pi., 
.gamma.).(4) 
where .beta..sub.0 denotes the view (rotational) angle at which the x-ray 
beam coincides with the slice to be constructed. 
More particularly, weighting function, w.sub.cs, is applied to three 
dimensional cone bean projection data to enable reconstruction of a slice 
from only one rotation worth of data. Therefore, the dead regions are 
reduced to a half-helical-pitch in length and slice profile broadening is 
reduced. 
In this embodiment of the present invention, images are reconstructed by 
interpolating data approximately 180.degree. apart. Specifically, one 
rotation worth of projection data is divided into two groups. Two 
corresponding rays from the two groups are then combined to compensate for 
errors caused by the table translation. Particularly, the two 
corresponding rays passing through the point I on the plane P 
(.beta..sub.1,.gamma..sub.1,z.sub.1) and 
(.beta..sub.2,.gamma..sub.2,z.sub.2) satisfy the following relations: 
EQU .beta..sub.2 =.beta..sub.1 +.pi.-2.gamma..sub.1, and (5a) 
EQU .gamma..sub.2 =.gamma..sub.1. (5b) 
The focal spot positions of these two rays need not to be on both, i.e., 
opposing, sides of the plane P. 
Both projection data measurements from tiffs ray pair, 
P(.beta..sub.1,.gamma..sub.1,z.sub.1) and 
P(.beta..sub.2,.gamma..sub.2,z.sub.2), contribute to the reconstruction of 
the point I. Particularly, if the ray pair contribution to the point I on 
the plane P is denoted as .DELTA.R, then: 
EQU .DELTA.R=w.sub.cs1 P(.beta..sub.1,.gamma..sub.1,z.sub.1)+w.sub.cs2 
P(.beta..sub.2,.gamma..sub.2,z.sub.2) (6) 
where w.sub.cs1 and w.sub.cs2 are weighting functions. 
As is known, the further the distance, H, i.e., the distance from the focal 
spot to the plane to be reconstructed, the greater the inconsistency 
artifacts. Therefore, each ray is weighted according to such distance so 
that the measurements nearer the plane P will have a larger contribution. 
For example, let H.sub.1 and H.sub.2 be the distances from x-ray source 14 
to the plane P for these two rays. The weights are given by: 
EQU w.sub.cs1 =H.sub.2 /(H.sub.2 -H.sub.1), and (7) 
EQU w.sub.cs2 =H.sub.1 /(H.sub.2 -H.sub.1). 
Given Equation (7), the weights can be written as: 
EQU w.sub.cs1 =(.beta..sub.2 -.beta..sub.0)/(.beta..sub.2 -.beta..sub.1), and(8 
) 
EQU w.sub.cs2 =(.beta..sub.0 -.beta..sub.1)/(.beta..sub.2 -.beta..sub.1). 
For .beta..sub.0 =.pi.. 
Given Equations (5a) and (5b), the weights then become: 
EQU w.sub.cs (.beta.,.gamma.)=1+(.beta.-.pi.)/(.pi.-2.gamma.) for 
0&lt;.beta.&lt;.pi.-2.gamma., and (9) 
EQU w.sub.cs (.beta.,.gamma.)=1-(.beta.-.pi.)/(.pi.-2.gamma.) for 
.pi.-2.gamma.&lt;=.beta.&lt;2 .pi.. 
A feathering method can be used to smooth the discontinuity across the line 
.beta.=.pi.-2.gamma. in the weighting function. This 180.degree. 
interpolation method thus permits reconstruction of an image with fewer 
than two gantry rotations in a cone beam scan. When the helical pitch, 
.DELTA.Z, equals zero, cone beam helical reconstruction reduces to fan 
beam reconstruction and the cylinder in FIG. 4(a) collapses into circle 54 
in the middle. In this extreme case, the ray pair represents two redundant 
data measured from opposite directions. The 180.degree. interpolation 
method, as described above, thus permits reconstruction of an image by 
interpolating data only approximately 180.degree. apart. As a result, the 
dead regions and slice profile broadening are reduced. 
