Abstract:
Methods, systems and processes for providing efficient, accurate and exact image reconstruction using portable and easy to use C-arm scanning devices and rotating gantries, and the like, that combines both a circle and a curve scan. The invention can provide exact convolution-based filtered back projection (FBP) image reconstruction by combining two curved scans of the object. The curved scan can be less than or greater than a full circle about an object being scanned. The invention can be done by a first curve within a first plane followed by a second curve within a second plane that is transversal to the first plane.

Description:
This invention is a Continuation-In-Part of PCT (Patent Cooperation Treaty) Serial No. PCT/US04/12536 filed Apr. 23, 2004, which is a Continuation-In-Part of U.S. patent application Ser. No. 10/728,136, filed Dec. 4, 2003 now U.S. Pat. No. 7,010,079 which claims the benefit of priority to U.S. Provisional Application, Ser. No. 60/430,802 filed, Dec. 4, 2002, and is a Continuation-In-Part of U.S. patent application Ser. No. 10/389,534 filed Mar. 14, 2003, now U.S. Pat. No. 6,804,321, which is a Continuation-In-Part of Ser. No. 10/389,090 filed Mar. 14, 2003, now U.S. Pat. No. 6,771,733, which is a Continuation-In-Part of Ser. No. 10/143,160 filed May 10, 2002 now U.S. Pat. No. 6,574,299, which claims the benefit of priority to U.S. Provisional Application 60/312,827 filed Aug. 16, 2001. 

