Abstract:
Method and device for digital x-ray tomosynthesis. Tomographic and/or three-dimensional images of an object are obtained with an x-ray source and a digital x-ray image sensor. The source, object and sensor are positioned relative to each other and attenuation data is obtained for a large number of rays of x-radiation through the object. A special algorithm is provided to convert the data into images. To calculate the images the algorithm uses iterative processes with a least squares type technique but with generalized (as opposed to specific) functions. The algorithm solves for the functions which are the images. Preferred embodiments include a system having an x-ray point source with a cone of diverging x-rays, a two-dimensional digital x-ray image sensor, two linear translation stages to independently move both the x-ray source and the digital x-ray image sensor, two rotation mechanisms to rotate the two linear translation stages, a microprocessor to control the data acquisition, and a computer programmed with a special algorithm to calculate the tomographic images. A plurality of sets of digital data (representing x-ray algorithm images of an object) are acquired by the digital x-ray image sensor, with the x-ray source and the digital x-ray image sensor located at different positions and angles relative to the object. The digital data representing the x-ray attenuation images is stored in the computer. Special mathematical algorithms then compute multiple images of the object using the acquired digital data. These images could include multiple tomographic images, a three-dimensional image, or a multiple three-dimensional images.

Description:
This application claims the benefit of provisional patent application Ser. No. 60/552,429 filed Mar. 11, 2004 and provisional patent application Ser. No. 60/646,014 filed Jan. 22, 2005. This invention relates to digital x-ray systems, and in particular, digital x-ray tomosynthesis systems. 

