Patent Publication Number: US-2023140174-A1

Title: Method for obtaining a ct image of an object with heel effect compensation in image space

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The invention relates to a method for obtaining a CT (Computer Tomography) image of an object, with the following steps: 
     a) generating x-rays using an x-ray source comprising an angled anode, wherein the x-rays are generated within an anode material of the angled anode, and the x-rays exiting the anode material have travelled different distances within the anode material depending on their exit location with respect to a y-direction; 
     b) recording at least one set of 2D projections of the object or a part of the object located within a beam path of the generated x-rays with a 2D x-ray detector located in the beam path behind the object, with the x-ray source and the 2D x-ray detector being part of a measurement setup, and with the 2D projections of a respective set of 2D projections being recorded one by one in different rotation positions of the object relative to the measurement setup, in particular wherein the rotation of the object relative to the measurement setup is with respect to a turning axis parallel to the y-direction, and wherein for each respective set of 2D projections, a respective shifting position of the object relative to the measurement setup including the x-ray source and the 2D x-ray detector with respect to the y-direction is chosen; and 
     c) generating at least one 3D CT image of the object or said part of the object, wherein for each respective set of 2D projections a 3D CT image is generated, with the 3D CT image consisting of a plurality of voxels to which respective grey values are associated. 
     Such a method is known from US 2019/0244397 A1. 
     Description of the Related Art 
     Computer tomography (CT) imaging is a powerful tool for non-destructively investigating the interior of objects in a three dimensional (3D) way. CT imaging is used, for example, in industry for verifying the quality of components, such as for detecting pores or cracks inside the volume of a component&#39;s material, or for determining the dimensions of otherwise non-accessible internal structures of a complex component. In medicine, CT imaging may be used to investigate the condition of tissue and bones, for example for detecting and characterizing tumors or other pathological changes, in a living human or animal body. 
     For CT imaging, a plurality of two-dimensional (2D) x-ray images, i.e., 2D projections, of the object of interest or a part of it are recorded, wherein the object is imaged in a different rotational position in each 2D projection. From such a set of 2D projections, a 3D CT image of the object or said part can be generated, i.e., calculated. In case an object is larger than the field of view of the used measurement setup, multiple sets of 2D projections can be recorded, with each set of 2D projections imaging a part of the object, and wherein the object is shifted to different positions with respect to the measurement setup for the different sets; the resulting 3D CT images may be assembled to one overall 3D CT image of the complete object. A 3D CT image consists of a plurality of voxels having respective grey values, with the grey value representing the respective local x-ray attenuation of the object. At the borders of structures of the object, the grey values generally change, resulting in a visible contrast in the 3D CT image. Note that in practice, one or more 2D CT partial images are often extracted from the 3D CT image, in order to get to know the internal structure of the object in one or more particular planes. 
     For recording a 2D projection of the object (or part of it), an x-ray source generates x-rays, the object (or its part) is placed into the beam path of the x-rays, and a 2D x-ray detector behind the object detects the x-ray intensity. 
     A common type of x-ray source used in CT imaging applies a so-called reflection type target. Here, an electron beam originating from a cathode is directed onto a metal target acting as anode for the electrons, wherein the anode comprises a target surface which is inclined with respect to the x-ray beam. A corresponding anode is also referred to as “angled anode”. The electrons somewhat penetrate the anode material and are slowed down in the anode material, what generates x-rays in the form of bremsstrahlung, having a basically continuous spectral range. The generated x-rays used for x-ray imaging have a general propagation direction basically transverse to the electron beam direction. Due to the inclined target surface, different local fractions of the generated x-rays travel through different thicknesses of anode material before exiting the anode material and travelling further to the object and the 2D x-ray detector. The thickness of anode material passed depends on the emission angle (or propagation direction) of the x-rays and can be correlated to the exit location of the x-rays on the target surface with respect to a direction, here called y direction, that corresponds to the electron beam direction. The anode material attenuates the generated x-rays passing through it, what influences the characteristics of the generated x-ray beams; this phenomenon is called “heel effect”. 
     The heel effect has two aspects. First, generally a larger thickness of anode material to be passed leads to more absorption of x-rays and vice versa, so the local intensity of the x-rays varies along the y-direction (“intensity effect”). Second, x-rays having a shorter wavelength (“harder x-rays”) generally experience less absorption as compared to x-rays having a longer wavelength (“soft x-rays”) in the anode material. As a consequence, local fractions of the x-rays having travelled through a larger thickness of anode material become “harder” as compared to local fractions having travelled through a shorter thickness of anode material, i.e., the spectral composition of the x-rays changes along the y direction (“spectral effect”). 