   FIELD OF INVENTION 
   This invention relates to computer tomography, and in particular to processes, methods and systems for reconstructing three dimensional images from the data obtained by a circle and arc scan of an object, such as when the C-arm rotates around an object within a first plane, and then the C-arm rotates around the object within another plane that is transversal to the first plane. 
   BACKGROUND AND PRIOR ART 
   Over the last thirty years, computer tomography (CT) has gone from image reconstruction based on scanning in a slice-by-slice process to spiral scanning to also include non-spiral scanning techniques such as those performed with C-arm devices, with all techniques and devices experiencing problems with image reconstruction. 
   From the 1970s to 1980s the slice-by-slice scanning was used. In this mode the incremental motions of the patient on the table through the gantry and the gantry rotations were performed one after another. Since the patient was stationary during the gantry rotations, the trajectory of the x-ray source around the patient was circular. Pre-selected slices through the patient were reconstructed using the data obtained by such circular scans. 
   From the mid 1980s to present day, spiral type scanning has become the preferred process for data collection in CT. Under spiral scanning a table with the patient continuously moves at a constant speed through the gantry that is continuously rotating about the table. At first, spiral scanning has used one-dimensional detectors, which receive data in one dimension (a single row of detectors). Later, two-dimensional detectors, where multiple rows (two or more rows) of detectors sit next to one another, have been introduced. In CT there have been significant problems for image reconstruction especially for two-dimensional detectors. Data provided by the two-dimensional detectors will be referred to as cone-beam (CB) data or CB projections. 
   In addition to spiral scans there are non-spiral scans, in which the trajectory of the x-ray source is different from spiral. In medical imaging, non-spiral scans are frequently performed using a C-arm device, which is usually smaller and more portable than spiral type scanning systems. For example, C-arm scanning devices have been useful for being moved in and out of operating rooms, and the like. 
     FIG. 1  shows a typical prior art arrangement of a patient on a table that moves through a C-arm device, that is capable of rotating around the patient, having an x-ray tube source and a detector array, where cone beam projection data sets are received by the x-ray detector, and an image reconstruction process takes place in a computer with a display for the reconstructed image. 
   There are known problems with using C-arm devices to reconstruct data. See in particular for example, pages 755-759 of Kudo, Hiroyuki et al., Fast and Stable Cone-Beam Filtered Backprojection Method for Non-planar Orbits, IOP Publishing LTD, 1998, pages 747-760. The Kudo paper describes image reconstruction using C-arm devices for various shift-variant filtered back projection (FBP) structures, which are less efficient than convolution-based FBP algorithms. 
   For three-dimensional (also known as volumetric) image reconstruction from the data provided by spiral and non-spiral scans with two-dimensional detectors, there are two known groups of algorithms: Exact algorithms and Approximate algorithms, that each have known problems. Under ideal circumstances, exact algorithms can provide a replication of an exact image. Thus, one should expect that exact algorithms would produce images of good quality even under non-ideal (that is, realistic) circumstances. 
   However, exact algorithms can be known to take many hours to provide an image reconstruction, and can take up great amounts of computer power when being used. These algorithms can require keeping considerable amounts of cone beam projections in memory. 
   Approximate algorithms possess a filtered back projection (FBP) structure, so they can produce an image very efficiently and using less computing power than Exact algorithms. However, even under the ideal circumstances these algorithms produce an approximate image that may be similar to but still different from the exact image. In particular, Approximate algorithms can create artifacts, which are false features in an image. Under certain circumstances and conditions these artifacts could be quite severe. 
   To date, there are no known algorithms that can combine the beneficial attributes of Exact and Approximate algorithms into a single algorithm that is capable of replicating an exact image under the ideal circumstances, uses small amounts of computer power, and reconstructs the exact images in an efficient manner (i.e., using the FBP structure) in the cases of complete circle and arc and incomplete circle and arc scanning. 
   If the C-arm rotates 360 degrees around the patient, this produces a complete circle. If the C-arm rotates less than 360 degrees around the patient, this produces an incomplete circle. In what follows, the word circle covers both complete and incomplete cases. Here and everywhere below by the phrase that the algorithm of the invention reconstructs an exact image we will mean that the algorithm is capable of reconstructing an exact image. Since in real life any data contains noise and other imperfections, no algorithm is capable of reconstructing an exact image. 
   Image reconstruction has been proposed in many U.S. patents. See for example, U.S. Pat. Nos. 5,663,995 and 5,706,325 and 5,784,481 and 6,014,419 to Hu; U.S. Pat. Nos. 5,881,123 and 5,926,521 and 6,130,930 and 6,233,303 and 6,292,525 to Tam; U.S. Pat. No. 5,960,055 to Samaresekera et al.; U.S. Pat. No. 5,995,580 to Schaller; U.S. Pat. No. 6,009,142 to Sauer; U.S. Pat. No. 6,072,851 to Sivers; U.S. Pat. No. 6,173,032 to Besson; U.S. Pat. No. 6,198,789 to Dafni; U.S. Pat. Nos. 6,215,841 and 6,266,388 to Hsieh. Other U.S. patents have also been proposed for image reconstruction as well. See U.S. Pat. No. 6,504,892 to Ning; U.S. Pat. No. 6,148,056 to Lin; U.S. Pat. No. 5,784,481 to Hu; U.S. Pat. No. 5,706,325 to Hu; U.S. Pat. No. 5,170,439 to Zeng et al. 
   However, none of the patents overcome all of the deficiencies to image reconstruction referenced above. The inventor is not aware of any known processes, methods and systems that combines the beneficial attributes of Exact and Approximate algorithms into a single algorithm that is capable of replicating an exact image under the ideal circumstances, uses small amounts of computer power, and reconstructs the exact images in an efficient manner (i.e., using the FBP structure) in the cases of complete circle and arc and incomplete circle and arc scanning. 
   SUMMARY OF THE INVENTION 
   A primary objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects that have been scanned with two-dimensional detectors. 
   A secondary objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that is able to reconstruct an exact image and not an approximate image. 
   A third objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that creates an exact image in an efficient manner using a filtered back projection (FBP) structure. 
   A fourth objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that creates an exact image with minimal computer power. 
   A fifth objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that creates an exact image utilizing a convolution-based FBP structure. 
   A sixth objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that is CB projection driven allowing for the algorithm to work simultaneously with the CB data acquisition. 
   A seventh objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that does not require storing numerous CB projections in computer memory. 
   An eighth objective of the invention is to provide improved processes, methods and systems for reconstructing images of objects scanned with a circle and arc x-ray source trajectory that allows for almost real time imaging to occur where images are displayed as soon as a slice measurement is completed. 