   BACKGROUND OF THE INVENTION 
   The prior art x-ray imaging systems include projection radiography, geometric tomography, projection radiography, computed axial tomography, and digital x-ray tomosynthesis methods, as described here. These x-ray systems have many applications, especially medical imaging and security applications such as baggage imaging. Projection radiography is a imaging technique involving an x-ray point source that emits a cone beam of x-rays through an object, and a two-dimensional x-ray image sensor (i.e. x-ray film or digital x-ray image sensor, for example) that measures the spatially varying attenuation of the x-ray cone beam after is passes through the object. 
   Geometric Tomography 
   Geometric tomography (GT), invented in the 1930s, involves a method for using a conventional film-based x-ray imaging system to provide radiographic images in tomographic slices. This method, displayed in  FIG. 1 , incorporates the co-motion of x-ray source  10  and image receptor  40  (i.e. x-ray film, for example) during the x-ray exposure. The co-motion of x-ray source  10  and film  40 , relative to imaged object  25 , produces an image on x-ray film  40  with a sharp focus at image plane  35  containing the fulcrums of motion  30 ,  31  and  32 . The spatially varying x-ray attenuation above and below image plane  35  is essentially “blurred” out by the relative co-motion. Different image planes  35  can be imaged on different sheets of film  40 , by varying the different velocities of x-ray source  10  and film  40  relative to imaged object  25 . The GT imaging method showed potential for improved diagnostic efficacy compared to conventional projection radiography, however, this method required much higher x-ray doses, mainly because each GT image required an equivalent x-ray dose as a projection radiograph. The dose issue resulted in limited clinical deployment of the GT method. 
   Computed Axial Tomography 
   During the 1970s, the development of computed axial tomography (CAT) methods provided a revolution in diagnostic radiography and a widespread clinical deployment of the CAT systems. The process is referred to as a “CAT scan.” A typical CAT system features an x-ray point source and spatial filtration (i.e. a slit) so that the x-ray source emits a fan beam of x-rays. A linear (i.e. one-dimensional) pixelated array of high performance x-ray detectors measures the attenuation of the x-ray fan beam after it passes through an object, such as a human body, for example. This system rotates 360 degrees around an object and provides x-ray attenuation data in a plurality of planes (planes of rotation) at a plurality of regularly spaced angles. The computer calculates a digital tomographic image of the object for each of the planes of the rotation. To do this the system moves to a plurality of positions perpendicular to the planes of rotation and repeats the imaging procedure to provide the plurality of tomographic images of the object. These tomographic images can be viewed separately, or can be processed by the computer to provide three-dimensional images of the object. With the CAT scan technique, all data is obtained with the x-ray source and the detectors in fixed positions relative to the object being imaged so there is no intentional blurring of any of the image information. 
   Digital X-ray Tomosynthesis 
   In the 1960–1970s, the development of fluoroscopic cameras, comprised of image intensifier tubes coupled to video sensors, enabled the emergence of digital x-ray tomosynthesis methods. A digital projection radiograph involves an x-ray point source that emits a cone beam of x-rays, and a digital x-ray image sensor comprised of a two-dimensional array of x-ray detectors (fluoroscopic camera, for example) that measures the spatially varying attenuation of the x-ray cone beam after is passes through an object. Digital x-ray tomosynthesis (DXT) involves the acquisition of a plurality of digital projection radiographs of an object with the x-ray source and the x-ray image sensor located at different positions and angles relative to the object. A computer then uses the digital data to compute a plurality of tomographic images of the object. 
   The DXT method, in the simplest sense, provides x-ray attenuation data and calculations that emulate the motional blurring of the GT method to visualize the single image plane at the fulcrum of motion of the x-ray source and image sensor. However, the DXT method provides a much more dose efficient radiographic modality than the earlier GT method because the DXT method enables the computation of a plurality of tomographic images from a single set of multi-positional projection radiographs. In contrast, the GT method requires a complete set of multi-positional projection radiographs for each tomographic image. A recent review of both the GT and DXT methods is provided in J. Dobbins, D. Godfrey, Phys. Med. Biol. 48 (2003), R65–R106. This review discusses the prior art of the DXT mathematical reconstruction algorithms; these algorithms will be compared to the present invention later in this specification. 
   Although the initial development of DXT methods showed potential for clinical applications, the clinical deployment has been limited due to the relative immaturity of the digital x-ray image sensors. The image performance of earlier image intensifier tubes was limited in spatial resolution and detective quantum efficiency (DQE). In the late 1990s to present, however, the emerging technological developments in flat panel digital x-ray image sensors have enabled high performance digital x-ray imaging capability in a two-dimensional pixelated array format. These developments have enabled resurgence in DXT development, with ongoing clinical investigation of DXT imaging for chest radiography and mammography, for example. However, to date, the DXT method has still not seen widespread clinical deployment. 
   Portable X-ray Vents 
   In the last few years, the digital x-ray image sensor technology has advanced to the point where some of the sensors have become truly portable. Also, high-speed computing and digital display technologies have become available in lightweight, portable packages. One of the applicants is a co-inventor of such a unit. 
   Least Squares Techniques 
   Least squares type techniques (also referred to by names such as chi-squared fitting) are well known techniques for fitting large amounts of data to known functions. 
   What is needed are better DXT systems, especially a portable DXT radiographic systems for use in remote medical applications, such as emergency trauma or combat casualty care, for example. In addition, there is a growing need for a portable, low dose, DXT radiographic system that will provide detection and characterization of explosives devices in packages and luggage. 
   SUMMARY OF THE INVENTION 
   This invention provides the methods and devices for digital x-ray tomosynthesis. Tomographic and/or three-dimensional images of an object are obtained with an x-ray source and a digital x-ray image sensor. The source, object and sensor are positioned relative to each other and attenuation data is obtained for a large number of rays of x-radiation through the object. A special algorithm is provided to convert the data into images. To calculate the images the algorithm uses iterative processes with a least squares type technique but with generalized (as opposed to specific) functions. The algorithm solves for the functions which are the images. Preferred embodiments include a system having an x-ray point source with a cone of diverging x-rays, a two-dimensional digital x-ray image sensor, two linear translation stages to independently move both the x-ray source and the digital x-ray image sensor, two rotation mechanisms to rotate the two linear translation stages, a microprocessor to control the data acquisition, and a computer programmed with a special algorithm to calculate the tomographic images. A plurality of sets of digital data (representing x-ray algorithm images of an object) are acquired by the digital x-ray image sensor, with the x-ray source and the digital x-ray image sensor located at different positions and angles relative to the object. The digital data representing the x-ray attenuation images is stored in the computer. Special mathematical algorithms then compute multiple images of the object using the acquired digital data. These images could include multiple tomographic images, a three-dimensional image, or a multiple three-dimensional images. 
   Features of preferred embodiment of the present invention include: 1) technique for positioning of the x-ray source to reduce total number of digital projection radiographs as compared to prior art devices to minimizing total x-ray dose, and image reconstruction with reduced image aliasing artifacts; 2) positioning of the digital x-ray image sensor to increase the field of view; 3) special linear reconstruction algorithms providing image reconstruction of the tomographic images as well as determination; 4) computationally efficient image reconstruction algorithms to provide rapid image reconstruction; 5) applications of nonlinear techniques, such as wavelet transforms and filtering of the acquired data, in order to provide image reconstruction of tomographic slices that are sparsely surrounded by spatially “cluttered” image data and system configurations that permit the invention to be applied to portable units. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a drawing that shows the geometric tomography technique using co-moving x-ray source and x-ray film during x-ray exposure. 
       FIG. 2  is a drawing that shows the side view of the mechanical features of the preferred embodiment. 
       FIG. 3  is a drawing that shows a geometrical description of the parameters used in the image reconstruction algorithm. 
       FIG. 4  is a drawing that shows the optimal x-ray source positions along one dimension. 
       FIG. 5  is a drawing that shows the two-dimensional positioning of the x-ray source. 
     FIGS.  6 A( 1 ) through  6 C( 2 ) are drawings that shows the optimal positioning of the x-ray image sensor for different positions of the x-ray source in order to maximize the field of view. 
       FIG. 7  is a computer simulation that shows five digital projection radiographs of a thick walled cone with an “x” in the middle, each radiograph simulated with the x-ray source at a different position. 
       FIGS. 8–10  are twelve reconstructed tomographic images of the thick walled cone with an “x” in the middle. 
       FIGS. 11 and 12  show the side view and front view of a luggage screening system that incorporates the invention. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   Hardware 
   Preferred embodiments of the DXT system hardware are displayed in  FIG. 2 . X-ray source  120  is attached to linear translation stage  85  which is attached to pivot assembly  90 . This enables the x-ray source  120  to be selectively positioned along a plane at the top of the DXT system so that x-ray source assembly  121  can be move left and right from its central position show in  FIG. 1 . In addition, the x-ray source  120  can be selectively tilted about pivot point  119  for each position of the x-ray source along the dimension of the translation stage  85 , in order to direct the cone of x-rays directly at the object to be imaged. Digital x-ray image sensor  115  is positioned on linear translation stage  105  which is attached to pivot assembly  110 . This enables the digital x-ray image sensor  115  to be selectively positioned along a plane at the base of the DXT system. The source-to-image distance (SID) is adjustable in the range of 50–100 cm by the use of telescopic pillars  95  supplied by SKF Magnetics with offices in Bethlehem, Pa. The object to be imaged is placed on platform  100  that is transparent to x-rays. Platforms can be placed at any of five vertical levels and then positioned vertically at different heights (0–50 cm) with pillars  95  relative to the digital x-ray image sensor  115 . 
   The preferred x-ray sensor  115  is an indirect detection sensor based on a thin-film transistor (TFT) technology that involves the fabrication of a two-dimension array of amorphous silicon-based electronic circuit pixels on large (30 cm×40 cm, for example) glass sheets. The TFT array is sensitive to visible radiation (400 nm–700 nm). This array is coated or covered with a material that converts incident x-rays to visible light and the visible light is recorded by the pixelated TFT array and digitized to provide a two-dimensional digital image of the incident x-rays. Typical x-ray to light converters include gadolinium oxisulfide (Gd 2 O 2 S:Tb) phosphor screens or dendritic cesium iodide scintillator coatings. Preferred x-ray to light converters include high x-ray attenuation characteristics (PI-200 Gd 2 O 2 S:Tb, 436 microns thick, 200 mg/cm 2 ; Kasei Opthonix, for example), these converters provide very good detective quantum efficiency (DQE) and very good special resolution. Alternate sensor embodiments include direct detection x-ray sensors; such as amorphous selenium coated TFT arrays, for example. Incident x-ray photons are converted directly into electronic charge in the amorphous selenium layer, and the charge is collected at each pixel. Indirect detection x-ray sensors available from Canon, Thales, and Varian and direct detection x-ray sensors are available from General Electric and Hologic. 
   The preferred x-ray source  120  features a tungsten non-rotating anode with a 50–160 kVp tube potential range, and a 0–0.5 mA tube current. A microprocessor subsystem controls the position of x-ray source  120  and x-ray sensor  115  by controlling translation stages  85  and  105  and rotation stages  90  and  110 . The microprocessor subsystem also controls the technique factors (tube voltage, tube current, and exposure time) of x-ray source  120 . The DXT system incorporate a computer and software to acquire, store, and display projection radiographs; provide digital tomosynthesis image reconstruction calculations; and display tomographic images. 
   Novel Features of the Invention 
   The preferred geometrical arrangement for the invention is displayed in  FIG. 3 . The positioning of x-ray source  120  is constrained to positions  210  (i.e., x i , y i , h) on x-ray source plane  200 , and the positioning of x-ray sensor  115  is constrained to x-ray sensor plane  230  that is parallel to x-ray sensor plane  200  at a distance h=SID. The system produces tomographic images located in tomographic image plane  220  that is parallel to x-ray sensor plane  120  at distances h and a tomographic slice thickness dh. A plurality of tomographic images are calculated, each image at different distances h from x-ray sensor plane  210 . Typical X-ray  223  is emitted at angle (θ,φ) [representing dimensions (elevation,azimuth)] by x-ray point source  120  that is located at position (X i , Y i , h=SID) in x-ray source plane, and travels a straight line trajectory that intercepts tomographic image plane  220  at position (x tomo , y tomo , h) and is then incident on x-ray sensor  115 , located at x-ray sensor plane  120 , at pixel position (x,y, h=0). 
   The primary objective of digital tomosynthesis is to provide an x-ray attenuation image that is isolated to a single tomographic slice, so that spatially varying x-ray attenuation from other slices does not clutter the image. A second objective is to help identify features by finding the three-dimensional positions of the features. Previous DXT methods have implemented a simple shift-and-add technique that basically emulates the motional blurring described by the GT method, in order to visualize the single image plane at the fulcrum of motion of the x-ray source and image sensor. We describe the important features of our method which include positioning of the source and sensor relative to the object being imaged and a special algorithm for constructing images using the acquired data:
     1) Optimal positioning of the x-ray source: This feature describes a preferred total number of digital projection radiographs, and total x-ray dose to produce image reconstruction with minimal image aliasing artifacts. These constraints depend on aliasing and spatial frequency range considerations, as well as the source-to-image distance (SID).   a) Aliasing: There is a minimum preferred angular positioning step of the x-ray source required to discriminate planes for desired spatial frequencies of the image. This step is such that the spatial shift of the highest spatial frequency from the top slice to the bottom slice is approximately 1 line pair. If a larger minimum step were to be used, there would be some ambiguity in the reconstruction as to which plane the feature was in (known as aliasing) which would increase the noise in the image. For the preferred system, we specify f max =1.5 lp/mm, object thickness=300 mm, which results in a minimum step of 2 milliradians (0.23 degrees).   b) Spatial Frequency Range: A large angular positioning range results in a reduction of the image slice thickness for the lowest spatial frequencies. The angular positioning range is approximately the ratio of this low spatial frequency to the image slice thickness. Very low spatial frequencies cannot be well localized without scanning through large angles. The preferred angular positioning range is 1.0 radian (57.32 degrees), depending on the features of interest.   c) Source-to-image distance (SID): The preferred SID is chosen based on geometrical requirements, clearance requirements, and spot size requirements. If a 600 micron diameter x-ray spot tube is used, then a reasonable specification of 200 micron blur requires a 3:1 ratio of SID to subject depth. This translates into a preferred SID=900 mm for a 300 mm typical subject. A 300 micron diameter spot tube will have lower x-ray exposure capability but places less constraints on the SID. From a geometrical perspective, there is a magnification effect for objects away from the detector, which becomes more extreme for small SIDs. A ratio of 2:1 magnification or less should cause little impact on the reconstruction process. The preferred range of SID is between 750 mm and 1000 mm.   d) One dimensional positioning of the x-ray source: The key to simultaneously resolving the constraints of aliasing and range is to let the positioning step length of the x-ray source be small near the center of the positioning range (so that small features do not alias) and larger as the tube moves away from the center (so that the low spatial frequencies are well localized). If constant sized steps were used, the number of image angles taken would be impractical. A preferred positioning of the x-ray source, in one scan dimension is displayed in  FIG. 4 . The positioning is symmetric around (X center , Y center ), defined as the center of x-ray source plane  200 , with 21 x-ray source positions, and a SID=900 mm. The positions of x-ray source  120  are described by   

