CT Scanner with improved processing efficiency 180 degrees+ fan angle reconstruction system

An x-ray source (14) is rotated along a non-circular path (18) to irradiate a subject in an examination region (10) with a fan beam (16) of radiation. Radiation detectors convert rays of the fan beam which have traversed the examination region into electronic data which is stored as data fans in an initial data memory (22). Each data fan is zero-filled (24) and convolved by a convolver (30). Preferably, the convolver transforms each data fan into Fourier-space (32), filters each data fan in Fourier-space with a roll-off filter (36), and converts the filtered data fan back from Fourier-space (38). Each fan is weighted (42) by a 1/cos weighting function and stored in a weighted data memory (44). Rays which are redundant in a 180.degree.+ fan reconstruction are removed (46) from the convolved data fans. A pixel driven backprojector (50) backprojects each convolved and redundant ray removed data fan into an image memory (56). The backprojector weights each data value of each ray in proportion to 1/R, where R is a distance from the apex of the backprojected convolved fan to each corresponding pixel of the resultant image. A video processor (60) converts the electronic image representation stored in the image memory (56) into appropriate format for display on a video monitor (62).

BACKGROUND OF THE INVENTION 
The present invention relates to the diagnostic imaging arts. It finds 
particular application in conjunction with computer tomographic (CT) 
scanners and will be described with particular reference thereto. However, 
it is to be appreciated that the present invention is also applicable to 
rotating nuclear or gamma cameras, and the like. 
Early CT scanners were of a traverse and rotate type. That is, a radiation 
source and oppositely disposed radiation detector traversed together along 
linear paths on opposite sides of the subject. The detector was repeatedly 
sampled during the traverse to create a plurality of data values 
representing parallel rays through the subject. After the traverse, the 
entire carriage was rotated a few degrees and the source and detector were 
traversed again to create a second data set. The plurality of parallel ray 
data sets at regular angular intervals over 180.degree. were reconstructed 
into a diagnostic image. Unfortunately, the traverse and rotate technique 
was very slow. 
One technique for speeding traverse and rotate scanners was to replace the 
radiation source and single detector with a radiation source that 
projected radiation along a narrow fan beam and to provide several 
detectors such that a plurality of parallel ray data sets at different 
angles were collected concurrently. In this manner, several of the data 
sets could be collected concurrently. This was several times faster, but 
still very slow. 
Rather than traversing the source and detector, it was found that the 
radiation source could be rotated only. That is, the radiation source 
projected a fan of data which spanned the examination region or scan 
circle. An arc of radiation detectors received the radiation which 
traversed the examination region. The radiation source was rotated around 
the subject. In a third generation scanner, the arc of detectors or an 
entire ring of stationary detectors rotated with the source. In a fourth 
generation scanner, an entire ring of stationary detectors was provided. 
In either type, fan beam data sets were sampled at a multiplicity of 
apexes around the subject. The data from the different angles within the 
fans at different angular orientations of the fan were sorted into 
parallel ray data sets. It was found that a complete set of parallel ray 
data sets could be generated by rotating the source 180.degree.+ the fan 
angle. Although much faster than the traverse and rotate technique, a 
large amount of data processing was required to sort or rebin the rays 
into the parallel ray data sets and to interpolate, as necessary, in order 
to make the rays within each data set more parallel. Although the data 
collection time was much faster, the image processing was slow. 
Rather than sorting the data into parallel ray data sets, it was found that 
the fan beam data sets could be reconstructed directly into an image 
representation by convolution and backprojection. The original convolution 
reconstruction technique for CT scanners is generally credited to A. V. 
Lakshminarayanan. His convolution technique is described, for example, in 
"Convolution Reconstruction Techniques for Divergent Beams", Herman 
Lakshminarayanan, and Naparstek, Comput. Biol. Med. Vol. 6, pp. 259-271 
(1976). Although the convolution and backprojection technique required 
significantly less processing hardware and time than the rebinning 
technique, the data collection was slower. In particular, the 
Lakshminarayanan algorithm required the apexes of the data fans to span a 
full 360.degree., not just the 180.degree.+ fan angle. 
