WO2005004063A2

WO2005004063A2 - Method using cone-beam computer tomography

Info

Publication number: WO2005004063A2
Application number: PCT/IB2004/050995
Authority: WO
Inventors: Class Bontus; Thomas Köhler
Original assignee: Philips Intellectual Property & Standards Gmbh; Koninklijke Philips Electronics N.V.
Priority date: 2003-07-04
Filing date: 2004-06-24
Publication date: 2005-01-13
Also published as: WO2005004063A3

Abstract

The invention relates to a computer tomography method in which a radiation source, which emits a cone-shaped beam, moves relative to an investigation area in which an object is arranged, in particular on a helical or circular trajectory. Measured values are acquired with a detector unit, wherein a CT image of the investigation area is reconstructed from these measured values using an nPI reconstruction method, in particular using an exact reconstruction method. In addition, a corrective image is reconstructed from the measured values using an approximative reconstruction method, and added voxel-wise to the CT image.

Description

Method using cone-beam computer tomography

The invention relates to a computer tomography method in which an investigation area, in which an object is arranged, is examined radiographically with a cone beam. The invention also relates to a computer tomograph for implementing this method and to a computer program to control the computer tomograph. In known methods of the type specified above, a radiation source moves on a helical or circular trajectory relative to the investigation area. Measured values are hereby acquired, from which a CT image, i.e. an absoφtion distribution in the investigation area, can be reconstructed. Frequently used for the reconstruction are «PI reconstruction methods, in which a voxel of the investigation area is reconstructed exclusively with measured values whose associated rays, projected onto a plane oriented peφendicular to the axis of rotation, known as the rotation plane, cover an angular range of «180° («=1,3,5,...). Examples of «PI reconstruction methods are exact reconstruction methods, such as the method described by Katsevich in "Analysis of an Exact Inversion Algorithm for Spiral Cone-Beam CT", Physics Medicine and Biology, vol. 47, pp. 2583-2597 (El), which, in comparison with approximative methods, have the advantage that the quality of the CT image does not deteriorate as the cone angle increases. The cone angle is hereby defined as the angle that a ray encloses with a plane containing the radiation source, which is oriented peφendicular to a rotation axis of the computer tomograph. However, nPI reconstruction methods have the disadvantage that they react extremely sensitively to object movements, so that even slight movements of the object lead to interfering motion artifacts in the CT image. It is an object of the present invention to specify a computer tomography method with an wPI reconstruction method in which the motion artifacts are reduced. This object is achieved in accordance with the invention by a computer tomography method with the following steps: a) Generation of a cone beam permeating an investigation area and an object located within it, using a radiation source. b) Generation of a relative movement between the radiation source on the one hand and the investigation area on the other, comprising at least one rotation about a rotation axis and having, in particular, a helical or circular form. c) Acquisition of measured values, which depend on the intensity in the beam beyond the investigation area, with a detector unit during the relative movement. d) Reconstruction of a CT image of the investigation area from the measured values using an «PI reconstruction method, in particular using an exact reconstruction method. e) Generation of a corrective image from the measured values using an approximative reconstruction method. f) Addition of the corrective image to the CT image. The invention is based on the recognition that reconstruction methods react especially sensitively to movements if, in order to reconstruct a voxel, use is exclusively made of measured values whose associated rays, projected onto a rotation plane, cover an angular range of «180° («=1,3,5,...), as is the case with «PI reconstruction methods. However, approximative methods, i.e. methods in which the quantity of measured values utilized can be varied, exhibit fewer motion artifacts. Therefore, combining an «PI reconstruction method with an approximative method by adding an approximatively reconstructed corrective image to a CT image initially reconstructed using an «PI reconstruction method produces CT images in which the motion artifacts are reduced. Claims 2, 3, 6, 7 and 8 describe preferred «PI reconstruction methods, which produce especially good image qualities. Claims 4 and 5 define embodiments in which, in order to reconstruct a corrective image by a filtered back projection, use is exclusively made of measured- value pairs, wherein the projections of the two rays assigned to a measured- value pair are oriented onto the rotation plane in inverse parallel to one another. Before the back projection, each measured value is multiplied by a weighting factor. The weighting factors are hereby selected such that the sum of the two weighting factors of a measured-value pair equals zero. Since the rays of a measured- value pair are oriented approximately in inverse parallel with each other, the values of a measured pair differ only slightly, so the corrective image without object movement exhibits only few values other than zero. If the object moves, the rays of a measured-value pair deliver greatly differing measured values since they have been detected at different moments, leading to image values that are brought about by the object movement and differ from zero. An addition of this corrective image to the CT image reconstructed with an nPI reconstruction method reduces the motion artifacts to an even greater extent. A computer tomograph for implementing the method in accordance with the invention is described in claim 9. Claim 10 defines a computer program to control a computer tomograph as claimed in claim 9.

The invention will be further described with reference to examples of embodiments shown in the drawings, to which, however, the invention is not restricted. Fig. 1 shows a computer tomograph in accordance with a particular embodiment, with which the method in accordance with the invention may be implemented in accordance with a particular embodiment. Fig. 2 shows a flow chart of the method in accordance with the particular embodiment. Fig. 3 shows a flow chart for an exact reconstruction of a CT image. Fig. 4 shows a perspective view of a PI straight line and a PI interval for a point in the investigation area. Fig. 5 shows the PI straight line and the PI interval for a point in the investigation area, projected into a plane peφendicular to the rotation axis. Fig. 6 shows a perspective view of parallel rays with different ray positions. Fig. 7 shows a perspective view of a κ-plane and a κ-line. Fig. 8 shows a perspective view of fans of rays formed by rebinning in parallel planes. Fig. 9 shows a flow chart for generating a corrective image. Fig. 10 shows a weighting function for measured values during an exact reconstruction. Fig. 11 shows a weighting function for measured values during a reconstruction of the corrective image. Fig. 12 shows a flow chart of a quasi-exact reconstruction of a CT image. Fig. 13 shows a perspective view of a helical trajectory of a focus-centered and a planar detector. Fig. 14 shows a perspective view of a helical trajectory of a detector, a κ-plane and a filtration line. Fig. 15 shows a flow chart of the determination for reconstruction of suitable κ-vectors. Fig. 16 to Fig. 20 show filtration lines and filtration directions on the planar detector. Fig. 21 shows a weighting function for measured values during the quasi-exact reconstruction. Fig. 22 shows a weighting function for measured values during the reconstruction of a corrective image. Fig. 23 shows a further weighting function for measured values during the quasi-exact reconstruction. Fig. 24 shows a further weighting function for measured values during the reconstruction of a corrective image.

