preferred embodiments of the invention will be set forth in detail with reference to the drawings , in which like reference numerals refer to like elements or steps throughout . the fdk algorithm expands upon the fan beam algorithm by summing the contribution to the object of all the tilted fan beams . the reconstruction is based on filtering and back projecting a single fan beam within the cone . based on the cone beam geometry in fig1 , which shows the relative locations of the cone beam emitter or other x - ray source 102 and the plane of the flat - panel detector 104 , the formula of the fdk is : the * sign denotes the convolution ; so : the distance from the x - ray source to the origin ; n , m : integer value where n = 0 and m = 0 corresponds to the central ray passing through the origin ; β : the projection angle defined in the scanning plane ; p : the virtual detector sampling interval along the t axis ; ξ : the virtual detector sampling interval along the z axis ; r β ( np , mξ ): the actual discrete 2 - d projection data ; h ( np ): the discrete one - dimensional ramp filter impulse response along the t axis . this means that fdk projects the off - scanning plane projection data into the scanning plane and then follows the 2 - d fan beam reconstruction algorithm . in equation ( 1 ), the factor of ½ in front of the integral is used to cancel the projection redundancy when a full circular scanning is conducted . this implies that the off - scanning plane projection data has the same redundancy as the projection data in the scanning plane . the cone beam half - scan scheme is also the extension of the fan beam half - scan combined with the fdk , in which the weighting coefficients calculated from the scanning plane geometry are applied to all projection rows as follow , this is the fdkhsfw scheme , where δ is half the full fan angle of the central scanning plane along the t axis . the off - scanning plane projection data are still treated as they have the same redundancy . ω ( β , np ) is the discrete weighting coefficient , calculated based on the scanning plane geometry , and can be represented by parker &# 39 ; s weighting function or any other weighting function as long as it can make a smooth transition between the doubly and singly sampled regions to avoid discontinuities at the borders of these regions . undoubtedly , fdkhsfw holds all the properties that the fdk full scan scheme does . for cone beam projection data off the scanning plane , however , it is impossible to obtain completely doubly sampled projections for a single circular orbit acquisition , even if projections are sampled over 360 °. in other words , the projection redundancy becomes less and less when projection rows get further away from the scanning plane . if the fdk algorithm had been directly applied to un - weighted half - scan projection data , the reconstructed images would unavoidably have artifacts . one way to handle the weighting on the less redundancy projection row data away from scanning plane , is proposed as follows : β ′ is the cone - weighting angle and is defined between the two lines o ′ s and oy when the cone - shaped curve plane o ′ sy is flattened . β ′ is dependent on the position of the row projection data in the z direction ( rotation axis ). δ ′ is half of the titled fan angle that is adopted from g . t . gullberg and g . l . zeng , “ a cone - beam filter backprojection reconstruction algorithm for cardiac single photon emission computed tomography ,” ieee trans . med . imaging 11 , 91 - 101 ( 1992 ). notice that when m is zero , this weighting function is actually the parker &# 39 ; s weighting function for fan - beam . by incorporating the cone - beam weighting function with the fdk , the fdkhscw is obtained as follows : please note that the projection data must be weighted prior to being filtered . since fdkhsfw is the commonly acknowledged scheme for half - scan reconstruction , the requirement for fdkhscw is that it should produce no more artifacts than fdkhsfw . further investigation of half scan cone beam weighting will now be disclosed . in a circular fan - beam half - scan , there are two redundant regions in the scanning plane in terms of the projection angle β . fig2 shows that the projecting ray data acquired in region i will have a conjugate ray data in region ii . in these two regions , the projection ray data is wholly or partly redundant . if half of the full fan angle is δ degrees , the half scan range in terms of projection angle defined in the scanning plane is from 0 ° to 180 °+ 2δ . the first and second redundant region is from 0 ° to 4δ and from 180 °− 2δ to 180 °+ 2δ respectively . in the traditional fdk cone - beam half scan scheme , all the row projection data are weighted by the same set of coefficients