U.S. Pat. No. 4,293,912 of R. Walters describes an improved convolution and 
backprojection technique in which the apexes of the data fans need only 
span 180.degree. plus the fan angle. Although the Walters technique has 
proved successful and even today is among the most widely used CT 
reconstruction algorithms, it still has several drawbacks. One of the 
drawbacks of the Walters technique is that it performs an estimation of 
line integral data which is computationally complex and time consuming. 
Another disadvantage is that in the backprojection, the data is weighted 
as 1/R.sup.2 which requires a squaring operation for every projection of 
every data point. This squaring operation is again time consuming and 
delays the processing time. 
The present application contemplates a CT scanner with a new and improved 
image reconstruction system which overcomes the above-referenced problems 
and others. 
SUMMARY OF THE INVENTION 
In accordance with one aspect of the present invention, a plurality of data 
fans are collected and zero filled. That is, the length of each data fan 
is increased, e.g. doubled, by adding zeroes to the end to eliminate or 
minimize edge error. The zero-filled fans are then convolved and 
backprojected into an image representation. 
In accordance with another aspect of the present invention, each of the 
data fans is convolved. Redundant rays are zeroed out of the convolved 
data sets. The convolved data sets are then backprojected into an image 
representation. 
In accordance with another aspect of the present invention, each of the 
data fans is Fourier transformed and filtered in Fourier-space with a 
roll-off filter. The data is inverse transformed back from Fourier-space 
and weighted in accordance with the distance between the fan apex and the 
origin of the coordinate system. The convolved data sets are then 
backprojected into an image representation. 
In accordance with another aspect of the present invention, fan data is 
convolved. The convolved fan beam data sets are backprojected with a pixel 
driven backprojector into an image representation. Each of the convolved 
data fans is weighted with the selected weighting by image pixel being 
generally in accordance with 1/R, where R is the distance from the fan 
apex to the pixel. 
One advantage of the present invention resides in its processing 
efficiency. 
Another advantage of the present invention is that the apex of the data 
fans need not be constrained to a circle. 
Another advantage of the present invention is that it enables data 
collected from non-circular CT scanners, e.g. CT scanners which are oval 
like the human body, to be reconstructed efficiently. 
Still further advantages of the present invention will become apparent to 
those of ordinary skill in the art upon reading and understanding the 
following detailed description of the preferred embodiments.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
With reference to FIG. 1, a CT scanner or other non-invasive examination 
apparatus examines an interior region of a subject in an examination 
region 10 and generates data indicative thereof. The CT scanner includes a 
subject support 12 for supporting a workpiece or patient to be examined in 
the examination region. An irradiating means 14, such as an x-ray tube, 
irradiates the patient with a fan beam 16 of x-rays or other penetrating 
radiation. The irradiating means 14 is mounted such that an apex of the 
fan beam of radiation rotates around the examination region 10. In the 
illustrated embodiment, the x-ray source moves around an oval track 18 
which generally parallels the cross-section of the subject. Rays of the 
fan-shaped beam of radiation which have traversed the examination region 
are received by a ring of radiation detectors 20. In the illustrated 
embodiment, the ring of radiation detectors extends along an oval path 
which generally parallels the cross-section of the subject. Alternately, 
the x-ray detectors can be mounted to the track or x-ray tube such that 
the detectors and x-ray beam rotate together with the detectors remaining 
across the examination region from the x-ray source. 
A fan beam data memory means 22 stores original electronic data fans from 
the CT scanner. Each data fan S(n) has 2N+1 data values representing 2N+1 
rays along the fan. More specifically, an initial fan has data values from 
S.sub.0 (-N) through S.sub.0 (N). M fans of data are collected with the 
M-th fan having 2N+1 data values S.sub.M-1 (-N) through S.sub.M-1 (N). A 
zero-filling means or step 24 pads zero data values to the end of each 
data fan. In the preferred embodiment, the zero-filler 24 adds 2N+1 zeroes 
to the end of each fan to protect against edge error. A zero-filled data 
memory 26 stores at least one of the zero-padded or zero-filled data fans. 