                     X   i     -     X   center       =     ±     d   [         exp   ⁡     (     ⅈ   ⁢           ⁢   a     )       -   1     a     ]               Eq   .           ⁢     (   1   )                 
where d=5 mm, a=0.15, and i=0, . . . , 10; so that the positions X i -X center =0, ±5.4 mm,±11.7 mm, ±18.9 mm, ±27.4 mm, ±37.2 mm, ±48.7 mm, ±61.9 mm, ±77.3 mm, and ±95.1 mm. The total positioning range for the preferred embodiment is 2* 116.1 mm=232.2 mm and degrees.
     e) Two-dimension positioning of the x-ray source: Most prior implementations of digital tomosynthesis so far have used positioning of the x-ray source in only one dimension, as shown in  FIG. 4 . While acceptable for some applications, there is a serious drawback, namely, it is impossible to localize the slice height of objects running parallel to the positioning direction. The best way around this is to position x-ray source  120  in two dimensions, thereby removing any sensitivity to object orientation. The preferred embodiment for this two dimension positioning, displayed in  FIG. 5 , is   
                       X   i     -     X   center       =       ±     d   ⁡     [         exp   ⁡     (     ⅈ   ⁢           ⁢   a     )       -   1     a     ]         ⁢           ⁢   and       ⁢     
     ⁢         Y   i     -     Y   center       =     ±     d   ⁡     [         exp   ⁡     (     ⅈ   ⁢           ⁢   a     )       -   1     a     ]                   Eq   .           ⁢     (   2   )                 
where d=5 mm, a=0.15, and i=0, . . . , 9. Eq. (2) gives the positions X i −X center =0, ±5.4 mm, ±11.7 mm, ±18.9 mm, ±27.4 mm, ±37.2 mm, ±48.7 mm, ±61.9 mm, ±77.3 mm, ±95.1 mm, and ±116.1 mm, with the y-axis of x-ray source  120  positioned at Y center ; and positions Y i −Y center =0, ±5.4 mm, ±11.7 mm, ±18.9 mm, ±27.4 mm, ±37.2 mm, ±48.7 mm, ±61.9 mm, ±77.3 mm, and ±95.1 mm, with the x-axis of x-ray source  120  positioned at X center . The total number of positions of x-ray source  120  is 41. The total angular positioning range is θ=2*arc tan(232.2 mm/900 mm)=0.52 radians=30 degrees.
     f) X-ray Source Positioning Accuracy: The required accuracy of the translation stage is a fraction of the spot size of x-ray source  120 . This translates to roughly 200 micron positioning accuracy for a spot size of 600 microns for x-ray source  120 , which can be reasonably achieved.   2) Optimal positioning of the x-ray sensor: FIGS.  6 A( 1 ) and ( 2 ) show that the preferred x-ray sensor  115  is positioned, for each x-ray source  120  position, so that the field of view of the object is maximized. This is accomplished by positioning x-ray sensor  115  in a direction opposite to x-ray source  120  position so that a line  300  between x-ray source  120  and the center of x-ray sensor  115  has a virtual fulcrum of motion  310  that is approximately 20 cm above x-ray sensor  115 ; this provide a 30 cm field-of view. FIG.  6 B( 1 ) and ( 2 ) show that a 42 cm fulcrum of motion  310  only provides a 12 cm field of view and FIG.  6 C( 1 ) and ( 2 ) show that a 0 cm fulcrum of motion  310  only provides a 22 cm field of view.   3) Linear image reconstruction algorithm: A preferred reconstruction algorithm takes input digital projection radiograph image data of an object with the x-ray source at multiple positions and angles relative to the object, and transforms the input data into tomographic images. A number of techniques have been demonstrated to date, such as the simple shift-and-add approach and iterative techniques. We have developed an image reconstruction algorithm that we believe outperforms the other approaches while being computationally manageable.   a) Mathematical Description of Reconstruction Algorithm: The geometry of the mathematical problem is displayed in  FIG. 3 . X-ray point source  120  (STET) is positioned at different N separate positions  210  (X i ,Y i ); i=1, N in x-ray source plane  200  located at h=SID. The x-ray sensor is located in the x-ray sensor plane located at h=0. The The three-dimensional object that is imaged is represented by a scalar function d(x tomo , y tomo , h) in the dimension of Hounsfield units we calculate d(xyh). The Hounsfield unit is the common metric for computer axial tomography (CAT) scan images. One Hounsfield unit equals a 0.1% difference in the density of water and the range of Hounsfield units is from −1000 (air) to 1000 (bone) with 0 Hounsfields as the density of water. The attenuation of x-rays directed along the line  223  (trajectory s) is given by   
                     A   i     ⁡     (   s   )       =       A   i     ⁡     [       ∫   0   s     ⁢       d   ⁡     (       x   tomo     ,     y   tomo     ,   h     )       ⁢     ⅆ     s   ′           ]               Eq   .           ⁢     (   2   )                 
where (x tomo , y tomo ) are the coordinates where the trajectory s crosses the tomographic plane at z=h. The functional form of A i (s) is nearly exponential and depends on the x-ray spectrum of the x-ray source. The coordinates
 
                   (       x   tomo     ,     y   tomo       )     =     (         x   ⁡     [     1   -     h   SID       ]       +       hX   i     SID       ,       y   ⁡     [     1   -     h   SID       ]       +       hY   i     SID         )             Eq   .           ⁢     (   3   )                 
and the line element
 
   
     
       
         
           
             
               
                 
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 m   i ( x,y )= m 0 i ( x,y )+ n   i ( x,y )  Eq. (7)
 
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                   9 
                   ) 
                 
               
             
           
         
       
     
   
   We need to invert equation 9 to express the tomographic slice image data d(x,y,h) in terms of the acquired noise free data m 0 (x,y). We invert equation 9 by minimizing the following generalized chi-squared function versus the function d(x tomo , y tomo , h) 
                   χ   2     =         ∑   i     ⁢     ∫     ∫           (         m   i     ⁡     (       x   o     ,     y   o       )       -       m0   i     ⁡     (       x   o     ,     y   o       )         )     ⁡     [       (       n   ⁡     (       x   0     ,     y   0       )       ⁢     n   ⁡     (       x   1     ,     y   1       )         )     avg     ]         -   1       ⁢     (         m   i     ⁡     (       x   1     ,     y   1       )       -     m0   ⁡     (       x   1     ,     y   1       )         )     ⁢     ⅆ     x   o       ⁢     ⅆ     y   o       ⁢     ⅆ     x   1       ⁢     ⅆ     y   1               +     ɛ   ⁢     ∫     ∫         d   ⁡     (     x   ,   y   ,   h     )       2     ⁢     ⅆ   x     ⁢     ⅆ   y     ⁢     ⅆ   h                       Eq   .           ⁢     (   10   )                 
where e is a regularization parameter chosen so that the first term averages 1 per measurement and m 0  is implicitly a function of d.
 