The computer tomograph shown in Fig. 1 comprises a gantry 1 , which can rotate about a rotation axis 14 running parallel to the z-direction of the system of coordinates shown in Fig. 1. To this end, the gantry 1 is driven by a motor 2 with a preferably constant, although adjustable, angular velocity. Secured to the gantry 1 is a radiation source S, for example an X-ray tube. This is equipped with a collimator configuration 3, which extracts from the radiation generated by the radiation source S a cone-shaped beam 4, i.e. a beam which, both in the z-direction and in a direction peφendicular to this (i.e. in a plane peφendicular to the rotation axis), has a finite expansion different from zero. The beam 4 penetrates a cylindrical investigation area 13, in which an object, e.g. a patient on a patient examination table (neither of which is shown) or a technical object, may be located. Once it has permeated the investigation area 13, the beam 4 hits a detector unit 16 with a detector surface, which is secured to the gantry 1 and which comprises a plurality of detector elements which, in this embodiment, are arranged in rows and columns in the form of a matrix. The detector columns run parallel with the rotation axis 14. The detector rows are located in planes peφendicular to the rotation axis, on an arc about the radiation source S in this embodiment (focus-centered detector). In other embodiments, however, they may take different shapes, e.g. may describe an arc about the rotation axis 14 or may be straight-line. Each detector element hit by the beam 4 supplies, in each position of the radiation source, a measured value for a ray from the beam 4. The spread angle of the beam 4, designated with α_max , determines the diameter of the object cylinder within which the object under investigation is located during the acquisition of the measured values. The spread angle is hereby defined as the angle that a ray lying in a plane peφendicular to the rotation axis 14 at the edge of the beam 4 encloses with a plane defined by the radiation source S and the rotation axis 14. The investigation area

13 and the object or patient examination table may be moved parallel with the rotation axis

14 and with the z-axis by means of a motor 5. Equivalent to this, however, the gantry could also be moved in this direction. If the object is, for example, not a living object, the object may be rotated during an investigation whilst the radiation source S and the detector unit 16 remain stationary. If the motors 2 and 5 are running simultaneously, the radiation source S and the detector unit 16 describe a helical trajectory relative to the investigation area 13. If, conversely, the motor 5 is stationary for the advance in the direction of the rotation axis 14, and the motor 2 causes the gantry to rotate, a circular trajectory arises for the radiation source S and the detector unit 16 relative to the investigation area 13. Only the helical trajectory will be considered below. The measured values acquired by the detector unit 16 are supplied to a reconstruction computer 10, which is connected to the detector unit 16 by, for example, a non-contact operating data transmission system (not shown). The reconstruction computer 10 reconstructs the CT image and the corrective image, adds these images voxel-wise and displays them, e.g. on a monitor 11. The two motors 2 and 5, the reconstruction computer 10, the radiation source S and the transfer of the measured values from the detector unit 16 to the reconstruction computer 10 are controlled by a control unit 7. In other embodiments, the acquired measured values may initially be supplied for the reconstruction to one or more reconstruction computers, which forward the reconstructed data to the reconstruction computer, e.g. via an optical-fiber cable. Fig. 2 shows the sequence of a measurement and reconstruction method that can be implemented with the computer tomograph as shown in Fig. 1. Following the initialization at step 101, the gantry rotates at an angular velocity that is constant in this embodiment example. It may, however, also vary, e.g. as a function of time or of the position of the radiation source. At step 103, the investigation area and the object or the patient examination table are moved in parallel with the rotation axis, and the radiation from the radiation source S is activated so that the detector unit 16 can detect the radiation from a plurality of angular positions. In this embodiment example, the table advance per rotation is selected such that, from every location x in the investigation area, the radiation source S is visible over an angular range of at least 180°. The recording of measured values with this table advance is designated PI acquisition. At step 105, the CT image is reconstructed with an exact method. The individual reconstruction steps are shown in Fig. 3. The following equation from El is cited to aid understanding of the exact reconstruction:

This equation describes an exact reconstruction of the absoφtion by back- projection of the measured values. Here, (x) designates the spatial absoφtion distribution in the investigation area at location x and /_PI (x) designates the part of the helix enclosed by a PI straight line 31. The PI straight line 31 of an object point at location x in the investigation area and 7_PI (x) are shown in Fig. 4 and Fig. 5 and are explained below. The radiation source moves relative to location x in the investigation area on a helical path 17. The PI straight line 31 is hereby the line that intersects both the helix, at two locations, and location x, wherein the helix section I_n (x) enclosed by the straight line covers an angle of less than 2π. The helix section I_n (x) is designated below as the PI interval. Rays that pass through location x and originate from a location within the helix section 7_PI (x) , as well as their measured values, are deemed to lie within the PI interval. Further, in equation (1) s is the angular position of the radiation source S on the helix referred to an arbitrary but fixed reference-angle position, and y(s) is the position of the radiation source in the three-dimensional space. The measured value D_f (y,Θ) can be described by the following line integral:

D_f (y(s),Θ) = dlf(y + lΘ) (2) o The unit vector© hereby indicates the direction of the ray associated with the measured value. At step 201, the measured values in accordance with equation (1) are partially derived in accordance with q, i.e. in accordance with the angular position of the radiation source at the location q = s. It should be noted here that only y, and not© , depends on q, so measured values of parallel rays have to be taken into account for the derivation in each case. With the focus-centered detector 16 used here, since parallel rays exhibit the same cone angle, the parallel rays 51 hit the same detector row 53, as shown in Fig. 6. The cone angle of a ray is hereby the angle enclosed by this ray with a plane peφendicular to the rotation axis 14. For the partial derivation, the measured values may first be re-sorted. To do this, measured values belonging to parallel rays, i.e. to the same detector row 53 but at different angular positions s_a , s_b , s_c of the radiation source, are combined to form a quantity. The measured values of each quantity are derived e.g. numerically with the known method of finite differences according to the angular position of the radiation source. The unit vector© depends on the κ-angle γ which may be described using so- called κ-planes 55 (see Fig. 7). The κ-planes 55 are explained below. To determine a K-plane 55, a function:

is introduced, which depends on non-negative integer values n and m, n > m . In this embodiment example, n = 2 and m = 1 are selected. Other values may, however, also be selected for n, m. Equation (1) would, nevertheless, remain exact and only the position of the κ-planes 55 would change. Furthermore, the vector function:

and the unit vector: β(*, ) = ^ (5) |^χ-yO)| are defined. The vector β points from the position of the radiation source y(s) to position x. To determine the κ-plane, a value of s₂ e I_P1 (x) is now selected, so that y(s), y(sι(s,s₂)), y(s ) and x are lying in one plane. This plane is designated the κ-plane 55, and the intersection line between the κ-plane 55 and the detector surface 74 is designated the κ-line 57. The detector 74 is bounded in Fig. 7 by two successive turns of the helical trajectory 17, and exhibits the curvature of the helix 17. This detector has been used here and in Figs. 8 and 14 as an illustrative example. For other detectors, such as the focus-centered or the planar detector, appropriate intersection lines 57 may be determined. Fig. 7 shows a fan-shaped part of a κ-plane. The edges of the fan meet at the location of the radiation source. This definition of the κ-plane 55 is equivalent to the solution to the equation: (x - y(s)) ^■ u(s, s₂) = 0, s₂ /_Pl(x) (6) in accordance with s₂. Thereby, u is the normal vector of the κ-plane 55. To determine the vector function Θ(s, x,γ) , the vector: e(s, \) = β(s, x) x u(s,x) (1) is defined. With the definitions for β and e, the vector function Θ(s,x,χ) can be quoted as follows: Θ(s, x, γ) = cos γ • β(s, x) + sin γ • e(s, x) (8) Since both vectors, β and e, are oriented peφendicular relative to u, the K- angle γ indicates the direction of the vector ©., and thereby the direction of a ray, within a K- plane. The κ-planes and κ-lines are described in detail in El, to which you are herewith referred. At step 203, the derived measured values are, in accordance with equation (1), multiplied along κ-lines by a weighting factor that decreases as the sine of the κ-angle y increases and is, in particular, equal to the reciprocal of the sine of the κ-angle, and are then added up. To this end, a κ-line is determined for every location x in the investigation area and for every radiation-source position s, wherein, as explained above, a value s₂ e I_P] (x) is selected in such a way that y(s), y(sι(s,s₂)), y(s₂) and x are lying in one plane, the κ-plane. The κ-line is then established as an intersection line between the κ-plane and the detector surface. The multiplications by the weighting factor and the integrations or additions may, for example, be undertaken using a Fourier transform. The derived and integrated measured values may be represented by the following equation: p(y(s),Φ(s,x)) = )^-^D_f (y(q),Θ(s,x,y)) \_q_ (9) _„sιn γ oq Here, p(y(s),Φ(s,x)) designates the derived, integrated measured values, and Φ(s, x) designates a unit vector pointing from the radiation-source position y(s) in the direction of location x in the investigation area. The still missing integration step in equation (1), or the back projection of the measured values, may be described by the equation: ^m=-^ øX ^piyω-^φ^^{)) (10)} Before the back projection, a rebinning of the measured values may take place at step 207. As a result of the rebinning, the measured values are re-sorted and re-inteφolated as if they had been measured with a different radiation source (an expanded radiation source arranged on a section of a helix, which can emit fans of rays that are always parallel with one another) and with a different detector (a planar, rectangular "virtual detector" containing the rotation axis 14). This is explained more fully with reference to Fig. 8. Reference number 17 designates the helical trajectory from which the radiation source penetrates the investigation area. Reference number 43 designates a fan-shaped beam, which starts from the radiation- source position So and the rays of which travel in a plane containing the rotation axis 14. The cone-shaped beam emitted from the radiation source in position S_o may be imagined as being composed of a plurality of planar ray fans, which are located in planes parallel with the rotation axis 14 and which intersect in the radiation-source position So. Of these ray fans, Fig. 8 shows just one, namely the ray fan 43. Also shown in Fig. 8 are further ray fans 41, 42 and 44, 45, which are parallel with the ray fan 43 and which lie in planes parallel with one another and with the rotation axis 14. The associated radiation-source positions S_₂ , S_, and S, , S₂ are taken from the radiation source S before or after it has reached the radiation-source position So respectively. All rays in the ray fans 41 to 45 have the same projection angle. The ray fans 41 to 45 define a beam 70 with a tent-like shape. For each group of ray fans, a rectangular virtual detector 160 is defined, which lies in a plane that contains the rotation axis 14 and is oriented peφcndicular to the parallel ray fans of a group. The corner points of the virtual detector 160 are the penetration points through this plane of the rays which, from the outer radiation-source positions, hit against the opposite section of the helix. For the beam 70 in Fig. 8, these are the intersection points of the ray fans 41 and 45 with the helix. Defined on the rectangular detector 160 are detector elements arranged in cartesian fashion, i.e. rows and columns. Measured values assigned to these detector elements may be determined by inteφolation of the measured values whose associated rays penetrate the rectangular detector in the vicinity of the particular detector element. For example, a measured value of a detector element on the rectangular detector 160 could be determined by linear inteφolation of four measured values whose associated rays exhibit penetration points on the detector 160 that are at a distance from the particular detector element that is smaller than the corresponding distance from penetration points of other rays. The measured values determined after the rebinning are then used for reconstructing the absoφtion distribution in the investigation area by back projection, which, in this embodiment example, is in accordance with equation (10). To this end, a voxel V(x) is determined at step 209 within a specifiable area

(Field Of View - FOV) in the investigation area, and a PI interval I_P1 (x) is determined for this voxel. Then at step 211, an angular position s is specified within the interval 7_PI (x) . At step 213, a check is made as to whether a measured value whose ray runs through the center of the voxel V(x) is present for the angular position s. If a ray of this kind cannot be found, the location at which a central ray would have hit against the detector surface is determined. The associated measured value is then calculated by inteφolation of the measured values of adjacent rays. The measured value that can be assigned to the ray matching the voxel, or the measured value obtained by inteφolation, is multiplied at step 214 by a weighting factor which becomes smaller as the distance of the radiation source y(s) from the location x to be reconstructed in the investigation area increases. In this embodiment example, this weighting factor is, in accordance with equation (10), equal to 1 /|x - y(s)| . At step 215, the weighted measured value is accumulated onto the voxel V(x) . At step 217, a check is made as to whether all angular positions s in the interval /_P,(x) have been taken into account. If this is not the case, the flow chart branches off to step 211. Otherwise, a check is made at step 219 as to whether all voxels F(x)in the FOV have been covered. If this is not the case, continuation is at step 209. If, conversely, all voxels V(x) in the FOV have been covered, the absoφtion in the entire FOV has been established, and so, therefore, has the CT image, so the exact reconstruction of the CT image is terminated at step 221. At step 107, continuation is with the generation of a corrective image in accordance with the flow chart shown in Fig. 9. To this end, at step 301, the measured values are firstly regrouped by rebinning, as described at step 207. Then, at step 303, the measured values associated with the rays are each multiplied by a weighting factor corresponding to the cosine of the cone angle of the particular ray. If the cone angle is small, the cosine is practically always equal to 1, so step 303 may be dispensed with. At step 305, a uni-dimensional filtration is applied to the measured values arising from the rebinning with a transmission factor increasing ramp-like with the spatial frequency. To this end, successive values in one direction parallel to the rotation plane, i.e. along one row of the virtual detector, are used. This filtration is undertaken along each row of the virtual detector for all projection angles. At step 307, a voxel V(x) is determined within the FOV specified at step 209. Then, at step 309, the quantity of projections that have penetrated this voxel during the acquisition is determined. A projection is the quantity of all rays emanating from a radiation-source position. The projection angle is the angle enclosed by the ray of the projection that intersects the rotation axis with a specified reference ray that also intersects the rotation axis. At step 311, a weighting function is determined for the voxel V(x) , indicating the weighting factors by which the measured values assigned to the rays have to be multiplied before the back projection. To do this, the weighting factors of the particular rays, or measured values, that contributed to the reconstruction of the voxel K(x)at steps 201 to

221 are required. Since only projections whose rays lie within the PI interval 7_pι(x)have been taken into account with the same weighting in this embodiment example, the characteristic shown in Fig. 10. arises as the weighting function 71 for the reconstruction of the CT image at step 105. Here, w_k is a constant weighting factor, which, in this embodiment example, equals 1. To reduce motion artifacts in the exactly reconstructed CT image, measured values from specified corrective-angle areas KW1 , KW2, which adjoin the PI interval 7_P1 (x) , are also to be taken into account in order to reconstruct the corrective image. These corrective-angle areas KW1, KW2 amount to 90° maximum and preferably equal 10°. Also for the reconstruction, a so-called PI partner within the PI interval 7_PI (x) is determined for each measured value from a corrective-angle area KW1, KW2. Two rays assigned to the particular measured value are PI partners if, when projected onto a plane oriented peφendicular to the rotation axis 14, they are oriented in inverse parallel to one another. Two PI partners form a measured-value pair. The weighting function is determined in such a way that the sum of the weighting factors of two PI partners equals zero. Each ray assigned to a measured value that does not lie in one of the corrective-angle areas KW1, KW2 or lies within the PI interval 7_PI (x) but has no PI partner in any of the corrective-angle areas is assigned a weighting factor of zero. A preferred weighting function in accordance with the invention is shown in Fig. 11. Here, the measured values in the corrective-angle area KW1 are multiplied by a weighting factor that, starting from 0 up to the boundary of the PI interval 7_PI (x) , increases linearly to w_k 12 . Starting from w_k 12 at the boundary of the PI interval, the measured values in the corrective-angle area KW2 fall linearly to zero. The corresponding PI partners of the rays from the corrective-angle area KW1 are located within the PI interval 7_PI (x) in the area PI, and the PI partners of the rays from the corrective-angle area KW2 are located in area P2. At step 313, an arbitrary projection is selected from the quantity of projections determined at step 309. Then, at step 315, a check is made as to whether a ray of the projection is running through the center of the voxel V(x) . If no ray of the projection passes the center of the voxel, the associated value has to be determined by inteφolation of the measured values of adjacent rays. The measured value that can be assigned to the ray passing the voxel, or the measured value obtained by inteφolation, is multiplied at step 317 by the weighting factor determined at step 311, and accumulated onto the voxel V(x) . At step 319, a check is made as to whether all projections from the quantity determined at step 309 have been taken into account. If this is not the case, the flow chart branches off to step 313. Otherwise, a check is made at step 321 as to whether all voxels in the FOV have been covered. If this is not the case, continuation is at step 307. If, on the other hand, all voxels in the FOV have been covered, the corrective image in the entire FOV has been determined and the reconstruction of the corrective image is terminated at step 323. Once a CT image and a corrective image have been determined in steps 105 and 107 respectively, these two images are added, voxel by voxel, at step 109. The resultant image represents a CT image with reduced motion artifacts. Below, a further particular embodiment of the invention is described, in which the measured values are detected by means of a 3PI acquisition and a CT image is reconstructed quasi-exactly. To this end, there first follows, to aid understanding, a mathematical description of the quasi-exact reconstruction method. The spatial absoφtion distribution or object function βx) may be represented by means of its Fourier transform Ff (ξ) :

