the present invention utilizes an optimal , minimum cost two - dimensional detector geometry , characterized by an exposure window which is limited vertically by the two nearest turns of the helical source trajectory . both the motivation for and the exploitation of this detector window differs greatly from the ones given in [ tam95 ] and [ eber95 ]. to explain the specific virtue of this exposure window , we refer again to fig1 which shows a perspective view of a source s , a detector 11 wrapped around the helix cylinder 12 and inside this an object cylinder 13 . in the sequel , unless stated otherwise , we assume that the object cylinder is rotating counter - clock - wise as shown around the z - axis and translated upwards in a right - handed helix , while the source s and the detector 11 are fixed in the space ( x , y , z ). fig6 shows the arrangement as seen from above , while fig7 shows the detector window unwrapped and rolled out on a plane . note that fig1 and 7 are consistent only if the rays in fig7 are understood to be coming from the source towards the viewer . fig2 shows the detector placed on the source cylinder 41 centered in s and having a radius which is twice as large as the helix cylinder 12 . fig1 is a pictorial representation of a two - dimensional detector and point - shaped ray source moving synchronously around an object in a helical trajectory ; fig2 is a depiction of a detector wrapped onto the surface of the source cylinder , centered in s ; fig3 is a depiction of a vertical section of the parallel scanning system described herein ; fig5 is a depiction of a parallel projection unwrapped and rolled out onto a 2 - d sheet ; fig6 is a depiction of the arrangement of fig1 as seen from above ; fig7 is a detector surface unwrapped and rolled out on a plane of the detector of fig1 ; fig8 is a straight side view of the depiction of fig1 ; fig9 is a straight top view of the depiction of fig1 ; fig1 a , 10 b are depictions of a rebinning parallel projection ; fig1 a , 12 b and 12 c depict 3 orthogonal views a , b and c of the parallel projection system of the invention ; fig1 is a view from above the fig1 representation where the object is fixed and the source and detector are rotating ; fig1 depicts a detector window of the helix cylinder rolled out on a plane of a sheet ; and fig1 is a depiction of the detector in fig1 reduced in height to a single row of detector elements . as mentioned , the 2d - detector 11 in fig1 is wrapped onto the helix cylinder 12 . unwrapped and rolled out on the plane of the sheet , the same detector surface 11 in fig7 is seen to be bounded by four straight lines , two vertical ones 31 and 32 , and two slanted ones 33 and 34 . within this area the object 17 is projected , i . e ., rays from the cone - beam source reaches active detector elements . horizontally , this area has to be extended to cover the object cylinder 13 , which translates to a certain width , or fan angle γ max , as seen from the source . as an example we have assumed that this object cylinder has a radius r = r 2 where r is the radius of the helix cylinder 12 . this means that horizontally on 12 the detector covers a rotation angle of 180 degrees out of 360 , and that seen from the source the detector 11 covers a fan angle from − 45 to + 45 degrees . in principle the detector may be extended to a full turn which then has a fan - angle from − 90 to + 90 degrees and would allow for an object cylinder that extends all the way to the helix . the slanted lines 33 and 34 are intersecting the cylinder surface 12 at the slope tan   ɛ = v ω   r = h 2  π   r ( 1 ) where v is the vertical translation velocity , w is the angular velocity for the rotation , and h is the pitch of the helix . at the core of the invention is the following property of the detector - exposure window . every point in a cylindrical long object , with a radius that fits inside the boundaries of the detector window , will be exposed ( projected ) during a rotation angle which is exactly 180 degrees , seen from the actual point in the object . a conjecture of this new sufficiency condition is that as soon as one point or a set of points ( i . e . a part of the long object ) has been fully exposed in the above sense , the reconstruction of this part can take place . this is in contradiction to the situation in [ tam95 ] and [ eber95 ]] where the whole roi has to be exposed to make the radon space complete before the actual reconstruction is commenced . an example of this 180 degree exposure is the line 18 in fig1 . it contains the three object points q 1 − q − q 2 and it is shown in two positions where the exposure starts and ends , respectively . note that the end points of this line is sliding and touching the outer cylinder so that during the rotation , both ends will coincide with the source s . any such line will be called a − line this line is also shown in fig6 in the same two positions . assume as before that the object is moving upwards and rotating counter - clockwise when seen from above . in the detector window of fig7 the line q 1 − q − q 2 crosses the lower boundary 34 as a single point at q in . after a rotation with the angle π + 2γ around the axis 14 this line will be seen as a single point again from the source leaving the detector at q out on the upper boundary 33 clearly , between entrance and exit the source has rotated exactly 180 degrees as seen from any point on this line . since we have chosen this line quite arbitrarily , the same thing is true for all points in the object which belong to fully exposed − lines . in fig6 and fig7 but not in fig1 we have inserted another − line p 1 − p − p 2 . in the fixed source - detector system of fig2 this line p enters and exits in positions which are exactly the reverse of the corresponding positions for the line q . the line p is therefore closer to the source than line q during its exposure , which takes place during a rotation angle of π − 2γ around the axis 121 . the points on line p travels over the detector surface along different and shorter curves as shown in fig7 but seen from any of these points , the source rotates around them exactly 180 degrees . every object point belongs to one and only one line . therefore , the detector system in fig . 