   A preferred embodiment of the invention uses a seven overall step process for reconstructing the image of an object under a circle and arc scan. In a first step a current CB projection is measured. Next, a family of lines is identified on a detector according to a novel algorithm. Next, a computation of derivatives between neighboring projections occurs and is followed by a convolution of the derivatives with a filter along lines from the selected family of lines. Next, using the filtered data, the image is updated by performing back projection. Finally, the preceding steps are repeated for each CB projection until an entire object has been scanned. This embodiment works with keeping several (approximately 2 to approximately 4) CB projections in memory at a time and uses one family of lines. 
   The invention is not limited to an object that undergoes a scan consisting of a single circle and a single arc. The invention can be applied to trajectories consisting of several circles and arcs by applying it to various circle and arc pairs, and then combining the results. 
   The circle and arc scanning can include a partial planar curve scan before or after an arc scan. The planar curved scan can be less than a full circle and even greater than a full circle. Additional and subsequent circle and arc scans can be done consecutively after a first circle and arc scan. 
   Unlike the prior art, the subject invention does not require the patient having to be moved during the measurements. Patients already hooked up to intravenous feeding lines and/or other machines are not easily moveable. Thus, the subject invention can be used as a portable device where the patient does not need to be moved while measurements are taking place. 
   Further objects and advantages of this invention will be apparent from the following detailed description of the presently preferred embodiments, which are illustrated schematically in the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE FIGURES 
       FIG. 1  shows a typical prior art view arrangement of a patient on a table that moves through a C-arm device, that is capable of rotating around the patient, having an x-ray tube source and a detector array, where cone beam projection data sets are received by the x-ray detector, and an image reconstruction process takes place in a computer with a display for the reconstructed image. 
       FIG. 2  shows an overview of the basic process steps of the invention. 
       FIG. 3  shows mathematical notations of the circle and arc scan. 
       FIG. 4  illustrates a stereographic projection from the current source position on to the detector plane used in the algorithm for the invention. 
       FIG. 5  illustrates finding of a filtering line for a reconstruction point x when the x-ray source is on the circle. 
       FIG. 6  illustrates finding of a filtering line for a reconstruction point x when the x-ray source is on the arc. 
       FIG. 7  illustrates a family of lines used in the algorithm of the invention corresponding to the case when the x-ray source is on the circle. 
       FIG. 8  illustrates a family of lines used in the algorithm of the invention corresponding to the case when the x-ray source is on the arc. 
       FIG. 9  is a three substep flow chart for identifying the set of lines, which corresponds to step  30  of  FIG. 2 . 
       FIG. 10  is a seven substep flow chart for preparation for filtering, which corresponds to step  40  of  FIG. 2 . 
       FIG. 11  is a seven substep flow chart for filtering, which corresponds to step  50  of  FIG. 2 .  FIG. 12  is an eight substep flow chart for backprojection, which corresponds to step  60  of  FIG. 2 . 
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Before explaining the disclosed embodiments of the present invention in detail it is to be understood that the invention is not limited in its application to the details of the particular arrangements shown since the invention is capable of other embodiments. Also, the terminology used herein is for the purpose of description and not of limitation. 
   This invention is a Continuation-In-Part of PCT (Patent Cooperation Treaty) Serial No. PCT/US04/12536 filed Apr. 23, 2004, which is a Continuation-In-Part of U.S. patent application Ser. No. 10/728,136, filed Dec. 4, 2003 which claims the benefit of priority to U.S. Provisional Application Ser. No. 60/430,802 filed, Dec. 4, 2002, and is a Continuation-In-Part of U.S. patent application Ser. No. 10/389,534 filed Mar. 14, 2003, now U.S. Pat. No. 6,804,321, which is a Continuation-In-Part of Ser. No. 10/389,090 filed Mar. 14, 2003, now U.S. Pat. No. 6,771,733, which is a Continuation-In-Part of Ser. No. 10/143,160 filed May 10, 2002 now U.S. Pat. No. 6,574,299, which claims the benefit of priority to U.S. Provisional Application 60/312,827 filed Aug. 16, 2001, all of which are incorporated by reference 
   As previously described,  FIG. 1  shows a typical prior art view arrangement of a patient on a table that moves through a C-arm device such as the AXIOM Artis MP, manufactured by Siemens, that is capable of rotating around the patient, having an x-ray tube source and a detector array, where CB projections are received by the x-ray detector, and an image reconstruction process takes place in a computer  4  with a display  6  for displaying the reconstructed image. For the subject invention, the detector array can be a two-dimensional detector array. For example, the array can include two, three or more rows of plural detectors in each row. If three rows are used with each row having ten detectors, then one CB projection set would be thirty individual x-ray detections. 
   Alternatively, a conventional gantry, such as ones manufactured by Siemens, Toshiba, General Electric, and the like, can be used, as shown by the dotted concentric lines, for the X-ray sources and detector array. The gantry can rotate partially, and up to a full circle, or greater than a full circle. 
     FIG. 2  shows an overview of the basic process steps of the invention that occur during the image reconstruction process occurring in the computer  4  using a first embodiment. An overview of the invention process will now be described. 
   A preferred embodiment works with keeping several (approximately 2 to approximately 4) CB(cone beam) projections in computer memory at a time and uses one family of lines. 
   In the first step  10 , a current CB (cone beam) projection set is taken. The next steps  20  and  30  identify sets of lines on a virtual x-ray detector array according to the novel algorithm of the invention, which will be explained later in greater detail. In the given description of the algorithm the detector array can be considered to be flat, so the selected line can be a straight tilted line across the array. 
   The next step  40  is the preparation for the filtering step, which includes computations of the necessary derivative of the CB projection data for the selected lines. 
   The next step  50  is the convolution of the computed derivative (the processed CB data) with a filter along lines from the selected family of lines. This step can also be described as shift-invariant filtering of the derivative of the CB projection data. In the next step  60 , the image of the object being computed is updated by performing back projection. In the final step  70  it is indicated that the above steps  10 - 60  are repeated, unless image reconstruction is completed or there are no more CB projections to process. 
   The invention will now be described in more detail by first describing the main inversion formula followed by the novel algorithm of the invention. 
   Referring to  FIG. 3 , we consider a source trajectory consisting of a circle C 1  and an arc C 2  attached to C 1 . Let y 0  be the point where they intersect.  FIG. 3  shows mathematical notations of the scan. Such a trajectory can be implemented with a C-arm by first rotating the C-arm around the patient in one plane, and then rotating the C-arm within a perpendicular plane. 
   Suppose the trajectory is described by the following parametric equations 1.
 