A convolver means 30 convolves each zero-padded data fan. The convolver 
means includes a one-dimensional Fourier transform means 32, such as a 
Fast Fourier transform chip, which performs a one-dimensional transform 
along each zero-padded data fan transforming it into Fourier-space. A 
Fourier-space buffer or memory means 34 stores the Fourier-space data 
fans. A roll-off filter, preferably the roll-off filter described in 
Equation (39) below, filters each Fourier-space data fan to sharpen the 
data. An inverse one-dimensional Fourier transform means 38 transforms 
each filtered data fan from Fourier-space back to data space. A filtered 
data fan buffer or memory means 40 stores each filtered data fan in data 
space. Optionally, the roll-off filter means 36 operates on each data fan 
in the time domain, eliminating the Fourier transform and the inverse 
Fourier transform means or steps 32, 38. However, with the preferred 
roll-off filter, the computational speed is accelerated by filtering in 
Fourier-space. 
A weighting means 42 weights each data value of each filtered data fan in 
accordance with a distance between the vertex of the fan and each data 
point. More specifically to the preferred embodiment, the weighting 
function weights across each data fan with a 1/cos weighting with the 
distance from the vertex to the center, preferably with the weighting 
function of Equation (44) below. Each weighted data fan is stored in a 
weighted data fan buffer or memory means 44. A redundant ray removal means 
46 eliminates redundant rays of each convolved data fan. Removing 
redundant rays after convolution eliminates the sharp discontinuities that 
typically occur when the redundant rays are removed prior to convolution. 
The redundant rays may be zeroed in accordance with Equation (32) below, 
or may be averaged in accordance with Equation (33) below. A redundant ray 
corrected convolved data buffer or memory means 48 stores the convolved, 
redundancy corrected data rays. 
A pixel driven backprojecting means 50 backprojects each of the convolved 
data fans to reconstruct an electronic diagnostic image representation. A 
weighting means 52 weights each data fan, image pixel by image pixel. The 
weighting is again proportional to the inverse of the first power of the 
distance R, i.e. 1/R, from the fan vertex to each image pixel. An adding 
means 54 adds each weighted data fan to appropriate rows of the electronic 
image representation assembled and stored in a reconstructed image memory 
56. 
A video processor 60 converts image data from the image memory 66 into 
appropriate format for display on a video monitor 62. Optionally, printers 
and other known devices may be utilized for converting the image data 
stored in the image memory 56 into a human-readable format. 
Looking at the present reconstruction technique from a more mathematical 
perspective, Radon's inversion formula provides the reconstruction of a 
function f from its line integrals 
##EQU1## 
Below is a simple derivation of the inversion formula. 
The projection theorem provides the relationship between the one 
dimensional Fourier transform of p.sub..theta. with the two-dimensional 
Fourier transform of the function f along the line of direction 
.THETA.=(cos .theta., sin .theta.). 
More precisely, 
EQU p.sub..theta. (.rho.)=f(.rho..THETA.). (2) 
Expressing the function f as the inverse Fourier transform of f, 
##EQU2## 
In polar coordinate, the above equation can be rewritten as, 
##EQU3## 
From Equation (2) of the projection theorem, 
##EQU4## 
The inner integral of the right hand side of the last equation is nothing 
but the convolution of p.sub..theta. with the convolution filter q which 
is the inverse Fourier transform of the function 
.vertline..rho..vertline., i.e. 
##EQU5## 
where, 
##EQU6## 
because the inverse Fourier transform of the function 
.vertline..rho..vertline. is FP(-1/2.pi..sup.2 l.sup.2). Equation (6) is 
usually referred to as the Radon inversion formula, or convolution 
back-projection algorithm. 