1 gives us a complete and perfectly balanced data capture for every point and hence also for the whole object . furthermore , from the conjecture above follows that it should be possible to reconstruct the object at the same pace as an incremental part ( each new set of − lines ) of the long object is fully exposed . the physical implementation and placement of the detector can of course be made in various ways as indicated in fig2 . for instance , it may be placed on the helix cylinder 12 itself , on the source cylinder 41 or on a plane 42 . in any case , the detected and utilized data must be restricted to the window defined by fig7 . in our invention , using the same detector data , the elaborate reconstruction in [ tam95 ] and [ eber95 ] will be replaced by a much simpler procedure . to describe this procedure , we do not have to limit the ongoing scanning and reconstruction to a predetermined roi , nor do we have to specify a 3d origin for the process . instead , scanning and reconstruction is like a constantly ongoing flow , in principle without beginning or end , where each new projection is absorbed and incorporated seamless to the previous result . for this purpose , the following is the general reconstruction procedure for every new projection . 3 . one - dimensional filtering with a ramp - filter across the detector ( where the filter design is dependent on rebinning and detector type ) 4 . back - projection along incoming ray direction with magnification factors , depending on type of rebinning as well as on detector type : plane , cylindrical , etc . a special case of this procedure is rebinning to parallel projections which we will describe in more detail . fig8 shows a straight side - view of fig1 with six rays 51 , 52 , 53 , 54 , 55 , and 56 coming from the source s positioned at the x - axis . fig6 shows a view from above where the object is fixed and the source and detector is rotating . with the source in the position s α we observe three fan - beams 61 , 62 , and 63 ( seen as rays in this view ), which comprises the six rays in fig8 and which produce the three projection sets t ( α , γ 1 ), t ( α , 0 ), t ( α ,− γ 1 ). the two outer rays are parallel to two other rays , 64 and 65 , coming from two other source positions which produce the projections t ( α + γ 1 , 0 ) and t ( α − γ 1 , 0 ) respectively clearly , we may resample our projection data so that data from such parallel fan - beams ( seen as rays ) are brought together . this can be done with either of the following two equivalent assignments . as shown in fig9 we are then free to see the data set p as generated by a parallel beam in the − direction . without loss of generality , this direction is horizontal in fig9 . perpendicular to these rays we place a virtual detector 72 on a vertical plane . the detector window 71 for the parallel projection in fig7 is unwrapped and rolled out into the sheet of fig5 . note that the complete detector positions for the parallel projection are put together from vertical lines 83 , 84 , and 85 each one stemming from different cone - beam detector positions . the resulting parallel beam detector area has the same slant as the cone - beam detector but is shortened with a factor of two in the − direction . the uppermost and lowermost part of the detector 81 and 82 in fig5 outlines another detector window included here for comparison only . to the best of our understanding , this window corresponds to the minimum size detector in [ scha96 ] and [ scha97 ] when mapped onto the helix cylinder 12 . for the given pitch = h and the given maximum fan angle γ max the height of this detector window is 2  v  ( π + γ max ) ω   r   cos   γ max = h cos   γ max  π + γ max π = h  ( 1 + γ max π cos   γ max ) ( 3 ) this formula indicates that the detector redundancy in [ scha96 ] and [ scha97 ] grows rather quickly for increasing fan - angles . the rays in the parallel projection emanate from a set of sources with vertical fan - beams , located on a specific section of the helix . rolled out in the plane of the sheet this part 73 of the source helix is superimposed on the detector 71 in fig5 . it takes the form of a line with the same slant as the detector but with opposite sign . because of this fact , in the present invention , the virtual detector 72 in the vertical mid - plane is bounded by a perfect rectangle with a width that equals the object cylinder diameter and a height which is exactly half the pitch = h / 2 . this is illustrated in fig1 , where an upward tilt of the source path 73 is exactly compensated for by a downward tilt of the detector . furthermore , since the distance