C 1 :={y ∈ R 3 :y 1 =cos s 1 ,y 2 =sin s 1 ,y 3 =0,0≦s 1 ≦s 1   mx },
 
C 2 :={y ∈ R 3 :y 1 =cos s 2 ,y 2 =0,y 3 =sin s 2 ,0≦s 2 ≦s 2   mx },   (1)
 
where
         C 1  denotes the circle,   C 2  denotes the arc,   s 1 ,s 2  are real parameters, and   y=(y 1 ,y 2 ,y 3 )=y(s) is a point on the trajectory representing the x-ray source position.       

   Without loss of generality we assume that both the circle and arc are of radius 1 and centered at the origin. Let ƒ be a function supported inside the cylinder x 1   2 +x 2   2 ≦r 2 , r&lt;cos s 2   mx &lt;1. The problem is to reconstruct ƒ inside a region of interest U. This set is the union of surfaces U=∪ s     2     ∈[0,s     2       mx     ) S(s 2 ), and each surface S(s 2 ) is formed by line segments connecting the point (cos s 2 ,0,sin s 2 ) to all points (cos s 1 , sin s 1 ,0)∈C 1  with 0≦s 1 ≦s 1   mx  and cos s 1 &lt;cos s 2  (see  FIG. 3 ). 
   Next, pick a reconstruction point x=(x 1 ,x 2 ,x 3 )∈U,x 3 ≧0. We say that x admits a π-line L π (x), if x belongs to a line segment one end-point of which is on C 1 , y(s 1 )∈C 1 , and the other—on C 2 ,y(s 2 )∈C 2  (see  FIG. 3 ). It is shown in the manuscript by A. Katsevich “Image reconstruction for the circle-and-arc trajectory”, submitted for publication, that all x=(x 1 ,x 2 ,x 3 )∈U,x 3 ≧0, admit unique π-lines. The π-line L π (x) determines two parametric intervals. The first one I 1 (x)⊂I 1  corresponds to the section of C 1  between y 0  and y(s 1 )∈C 1 . The second one I 2 (x)⊂I 2  corresponds to the section of C 2  between y 0  and y(s 2 )∈C 2 . 
   We use the following notations in equations 2 and 3 as follows: 
                       D   f     ⁡     (     y   ,   Θ     )       :=       ∫   0   ∞     ⁢       f   ⁡     (     y   +     Θ   ⁢           ⁢   t       )       ⁢     ⅆ   t           ,       Θ   ∈     S   2       ;             (   2   )                   β   ⁡     (     s   ,   x     )       :=       x   -     y   ⁡     (   s   )                x   -     y   ⁡     (   s   )                  ,     x   ∈   U     ,     s   ∈         I   1     ⁡     (   x   )       ⋃         I   2     ⁡     (   x   )       .                 (   3   )               
where
         S 2  is the unit sphere,   ƒ is the function representing the distribution of the x-ray attenuation coefficient inside the object being scanned,   Θ is a unit vector,   D ƒ (y,Θ) is the cone beam transform of ƒ,   β(s,x) is the unit vector from the focal point y(s) pointing towards the reconstruction point x.       
   Next, we suppose s∈I 1 (x). Project x,C 1 ,C 2  onto the detector plane DP(s) as shown in  FIG. 4 . We assume that for each y(s)∈C 1 ∪C 2 , DP(s) contains the origin O and is perpendicular to the line through y(s) and O. The shape of Ĉ 2  on DP(s) depends on s. If 0&lt;s&lt;π, then Ĉ 2  is bent to the right as shown in  FIG. 5 . If s=π, Ĉ 2  is a line segment. If π&lt;s&lt;s 2   mx , then Ĉ 2  is bent to the left. Here and everywhere below we use the convention that whenever a geometrical object is projected onto the detector plane (e.g., point x, circle C 1 , arc C 2 , etc.), the corresponding projection is denoted with hat (e.g., {circumflex over (x)}, Ĉ 1 , Ĉ 2 , etc.). For y(s)∈C 1  define the unit vector by equation 4. 
                       u   1     ⁡     (     s   ,   x     )       :=           y   .     ⁡     (   s   )       ×     β   ⁡     (     s   ,   x     )                    y   .     ⁡     (   s   )       ×     β   ⁡     (     s   ,   x     )                  ,     x   ∈   U     ,     s   ∈         I   1     ⁡     (   x   )       .               (   4   )               
u 1 (s,x) is the unit vector perpendicular to the plane through x, y(s), and tangent to C 1  at y(s). Here {dot over (y)}(s) is the velocity vector of the source at the current position.
 
   We now pick a source position y(s) on the arc y(s)∈C 2 ,s∈I 2 (x). Find a plane through x and y(s), which is tangent to C 1  at some y(s t (s,x)),s t (s,x)∈I 1 (x). This determines another unit vector by equation 5. 
                       u   2     ⁡     (     s   ,   x     )       :=         (       y   ⁡     (       s   t     ⁡     (     s   ,   x     )       )       -     y   ⁡     (   s   )         )     ×     β   ⁡     (     s   ,   x     )                  (       y   ⁡     (       s   t     ⁡     (     s   ,   x     )       )       -     y   ⁡     (   s   )         )     ×     β   ⁡     (     s   ,   x     )                  ,     x   ∈   U     ,     s   ∈         I   2     ⁡     (   x   )       .               (   5   )               
u 2 (s,x) is the unit vector perpendicular to the plane through x, y(s), and tangent to C 1  at y(s t (s,x)).
 