Equation (2) is inserted into equation (1), wherein the object function is substituted by its Fourier representation in accordance with equation (11) and the integration variables / and γ are transformed in accordance with: W_[ = /cos7, u₂ = lύn γ (12) This leads to: f(x) = jds {d ³ξ e^{2π,ξ y(s)} (ξ ■ y (s)) sgn(ξ ^■ φ, x))δ(ξ • (x - y (s)))Ff(ξ) . I_P, (") (13) Below, the vector ξ is shown in spherical coordinates:

Further, the Fourier Slice Theorem is known from "The Mathematics of Computerized Tomography", F. Natterer, Wiley, New York, USA, 1986: FRf(ξ,ω) = Ff(ξύn θ cosφ,ξsin θ sin φ,ξ cosθ) . (15) Rf(p,ω) is hereby the radon transform of the object function f(x) and FRf(ξ,ω) is the Fourier transform of the radon transform. They are expressed by the following:

FRf(ξ,ω) = jdp c ^2mpξRf(p,ω) . (17)

Finally, the insertion of equation (15) into equation (13) using the symmetry relation FRf(ξ,ω) = FRf(-ξ,-ω) leads to: f(x) = φsinθσ(x,ω)R"f(ω ■ x,ω) (18)

Here, R"f(p,ω) is the second derivative after p of the radon transform of the object function, and σ(x,ω) represents the following sum: σ(x,β>) =

5 e 7_ft (x) . (19) J The variables S_j = S_j (x,ω) designate those angular positions for which the equation: (x - y(_Sj)) - ω = 0 (20) is fulfilled. It has been shown in Εl that, for a PI acquisition and for the vectors e(s, x) defined in equation (7): σ(x,<y) = l (21) applies. Up to now, all equations have related to the PI acquisition. As already mentioned above, it can be seen, especially in equation (1), that the actual back projection, i.e. the integration via the angular position s , is restricted to the PI interval 7_PI (x) . Below, equation (1) is amended in such a way that the integration via the angular positions can be undertaken via an arbitrary interval 7_BP (x) of the helical trajectory. The interval 7_BP (x) is hereby to cover an area of the trajectory that is greater than 7_PI (x) , i.e. 7_PI (x) c I _BP (x) . An amendment to the integration interval from 7_PI (x) to 7_BP (x) leads in equation (18) only to an amendment of the function σ(x,ω)m^' accordance with the following equation: σ(x, <y) = ∑ sgn(-y • y(_?, )) • sgn(ω • e(s , x)), S_j e I _BP (x) . (22) J Since the interval 7_BP (x) is greater than the interval 7_PI (x) , more angular positions s could fulfill equation (20), so σ(x, ω) would no longer be constant. For the reconstruction in accordance with equation (18), however, this is a prerequisite, as can be seen from equation (21 ) above. Therefore, a new function: σ(x, ω) = ∑ sgn(ω • y(s_y ))[sgn(ω ^■ e, ($, , x)) + ... + sgn(ω • e_πe (s, , x))] with J

_Sj I_BP(x) (23) with new vectors, κ-vectors e_k (s_y , x) , k - 1,..., n_c is defined, wherein the new vectors e_k (s , x) are selected such that the function σ(x, ω) supplies a value that is independent of ω , but may be dependent on x : σ = N(x) . (24) With a choice of vectors of this kind, σ(x, ω) may be replaced by σ I N(x) in equation (18) without equation (18) losing its exactness. Once σ(x,ω) has been replaced by σ I N(x) in equation (18), a repetition of the calculation steps that have led to equation (18) in reverse order yields the following equation:

wherein Θ_k (s,x,r) = cosγ - β(s, x) + sm r - e_k (s, x), k = \,..., n_e (26) applies. By contrast with the filtered back projection in accordance with equation (1), the back projection, i.e. the integration via the angular positions s , takes place via the interval 7_BP (x) in accordance with equation (25). The κ-vectors e_k (s, x) are selected such that, for a given interval 7_BP (x) , equation (24) is fulfilled in the investigation area for all, or at least for the majority, of possible combinations of vectors ω and locations x . If equation (24) is fulfilled for all possible combinations, equation (18) applies exactly, even after σ(x,ω) has been replaced by σ I N(x) . Therefore, the back projection in accordance with equation (25) is designated "exact" in this case. If, conversely, equation (24) is fulfilled for only a majority of combinations, the filtered back projection in accordance with equation (25) is designated "quasi-exact". So, the filtered back projection is quasi-exact if equation (24) is fulfilled for more than 50% of the combinations of vectors ω and locations x in the investigation area, e.g. for 60%, 70%, 80% or 90% of these combinations. After this mathematical description of a quasi-exact reconstruction, there now follows a description of the sequence of one embodiment of a measurement and quasi-exact reconstruction method. Equation (25) is hereby a specification by which the following steps are governed. The individual steps are shown in Fig. 2, as was the case with the first embodiment. Following the above-described initialization at step 101, the investigation area, and the object or patient examination table, are shifted parallel with the rotation axis at step 103, and the radiation from the radiation source S is activated so that the detector unit 16 can detect the radiation from a plurality of angular positions. In this embodiment example, the pitch, i.e. the table advance per rotation, is selected such that, from every location x in the investigation area, the radiation source S is visible over an angular range of at least 540°. The interval 7_BP (x) may hereby be chopped, i.e. the areas of the helical trajectory in which the radiation source is visible from a location x may alternate with areas in which the radiation source is not visible from the location x . The important thing is that the projections of the areas of the helix in which the radiation source is visible from a location x in the investigation area together cover an angle greater than or equal to 540° on a plane peφendicular to the rotation axis 14, and that this condition is fulfilled for all locations in the investigation area to be reconstructed. An acquisition in which the table advance is selected in this manner is known as a 3PI acquisition. If, for example, the geometry of the acquisition is characterized by a fan angle of 52.1°, an extension of the detector in the z-direction of