from the virtual detector 72 to the source is everywhere identical to the distance to the real detector , the real detector height h is always demagnified to exactly h / 2 at the virtual detector . fig5 illustrates the second part of the rebinning - resampling procedure , namely from equidistant grid points in r to equidistant grid points iny = r sinγ and y are used as coordinates also for the rebinned parallel projection system .) the aforementioned property of the virtual detector area being a perfect rectangle is further illustrated in fig1 , which shows three orthogonal views a , b , and c of the parallel projection system . seven source positions are indicated . in a , b we can see the projection from one of the source positions s as a line d - e . clearly , in view b we see that all the three points s , d , and e are on the helix . furthermore , the plane of the virtual detector intersects the helix in two points which are exactly halfway between s and d at the upper ray 111 and halfway between s and e at the lower ray 112 . therefore the height of the vertical detector is h / 2 with its midpoint on the x - axis for any s . this proofs that the virtual detector is a rectangle with horizontal boundaries . thus , using the insight that there is a special detector window which delivers sufficient and non - redundant data , we capture cone - beam projection data on this detector and rebin them into parallel projection data to create an advantageous situation for the actual reconstruction . the complete procedure consists of the following three steps . 1 . rebinning to parallel projections as described by the fig6 , 8 , 9 , 10 , and 11 . 2 . filtering with a conventional ramp - filter along horizontal rows in the virtual detector plane . 3 . back - projection in the direction of the original rays using a constant magnification factor . in the present invention , after parallel rebinning , the one - dimensional filtering takes place along horizontal rows in the virtual detector 72 of fig5 . in contrast , in [ scha96 ] and [ scha97 ] the filtering takes place along horizontal rows of a real detector placed on the source cylinder , shown as the arc 41 in fig4 . fig4 shows this detector mapped onto the virtual detector plane 121 . the horizontal rows in the real detector are mapped onto curves in 121 which are neither horizontal nor straight . clearly , after filtering along such curves in the virtual detector plane rather than along straight horizontal rows as in the present invention the reconstruction result will be rather different . even so , step 3 in the above procedure may very well be replaced by the version . reconstruction of one horizontal slice from generalized projections . the simplification is due to the perfectly balanced data capture in the present invention . we know a priori that there is one and only one source position that contributes to each detector position in the generalized projections as shown in fig1 . hence , there is no need to keep track of multiple exposure contributions , since there are neither missing nor redundant data in any projection . the situation is different in [ scha97 ] which is illustrated in fig3 showing a vertical section of the parallel scanning system . the real detector 125 is much higher than in fig9 so that the virtual detectors 121 , 122 , and 123 for neighboring half turns overlap vertically . therefore , in a vertical plane ( such as the plane of the sheet ) a horizontal slice of the object is partially illuminated not from one but from three source positions on the trajectory . this irregularly distributed redundancy in exposure is also reflected in fig4 which shows the virtual detector window in [ scha97 ] for the minimum sized detector . the upper and lower boundaries 131 and 132 , respectively , are the same as 81 , 82 in fig8 although mapped onto the virtual planar detector . in the most likely physical embodiment of the 2d - detector arrangement proposed in this invention , the detector elements are placed onto the source cylinder 41 . see fig2 . for moderate cone angles the detector elements are then facing the incoming rays rather straight on . for detectors made to cover high cone angles it might be more appropriate to mount the detector elements on the inside of a sphere centered in s . this would guarantee or at least make it more easy to secure that all detectors are facing the incoming rays correctly . fig1 shows again the detector window 11 on the helix cylinder rolled out on the plane of the sheet . however , this time it is overlaid with the same the detector window mapped onto the source cylinder arc 41 . when rolled out on the sheet , this latter detector appears in fig1 outlined as 141 . considering the geometry of fig4 it might be more optimal to place the detector on the source cylinder arc 43 having the smallest possible radius close up to the object cylinder 13 . however , since the geometry of such a detector would conform exactly with 141 , we may discuss the geometry of 141 without loss of generality . the detector 141 coincides with 11 in the middle but varies with γ so that the top - most and bottom - most point of the detector are found at z top = v ω   r  π + 2  γ cos   γ  and   z bottom = v ω   r  π - 2  γ cos   γ , ( 4 ) respectively . the height h is then varying as h  ( γ ) = z top + z bottom = v ω   r  2  π cos   γ = h cos   γ ( 5 ) where h is the pitch as before . thus , data which are captured