   The detector plane DP(s) corresponding to the source y(s)∈C 2  with various points and lines projected onto it is shown in  FIG. 6 . 
   Using equations 4 and 5 we obtain the following reconstruction formula for ƒ∈C 0   ∞ (U) by equation 6: 
                         f   ⁡     (   x   )       =       ⁢       -     1     2   ⁢     π   2           ⁢       ∑     k   =   1     2     ⁢       ∫       l   k     ⁡     (   x   )         ⁢         δ   k     ⁡     (     s   ,   x     )              x   -     y   ⁡     (   s   )                                      ⁢           ∫   0     2   ⁢   π       ⁢       ∂     ∂   q       ⁢       D   f     ⁡     (       y   ⁡     (   q   )       ,       Θ   k     ⁡     (     s   ,   x   ,   γ     )         )           ⁢     |     q   =   s       ⁢         ⅆ   γ       sin   ⁢           ⁢   γ       ⁢     ⅆ   s         ,                   (   6   )               
where using equation 7
 Θ k ( s,x, γ):=cos γβ( s,x )+sin γ e   k ( s,x ),   e   k ( s,x ):=β( s,x )× u   k ( s,x ).   (7)         x is the cross-product of two vectors,   q is parameter along the trajectory,
 
and δ k  is defined by equation 8 as follows:
 
δ 1 ( s,x )=1, s∈I   1 ( x ), δ 2 ( s,x )=−sgn( u   2 ( s,x )· {dot over (y)} ( s )), s∈I   2 ( x );   (8)
       
   If C 2  is parameterized as in equation 1 (with {dot over (y)}(s) as shown in  FIG. 6 ), then δ 2 (s,x)=−1,s∈I 2 (x). If C 2  is parameterized in such a way that the source moves down along C 2  as s increases, then δ 2 (s,x)=1,s∈I 2 (x). In this case the motion of the source along C 1 ∪C 2  is continuous, so δ k (s,x) is continuous as well. 
   We now describe an efficient (that is, of the convolution-based FBP type) implementation of inversion formula equation 6. Pick the source on the circle y(s)∈C 1 . Then filtering is performed along the lines on the detector parallel to {dot over (y)}(s). The resulting family, which is one-parametric, is denoted L 1  in  FIG. 7 . Pick any l∈L 1 . 
   As follows from  FIG. 5 , all x whose projection belongs to l and appears to the right of Ĉ 2  share l as their filtering line. Suppose now the source is on the arc y(s)∈C 2 . For a point x∈U we have to find s t ∈I 1 (x) (see  FIG. 6 ). This determines the filtering line on the detector, which is tangent to Ĉ 1  at ŷ(s t ). The set of all such lines is a one-parametric family, which we denote L 2 . The range of s t  values, s t   mn ≦s t ≦s t   mx , depends on the region of interest (ROI) and is illustrated in  FIG. 8 . Pick s t ∈[s t   mn ,s t   mx ], and let l∈L 2  be the corresponding filtering line. Clearly, all x∈U which project onto l to the left of y(s t ) share l as their filtering line. Hence the proposed algorithm is of the convolution-based FBP type. One can perform filtering along lines on the detector (either L 1  or L 2 , depending on the source position), and then perform backprojection. 
   Let us describe this in more detail. It is clear that s t (s,x) actually depends only on s and β(s,x). Therefore, we can write below including equations 9 and 10. 
   
     
       
         
           
             
               
                 
                   
                     
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   Any source position y(s) and a filtering line from the corresponding family (either L 1  or L 2 , depending on whether y(s)∈C, or y(s)∈C 2 ), determine a plane. We call it a filtering plane. Since e(s,β)·β=0,|e(s,β)|=1, we can write equations 11 and 12.
 
β=(cos θ,sin θ);  e   k ( s, β)=(−sin θ,cos θ)   (11)
 
for all β, e k (s,β) confined to a filtering plane. Here θ denotes polar angle within the filtering plane. Therefore,
 
                     Ψ   k     ⁡     (     s   ,   β     )       =         ∫   0     2   ⁢   π       ⁢       ∂     ∂   q       ⁢       D   f     ⁡     (       y   ⁡     (   q   )       ,     (       cos   ⁡     (     θ   +   γ     )       ,     sin   ⁡     (     θ   +   γ     )         )       )           ⁢     |     q   =   s       ⁢       1     sin   ⁢           ⁢   γ       ⁢     ⅆ   γ                 (   12   )               
for all β confined to a filtering plane.
 