   We simplify equation 9 by first remapping d to the function D 
                   d   ⁡     (     x   ,   y   ,   h     )       =       D   ⁡     (       x   ⁢     SID     SID   -   h         ,     y   ⁢     SID     SID   -   h         ,     h   ⁢     SID     SID   -   h           )       ⁢       (     SID     SID   -   h       )     2               Eq   .           ⁢     (   11   )                 
and changing variables
 
   
     
       
         
           
             
               
                 z 
                 = 
                 
                   
                     SID 
                     
                       SID 
                       - 
                       h 
                     
                   
                   ⁢ 
                   h 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
   
   The integrals in equations 6 and 10 are then transformed to 
   
     
       
         
           
             
               
                 
                   
                     ∫ 
                     0 
                     h 
                   
                   ⁢ 
                   
                     
                       d 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     h 
                                     SID 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 hX 
                                 i 
                               
                               SID 
                             
                           
                           , 
                           
                             
                               y 
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     h 
                                     SID 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 hY 
                                 i 
                               
                               SID 
                             
                           
                           , 
                           h 
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       h 
                     
                   
                 
                 = 
                 
                   
                     
                       ∫ 
                       0 
                       h 
                     
                     ⁢ 
                     
                       
                         D 
                         ⁡ 
                         
                           [ 
                           
                             
                               x 
                               + 
                               
                                 
                                   
                                     X 
                                     i 
                                   
                                   ⁢ 
                                   h 
                                 
                                 
                                   SID 
                                   - 
                                   h 
                                 
                               
                             
                             , 
                             
                               y 
                               - 
                               
                                 
                                   
                                     Y 
                                     i 
                                   
                                   ⁢ 
                                   h 
                                 
                                 
                                   SID 
                                   - 
                                   h 
                                 
                               
                             
                             , 
                             
                               SIDh 
                               
                                 SID 
                                 - 
                                 h 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             SID 
                             
                               SID 
                               - 
                               h 
                             
                           
                           ) 
                         
                         2 
                       
                       ⁢ 
                       
                         ⅆ 
                         h 
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                     ⁢ 
                     
                       
                         D 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               + 
                               
                                 
                                   
                                     X 
                                     i 
                                   
                                   ⁢ 
                                   z 
                                 
                                 SID 
                               
                             
                             , 
                             
                               y 
                               + 
                               
                                 
                                   
                                     Y 
                                     i 
                                   
                                   ⁢ 
                                   z 
                                 
                                 SID 
                               
                             
                             , 
                             z 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         z 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
           
             
               and 
             
             
               
                   
               
             
           
           
             
               
                 
                   ∫ 
                   
                     ∫ 
                     
                       
                         ⅆ 
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               h 
                             
                             ) 
                           
                           2 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                       ⁢ 
                       
                         ⅆ 
                         y 
                       
                       ⁢ 
                       
                         ⅆ 
                         h 
                       
                     
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       ∫ 
                       
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     SID 
                                     
                                       SID 
                                       - 
                                       h 
                                     
                                   
                                 
                                 , 
                                 
                                   y 
                                   ⁢ 
                                   
                                     SID 
                                     
                                       SID 
                                       - 
                                       h 
                                     
                                   
                                 
                                 , 
                                 
                                   SIDh 
                                   
                                     SID 
                                     - 
                                     h 
                                   
                                 
                               
                               ) 
                             
                           
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               SID 
                               
                                 SID 
                                 - 
                                 h 
                               
                             
                             ) 
                           
                           4 
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                         ⁢ 
                         
                           ⅆ 
                           h 
                         
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 
                                   SIDh 
                                   
                                     SID 
                                     - 
                                     h 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 SID 
                                 
                                   SID 
                                   - 
                                   h 
                                 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                           ⁢ 
                           
                             ⅆ 
                             y 
                           
                           ⁢ 
                           
                             ⅆ 
                             h 
                           
                         
                       
                     
                     = 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                           ⁢ 
                           
                             ⅆ 
                             y 
                           
                           ⁢ 
                           
                             ⅆ 
                             z 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
   
   Equation 9 is then expressed as 
   
     
       
         
           
             
               
                 
                   m0 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       w 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       A 
                       ⁡ 
                       
                         [ 
                         
                           sec 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               θ 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 D 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       + 
                                       
                                         
                                           
                                             X 
                                             i 
                                           
                                           ⁢ 
                                           z 
                                         
                                         SID 
                                       
                                     
                                     , 
                                     
                                       y 
                                       + 
                                       
                                         
 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             Y 
                                             i 
                                           
                                           ⁢ 
                                           z 
                                         
                                         SID 
                                       
                                     
                                     , 
                                     z 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 z 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⊗ 
                     
                       psf 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
   
   and equation 9 is expressed as 
   
     
       
         
           
             
               
                 
                   
                     χ 
                     2 
                   
                   = 
                   
                     
                       ∑ 
                       i 
                     
                     ⁢ 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         o 
                                       
                                       , 
                                       
                                         y 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m0 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         o 
                                       
                                       , 
                                       
                                         y 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       n 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             x 
                                             0 
                                           
                                           , 
                                           
                                             y 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       n 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             x 
                                             1 
                                           
                                           , 
                                           
                                             y 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 avg 
                               
                               ] 
                             
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       ( 
                       
                         
                           
                             m 
                             i 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                               
                               , 
                               
                                 y 
                                 1 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           m0 
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                               
                               , 
                               
                                 y 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ⅆ 
                       
                         x 
                         o 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       
                         y 
                         o 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       
                         x 
                         1 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       
                         y 
                         1 
                       
                     
                   
                   + 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ɛ 
                     ⁢ 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                           ⁢ 
                           
                             ⅆ 
                             y 
                           
                           ⁢ 
                           
                             ⅆ 
                             h 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
   
   The noise correlation is generated from the noise power spectrum (NPS) function
 
( n   i ( x   0   , y   0 ) n   i ( x   1   ,y   1 )) avg   =∫NPS ( x   0   −x,y   0   −y,x   1   −x,y   1   −y,x,y ) dxdy    Eq. (17)
 
   We assume at the noise is slowly varying over the region integrated (a common approximation), so we can approximate 
                   ∫       NPS   ⁡     (         x   o     -   x     ,       y   o     -   y     ,       x   1     -   x     ,       y   1     -   y     ,   x   ,   y     )       ⁢     ⅆ   x     ⁢     ⅆ   y         =     
     ⁢       nps   ⁡     (         x   o     -     x   1       ,       y   o     -     y   1         )       ⁢         noise   i     ⁡     (       x   o     ,     y   o       )         ⁢         noise   i     ⁡     (       x   1     ,     y   1       )                   Eq   .           ⁢     (   18   )                 
where nps is the noise power spectrum correlation function normalized to 1 at 0 lp/mm, and “noise” is more slowly varying 0 lp/mm noise. The assumption restated is that the noise varies too slowly to matter much which parameter x is used in its argument.
 