In another aspect, the present invention enables increasing the table 
speed. More specifically, and in one embodiment, cone beam three 
dimensional rays are used to reconstruct an image slice with less than 
360.degree. worth of data. For example, and referring to FIG. 6, during a 
fan beam half-scan, x-ray source 14 projects x-rays 16 at a projection 
angle .beta. and a fan angle .gamma..sub.m. Detector array 18 receives the 
x-rays at a detector angle .gamma. and generates projection data, or 
attenuation measurements. Let .beta..sub.0 denote the projection angle at 
which x-ray source 14 coincides with the slice to be reconstructed. The 
projection measurements are sometimes redundant. Specifically, 
measurements along the same line in opposite directions contain same 
information, except for noise. The two rays on such line, or ray pair 
(.beta..sub.1, .gamma..sub.1), (.beta..sub.2, .gamma..sub.2), satisfy the 
equation: 
EQU .beta..sub.2 =.beta..sub.1 +.pi.+2.gamma..sub.1, and (10a) 
EQU .gamma..sub.2 =-.gamma..sub.1. 
In such circumstances, ray (.beta..sub.1, .gamma..sub.1) is the mirror ray 
of (.beta..sub.2, .gamma..sub.2). 
As shown in FIG. 7, fan beam projection data may be classified as either 
redundant, i.e., corresponding to a ray pair, or non-redundant. In FIG. 7, 
the redundant data is indicated by shaded regions 60 and 62. Specifically, 
FIG. 7 depicts fan-beam projection data relating to projection angle 
.beta., where 0&lt;.beta.&lt;.pi.+2.THETA., .THETA..gtoreq..gamma..sub.m. A ray 
in unshaded region 64 does not have a mirror ray in the fan-beam half-scan 
data set. To avoid gross shading due to this non-uniform sampling density, 
it is known to weight redundant rays in the fan-beam half-scan data set so 
that the sum of their weights equals unity: 
EQU w.sub.FBHS (.beta..sub.1, .gamma..sub.1)+w.sub.FBHS (.beta..sub.1 
+.pi.+2.gamma..sub.1, -.gamma..sub.2)=1, (11) 
where w.sub.FBHS (.beta.,.gamma.) is a weighting function applied to the 
fan-beam half-scan projection data. The weighting functions are generally 
smoothed to avoid producing artifacts. For example, in accordance with one 
known weighting method, the following function can be used: 
EQU w.sub.FBHS (x(.beta.,.gamma.))=3x.sup.2 -2x.sup.3, (12) 
where: 
##EQU1## 
Projection data for cone beam helical scans, however, is more complicated 
because of the multi-dimensional characteristics of detector array 18. 
Referring to FIG. 8, gantry 12 translates one helical pitch (.DELTA.Z) in 
each gantry rotation, with the trajectory of x-ray source 14 indicated by 
line 70. With cone beam helical scans, projection data is acquired, and a 
resulting image is constructed, using three dimensional rays (.beta., 
.gamma., Z). Therefore, the fan beam half-scan algorithm described above 
is not immediately applicable to cone beam helical scan projection data. 
One aspect of the present invention is to enable table speed to be 
increased without sacrificing image quality in a cone beam helical scan. 
If table 46 moves along the z-axis and through gantry opening 48 so that 
plane P is only supported by .THETA. worth of data, where 2 
.pi.&gt;.THETA.&gt;.pi.+fan angle, the focal spot trajectory of this portion of 
the cone-beam data is within the region outlined by a highlighted cylinder 
72. 
In accordance with one embodiment of the present invention, a cone beam 
half-scan weighting function, w.sub.CBHS, is applied to the cone beam 
projection data (.beta.,.gamma.,z). This cone beam half-scan weighting 
function is related to w.sub.FBHS according to the equation: 
EQU W.sub.CBHS (.beta.,.gamma.,z)=W.sub.FBHS (.beta.-.beta..sub.0 +.THETA./2, 
.gamma.). (14) 
By multiplying the weighting to the cone beam projection data as suggested 
above, an image may be reconstructed with less than one gantry rotation 
worth of data. Therefore, the table speed can be increased without 
sacrificing image quality. Although the above description assumes cone 
beam helical scanning, the method can also be applied to the cone beam 
step and shoot scanning. 
From the preceding description of various embodiments of the present 
invention, it is evident that the objects of the invention are attained. 
Although the invention has been described and illustrated in detail, it is 
to be clearly understood that the same is intended by way of illustration 
and example only and is not to be taken by way of limitation. For example, 
although the embodiments of the invention are described herein in the 
context of a cone beam helical scanning system, the present invention 
could be used in a cone beam step and shoot scanning system. Accordingly, 
the spirit and scope of the invention are to be limited only by the terms 
of the appended claims.