175.1 mm, a distance from the radiation source to the rotation axis of 570 mm and a distance from the radiation source to the detector center of 1040 mm, 57.6 mm per rotation can be selected as the pitch in order to enable a 3PI acquisition. In other embodiments that use a quasi-exact reconstruction, a different pitch would also be possible. At step 105, the CT image is reconstructed with the measured values detected by means of a 3PI acquisition. A corresponding flow chart is shown in Fig. 12. At step 401, the measured values are derived in accordance with equation (25) partially according to q, i.e. according to the angular position of the radiation source. The procedure may be as already described at step 201. At step 403, the derived measured values are projected along their rays onto a fictitious planar detector 60 (see Fig. 13). The planar detector 60 is rectangular, contains the rotation axis 14 and is oriented peφendicular to the ray that hits the rotation axis with a peφendicular orientation. It is bounded by rays 62, 64, 66, 68, which emanate from the radiation source and hit the corners of the actual, focus-centered detector 16. At step 405, filtration lines and filtration directions are determined, wherein the filtration lines, together with filtration directions, indicate the order in which the measured values are filtered, i.e. the order in which the integration via the κ-angle γ in equation (25) is undertaken. To this end, the relation existing between the filtration lines or filtration directions and the κ-vectors e_k (s, x) will first be explained. Together with the vector β(s, x) , each κ-vector e_k (s, x) spans a κ-plane. In accordance with equation (26), any change to the κ-angle γ leads to a change in the radiation direction Θ_k (s, x, γ) within the -plane. During the γ integration, for a given angular position s, a given location , and a vector e_λ (s,x) , the measured values on the detector are processed in the order in which the rays corresponding to the measured values change with the directions Θ_k (s, x,γ) with a varying γ . In other words, the measured values are filtered along the intersection line between the detector surface and the κ-plane that is defined by a vector e_k (s, x) for a given s and x . This is illustrated by way of example in Fig. 14. For a location x in the investigation area and an angular position s and radiation-source position y(s), the vector β(s, x) 73 and a κ-vector e_A (s, x) 72 are shown. The vectors β(s, x) and e_k (s, x) span a K- plane 70, which intersects the detector 74 in an intersection line 76. Here again, as in Fig. 7, the detector 74 is bounded by two successive turns of the helical trajectory 17 and exhibits the curvature of the helix 17. This detector has been used here as an example for illustration puφoses. Corresponding intersection lines 76 can be determined for other detectors, such as the focus-centered or the planar detector. The measured values that lie on the intersection line 76 are filtered in sequence. With a given radiation-source position s and a given location x in the investigation area, a vector e_k (s,x) defines a filtration line 76. The important elements for implementing the method in accordance with the invention are not then the κ-vectors e_k (s, x) , but the filtration lines resulting from the κ-vectors. As already described above, the κ-vectors e_k (s, x) are selected such that, with a given interval 7_BP (x) , equation (24) is fulfilled for all or at least for a majority of possible combinations of vectors ω and locations x in the investigation area. If equation (24) is fulfilled for all these combinations, equation (18) applies exactly, even after σ(x,ω) has been replaced by σ I N(x) . The back projection in accordance with equation (25) is therefore designated "exact" in this case. If equation (24) is fulfilled for only a majority of combinations, the filtered back projection in accordance with equation (25) is designated "quasi-exact". Any quantity of filtration lines that can be derived from κ-vectors e_k (S_j , x) that fulfill equation (24) exactly or quasi-exactly is applicable. The individual steps that have led in this embodiment example to κ-vectors e_k (S_j , x) that fulfill equation (24) quasi-exactly are described below, and shown in Fig. 15. Firstly, at step 500, values for σ and n are specified. Experiments with different values for σ and «_chave shown that, with the 3PI acquisition used here, the reconstruction can be undertaken with a relatively small computation input for σ = 3 and n_c = 3. These values are therefore selected for the determination of the κ-vectors. However, different values for σ and n_c could also be specified. It would also be possible to specify a value for n_c for each location x to be reconstructed. At step 502, an interval 7_BP (x) is determined for each location x to be reconstructed in the investigation area. Since, in this embodiment example, the measured values have been detected with a 3PI acquisition, the helix section 7_BP (x) projected onto a plane peφendicular to the rotation axis covers an angle of 540°. The helix section 7_BP(x) may, for example, be determined numerically by simulation of the acquisition, in particular of the motion of the radiation source S on the helical trajectory 17. At step 504, a location x in the investigation area is selected and at step 506, a vector ω is selected from a specified quantity of vectors ω . A preferred quantity of vectors ω , from which a vector is selected, may be created as follows. Firstly, planes distributed uniformly in the space, which all include the location x selected at step 504, are defined. Preferably, 100 to 1000 of these planes are defined. The quantity of vectors ω is then formed from those vectors that are located normal to the particular plane and point from the origin of a reference coordinate system to the particular plane. The reference coordinate system may be a cartesian coordinate system whose origin is, for example, a point on the rotation axis 14. Next, at step 508, all angular positions s within the interval 7_BP (x) , or all radiation-source positions y(S_j ) for which equation (20) is fulfilled, are calculated for the selected location x in the investigation area and the selected vector ω . The angular positions s_J are selected such that the connecting line from the radiation source y(s ) to the location x is oriented peφendicular to the vector ω . At step 510, the derivative y(s_y) is determined at the angular positions s_y that were determined at step 508, and the values for x, ω and y(s ) are inserted into equation (23). Since σ = 3 was selected at step 500, equation (23) represents a defining equation for the vectors e, (s , x) , e₂ (s , x) and e₃ (S_j , x) . At step 514, a check is made as to whether all vectors from the specified quantity have already been used for determination of the defining equations at step 510. If this is the case, continuation is at step 516. Otherwise, step 506 follows. At step 516, a check is made as to whether all locations x in the investigation area have been used for determination of the defining equations. If this is the case, step 518 follows. Otherwise, continuation is at step 504. At step 518, the system of defining equations determined at step 510 is solved numerically, so the vectors e, (s ,x) , e₂ (s , x) and e₃ (s ,x) fulfill the equation σ(x,ω) = 3 for at least a majority of combinations of vectors ω and locations x . The method just described for determining a quantity of κ-vectors e_k (s, x) should be regarded as simply an embodiment example. Any method for determining K- vectors may be applied if it enables κ-vectors e_k (s,x) to be determined for the 3PI acquisition that fulfill equation (24) for at least a majority of combinations of vectors ω and locations x in the investigation area. Next, filtration lines are determined using the vectors e_k (s, x) . To do this, for every angular position s of the radiation source and for every location x in the investigation area, the vector β(s, x) in accordance with equation (5), which points from the radiation source to the location x , is formed. Then, κ-planes that are spanned by the vectors e_k (s, x) and β are determined. So, for every combination of radiation-source position and location in the investigation area, i.e. for every measured value, and for every vector e_k (s,x) , a κ-plane is determined. The intersection lines between these κ-planes and the detector form the filtration lines. In other words, at least one filtration line is assigned to every measured value. To determine the filtration direction of a filtration line of a measured value, the direction in which the direction vector θ_k (s, x, γ) moves on the filtration line with the increasing κ-angle γ is investigated with the aid of equation (26). The direction of movement of the vector Θ_k (s,x,χ) for a given filtration line is the filtration direction. Examples of results of this numerical determination of filtration lines and filtration directions are shown in Figs. 16 to 20. They are explained below. The planar detector 60 is firstly divided into multiple areas. Area 92 is designated a PI window and is bounded by two PI lines 80 and 84. The PI lines 80 and 84 may be described mathematically by the following equations:

and

Here, w_pl and v_pl are coordinates on the planar detector 60 in accordance with the coordinate system in Fig. 16. For reasons of clarity, this coordinate system is shown below the planar detector 60. However, the origin of the coordinate system lies in the center of the detector. The PI window 92 has the following meaning. It is known that measured values located on the detector in the PI window 92 are also located in the PI interval 7_P1 (x) of a voxel. Furthermore, two 3PI lines 100 and 102 are introduced, being described by the following equations:

^VPI ("PI ) = (29)

and

with n = 3. (30) The area of the detector enclosed by the two 3PI lines 100, 102 is designated the 3PI window. Measured values located on the detector in the 3PI window are also located in the 3 PI interval. Assigned to every measured value that lies within the PI window are three filtration lines. The first filtration lines for measured values located in the PI window have been determined from the K- vectors e, (s , x) . One section of these filtration lines is shown in Fig. 16 on the planar detector 60, and runs either at a tangent to the PI line 80 (filtration lines 88, shown as broken lines) or at a tangent to the PI line 84 (filtration lines 89, shown as dotted lines). The sections of the filtration lines 88 are the sections of the tangents against the PI line 80 which, starting from the contact point in Fig. 16, run to the left. Conversely, the sections of the filtration lines 89 are the sections of the tangents against the PI line 84 which, starting from the contact point in Fig. 16, run to the right. The sections of the filtration lines 88 could also run from the contact point to the right, and the sections of the filtration lines 89 could also run from the contact point to the left. The only important thing here is that the progressions of the sections of the filtration lines 88, 89, starting from their respective contact points, are opposed. The PI window in Fig. 16 can be divided into two areas. One area is covered by the part shown of the filtration lines 88, and the other area is covered by the part shown of the filtration lines 89. The boundary between these areas is the line that runs asymptotically relative to the PI lines 80, 84 from the end of the PI line 80 shown on the left in Fig. 16 to the right-hand end of the PI line 84. Depending on the area on the detector 60 in which a measured value to be filtered is located, the corresponding filtration line will be assigned to the measured value. If, for example, a measured value is located at a point 85 on the detector 60, then the first filtration line 88 that is in contact with this measured value will be assigned to this measured value. In this embodiment example, the filtration direction along a filtration line 88, 89 corresponds to the direction 82, i.e. from left to right in Fig. 16. In a different embodiment, the direction 82 could also be oriented the opposite way. Although the filtration lines 88, 89 are not shown running over the entire detector, in the following step 111, filtration naturally takes place over the entire detector along a filtration line. This illustration has been selected to make it clear that none of the filtration lines 89 are assigned to measured values located in an area that, in Fig. 16, is covered by the filtration lines 88. If the filtration lines 88, 89 had been shown in full, there would have been areas on the detector in Fig. 16 in which both filtration lines 88 and filtration lines 89 were present. This could be confusing. The same applies to Figs. 10 and 13. The second filtration lines for measured values that are located in the PI window have been determined from the κ-vectors e₂(s ,x) . Sections of these filtration lines are shown in Fig. 17 on the planar detector 60, and run either tangentially relative to the PI line 80 (filtration lines 90, shown as broken lines) or tangentially relative to the PI line 84 (filtration lines 91, shown as dotted lines). The sections of the filtration lines 90 are the sections of the tangents against the PI line 80 which, starting from the respective contact point, by comparison with the sections of the filtration lines 88 shown in Fig. 16, run in the opposite direction. Likewise, the sections of the filtration lines 91 are the sections of the tangents against the PI line 84 which, starting from the respective contact point, by comparison with the filtration lines 89, run in the opposite direction. Again, in Fig. 17, the detector surface is divided into two areas. One area is covered by the sections of the filtration lines 90 and the other area is covered by the sections of the filtration lines 91. The boundary between these two areas is formed by the line that runs tangentially relative to both PI lines 80, 84 in Fig. 17 from the left-hand end of the PI line 80 to the right-hand end of the PI line 84. Depending on the area on the detector 60 in which a measured value to be filtered is located, the corresponding filtration line will be assigned to the measured value. So, for example, a measured value at point 85 on the detector 60 will have assigned to it the second filtration line 91 that is in contact with this measured value. The filtration direction 86 is largely opposite to the filtration direction 82 of the first filtration line. In Fig. 17, it runs from right to left. The third filtration lines 94, for measured values located in the PI window 92, have been determined from the κ-vectors e₃ (s , x) and run parallel with the projection of the vector y onto the planar detector 60. They run in direction 96 of the derivative of the radiation-source position y(s) on a helical trajectory in accordance with the angular position s projected onto the planar detector 60. These filtration lines 94 and the associated filtration direction 96 are shown in Fig. 18. Measured values located not in the PI window, but in the 3PI window, are filtered only along the filtration lines 94 in filtration direction 96. In other words, a filtration line 94 is assigned to each of these measured values. The filtration lines 94 are therefore also shown in the 3PI window in Fig. 18. By contrast with Figs. 16 and 17, the filtration lines in Fig. 18 are extended over the entire detector. It is known from "The M-PI-Method for Helical Cone-Beam CT", IEEE

Transactions on Medical Imaging, vol. 19, no. 9, pp. 848-863, 2000 that, in 3PI geometry, measured values for whose acquisition the radiation source was not located in the interval 7_BP (x) lie outside the 3PI window. These measured values are not taken into account for the back projection, i.e. the integration via the angular position s in equation (25). Therefore, no filtration lines were determined for these measured values. In another example of a result, the progression of the filtration lines for measured values located in the PI window is unchanged. However, the filtration lines for measured values that are not located in the PI window but are in the 3PI window do change. This area, designated the outer 3PI window, is in turn divided into multiple areas. One area 103 is formed by a quantity of lines parallel with y , wherein all parallel lines lie fully within the 3PI window (see Fig. 19). Assigned to a measured value that lies in the outer 3PI window in an area 103, which is covered by the parallel lines, are filtration lines 104, which point in direction 110 of the projection of y onto the planar detector 60. These filtration lines are shown in Fig. 19, again extending over the entire detector. However, they are assigned only to measured values lying in the area 103. Measured values lying in areas 106 within the outer 3PI window, which are not filled by the filtration lines 104, are assigned to filtration lines 112, which run tangentially relative to the 3PI line 100. Assigned to a measured value hereby as filtration line 112 is the particular tangent that runs through this measured value and whose point of contact is nearer to the center of the detector than the measured value itself. For measured values that lie within the 3PI window in the areas 108, filtration lines 114 have been determined in this example, which run tangentially relative to the 3PI line 102, wherein assigned to a measured value as filtration line 114 is the particular tangent that runs through this measured value and whose point of contact is nearer to the center of the detector than the measured value itself. The filtration lines 112 and 114 and their filtration directions 116 and 118 are shown in Fig. 20. The filtration lines and filtration directions described should be regarded only as examples. All filtration lines and filtration directions whose K-vectors fulfill equation (24), at least for a majority of combinations of x and ω may be utilized. When the filtration lines and filtration directions have been determined for the 3PI acquisition, they may be used for all subsequent reconstructions of measured values that have been acquired in this manner. If these filtration lines and filtration directions are known, therefore, step 405 may be dispensed with. Next, the measured values projected onto the planar detector 60 are filtered at step 407 in accordance with equation (25) along the filtration lines determined at step 405. To do this, a measured value and a filtration line associated with the measured value are first selected. Along this filtration line, the measured values are multiplied in the filtration direction by a weighting factor and added up. The weighting factor hereby decreases as the sine of the κ-angle increases. It is, in particular, equal to the reciprocal of the sine of the K-angle. The result of the summation is the filtered measured value. This is repeated for all filtration lines of this measured value so that, for any one measured value, a quantity of filtered measured values is determined that is equal to the quantity of filtration lines. These filtered measured values are added to form a measured value. Another, as yet unfiltered measured value is then selected, and the filtration along the filtration lines of this measured value is repeated. Once all measured values have been filtered, the filtration is complete. To determine the filtered measured values on a filtration line, the measured values are preferably re-inteφolated on the planar detector 60 in such a way that, in respect of the κ-angle, they are equidistant on this filtration line. The inteφolated measured values are then multiplied by the weighting factor in accordance with equation (25) along the filtration line and integrated, wherein the multiplication by a weighting factor and the integration may be undertaken in a known manner using a Fourier transform. The measured values may be filtered as follows with the filtration lines shown in Figs. 16 to 20. Every measured value located within the PI window is filtered three times, this being along the three filtration lines assigned at step 405. Subsequently, three filtered measured values exist for every unfiltered measured value, and these are added up to form one measured value. Every measured value located within the 3PI window is filtered once, this being along the filtration line assigned at step 405. The filtration has been executed here on the planar detector. It may, however, also be executed on any arbitrary detector. Where applicable, the measured values and the filtration lines would then have to be projected onto this detector. It is, in particular, useful to filter the measured values on the focus-centered detector. The projection of the measured values onto the planar detector undertaken at step 403 could then be dispensed with and, to determine the filtration lines at step 405, the intersection lines between the particular κ-planes and the focus-centered detector would have to be determined. The filtered measured values are then used for reconstruction of the CT image by a back projection, which, in this embodiment example, is in accordance with equation (25). To this end, at step 409, a voxel V(x) is determined within a specifiable area