   We next define an “integrated Hounsfield” measurement M and its noise free version M 0 , which is a transform of the raw measurement m 0   
   
     
       
         
           
             
               
                 
                   
                     M0 
                     i 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   ∫ 
                   
                     
                       D 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             + 
                             az 
                           
                           , 
                           
                             y 
                             + 
                             
                               β 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               z 
                             
                           
                           , 
                           z 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       z 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
           
             
               
                 
                     
                 
                 ⁢ 
                 
                   = 
                   
                     
                       1 
                       
                         sec 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             θ 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         A 
                         
                           - 
                           1 
                         
                       
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 m0 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                 
                                 ) 
                               
                             
                             
                               
                                 w 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ⊗ 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       psf 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   ) 
                 
                 ] 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     M 
                     i 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     
                       sec 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           θ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     A 
                     ⁡ 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 m 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                 
                                 ) 
                               
                             
                             
                               
                                 w 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ⊗ 
                         
                           [ 
                           
                             
                               psf 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
   
   By using a first order Taylor expansion (which is appropriate for the low noise regime we will operate in), and bringing the slowly varying white field term w i (x,y) outside of the convolution, the difference of M and M 0  is a transformed noise term 
                         M   i     ⁡     (     x   ,   y     )       -       M0   i     ⁡     (     x   ,   y     )         =     
     ⁢         (         m   i     ⁡     (     x   ,   y     )       -       m0   i     ⁡     (     x   ,   y     )         )     ⊗     (       psf     -   1       ⁡     (     x   ,   y     )       )         sec   ⁢           ⁢       θ   i     ⁡     (     x   ,   y     )       ⁢       w   i     ⁡     (     x   ,   y     )       ⁢       A   ′     ⁡     (         M   i     ⁡     (     x   ,   y     )       ⁢   sec   ⁢           ⁢       θ   i     ⁡     (     x   ,   y     )         )             ⁢     
     ⁢       where   ⁢           ⁢       A   ′     ⁡     (   v   )         =       ⅆ     ⅆ   v       ⁢       A   ⁡     (   v   )       .                 Eq   .           ⁢     (   21   )                 
Notice that the term in the denominator w i (x,y)A′(M i (x ,y)secθ i (x,y)) is the derivative of the signal versus thickness which we will call “dsignal” and which can calculated or measured. Using this expression
   m   i ( x,y )− m 0 i ( x,y )=[( M 0 i ( x,y )− M   i ( x,y ))dsignal i ( x,y )sec θ i ( x,y )]{circle around (×)} psƒ ( x,y )  Eq. (22) 
   We can now express the optimization function (equation 16) as 
   
     
       
         
           
             
               
                 
                   χ 
                   2 
                 
                 = 
                 
                   
                     ∑ 
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     i 
                                   
                                   - 
                                   
                                     M0 
                                     i 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   
                                     dsignal 
                                     i 
                                   
                                   ⁢ 
                                   sec 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   
                                     noise 
                                     i 
                                   
                                 
                               
                             
                             ) 
                           
                           ⊗ 
                           psf 
                           ⊗ 
                           
                             ( 
                             
                               nps 
                               
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           ⊗ 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           mtf 
                           ⊗ 
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     i 
                                   
                                   - 
                                   
                                     M0 
                                     i 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   
                                     dsignal 
                                     i 
                                   
                                   ⁢ 
                                   sec 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     i 
                                   
                                 
                                 
                                   
                                     noise 
                                     i 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         ] 
                       
                       
                         
                           x 
                           = 
                           0 
                         
                         , 
                         
                           y 
                           = 
                           0 
                         
                       
                     
                   
                   + 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ɛ 
                     ⁢ 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                           ⁢ 
                           
                             ⅆ 
                             y 
                           
                           ⁢ 
                           
                             ⅆ 
                             z 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
   
   The convolution in the middle is, by definition, the detective quantum efficiency
 
 dqe=psƒ {circle around (×)}( nps   −1 ){circle around (×)}psƒ  Eq. (24)
 
   We define the ratio 
                   noise     dsignal   2       =     Δ   ⁢           ⁢   t2             Eq   .           ⁢     (   25   )                 
which is the “thickness noise” squared. Since noise is a strong function of thickness, and a weak function of angle, we can treat Δt 2  as a function only of thickness. This function, like dsignal, can be calculated or measured.
 
   For simplicity, we define a noise function 
                     σ   i     ⁡     (     x   ,   y     )       =               noise   i     ⁡     (     x   ,   y     )             dsignal   i     ⁡     (     x   ,   y     )         ⁢   sec   ⁢           ⁢         θ   i     ⁡     (     x   ,   y     )         -   1         ⁢     
     ⁢           =     sec   ⁢           ⁢         θ   i     ⁡     (     x   ,   y     )         -   1       ⁢       Δ   ⁢           ⁢     t2   ⁡     (         M   i     ⁡     (     x   ,   y     )       ⁢   sec   ⁢           ⁢       θ   i     ⁡     (     x   ,   y     )         )                       Eq   .           ⁢     (   26   )                 
and equation 23 is expressed as
 
   
     
       
         
           
             
               
                 
                   χ 
                   2 
                 
                 = 
                 
                   
                     
                       ∑ 
                       i 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               
                                 
                                   M 
                                   i 
                                 
                                 - 
                                 
                                   ∫ 
                                   
                                     D 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           x 
                                           + 
                                           
                                             
                                               α 
                                               i 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         , 
                                         
                                           y 
                                           + 
                                           
                                             
                                               β 
                                               i 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         , 
                                         z 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   σ 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⊗ 
                           dqe 
                           ⊗ 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   M 
                                   i 
                                 
                                 - 
                                 
                                   ∫ 
                                   
                                     D 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           x 
                                           + 
                                           
                                             
                                               α 
                                               i 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         , 
                                         
                                           y 
                                           + 
                                           
                                             
                                               β 
                                               i 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         , 
                                         z 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   σ 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         
                           x 
                           = 
                           0 
                         
                         , 
                         
                           y 
                           = 
                           0 
                         
                       
                     
                   
                   + 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ɛ 
                     ⁢ 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                           ⁢ 
                           
                             ⅆ 
                             y 
                           
                           ⁢ 
                           
                             ⅆ 
                             z 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
   
   We can solve equation 27 by calculating the derivative of χ 2  versus D(x 0 ,y 0 ,z 0 ) and calculating where this derivative equals 0 
   
     
       
         
           
             
               
                 0 
                 = 
                 
                   
                     ∑ 
                     i 
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     ∫ 
                                     
                                       
                                         D 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               x 
                                               + 
                                               
                                                 
                                                   α 
                                                   i 
                                                 
                                                 ⁢ 
                                                 z 
                                               
                                             
                                             , 
                                             
                                               y 
                                               + 
                                               
                                                 
                                                   β 
                                                   i 
                                                 
                                                 ⁢ 
                                                 z 
                                               
                                             
                                             , 
                                             z 
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         ⅆ 
                                         z 
                                       
                                     
                                   
                                   
                                     
                                       σ 
                                       i 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         y 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               ⊗ 
                               
                                   
                                 dqe 
                                 ) 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 
                                   σ 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         
                           
                             x 
                             = 
                             
                               
                                 x 
                                 o 
                               
                               - 
                               
                                 
                                   α 
                                   i 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           , 
                           
                             y 
                             = 
                             
                               
                                 y 
                                 o 
                               
                               - 
                               
                                 
                                   β 
                                   i 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                       
                       + 
                       
                         ɛ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           D 
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 o 
                               
                               , 
                               
                                 y 
                                 o 
                               
                               , 
                               z 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
   
   The problem with this expression is that D is a continuous variable of z, and divisions into z-slices will lead to inaccuracy and/or increased computational burden. There is a solution to this problem, however, which is to let D be defined from a generator that is discrete 
   
     
       
         
           
             
               
                 
                   D 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∑ 
                     j 
                   
                   ⁢ 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           - 
                           
                             
                               α 
                               j 
                             
                             ⁢ 
                             z 
                           
                         
                         , 
                         
                           y 
                           - 
                           
                             
                               β 
                               j 
                             
                             ⁢ 
                             z 
                           
                         
                         , 
                         z 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
   
   Then equation 28 becomes 
   
     
       
         
           
             
               
                 0 
                 = 
                 
                   
                     ∑ 
                     i 
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   
                                     ∫ 
                                     