The above inversion formula is suitable for parallel beam geometry data 
collection. In practice, the geometry of the data collection is commonly a 
fan beam geometry. We shall establish a reconstruction formula which 
requires the data collection of 180 degree plus fan beam angle. To do 
this, we shall re-evaluate the convolution 
##EQU7## 
of Radon's inversion formula, Equation (6) using fan beam geometry. In 
fact, by replacing the expression P.sub..theta. by its definition, we 
obtain, 
##EQU8## 
The above integral is over the whole plane in cartesian coordinates. We 
shall express this integral using the polar coordinate system in order to 
derive a reconstruction algorithm from fan beam data. We are restricting 
ourselves to the case of an object with finite extent, i.e., the function 
f is of compact support, and we assume that the trajectory .PHI. of the 
vertex of the fan satisfies the following condition: 
For any given couple (X, .theta.), there exists a .psi. with the following 
properties: 
EQU r=&lt;X,.THETA.&gt;=&lt;.PHI.(.psi.),.THETA.&gt;, (10) 
and 
EQU &lt;.PHI.(.psi.),.THETA..sup..perp. &gt;.noteq.0. (11) 
Here we assume that the cure .PHI. is a function of the angle .psi.. 
To satisfy Equation (10) the vertex .PHI.(.psi.) must be on the line 
perpendicular to the direction .THETA. and going through the point X, see 
FIG. 2. To satisfy Equation (11), .PHI.(.psi.) must not be on the line 
going through the origin of the coordinate system, and parallel to 
.THETA.. 
For each couple (X, .theta.), we make a change of variables defined by the 
following equation: 
EQU l.THETA.+t.THETA..sup..perp. =.PHI.(.psi.)+.rho.A, (12) 
where .psi. satisfies Equation (10). Since the Jacobian of this 
transformation is equal to .rho., and since l=&lt;.PHI.(.psi.)+.rho.A, 
.THETA.&gt;, Equation (9) can be rewritten as, 
##EQU9## 
Defining the fan beam data at the vertex .PHI.(.psi.) as 
EQU g.sub..psi. (.alpha.)=.sub.].sub.0.sup..infin. f(.PHI.(.psi.)+.rho.A) 
d.rho., (14) 
we have the following equation: 
##EQU10## 
From Equation (10), we have 
##EQU11## 
By differentiating with respect to .theta. first, and by using Equation 
(11), Equation (15) can be rewritten as, 
##EQU12## 
Let .phi. denotes the angle (.PHI.(.psi.), .PHI.(.psi.)-X), .gamma. the 
angle between .PHI.(.psi.) and the ray parallel to the vector A going 
through .PHI.(.psi.), and .GAMMA. the half-fan angle, see FIG. 2. 
Expressing the angle .alpha. in terms of .gamma. and .psi., and the inner 
product of two vectors as the product of their modules with the cosine of 
their angle, we obtain, 
##EQU13## 
Consequently, the integral of the right hand side of the above equation is 
nothing but the convolution of the function g.sub..psi. with the function 
h defined by 
##EQU14## 
It should be noted that there is no assumption made on the trajectory of 
the vertex .PHI.(.psi.). In particular, the trajectory is not required to 
be a circle. It might further be noted that the vertex .PHI.(.psi.) does 
not have to be outside of the support of the function to be reconstructed. 
From Equations (6), (7), and (8), we have the following inversion formula 
for fan beam reconstruction 
##EQU15## 
where Q.sub..theta. (r) is given by Equation (18). 