defined in the scanning plane because the row projection data away from the scanning plane are expected to have the same redundancy as those in the scanning plane . the proposal of the circular cone - beam half scan weighting scheme is based on the idea that the weighting coefficients should be different for projection data in different rows , and for the row projection data furthest away from the scanning plane , it should be weighted less . as of this date , we have not seen any literature discussing this issue . we found that if we use β ′ = β ⁢ ⁢ 1 1 + m 2 ⁢ ξ 2 / so 2 as the weighting angle for different row projection data , then , the weighting coefficients in the first redundant region away from the scanning plane are not much different from those calculated in the scanning plane ; the biggest difference is below 0 . 2 percent if δ = 15 ° and the half cone angle is also 15 °. on the other hand , when β ′ is used as the weighting angle in the second redundant region , the weighting coefficients away from the scanning plane behave obviously differently from those in the scanning plane and different from each other at the different rows , thus resulting in the compensation for the density drop in the place away from the scanning plane in the reconstruction image . the weighting angle β ′ has two characteristics : first , it has row position dependence that is reflected by mξ , indirectly connected to the cone angle information ; second , it has less difference from β when β is in the first redundant region than when β is in the second redundant region . thus , it is beneficial to construct the cone angle dependent weighting coefficients in the second redundant region to achieve our scheme . in order to make computer simulation closer to the practical cbct configuration , geometric parameters are set in terms of physical length ( millimeter ) rather than normalized units . the distances from the x - ray source to the iso - center of the reconstruction and to the detector are 780 mm , and 1109 mm respectively . the full fan and cone angle are 30 degrees . the detector area is 595 × 595 mm 2 and has a 512 by 512 matrix size . the voxel size is 0 . 816 mm 3 . cartesian coordinate ( x , y , z ) is used to define the object , where z is the rotation axis . the sampling rate of projection angle is 0 . 8 ° with the total number of projection images of 450 for full scan and 262 for half scan . the low contrast shepp - logan phantom was used ( see ref . 7 for geometrical parameters ), all of its geometrical parameters are multiplied by 200 to simulate the physical length ( millimeter ) of the phantom . based on the scanning geometrical parameters defined above weighting coefficient distribution associated with fdkhsfw and fdkhscw are compared by picking up β = 46 ° in the redundant region i and β = 192 ° in the redundant region ii . fig3 a - 3d show , respectively , fdkhsfw ( β = 46 °), fdkhscw ( β = 46 °), fdkhsfw ( β = 192 °) and fdkhscw ( β = 192 °). fig4 a - 4c show the reconstructed sagittal image from different fdk schemes ( respectively , fdkfs , fdkhsfw and fdkhscw ) at x = 0 mm with the display window [ 1 . 005 1 . 05 ]. fig4 d shows the phantom . fig4 e and 4f show the profile comparison along the solid white vertical and horizontal lines in the phantom image of fig4 d . the ramp filter was used on the noise - free weighted projection data before back - projection . in order to test the performance of this new scheme over the quantum noise that is commonly encountered in practical cbct data acquisition , we generated quantum noise contaminated data . x - ray with 100 kvp was selected which corresponds to an effective photon fluence of 2 . 9972 * 10 7 photons / cm 2 · mr . the exposure level per projection was set to 4 mr , the total exposure levels for fdkfs and fdkhscw are 1800 mr and 1048 mr respectively . fig5 a - 5d show the reconstructed results under different noise levels and profile comparisons . a hamming window is used during filtering to suppress the noise . more specifically , fig5 a and 5b show , respectively , fdkfs with total exposure level of 1800 mr and fdkhscw with total exposure level of 1048 mr , while fig5 c and 5d show a profile comparison between fdkfs , fdkhscw , and the phantom along the solid vertical and horizontal lines in fig4 d . a variation of the first preferred embodiment will be presented . in terms of the radon domain , as fig6 shows , the dotted points in the circle ( zones 602 ) represent the radon value of the object acquired in a circular scan , while the shaded area in the circle ( zone 604 ) represents the missing radon points set which cannot be acquired through a circular scan . as argued by hu , in a circular scan , fdk uses only the redundant points inside the object circle and on the dotted arc