   Equation (12) is of convolution type and one application of Fast Fourier Transform (FFT) gives values of Ψ k (s,β) for all β confined to the filtering plane at once. Equations (10) and (12) would represent that the resulting algorithm is of the convolution-based FBP type. This means that processing of every CB projection consists of two steps. First, shift-invariant and x-independent filtering along a family of lines on the detector is performed. Second, the result is back-projected to update the image matrix. A property of the back-projection step is that for any point {circumflex over (x)} on the detector the value obtained by filtering at {circumflex over (x)} is used for all points x on the line segment connecting the current source position y(s) with {circumflex over (x)}. Since ∂/∂q in (12) is a local operation, each CB projection is stored in memory as soon as it has been acquired for a short period of time for computing this derivative at a few nearby points and is never used later. 
   Now we describe the algorithm in detail following the seven steps  10 - 70  shown in  FIG. 2 . 
   Step  10 . We load the current CB (cone beam) projection into computer memory. Suppose that the mid point of the CB projections currently stored in memory is y(s 0 ). The detector plane corresponding to the x-ray source located at y(s 0 ) is denoted DP(s 0 ). 
   Step  20 . Here we assume that the x-ray source is located on the circle C 1 . Referring to  FIG. 7 , we form a set of lines parallel to the projection of the circle that covers the projection of the region of interest (ROI) inside the object being scanned. 
   Step  30 .  FIG. 9  is a three substep flow chart for identifying the set of lines, which corresponds to step  30  of  FIG. 2 . Here we assume that the x-ray source is located on the arc C 2 . Referring to  FIG. 8 , the set of lines can be selected by the following substeps  31 ,  32 , and  33 . 
   Step  31 . Choose a discrete set of values of the parameter s t  inside the interval [s t   min ,s 1   max ]. 
   Step  32 . For each s t  chosen in Step  31  find a line tangent to the projected circle Ĉ 1 . 
   Step  33 . The collection of lines constructed in Step  32  is the required set of lines (see  FIG. 8  which illustrates the family of lines used in the algorithm of the invention). 
   Step  40 . Preparation for Filtering 
     FIG. 10  is a seven substep flow chart for preparation for filtering, which corresponds to step  40  of  FIG. 2 , which will now be described. 
   Step  41 . If the x-ray source is located on the circle C 1 , fix a filtering line l ƒlt ∈L 1  from the set of lines obtained in Step  20 . If the x-ray source is located on the arc C 2 , fix a filtering line l ƒlt ∈L 2  from the set of lines obtained in Step  30 . 
   Step  42 . Parameterize points on the said line by polar angle γ in the plane through y(s 0 ) and l 71 lt . 
   Step  43 . Choose a discrete set of equidistant values γ j  that will be used later for discrete filtering in Step  50 . 
   Step  44 . For each γ j  find the unit vector β j  which points from y(s 0 ) towards the point on l ƒlt  that corresponds to γ j . 
   Step  45 . Using the CB projection data D 71 (y(q),Θ) for a few values of q close to s 0  find numerically the derivative (∂/∂q)D ƒ (y(q),Θ)| q=s     0    for all Θ=β j . 
   Step  46 . Store the computed values of the derivative in computer memory. 
   Step  47 . Repeat Steps  41 - 46  for all lines l ƒlt . This way we will create the processed CB data Ψ(s 0 ,β j ) corresponding to the x-ray source located at y(s 0 ). 
   Step  50 . Filtering 
     FIG. 11  is a seven substep flow chart for filtering, which corresponds to step  50  of  FIG. 2 , which will now be described. 
   Step  51 . Fix a filtering line l ƒlt . If the x-ray source is located on the circle C 1 , we take l ƒt ∈L 1 . If the x-ray source is located on the arc C 2 , we take l ƒlt ∈L 2 . 
   Step  52 . Compute FFT (Fast Fourier Transform) of the values of the said processed CB data computed in Step  40  along the said line. 
   Step  53 . Compute FFT of the filter 1/sin γ 
   Step  54 . Multiply FFT of the filter 1/sin γ (the result of Steps  53 ) and FFT of the values of the said processed CB data (the result of Steps  52 ). 
   Step  55 . Take the inverse FFT of the result of Step  54 . 
   Step  56 . Store the result of Step  55  in computer memory. 
   Step  57 . Repeat Steps  51 - 56  for all lines in the said family of lines. This will give the filtered CB data Φ(s 0 ,β j ). 
   By itself the filtering step can be well known in the field and can be implemented, for example, as shown and described in U.S. Pat. No. 5,881,123 to Tam, which is incorporated by reference. 
   Step  60 . Back-Projection 
     FIG. 12  is an eight substep flow chart for backprojection, which corresponds to step  60  of  FIG. 2 , which will now be described. 
   Step  61 . Fix a reconstruction point x, which represents a point inside the patient where it is required to reconstruct the image. 
   Step  62 . If s 0  belongs to I 1 (x)∪I 2 (x), then the said filtered CB data affects the image at x and one performs Steps  63 - 68 . If s 0  is not inside I 1 (x)∪I 2 (x), then the said filtered CB data is not used for image reconstruction at x. In this case go back to Step  61  and choose another reconstruction point. 
   Step  63 . Find the projection {circumflex over (x)} of x onto the detector plane DP(s 0 ) and the unit vector β(s 0 ,x), which points from y(s 0 ) towards x. 
   Step  64 . Identify filtering lines l ƒlt ∈L 1  or  1   ƒlt ∈L 2  (depending on where the x-ray source is located) and points on the said lines that are close to the said projection {circumflex over (x)}. This will give a few values of Φ(s 0 ,β j ) for β j  close to β(s 0 ,x). 
   Step  65 . With interpolation estimate the value of Φ(s 0 ,β(s 0 ,x)) from the said values of Φ(s 0 ,β j ) for β j  close to β(s 0 ,x). 
   Step  66 . Compute the contribution from the said filtered CB data to the image being reconstructed at the point x by multiplying Φ(s 0 ,β(s 0 ,x)) by −δ k (s 0 ,x) /(2π 2 |x−y(s 0 )|). The quantity δ k (s 0 ,x) is defined by equation 8). 
   Step  67 . Add the said contribution to the image being reconstructed at the point x according to a pre-selected scheme (for example, the Trapezoidal scheme) for approximate evaluation of the integral in equation (10). 
   Step  68 . Go to Step  61  and choose a different reconstruction point x. 
   Step  70 . Go to Step  10  ( FIG. 2 ) and load the next CB projection into computer memory. The image can be displayed at all reconstruction points x for which the image reconstruction process has been completed (that is, all the subsequent CB projections are not needed for reconstructing the image at those points). Discard from the computer memory all the CB projections that are not needed for image reconstruction at points where the image reconstruction process has not completed. The algorithm concludes when the scan is finished or the image reconstruction process has completed at all the required points. 
   The invention is not limited to an object that undergoes a scan consisting of a single circle and a single arc. The algorithm can be applied to trajectories consisting of several circles and arcs by applying it to various circle and arc pairs, and then combining the results. The algorithm can be applied to trajectories in which a planar curve is not necessarily a circle, but, for example, an ellipse, and the like. 
   Other Embodiments of the invention are possible. For example, one can integrate by parts in equation (6) as described in the inventor&#39;s previous U.S. patent application Ser. No. 10/143,160 filed May 10, 2002 now U.S. Pat. No. 6,574,299, now incorporated by reference, to get an exact convolution-based FBP-type inversion formula which requires keeping only one CB projection in computer memory. The algorithmic implementation of this alternative embodiment can be similar to and include the algorithmic implementation of Embodiment Two in the inventor&#39;s previous U.S. patent application Ser. No. 10/143,160 filed May 10, 2002 now U.S. Pat. No. 6,574,299, now incorporated by reference. The corresponding equations will now be described. Introduce the following notations in equation 13: 
                         Ψ     1   ⁢   k       ⁡     (     s   ,   β     )       =       ∫   0     2   ⁢   π       ⁢         D   f     ⁡     (       y   ⁡     (   s   )       ,       Θ   k     ⁡     (     s   ,   β   ,   γ     )         )       ⁢       ⅆ   γ       sin   ⁢           ⁢   γ             ,     
     ⁢         Ψ     2   ⁢   k       ⁡     (     s   ,   β     )       =       ∫   0     2   ⁢   π       ⁢       (       ∇       u   k     ⁡     (     s   ,   β     )         ⁢     D   f       )     ⁢     (       y   ⁡     (   s   )       ,       Θ   k     ⁡     (     s   ,   β   ,   γ     )         )     ⁢     cot   ⁡     (   γ   )       ⁢     ⅆ   γ           ,     
     ⁢         Ψ     3   ⁢   k       ⁡     (     s   ,   β     )       =       ∫   0     2   ⁢   π       ⁢       (       ∇       u   k     ⁡     (     s   ,   β     )         ⁢     D   f       )     ⁢     (       y   ⁡     (   s   )       ,       Θ   k     ⁡     (     s   ,   β   ,   γ     )         )     ⁢     ⅆ   γ             ⁢     
     ⁢           Ψ     4   ⁢   k       ⁡     (     s   ,   β     )       =       ∫   0     2   ⁢   π       ⁢       (       ∂     ∂   γ       ⁢       D   f     ⁡     (       y   ⁡     (   s   )       ,       Θ   k     ⁡     (     s   ,   β   ,   γ     )         )         )     ⁢       ⅆ   γ       sin   ⁢           ⁢   γ             ,     
     ⁢     k   =   0     ,   1   ,   2.             (   13   )               
Here ∇ u D ƒ  denotes the derivative of D ƒ  with respect to the angular variables along the direction u by equations 14 and 15:
 