A natural and economical implementation of the above equation is to 
back-project each convolved view to every pixel of the image in the 
reconstruction field. The reconstructed image is simply the accumulating 
sum of the back-projections of convolved fan-beam views. For this purpose, 
we shall define a fan view s.sub..psi. for each angle .psi. associated to 
a vertex .PHI.(.psi.) by the following equation: 
EQU s.sub..psi. (.gamma.)=g.sub..psi. (.gamma.+.psi.+.pi.). (21) 
We also define the convolution q.sub..psi. of the view s.sub..psi. with 
the filter h by 
##EQU16## 
We have the following relation 
EQU q.sub..psi. (.PHI.)=Q.sub..theta. (r), (23) 
if 
##EQU17## 
The following important observations can be made: 1. For any point X along 
a ray joining the vertex of the starting view, we have sufficient data to 
reconstruct f(X) as soon as we have collected data all the way to the 
intersection of the trajectory .PHI. with the ray. In fact, if we assume 
that the first fan view starts at the angle .psi., then the angle 
associated to the case when the vertex is at the intersection of the 
trajectory .PHI. and the ray is given by (See FIG. 3) 
EQU .psi.'=.psi.+(.pi.-.phi.'+.phi.). (25) 
Note that the above relation is based solely on the fact that the sum of 
angles of a triangle is 180 degree, and no assumption is made on the 
trajectory .PHI.. 
According to Equation (24), we have 
##EQU18## 
For this reason, with the current geometry where both the trajectory .PHI. 
and the scan field are concentric circles, we need 180 degree plus fan 
angle in order to be able to reconstruct pixels of the image along the 
second tangent to the reconstruction field, see FIG. 4. 
In order to express the right hand side of Equation (20) in terms of the 
angle .psi., we compute d.theta.. Based on Equation (24), we have 
##EQU19## 
If u and v are the coordinates of the point X in the new coordinate system 
with O .PHI.(.psi.) being the second axis, see FIG. 5, 
EQU u=d sin(.phi.), (28) 
where d is the distance from the point X to .PHI.(.psi.). Consequently, by 
differentiating both sides of the above equation with respect to the 
variable .psi., we obtain 
##EQU20## 
Hence, Equation (27) can be rewritten as 
##EQU21## 
From the above relations, the reconstruction Equation (20) becomes 
##EQU22## 
where, a) .psi..sub.X is the angle of the vertex which is the intersection 
of the trajectory .PHI. with the ray joining the starting view 
.PHI.(.psi..sub.0) and the point X, and 
b) .phi.(X) is the ray joining the vertex .PHI.(.psi.) and the point X. 
Note that in a numerical implementation of the above formula, the total 
number of views involved in the back-projection depends on the point X. 
However, this number is constant for all points along a line emanating 
from the vertex of the first view. 
If we zero out the convolved view q.sub..psi., from the last ray of the fan 
to the ray (of angle .phi..sub.0 (.psi.)) joining the vertex .PHI.(.psi.) 
and the vertex of the starting view, i.e. if we define q.sub..psi. ' by 
##EQU23## 
or, alternatively, if we multiply the convolved data along the two 
opposite rays with weights whose sum is equal to one, i.e. 
##EQU24## 
where w.sub..psi. is a differentiable function which is equal to 1 if 
.phi.&lt;.phi..sub.0 (.psi.), and which rolls down to zero at .phi.=.GAMMA., 
we obtain the following 180 degree plus fan reconstruction formula 
##EQU25## 
From the above observation, it follows that: 2. If the trajectory .PHI. is 
such that every line going through the reconstruction field intersects 
.PHI. at two points (e.g. a circle), then we can back-project all views 
for the complete 360 degree and reconstruct 2f(X), so that 
##EQU26## 
Here we address the case when there is a finite number of views and rays. 
We assume further that, 
1. the total number of views is M, and the angle between two consecutive 
views is constant and equal to .DELTA..psi., 
2. each view has (2N+1) equal angular rays with .DELTA..phi.=.GAMMA.N being 
the angle between two consecutive rays. 
Under these assumptions, the numerical evaluation of Equations (34) and 
(35) can be carried out as follows. 