boundaries , but does not use the non - redundant points on the solid arc boundaries within the object circle . thus , the following correction term is used : if the above reconstruction is designated f 1 ( t , s , z ), then the final reconstruction is a second preferred embodiment , which is particularly applicable to breast imaging , will now be disclosed . based on the geometric parameters of current cbbct , we designed a new scanning scheme for cbbct . as fig7 a and 7b show , the position of the x - ray source 102 is at z = 0 cm during the circular scan . after the circular scan , the x - ray source 102 and detector 104 lower down simultaneously while they are still rotating . when the x - ray source gets to the point where z = 10 , it starts to shoot and keeps shooting between the interval δl till it finishes eight shoots in this helical scan . for each shoot during the helical scan , the x - ray source still maintains the same collimation as it does in circular scan . this scheme is called sparse spiral scan . in practice , to efficiently implement sparse spiral scan , the system is required to comprises a slip ring on the gantry frame for providing the connection to the electrical power to the on - gantry components and the communication between the two - dimensional detector and the computer system and a motor to move the gantry or the support of the object on which the object rests up or down along the rotation axis ; the projection angles associated with helical line scan are illustrated by fig7 b . there are eight shoot points during helical scan which cover the angular range of 4 π , and the movement in z direction is from 48 mm to 128 mm with the increment interval of 10 mm based on the size of the simulated breast phantom . some of the radon data points acquired from this additional scanning trajectory still can be acquired through the circular scan ; this is what is called redundancy sampling points in radon domain , and can be efficiently eliminated by the redundant window function ( rwf ), and the geometric setup of the collimator as during the hl scanning can avoid the redundant sampling radon points between those shooting points on this helical line . since the collimation during helical line scanning unavoidably induces the longitudinal truncation , a geometric dependent truncation window function ( twf ) will be used to handle this case . composite reconstruction framework is probably the most preferable algorithm for the cbctbi . the reconstructed object is f ( r ), and can be mathematically described by the following equation : f 1 ( r ): reconstructed object from single circular scan , f 2 ( r ): reconstructed object from hui &# 39 ; s term based on single circular scan , f hl ( r ): reconstructed object from sparse helical line scan [ 5 ], based on fig7 a , the mathematic formula of the f 1 ( r ), and f 2 ( r ) can be described as the reconstruction term for the helical line scan cab be formatted as a type of filtered backprojection ( fbp ): w z i ( l , φ ) is the window function used to block out the radon data acquired through helical line scan but have already been accessed by circular scan . based on the scanning design , the helical line scan will unavoidably encounter the longitudinal truncation during the scanning . some radon points it acquires do not reflect the actual radon data and should be eradicated during the back - projection . window w d zi ( l , φ ) is used to accomplish this purpose . while preferred embodiments and variations thereon have been disclosed above , those skilled in the art who have reviewed the present disclosure will readily appreciate that other embodiments can be realized within the scope of the invention . for example , numerical values are illustrative rather than limiting . also , the invention can be implemented on any suitable scanning device , including any suitable combination of a beam emitter , a flat panel or other two - dimensional detector or other suitable detector , and a gantry for relative movement of the two , as well as a computer for processing the image data to produce images and a suitable output ( e . g ., display or printer ) or storage medium for the images . software to perform the invention may be supplied in any suitable format over any medium , e . g ., a physical medium such as a cd - rom or a connection over the internet or an intranet . therefore , the present invention should be construed as limited only by the appended claims .
a cone - beam scanning system scans along a half circle . the reconstruction uses a weighting function which decreases for rows farther from the scan plane to take the redundancy of the projection data into account . another embodiment uses a circle plus sparse helical scan geometry . image data can be taken in real time .