                       (       ∇   u     ⁢     D   f       )     ⁢     (       y   ⁡     (   s   )       ,   Θ     )       =         ∂     ∂   t       ⁢       D   f     ⁡     (       y   ⁡     (   s   )       ,           1   -     t   2         ⁢   Θ     +   tu       )         ⁢     |     t   =   0           ,     Θ   ∈       u   ⊥     .               (   14   )               
Here u ⊥  denotes the set of unit vectors perpendicular to u. Denote also
 
                       μ     1   ⁢   k       ⁡     (     s   ,   x     )       =       ∂     ∂   s       ⁢     1          x   -     y   ⁡     (   s   )                    ,         μ     2   ⁢   k       ⁡     (     s   ,   x     )       =           β   s     ⁡     (     s   ,   x     )       ·       u   k     ⁡     (     s   ,   x     )                x   -     y   ⁡     (   s   )                  ,     
     ⁢         μ     3   ⁢   k       ⁡     (     s   ,   x     )       =           (     e   k     )     s     ⁢       (     s   ,   x     )     ·       u   k     ⁡     (     s   ,   x     )                  x   -     y   ⁡     (   s   )                  ,         μ     4   ⁢   k       ⁡     (     s   ,   x     )       =             β   s     ⁡     (     s   ,   x     )       ·       e   k     ⁡     (     s   ,   x     )                x   -     y   ⁡     (   s   )                .               (   15   )               
Here β s =∂β/∂s and e s =∂e/∂s.
 
Let s bk (x) and s tk (x) denote the end-points of I k (x), k=1,2. More precisely, I k (x)=[s bk (x),s tk (x)].
 