on the source cylinder have to be resampled from the unevenly sloping detector area in fig1 to the grid of the detector ( also shown in fig1 ), defined by vertical lines and evenly sloping lines with rhombus shaped detector elements . when projection data are resampled once more into parallel projections on the planar virtual detector in fig5 the final grid pattern will be perfectly rectangular . an important special case for the present invention is when the detector 141 ( and the pitch ) of fig1 is reduced in height to a single row 150 of detector elements 151 , which is shown in fig1 . we note that also in this special case will the height of the detector element increase with increasing fan angle as predicted by the above formula ( 5 ). normally , the detector array in fig1 would no longer be considered as a two - dimensional detector but a one - dimensional array detector . one - dimensional array detectors are used in existing helical fan - beam tomographs for which the state - of - the - art is represented by [ king93 ]. the detector is normally placed on the surface of a source cylinder 41 although not designed as the one in fig1 . instead , the detector elements are of constant height and they are not placed in a slanted fashion but horizontally straight on the source cylinder surface . as a consequence , to secure sufficient data , either the height of the detector elements have to be increased , as in formula ( 3 ) which decreases the resolution in the z - direction , or the pitch of the helix has to be decreased with the same factor , which reduces the scanning efficiency and increases the dose compared to the present invention . the scanning will also acquire much redundant data so that the accompanying reconstruction procedure has to employ elaborate weighting factors to compensate for multiple exposure . using the present invention with a detector designed and arranged accordingly , for instance as in fig1 , the data capture will be complete and free of redundancy and the reconstruction procedure can be simplified to contain the three steps rebinning , one - dimensional ramp filtering , and backprojection with constant magnification factor . all references cited herein are incorporated herein by reference in their entirety and for all purposes to the same extent as if each individual publication or patent or patent application was specifically and individually indicated to be incorporated by reference in its entirety for all purposes . [ dan97a ] p . e . danielsson , “ förfarande och anordning för tomografering ”, swedish patent application no 9700072 - 3 , filed jan . 14 , 1997 . [ dan97b ] p . e . danielsson , paul edholm , jan eriksson , maria magnusson seger , “ towards exact 3 d - reconstruction for helical scanning of long objects ” , conf . record from 1997 int . meeting on fully three - dimensional image reconstruction , nemacollin , p a , jun . 25 - 28 , 1997 . [ feld84 ] l . a . feldkamp , l . c . davis , j . w . kress , “ practical cone beam algorithms ” , journal of optical soc . am . vol . a6 , pp . 612 - 619 , 1984 . [ wang93 ] g . wang , t . h . lin , p . c . cheng , d . m . shinozaki , “ a general cone - beam reconstruction algorithm ” , ieee trans . on medical imaging , vol . 12 pp . 486 - 496 , 1993 . [ scha96 ] s . schaller , t . flohr , p . steffen , “ a new approximate algorithm for image reconstruction in cone - beam spiral ct at small cone angles ” , conference record , ieee medical imaging conference , pp . 1703 - 1709 , november 1996 , anaheim , calif . [ scha97 ] s . schaller , t . flohr , p . steffen , “ new efficient fourier - reconstruction method for approximate image reconstruction in spiral cone - beam ct at small cone angles ” , to be published in proc . spie med . imaging conf ., newport beach , calif ., feb . 22 - 28 , 1997 . [ tam95 ] k . c . tam , “ three - dim . computerized tomography scanning method and system for large objects with smaller area detectors ” , u . s . pat . no . 5 , 390 , 112 , feb . 14 , 1995 . [ eber95 ] j . w . eberhard ; k . c . tam , “ helical and circle scan region of interest computerized tomography ” , u . s . pat . no . 5 , 463 , 666 , oct . 31 , 1995 . [ gra87 ] p . grangeat , “ mathematical framework of cone - beam 3 d reconstruction via the first derivative of the radon transform ” , in “ mathematical methods in tomography ” , g . t . herman , a . k . luis , f . natterer ( eds ), lecture notes in mathematics , springer , 1991 . [ king93 ] k . f . king , a . h . lonn , c . r . crawford , “ computed tomographic image reconstruction method for helical scanning using interpolation of partial scans for image construction ” , u . s . pat . no . 5 , 270 , 923 , dec . 14 , 1993 .
complete helical cone - beam scanning and non - redundant data acquisition are obtained for three - dimensional tomographic imaging of arbitrary long objects . the minimum sized two - dimensional detector window is bounded by two consecutive turns of the helix . the ray source exposes all object points during a rotation of exactly 180 degrees when seen from the points themselves . only one - dimensional filtering is employed in the reconstruction . rebinning to parallel beams , as seen along the axis of rotation , allows for especially simple procedures without any need for pre - weighting or magnification factors . as a special case , the invention is applicable to helical fan - beam scanning with one - dimensional detector arrays .