The convolved view q.sub..psi. given by Equation (22) can be evaluated 
using Simpson's method to compute an integral. From the definition of FP 
of an integral with kernel 1/(l.sup.2), q.sub..psi. can be rewritten as 
##EQU27## 
Defining S.sub.m and Q.sub.m by 
##EQU28## 
for m=0, 1, . . . , M-1, and n=-N, . . . , N, the convolved view 
q.sub..psi. is approximated by taking .epsilon.=.DELTA..PHI., yielding 
##EQU29## 
for j=-N, . . . , N. 
Beside the weight, the right hand side of the above equation is the 
discrete convolution of S.sub.m with the kernel K defined by 
##EQU30## 
The last step of the reconstruction involves the numerical evaluation of 
the integral either in Equation (34) or (35). This integration is known as 
the back-projection, i.e. the summation of weighted convolved views along 
rays going through the point X. This back-projection involves two steps. 
The first step is to estimate the expression q.sub..psi. '(.phi.(x)) in 
Equation (34) or q.sub..psi. (.phi.(x)) in Equation (35). The second step 
is to weight such an estimate and then to integrate the product from the 
starting fan to the last fan. Again, Simpson's method will be used to 
estimate an integral. 
For the first step of this numerical evaluation, an array Q.sub.m ' is 
formed from Q.sub.m based on Equation (32) or (33) in the case of 180 
degree plus fan angle reconstruction. The expression q.sub..psi. 
'(.phi.(x)) in Equation (34) is estimated by an interpolation I as a 
function of the array Q.sub.m ' and the point X, i.e. 
EQU q.sub..psi. '(.phi.(X)).congruent.I(Q.sub.m ',X). (40) 
In the case of linear interpolation, for example, the j.sup.th ray is 
located so that the point X lies within the sector defined by the rays j 
and (j+1). From this, the coefficient of proportionality .lambda..sub.m,X 
is calculated, and 
EQU I(Q.sub.m ',X)=Q.sub.m '(j)+.lambda..sub.m,X (Q.sub.m '(j+1)-Q.sub.m '(j)). 
(41) 
Finally, the value f(X) is estimated based on Equation (34) as 
##EQU31## 
with 
EQU .psi.=m.DELTA..psi.+.psi..sub.0. (43) 
The numerical evaluation of Equation (35) in terms of Q.sub.m can be 
carried out in a fashion similar to the above described procedure. 
The reconstruction procedure described below assumes input data in the form 
of line integrals {S.sub.m (n), m=0, . . . , M-1; n=-N, . . . , N} as 
given in Equation (37). From these input data, the following steps are 
carried out: 
1. Initialize m to zero, and set F(X)=0 for all pixels X. 
2. Compute the convolved view Q.sub.m of the view S.sub.m according to 
Equation (38). This computation may be broken up as follows: 
a) compute the discrete Fourier transform of S.sub.m after padding it with 
0, 
b) multiply the previous result with the discrete Fourier transform of the 
kernel K which is given by Equation (39), or alternatively with the 
product of the discrete Fourier transform of K with some window function, 
such as Hanning window, for a smoother reconstructed image, 
c) compute the discrete inverse Fourier transform of the output from the 
previous step, 
d) multiply the obtained result with the weight w.sub.m defined by 
##EQU32## 
3. Generate Q.sub.m ' by zeroing out the convolved view Q.sub.m along the 
appropriate rays based on the condition specified in Equations (32), or by 
weighting Q.sub.m based on Equation (33). 
4. Back-project Q.sub.m ' by generating the accumulating sum 
##EQU33## 
for all pixels X. 5. Increment m by 1, and go to step 2 if m&lt;M. 
When m=M-1, the accumulating sum F(X) is an estimate of f(X) for all pixels 
X. 
The invention has been described with reference to the preferred 
embodiment. Obviously, modifications and alterations will occur to others 
upon reading and understanding the preceding detailed description. It is 
intended that the invention be construed as including all such 
modifications and alterations insofar as they come within the scope of the 
appended claims or the equivalents thereof.