                                       
                                         ∑ 
                                         j 
                                       
                                       ⁢ 
                                       
                                         
                                           G 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 - 
                                                 
                                                   
                                                     α 
                                                     j 
                                                   
                                                   ⁢ 
                                                   z 
                                                 
                                                 + 
                                                 
                                                   
                                                     α 
                                                     i 
                                                   
                                                   ⁢ 
                                                   z 
                                                 
                                               
                                               , 
                                               
                                                 y 
                                                 - 
                                                 
                                                   
                                                     β 
                                                     j 
                                                   
                                                   ⁢ 
                                                   z 
                                                 
                                                 + 
                                                 
                                                   
                                                     β 
                                                     i 
                                                   
                                                   ⁢ 
                                                   z 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           ⅆ 
                                           z 
                                         
                                       
                                     
                                   
                                   - 
                                   
                                     M 
                                     i 
                                   
                                 
                                 
                                   
                                     σ 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       x 
                                       , 
                                       y 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             ⊗ 
                             dqe 
                           
                           ) 
                         
                         ⁢ 
                         
                           
                               
                             
                               1 
                               
                                 
                                   σ 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           
                             
                               x 
                               = 
                               
                                 
                                   x 
                                   o 
                                 
                                 - 
                                 
                                   
                                     α 
                                     i 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             , 
                             
                               y 
                               = 
                               
                                 
                                   y 
                                   o 
                                 
                                 - 
                                 
                                   
                                     β 
                                     i 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         ɛ 
                         ⁢ 
                         
                           
                             ∑ 
                             i 
                           
                           ⁢ 
                           
                             G 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     x 
                                     o 
                                   
                                   - 
                                   
                                     
                                       α 
                                       i 
                                     
                                     ⁢ 
                                     
                                       z 
                                       o 
                                     
                                   
                                 
                                 , 
                                 
                                   
                                     y 
                                     o 
                                   
                                   - 
                                   
                                     
                                       β 
                                       i 
                                     
                                     ⁢ 
                                     
                                       z 
                                       o 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
   
   Equation 30 can only be true if the individual elements in the i-summation are identically equal to zero 
   
     
       
         
           
             
               
                 0 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               
                                 ∫ 
                                 
                                   
                                     ∑ 
                                     j 
                                   
                                   ⁢ 
                                   
                                     
                                       G 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             x 
                                             - 
                                             
                                               
                                                 α 
                                                 j 
                                               
                                               ⁢ 
                                               z 
                                             
                                             + 
                                             
                                               
                                                 α 
                                                 i 
                                               
                                               ⁢ 
                                               z 
                                             
                                           
                                           , 
                                           
                                             y 
                                             - 
                                             
                                               
                                                 β 
                                                 j 
                                               
                                               ⁢ 
                                               z 
                                             
                                             + 
                                             
                                               
                                                 β 
                                                 i 
                                               
                                               ⁢ 
                                               z 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       z 
                                     
                                   
                                 
                               
                               - 
                               
                                 M 
                                 i 
                               
                             
                             
                               
                                 σ 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ⊗ 
                         dqe 
                       
                       ) 
                     
                     ⁢ 
                     
                       1 
                       
                         
                           σ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                       
                     
                   
                   + 
                   
                     ɛ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         G 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
   
   We can simplify equation 31 further by defining a function C as 
   
     
       
         
           
             
               
                 
                   
                     C 
                     ij 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     0 
                     
                       z 
                       max 
                     
                   
                   ⁢ 
                   
                     
                       δ 
                       ⁡ 
                       
                         [ 
                         
                           
                             x 
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     α 
                                     i 
                                   
                                   - 
                                   
                                     α 
                                     j 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               z 
                             
                           
                           , 
                           
                             y 
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     i 
                                   
                                   - 
                                   
                                     β 
                                     j 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               z 
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       z 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
   
   In equation 31, z max  is determined by the height of the object that is imaged, and the integral excludes absorption below the detector or above z max  in the solution. Equation 31 can be expressed as 
   
     
       
         
           
             
               
                 
                   dqe 
                   ⊗ 
                   
                     ( 
                     
                       
                         M 
                         i 
                       
                       
                         σ 
                         i 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     dqe 
                     ⊗ 
                     
                       ( 
                       
                         
                           
                             C 
                             ij 
                           
                           ⊗ 
                           
                             G 
                             j 
                           
                         
                         
                           σ 
                           i 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     ɛ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       σ 
                       i 
                     
                     ⁢ 
                     
                       G 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sum 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         over 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
   
   This is the final equation to be solved for G, then we get D from G and finally d from D. If s were constant, then equation 33 could be inverted using Fourier transforms because the convolutions become products
 
ℑ( dqe )ℑ( M   i )=ℑ( dqe )ℑ( C   ij )ℑ(G j )+εσ i   2 ℑ( G   i )  Eq. (34)
 
where ℑ denotes a Fourier transform. Equation 33 is then expressed as
 
ℑ( G   i )=(ℑ( dqe )ℑ( C   ij )+εσ i   2 δ ij ) −1 ℑ( dqe )ℑ( M   i )  Eq. (35)
 
   In equation 35, the inverses are performed individually over each spatial frequency. Note that these inverse matrices can be computed once and stored as a look-up table for improved computational speed. The stored data is required for each noise level and each value of z max , so this data will require a large storage capacity. The technique that we use to solve equation 33 for a non-constant s involves treating the problem in multiple iterations with a constant σ trial , solving equation 33 using the Fourier transform method (equation 34), calculating an error term, then iterating by reconstructing for the error term but using successively different values for the trial σ value σ trial . First calculate the error term 
   
     
       
         
           
             
               
                 error 
                 = 
                 
                   
                     dqe 
                     ⊗ 
                     
                       ( 
                       
                         
                           M 
                           i 
                         
                         
                           σ 
                           i 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     dqe 
                     ⊗ 
                     
                       ( 
                       
                         
                           
                             C 
                             ij 
                           
                           ⊗ 
                           
                             G 
                             j 
                           
                         
                         
                           σ 
                           i 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     ɛ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       σ 
                       i 
                     
                     ⁢ 
                     
                       G 
                       j 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
   
   Then, invert the error term using a constant σ
 
 dqe {circle around (×)}( C   ij {circle around (×)}Δ G   i )+εσ trial   2   ΔG   i =error  Eq. (37)
 
   Then, update the reconstructed generator coefficients
 
G i ←G i +ΔG i   Eq. (38)
 
   Then repeat with a different value of σ trial . Eventually, the “error” becomes sufficiently small and we have our solution. We start with the largest value s for s, and then gradually decrease. Boundary conditions are handled by reflection conditions with tapering. The minimization function in this terminology is 
   
     
       
         
           
             
               
                 
                   χ 
                   2 
                 
                 = 
                 
                   
                     
                       
                         ∑ 
                         i 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       ij 
                                     
                                     ⊗ 
                                     
                                       G 
                                       j 
                                     
                                   
                                   
                                     σ 
                                     j 
                                   
                                 
                                 - 
                                 
                                   
                                     M 
                                     i 
                                   
                                   
                                     σ 
                                     i 
                                   
                                 
                               
                               ) 
                             
                             ⊗ 
                             dqe 
                             ⊗ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       ik 
                                     
                                     ⊗ 
                                     
                                       G 
                                       k 
                                     
                                   
                                   
                                     σ 
                                     i 
                                   
                                 
                                 - 
                                 
                                   
                                     M 
                                     i 
                                   
                                   
                                     σ 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         
                           
                             x 
                             = 
                             0 
                           
                           , 
                           
                             y 
                             = 
                             0 
                           
                         
                       
                     
                     + 
                     
                       ɛ 
                       ⁡ 
                       
                         ( 
                         
                           
                             G 
                             k 
                           
                           ⊗ 
                           
                             C 
                             kj 
                           
                           ⊗ 
                           
                             G 
                             j 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     | 
                     
                       
                         x 
                         = 
                         0 
                       
                       , 
                       
                         y 
                         = 
                         0 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
   
   So, the first term should equal (# of tube positions)×(# of pixels) when ε is set correctly. 
   To summarize, the steps are:
         1) Transform the measurements m into the “integrated Hounsfield” form M   2) Repeat on:
           a. Calculate error term using non-constant s   b. Select trial value strial   c. Invert the error term with strial   d. Update G   e. Change strial   
           3) For a given slice selection, calculate D from G   4) Calculate d from D
 
b) Computational Description of Reconstruction Algorithm
       

   This section provides the computational flow chart for the linear image reconstruction algorithm. 
   Raw data mraw i (x,y):
     Data taken over all of the pixels, one shot for each tube location i. Nominal 41 tube locations, 2304 by 3072 pixels.   