   Integrating by parts with respect to s in equation (6) and using that δ k (s, x) is a constant with respect to s within each I k (x) (so it can be replaced by δ k (x)) we get similarly to equation (17) in U.S. Pat. No. 6,574,299 to Katsevich and the same assignee of the subject invention which is incorporated by reference: 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   While the invention has been described with rotating C-arm type devices, the invention can be used with rotating gantry devices. 
   Furthermore, the amount of rotating can include a single rotational curve of at least approximately 5 degrees up to approximately 360 degrees or greater. Theoretically, there is no limit on the minimum range of rotation. Under realistic practical circumstances, a minimum range of rotation is between approximately 10 and approximately 20 degrees. 
   The circle and arc scanning of an object can have an arc scanning before or after a single rotational curve scan as defined above. 
   Subsequent circle and arc scanning can occur as needed for image reconstruction. 
   The invention can allow for two scanning curves to be used which are transversal to one another. One of the scanning curves can be sized anywhere from an arc up to a full circle, and the other scanning curve can also be sized anywhere from an arc up to a full circle. 
   In medical applications, the length of one curve is generally shorter than the length of the other curve. The planes of the curves can cross each other at a tranversal angle of approximately 45 to approximately 135 degrees. Alternatively, the transversal angle can be in the range of approximately 80 to approximately 100 degrees. Still furthermore the second curve can be approximately perpendicular to the first curve. 
   Although the preferred embodiments describe applications of using x-ray sources for creating data for image reconstruction, the invention can be applicable with other sources that create line integral data for image reconstruction, such as but not limited to early arriving photons. 
   Other computational schemes for evaluating the integrals in equations 9, 10, 12, 13, and 16 are possible. For example, one can choose to perform filtering using native geometries, as explained in the paper by F. Noo, J. Pack, D. Heuscher “Exact helical reconstruction using native cone-beam geometries”, published in the journal “Physics in Medicine and Biology”, 2003, volume 48, pages 3787-3818. 
   Introduce the coordinates in the detector plane as follows. u is the horizontal coordinate, and w is the vertical coordinate. If the detector is flat, then computation of ƒ using equations 9-12 can be replaced by the following steps. 
   Step  1 . Distance weighting. 
                     g   1     ⁡     (     s   ,   u   ,   w     )       =         1         u   2     +     w   2     +   1         ⁢     ∂     ∂   q       ⁢       D   f     ⁡     (     q   ,   u   ,   w     )         ⁢     ❘     q   =   s       .             (   17   )               
Step  2 . Hilbert transform filtering of g 1  along filtering lines on the detector:
 
                       g   2     ⁡     (     s   ,   u   ,   ρ     )       =       ∫     -   ∞     ∞     ⁢         K   H     ⁡     (     u   -     u   ′       )       ⁢       g   1     ⁡     (     s   ,     u   ′     ,     w   ⁡     (     ρ   ,     u   ′       )         )       ⁢     ⅆ     u   ′             ,           (   18   )               
where
 
   K H  is the Hilbert transform kernel, 
   ρ is the variable that parametrizes filtering lines, and 
   w(ρ,u t ) is the w-coordinate of the point, whose u-coordinate equals u t , and 
   which belongs to the filtering line determined by the parameter ρ. 
   Step  3 . Backward height rebinning.
 
 g   3 ( s,u,w )= g   2 ( s,u, ρ( u,w )),   (19)
 
where
 
   ρ(u,w) is the function that gives the value of the parameter ρ, corresponding to the filtering line that contains all x projected onto the point on the detector with coordinates (u,w). 
   Step  4 . Backprojection. Evaluate the integrals 
                     ∫       I   k     ⁡     (   x   )         ⁢       1       V   *     ⁡     (     s   ,   x     )         ⁢       g   3     ⁡     (     s   ,       u   *     ⁡     (     s   ,   x     )       ,       w   *     ⁡     (     s   ,   x     )         )       ⁢     ⅆ   s         ,     k   =   1     ,   2   ,           (   20   )               
where
 
   u*(s,x),w*(s,x) are the coordinates of the point x projected onto the detector plane DP(s), 
   V*(s,x)=1−(x 1 y 1 (s)+x 2 y 2 (s)+x 3 y 3 (s)) if y(s) is on the arc, 
   V*(s,x)=1−(x 1 y 1 (s)+x 2 y 2 (s)) if y(s) is on the circle. 
   Step  5 . Multiply the integrals in (20) by δ k (s,x)/(2π),s∈I k (x), and add them to obtain ƒ(x). 
   Integrating by parts with respect to s in equation 20 we can obtain another set of formulas, in which the derivative along the source trajectory is avoided. In this case the filtering step will involve computing the following quantities: 
   
     
       
         
           
             
               
                 
                   
                     
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   where 
   g t   1,w  is the derivative of g 1  with respect to w. 
   Also, equations analogous to 17 through 21 can be obtained if the detector is curved, and not flat. 
   While the invention has been described, disclosed, illustrated and shown in various terms of certain embodiments or modifications which it has presumed in practice, the scope of the invention is not intended to be, nor should it be deemed to be, limited thereby and such other modifications or embodiments as may be suggested by the teachings herein are particularly reserved especially as they fall within the breadth and scope of the claims here appended.