(Field Of View - FOV) in the investigation area, Then, at step 411, an angular position s within the interval 7_BP (x) is specified. At step 413, a check is made as to whether a measured value whose ray runs through the center of the voxel V(x) is present for the angular position s . If a ray of this kind cannot be found, the location at which a central ray would have hit against the detector surface is determined. The associated filtered measured value is then calculated by inteφolation of the measured values of adjacent rays. The measured value that can be assigned to the ray matching the voxel, or the measured value obtained by inteφolation, is multiplied at step 415 by a weighting factor which becomes smaller as the distance of the radiation source y(s) from the location x to be reconstructed in the investigation area increases. In this embodiment example, this weighting factor is, in accordance with equation (25), equal to 1 /|x - y(s)\ . At step 417, the weighted measured value is accumulated onto the voxel V(x). At step 419, a check is made as to whether all angular positions s in the interval 7_BP (x) have been taken into account. If this is not the case, the flow chart branches off to step 411. Otherwise, a check is made at step 421 as to whether all voxels V(x) in the FOV have been covered. If this is not the case, continuation is at step 409. If, conversely, all voxels V(x) in the FOV have been covered, the absoφtion in the entire FOV has been established, and so, therefore, has the CT image, so the quasi-exact reconstruction method terminates at step 423. At step 107, a corrective image is generated. As was the case for the first embodiment example, the flow chart for the generation of the corrective image is shown in Fig. 9. The above-described steps 301 to 309 may be adopted unchanged. At step 311 , a weighting function is determined for the voxel V(x) , indicating the weighting factors by which the measured values assigned to the rays have to be multiplied before the back projection. To do this, the weighting factors of the particular rays, or measured values, that contributed to the reconstruction of the voxel ^(x)at steps 401 to 415 are required. Since only rays whose assigned radiation-source positions lie within the area 7_BP (x) have been taken into account with the same weighting in this embodiment example, the characteristic shown in Fig. 21 arises as the weighting function 81 for the reconstruction of the CT image at step 105 if the 3PI interval 7_BP(x) is not chopped. Here, vv_k is a constant weighting factor, which, in this embodiment example, equals 1/3. The horizontal axis in Fig. 21 designates the helix section. The fact that the radiation source is visible from location x over an angular range of 540° means that the area of the helical trajectory on which the radiation source is moving whilst the voxel F(x)is located in the cone-beam, projected onto a rotation plane, equals 540°. To reduce motion artifacts in the quasi-exactly reconstructed CT image, measured values from specified corrective-angle areas KW1 ', KW20 which adjoin the 3PI interval, are also to be taken into account in order to reconstruct the corrective image. These corrective-angle areas KW1 ', KW2' preferably equal 10°. Also for the reconstruction, a so- called 3PI partner within the 3PI interval is determined for each measured value from a corrective-angle area KW1 ', KW2'. Two measured values and their associated rays are 3PI partners if the radiation-source positions y(s) assigned to the rays differ in terms of the angular position s by at least 3π and not more than 4π and if the projections of these rays onto a plane oriented peφendicular to the rotation axis are oriented in inverse parallel to one another. Two 3PI partners form a measured-value pair. The weighting function is determined in such a way that the sum of the weighting factors of two 3PI partners equals zero. Each ray assigned to a measured value that does not lie in one of the corrective-angle areas KW1 ', KW2' or lies within the 3PI interval but has no 3PI partner in any of the corrective-angle areas is assigned a weighting factor of zero. A preferred weighting function in accordance with the invention is shown in Fig. 22. Here, the measured values in the corrective-angle area KW1 ' are multiplied by a weighting factor that, starting from 0 up to the boundary of the 3PI interval 7_BP(x) , increases linearly to w_k 1 . Starting from w_k 12 at the boundary of the PI interval, the measured values in the corrective-angle area KW2' fall linearly to zero. The corresponding 3PI partners of the rays from the corrective-angle area KW1 ' are located within the 3PI interval 7_BP (x) in the area PT, and the 3PI partners of the rays from the corrective-angle area KW2' are located in area P2'. As already described above, the 3PI interval 7_BP(x) may be chopped, i.e. the voxel V(x) is not continuously illuminated by the radiation source, but enters and exits the cone-beam multiple times. Here again, to determine the weighting function of the corrective image, the weighting function of the quasi-exact reconstruction first has to be determined. To this end, the interval 7_BP(x) for the voxel V(x) must be known. If the interval is not known, the individual helix sections on which the radiation source is moving while the voxel V(x) is illuminated may, with known dimensions of the computer tomograph and the cone-beam, be determined by, for example, simulation of the acquisition. A part of an example of a weighting function determined in this manner is shown in Fig. 23. Here, II, 12, 13 designate helix sections that are partial sections of the 3PI interval 7_BP (x) . To reduce motion artifacts, measured values located in corrective-angle areas Kl ... K6, which adjoin the individual sections of the 3PI interval 7_BP (x) , are used in order to reconstruct the corrective image (see Fig. 24). The weighting factors in areas Kl ... K6 increase to a value of w_k 12 , starting from zero up to the boundary of the particular section of the 3PI interval 7_BP(x) . Every measured value in any one of the corrective-angle areas Kl ... K6 has a 3PI partner within the sections II, 12, 13. The 3PI partners of the measured values from the corrective-angle area Kl are located in area Bl, the 3PI partners of the measured values from K2 are located in area B2, the 3PI partners of the measured values from K3 are located in area B3, etc. Measured values that are located neither in the corrective-angle areas Kl ... K6 nor in the areas B 1 ... B6 are weighted with zero. The resultant weighting function for reconstruction of the corrective image is shown in Fig. 24. The following steps 313 to 323, which describe the reconstruction of the corrective image using the weighting function, can be taken from the above description of the first embodiment example. Once a CT image and a corrective image have been determined in steps 105 and 107 respectively, these two images are added, voxel by voxel, at step 109. The resultant image represents a CT image with reduced motion artifacts. LIST OF REFERENCE CHARACTERS α_max Spread angle λ,λ,,λ₂,λ_a,λ_b,λ_c Angular positions of radiation source on helical trajectory X Location in the investigation area ^wk Weighting factor Λ>lO)JBp(^X) Helix sections Bl ...B6 Areas within the intervals 11, 12, 13 Kl ...K6 Corrective-angle areas KW1.KW2, KWl' , KW2' Corrective-angle areas 11,12,13 Partial intervals of the interval 7_BP (x) P1,P2 Areas within the interval 7_PI (x) P10P2' Areas within the interval 7_BP (x) s Radiation source "->_₂ ,!->__! ,ύ₀, _lX₂ Radiation-source positions