   Dead pixel map dead(x,y):
         Map of dead pixels, assumed constant for all i. If not, then must have a map for each i.       

   Binning function Bin(image,binsize):
         Bins image data into N×N units. Nominal choices are N=4 (fine) and N=8 (coarse).       

   Compute binned raw data mbin i (x,y):
 
 m bin i ( x,y )=Bin(mraw i ( x,y )*dead( x,y ),  N )/Bin(dead( x,y ),  N )
 
   Dark field dark(x,y):
         Dark field, assumed constant for all i. This assumption must be verified, may not be constant.       

   White field white i (x,y):
         White field, definitely not constant vs. i, due to geometrical, grid, and other effects. Needs to be recalibrated every time tube settings are changed, either in output or position.
 
white i ( x,y )=mbin i ( x,y )−dark( x,y ) for an exposure with no subject (or a small amount of plastic plate).
       

   Compute calibrated measurements mcal i (x,y):
 
 m cal i ( x,y )=( m bin i ( x,y )−dark( x,y ))/white i ( x,y )
 
   Linearization calibration:
         Compute mcal i (x,y) for various thicknesses t of water equivalent plastic plate, to give mthick i (x,y,t). Fit to the following function:
 
 m thick i ( x,y,t )= C 0 i ( x,y )*exp ( t *( C 1 i ( x,y )+ C 2 i ( x,y )* t )/( C 3 i ( x,y )+ t )
   where C 0 , C 1 , C 2 , and C 3  are slowly varying functions, probably a low order polynomial fit in x and y for each i.       

   Linearize the measurement:
 
 m lin i ( x,y )={− C 1+ln( m cal/ C 0)+[(ln( m cal/ C 0)− C 1) 2 +4* C 2* C 3*ln( m cal/ C 0)] 1/2 }/(2* C 2)
         where I have dropped the (x,y) and i notation for simplicity.   This parameter was called M i  in solver algorithm write-up. The 1/secθ i (x) term in the write up is implicitly included in our linearization calibration because we use flat plates, which already have the effect built in.       

   Secant function used to derive σ:
         Let source be at (xs i , ys i , SID) and detector be at (x,y, 0) then
 
secθ i ( x,y )=[( xs−x ) 2 +( ys−y ) 2   +SID   2 ] 1/2   /SID  (the ratio of pixel distance from source to  SID )
       

   Noise function:
         Functional relationship that estimates noise from thickness of water traversed, found by modeling:
 
Δ t 2 e ( t )=1.487*10 −5  mm 4 *exp([( t /4.385 cm)*38.6 cm+( t /5.481 cm)* t ]/[38.6 cm+ t ])/(pixel area)
   The output has units of length 2 , which corresponds to the error squared of the length estimate.       

   Compute Sigma:
 
σ i ( x,y )={Δ t 2 e[m lin i ( x,y )*secθ i ( x )]} 1/2 /secθ i ( x )
         This is the estimate of the error in mlin i (x,y). It should be very close to a computation of the variance from the region around each pixel for uniform subjects, and can be checked in this way.   As an alternative, we can try using a 3×3 nearest neighbor box and computing the variance within this box as an estimate for σ i (x,y). The output of this method should be clipped on the low side by a minimum value.       

   Pick eps:
         This is a global constant parameter that determines how smooth the solution is. We will try reconstructions for different values until we understand what is preferable.       

   Compute the C i,j (k x ,k y ) Cholesky matrices:
         These will be described separately, as it is the most involved portion of the algorithm.       

   Generators:
         We will compute generators G i (x,y) as the output to the main algorithm, which is then put into a simple quick slice algorithm to recover the slice image. The generators are computed on an array the size of mbin, but are imbedded in a larger padded array, and use reflection or zero padding as boundary conditions.       

   Height:
         The height from the detector is described by h, but the algorithm uses a modified height variable that I have called z, but will start calling z eff  to prevent confusion. The relationship is as follows:
 
 z   eff =( SID*h )/( SID−h ), with inverse  h =( SID*z   eff )/( SID+z   eff )
       

   Pivot point:
         We use h max  to denote the maximum object height from the detector. The algorithm assumes that there is nothing above this height. The corresponding z max =(SID*h max )/(SID−h max ). The pivot point is placed at h piv =1/(2/h max −1/SID). The corresponding z piv =z max /2.       

   Tube angle parameters
         Each tube position has angles (Θx i  and Θy i  associated with it. The formula for these angles is:
 
Θ x   i   =xs   i   /SID, Θy   i   =ys   i   /SID. 
       

   Slice image:
         The images are first computed for a scaled slice that we call D:
 
 D ( x,y,z   eff )=sum i   {G   i   [x−Θx   i *( z   eff   −z   piv ),  y−Θy   i *( z   eff   −z   piv )]}
   Or in other words, the generator images are shifted by an amount that depends on height, then added together. The value of z eff  given height is described above.   To recover the actual image d(x,y,h), we must zoom and scale D:
 
 d ( x,y,h )= D ( x *zoom,  y *zoom,  z   eff )*zoom 2 , where zoom= SID /( SID−h )= z   eff   /h. 
   This removes the geometrical magnification effect.       

   We are ignoring the DQE effects for this first version of the algorithm. 
   Computing the generators:
         The following is iterated:
           First compute an error value:
 
err i ( x,y )= m lin i ( x,y )−sum j   {C   ij ( x,y ){circle around (×)} G   j ( x,y )}−eps*σ i ( x,y ) 2   *G   i ( x,y )
   The error is computed over the detector area.   The convolution operation {circle around (×)} will be described later. The convolution requires that the array be imbedded in a larger working area, and before the convolution takes place the data is padded either by reflection or just zeroing outside the detector area. The choice of boundary conditions must be the same as that used to generate the slice images.   Next invert this error value:
 
Δ G   i ( x,y )=[ C   ij ( x,y )+eps*σ0 2 *δ( x,y )*δ ij ] −1   {circle around (×)}err   i ( x,y )
   Where σ 0  is a constant, at least for a particular iteration. Then
 
G i (x,y)←G i (x,y)+ΔG i (x,y)
   Again, this is computed over the detector area.   
           Each iteration can either use the same choice for σ 0  or can vary to improve convergence. A good starting value is the maximum of σ i (x,y). The total rms of [err i (x,y)/σ i (x,y)] is used to decide if convergence is adequate.       

   Convolution operation:
         In order to efficiently compute the forward and backward convolutions while minimizing data storage, a number of complications must be introduced:
           1) The convolutions, as is usual are computed using FFT&#39;s   2) Since the matrix operations take place in Fourier space, C ij (x,y) is never actually computed or stored but rather C ij (k x ,k y ).   3) In Fourier space the convolutions become just simple matrix multiplications.   4) Since the generator functions are real, and their Fourier transform are symmetrical, a normal FFT has a factor of 2 redundancy. We therefore use a 2d Real FFT instead.   5) Due to the fact that C ij (x,y) is real, C ij (k x ,k y ) is symmetrical. We therefore only need to compute for half of the spatial frequencies, say k x ≧0.   6) Since we picked the pivot point z piv  at exactly half of z max , the C ij (x,y) turn out to be symmetrical (in x,y), and therefore C ij (k x ,k y ) is real.   7) The C ij (k x ,k y ) are also symmetrical, and in fact positive definite, in i,j. The positive definiteness follows from the definition of C, which will be discussed later. This allows C to be factored into a Cholesky decomposition, that is into a product of a lower triangular matrix L with itself C=L*L T .   8) Only L is stored, and only the triangle of data values are saved and not the zeros.   9) To be more precise, we use L*L T =C ij (k x ,k y )+eps*σ 0   2 *δ i,j      10) In the forward operation we need to subtract σ 0   2  back out from the σ i (x,y) 2  because we added it into the Cholesky matrices   11) The inverse operation can be computed almost as efficiently as the forward operation due to the properties of Cholesky matrices. Therefore we do not need to save separate matrices for each operation.   12) We need to write a special code to take the packed L matrices and perform the forward and inverse vector multiply operation. This is pretty easy, I will write it up.   13) We need to get a Real FFT code and implement it. A site called fftw.org seems to have this for free, but we will need to change it to single precision.   14) The FFT&#39;s use padded arrays, which need to approximately add at least the size of the convolution kernels of C ij (x,y). The array plus padding also must be a nice size for the FFT, such as a power of 2 times some small prime number (2,3,5).   15) The padding rules will have to be evaluated, but either zero padding or reflection padding will probably be best. It should make the most difference when the subject is sticking off the edge.   
               c) Numerical operation count: This section provides computational operation count as follows:
       Assumptions: 41 measurements, each with 2000×2000 pixels, and reconstruct 20 slices   Input FFT&#39;s: 28*10^9 real multiplies   Matrix operations: 13*10^9 real multiplies   Output FFT&#39;s: 14*10^9 real multiplies   Total: 55 Gops   
       

   The computation time is therefore 1 min on a 1 Gflop processor. Note the assumptions are conservative and much faster processing times are possible with smaller data sets.
     d) Rationale for image reconstruction algorithm: The preferred linear image reconstruction algorithm, described by the χ 2  minimization equation (10), is derived using equation (18.4.9) and (18.4.11) in Numerical Recipes in Fortran 77; the Art of Scientific Computing”, Chapter 18.4 “Inverse Problems and the Use of A Priori Knowledge”, Cambridge University Press, pg. 795–799 (1986–1992).   

                   χ   2     =         ∑   ij     ⁢       [       c   i     -       ∑   μ     ⁢       R     i   ⁢           ⁢   μ       ⁢     u   ⁡     (     x   μ     )             ]     ⁢       S   ij     -   1       [       c   j     -       ∑   μ     ⁢       R     j   ⁢           ⁢   μ       ⁢     u   ⁡     (     x   μ     )             ]         +     λ   ⁢       ∑   μ     ⁢       u   2     ⁡     (     x   μ     )                     Eq   .           ⁢     (   40   )                 
where c i ≡m i (x 0 ,y 0 ) are the measurements,
 
               ∑   μ     ⁢       R     i   ⁢           ⁢   μ       ⁢     u   ⁡     (     x   μ     )           ≡       m   o     ⁡     (       x   o     ,     y   o       )             
are the unknown model data convolved with the response function of the measurements system, and S ij ≡(n(x 0 ,y 0 )n(x 1 ,y 1 )) avg  is the covariance noise function. Eq. (40), and therefore Eq. (10), is the functional generalization of the conventional chi-squared fitting algorithm that fits a straight line, for example, to a measured set of data points (x i ,y i ); the best fit seeks to minimize the function (Eq. 15.1.5 in Numerical Recipes)
 
                   χ   2     =       ∑   i     ⁢     (         y   i     -     y   ⁡     (         x   i     ;     a   1       ,   …   ⁢           ,     a   m       )           σ   i       )               Eq   .           ⁢     (   41   )                 
where σ i  is the uncertainty, or noise, associated with each data point. The fit is constrained to functional form (straight line or quadratric function, for example) and constants (a 1 , . . . , a m ) are determined by the fit. Equations (40) and (10) seek to minimize an equivalent χ 2  function but do not constrain the problem to a specific functional form. The optimal three-dimensional function d(x tomo ,y tomo , h), representing the tomographic images, is calculated by the minimization routine. Equation (40), and equation (10), is constrained by the second term that imposes a certain amount of “smoothness” to the solution, depending on the value of the regularization paerameter λ (or ε). The preferred embodiment uses a value ε=0.01 for regularization parameter.
     e) Non-linear reconstruction algorithms: We assume that optimal reconstruction algorithm is the best that can be done with no assumptions about the nature of the subject being imaged. The proposed DXT system incorporates non-linear algorithms that incorporate additional information:
       Minimization of the effect of overlaying tissue by tailoring sidelobe artifacts to be lower from regions with strong features and higher from featureless areas. Techniques developed for sonar and radar may be applicable. These basically work by weighing projections more heavily which pass through “windows” in the subject.   Sharpening of slice thickness by “concentrating” features. Some spatial frequencies can be located in depth better than others, so we can increase the probability that a feature at one spatial frequency lies together with the same feature measured at a different spatial frequency.   Imposing an outer boundary of the subject, so we can numerically constrain zero tissue outside of this boundary.   
       
   Simulation of the Reconstruction Algorithm 
   A computer simulation of a preferred linear reconstruction algorithm is displayed in  FIGS. 7 and 8 . The object is a thick walled cone with an “x” in the very center of the object; this object was taken to be a combination of low and high spatial frequencies to demonstrate various aspects of the algorithm, and present a reasonably challenging case. The simulated cone is made up of 100 separate slices to ensure accuracy. Multiple digital projection radiographs of the cone viewed from various positions of the x-ray source  120  are displayed in  FIG. 7 .  FIGS. 8 through 10  displays original tomographic images (left) and images reconstructed from the digital projection radiographs of the cone model (right) in steps of 10% of the thickness from 0% to 100%.  FIG. 8  shows the bottom 4 slices 0%, 10%, 20%, and 30%.  FIG. 9  shows slices 40%, 50%, 60%, and 70%.  FIG. 10  shows slices 80%, 90%, and 100%. Notice that there is only a faint hint of the center 50% slice cross which spilled over into the 40% and 60% reconstruction slices. Notice also the preservation of both the high resolution from the center cross and the lower spatial frequencies from the wall of the cone. 
   Alternate Embodiments 
   An alternate embodiment of the invention involves the use of multiple x-ray sources in order to provide faster imaging times, and therefore higher throughput.  FIG. 11  displays the side views and  FIG. 12  the front views of a DXT system that incorporates a line of five x-ray sources  350  in a line that sequentially expose digital x-ray sensor  365  and provide five separate projection radiographs. Conveyor belt  370  re-positions object  360  in a direction perpendicular to the line of x-ray sources  350  to a plurality of positions (8 positions preferred) where x-ray sources  350  provide five digital projection radiographs at each position of object  360  in order to provide a total of 40 projection radiographs; this data is used to reconstruct tomographic images of object  360 . This system is useful for luggage inspection. It uses the same basic algorithm as described above. 
   While there have been shown what are presently considered to be preferred embodiments of the present invention, it will be apparent to those skilled in the art that various changes and modifications can be made herein without departing from the scope and spirit of the invention. For example, prior art techniques can be incorporated for fast readout of data to enable good images of moving parts such as the heart and lungs of people. The invention can be applied to a variety of uses in addition to medical imaging and luggage screening. The invention can be made portable for use by emergency teams and battlefield conditions. Known techniques can be incorporated for automatic recognition of items such as weapons and explosives based on shape and attenuation information. Many modifications could be made to the specific algorithm which has been described in detail without departing from the basic concepts of the present invention. For example other least square techniques other than the one specifically described can be used with generalized functions to turn x-ray data into images. 
   Thus, the scope of the invention is to be determined by the appended claims and their legal equivalents.