1 Gantry 2,5 Motor 3 Collimator assembly 4 Beam 7 Control unit 10 Image-processing computer 11 Monitor 13 Investigation area 14 Rotation axis 16 Detector unit 17 Helical trajectory 31 PI straight line 41,42,43,44,45 Fans of rays parallel with one another 51 Parallel rays 53 Detector row , 70 κ-plane κ-line, 160 Planar detector, 64, 66, 68 Rays Beam Weighting function κ-vector β-vector Detector,88,89,90,91,94,104,112,114 Filtration lines, 84 PI lines, 86, 96, 110, 116, 118 Filtration directions Location on the detector PI window0, 102 3PI lines3, 106, 108 Detector areas

Claims

CLAIMS:

1. A computer tomography method with the following steps: a) Generation of a cone-shaped beam permeating an investigation area and an object located within it, using a radiation source, b) Generation of a relative movement between the radiation source on the one hand and the investigation area on the other, comprising at least one rotation about a rotation axis and having, in particular, a helical or circular form, c) Acquisition of measured values, which depend on the intensity in the beam beyond the investigation area, with a detector unit during the relative movement, d) Reconstruction of a CT image of the investigation area from the measured values using an «PI reconstruction method, in particular using an exact reconstruction method, e) Generation of a corrective image from the measured values using an approximative reconstruction method, f) Addition of the corrective image to the CT image.

2. A computer tomography method as claimed in claim 1, characterized in that, at step b) the relative movement takes the form of a helix, and at step d) the reconstruction of a CT image using an «PI reconstruction method exhibits the following steps: partial derivation of measured values to which parallel rays with different radiation-source positions are assigned, in accordance with an angular position of the radiation source on the helix, as assigned to the particular measured value, filtering of the derived measured values along κ-lines, reconstruction of the CT image by back projection of those filtered measured values that are located in a PI interval.

3. A computer tomography method as claimed in claim 2, characterized in that the filtering of a derived measured value exhibits the following steps: determination of a κ-line for the measured value, multiplication of the measured values located on the κ-line by a weighting factor that increases with the reciprocal of the sine of the κ-angle and is, in particular, equal to this reciprocal, addition of the weighted measured values located on the κ-line, and equating of the resultant sum with the filtered measured value, so that the resultant sum is the filtered measured value.

4. A computer tomography method as claimed in claim 1, characterized in that the approximative reconstruction method for reconstruction of the corrective image exhibits a filtered back projection in which only measured-value pairs are used whose assigned rays, when projected onto a plane oriented peφendicular to the rotation axis, are oriented in inverse parallel to one another, and of which only one measured value was used in each case for the HPI reconstruction at step d), wherein, before the filtered back projection, every measured value is multiplied by a weighting factor and the weighting factors are selected such that the sum of the weighting factors of a measured- value pair equals zero.

5. A computer tomography method as claimed in claim 4, characterized in that the approximative reconstruction method at step e) exhibits the following steps: rebinning of the measured values to form a quantity of groups, wherein each group comprises a plurality of planes parallel with one another and with the rotation axis, in each of which a ray fan is located, uni-dimensional filtering of the measured values generated by the rebinning for each group in the direction peφendicular to the direction of the planes, multiplication of the filtered measured values by a weighting factor that increases with the cosine of the cone angle of the particular ray that is assigned to the particular measured value, and, in particular, is equal to the cosine of the cone angle, back projection of the measured-value pairs, wherein, before the back projection, every measured value is multiplied by a weighting factor and the weighting factors are selected such that the sum of the weighting factors of a measured-value pair equals zero.

6. A computer tomography method as claimed in claim 1, characterized in that, at step b), the relative movement takes the form of a helix and at step d) the reconstruction of a CT image using an nPI reconstruction method exhibits the following steps: partial derivation of measured values to which parallel rays with different radiation-source positions are assigned, in accordance with an angular position of the radiation source on the helix, as assigned to the particular measured value, filtering of the derived measured values along filtering lines, wherein at least some of the measured values have multiple filtering lines assigned to them so that these measured values are filtered multiple times, reconstruction of the CT image by back projection of those filtered measured values that are located in an «PI interval.

7. A computer tomography method as claimed in claim 6, characterized in that the filtering of a derived measured value exhibits the following steps: provision of at least one filtering line assigned to the derived measured value, wherein every filtering line has a filtering direction assigned to it, multiplication of measured values along every filtering line assigned to the measured value by a weighting factor, which increases with the reciprocal of the sine of the κ-angle and, in particular, corresponds to this reciprocal, formation of sums by addition of all weighted measured values along every filtering line of the measured value in the filtering direction that has been assigned to the particular filtering line, addition of the sums to form a filtered measured value.

8. A computer tomography method as claimed in claim 7, characterized in that the provision of multiple filtering lines exhibits the following steps: determination of κ-vectors that enable an exact reconstruction, wherein at least one κ-vector is assigned to every combination of radiation-source position and a location in the investigation area to be reconstructed, determination of a κ-plane for every combination of radiation-source position and the location in the investigation area to be reconstructed and every κ-vector assigned to the particular combination, determination of an intersection line for every κ-plane between the particular κ-plane and the detector surface, wherein every intersection line represents a filtering line, determination of a filtering direction for every filtering line, and assignment of the particular filtering direction to the corresponding filtering line, assignment of every filtering line to a measured value corresponding to the combination, associated with the filtering line, of location in the investigation area to be reconstructed and radiation-source position .

9. A computer tomograph for implementing the method as claimed in claim 1 with: a radiation source (S) for generating a cone-shaped beam (4) penetrating an investigation area (13) and an object located therein, a drive assembly (2, 5) in order to rotate an object contained in the investigation area (13) and the radiation source (S) relative to one another about a rotation axis (14) and shift them parallel to the rotation axis (14), a detector unit (16) with a detector surface, which is coupled with the radiation source (S) for the acquisition of measured values, a reconstruction unit (10) for reconstructing the CT image using an «PI reconstruction method, in particular an exact reconstruction method, and for reconstructing the corrective image using an approximative reconstruction method within the investigation area from the measured values acquired by the detector unit (16), a control unit (7) to control the radiation source (S), the detector unit (16), the drive assembly (2, 5) and the reconstruction unit (10) in accordance with the following steps: a) Generation with the radiation source of a cone-shaped beam penetrating an investigation area and an object located within it, b) Generation of a relative movement between the radiation source on the one hand and the investigation area on the other, comprising at least one rotation about a rotation axis and taking, in particular, the form of a helix or a circle, c) Acquisition of measured values, which depend on the intensity in the beam beyond the investigation area, with the detector unit during the relative movement, d) Reconstruction of a CT image of the investigation area from the measured values using an nPl reconstruction method, in particular using an exact reproduction method, e) Generation of a corrective image from the measured values using an approximative reconstruction method, f) Addition of the corrective image to the CT image.

10. A computer program for a control unit to control a radiation source, a detector unit, a drive assembly and a reconstruction unit of a computer tomograph for implementing the method as claimed in claim 1 in accordance with the following sequence: a) Generation with a radiation source of a cone-shaped beam penetrating an investigation area and an object located within it, b) Generation of a relative movement between the radiation source on the one hand and the investigation area on the other, comprising at least one rotation about a rotation axis and taking, in particular, the form of a helix or a circle, c) Acquisition of measured values, which depend on the intensity in the beam beyond the investigation area, with the detector unit during the relative movement, d) Reconstruction of a CT image of the investigation area from the measured values using an «PI reconstruction method, in particular using an exact reproduction method, e) Generation of a corrective image from the measured values using an approximative reconstruction method, f) Addition of the corrective image to the CT image.