     When recording a 2D x-ray image, i.e., 2D projection, with the x-rays generated with a reflection type target, the “intensity effect” leads to a general intensity gradient along the y-direction, distorting the x-ray 2D projection. The “intensity effect” can be compensated for by the so called “flat field correction”. For this purpose, a 2D projection is recorded without any object. The local measured intensities at the 2D x-ray detector without an object may be inverted to obtain local correction factors to be applied to the measured intensities of 2D projections with an object. Flat field correction compensates for different distances to the x-ray source (i.e., the electron spot on the target) and different incident angles of x-rays of local parts of the 2D x-ray detector surface as well as possible variance in local detector sensitivity, but it also compensates for the “intensity effect” aspect of the heel effect. 
     However, the other aspect of the Heel effect, namely the “spectral effect”, still remains even after flat field correction. When a 2D projection of an object (or a part of it) is recorded, the object is radiated with x-rays wherein their spectral composition varies with location. More precisely, the hardness of the x-rays reaching the object depends on the position in y-direction. Note again that harder x-rays are generally less absorbed than softer x-rays, what also applies to the object material. For this reason, the attenuation of the x-rays by the object not only depends on the characteristics of the object (i.e., its geometry and material composition), but also depends on the location in y-direction. When for example recording a 2D projection of an object which is uniform in y direction, the measured x-ray intensity at the x-ray detector is larger in regions of the object where the local x-rays reaching the object were harder as compared to regions where the local x-rays were softer, since the harder x-rays were less absorbed by the object; this causes a grey value gradient of the object in the 2D projections and a resulting 3D CT image. This becomes particularly apparent when combining 3D CT images of an object larger than the field of view of the measurement setup (in y-direction, “vertical field of view”) into an overall 3D CT image. At the transitions of 3D CT images combined, an abrupt change of contrast appears in the object; this image artefact is sometimes referred to as “bamboo effect”. 
     US 2019/0244397 A1 suggests a heel effect correction factor to be applied on the detector output in the 2D projections, taking into account the spectral absorption behaviour of a tungsten anode material at every emission angle, and taking into account the path length travelled in the tungsten anode material. Further, the spectral absorption behaviour of the object material and the spectral detector absorption efficiency is taken into account. This procedure requires a lot of calculation efforts and detailed knowledge about the characteristics of the anode material, the object material and the detector. 
     K. Liu et al. (K. Liu, R. Boardman, M. Mavrogordato, F. A. Loveridge, W. Powrie, “The Importance of the Heel Effect in X-Ray Ct Imaging of Soils”, Environmental Geotechnics, 2020, pp. 1-15, ISSN 2051-803X) suggest application of a self-wedge correction process, wherein an average projection of the specimen is determined from 2D projections of all rotation angles, a minimum grey value for the average projection is determined and subtracted from the average projection, a median filter is applied, and the filtered average projection is used for correcting the original 2D projections by subtracting it in each case. The corrected 2D projections can be used for generating a 3D CT image of the specimen. While this approach seems to have reduced the heel effect, it requires still quite some calculation effort, in particular in each 2D projection. Further, improvement in contrast seems to be somewhat limited. 
     SUMMARY OF THE INVENTION 
     The present invention provides a method for obtaining 3D CT images of objects, wherein heel effect artefacts are reduced in a simple way. This is achieved by a method as introduced in the beginning, characterized in that the method further comprises a step of: 
     d) for each generated 3D CT image, correcting the 3D CT image into a corrected 3D CT image or correcting a 2D CT partial image of the 3D CT image into a corrected 2D CT partial image, wherein for each slice of voxels of the 3D CT image, a scaling factor is determined, wherein each slice of voxels comprises those voxels of the 3D CT image having an identical position in y-direction in the 3D image, wherein the grey values of each voxel of a respective slice of voxels of the 3D CT image or the 2D CT partial image are multiplied with the scaling factor determined for the respective slice of voxels, resulting in corrected grey values for the voxels of the corrected 3D CT image or the corrected 2D CT partial image, wherein the scaling factors for the slices of voxels are determined with at least one 3D CT calibration image measured with the measurement setup, wherein the at least one 3D CT calibration image pictures similar or identical object structures of a calibration object placed within the beam path of the x-rays in different regions of a field of view of the measurement setup with respect to the y-direction, and wherein for the at least one 3D CT calibration image a grey value contribution to the grey values of voxels belonging to the similar or identical object structures in said different regions attributable to the slice position in y direction is determined at least approximately, and the scaling factor for a respective slice of voxels is chosen such that it compensates for the determined grey value contribution for that slice. 
     The inventive method applies an at least approximate compensation of the heel effect, with respect to its spectral effect aspect, on the 3D CT image level of CT imaging (“image space”). According to the invention, the reconstructed 3D CT images undergo a correction by means of a scaling factor that is determined for each slice of voxels in the 3D CT image, and which is directly applied to the grey values of the voxels of the 3D CT image or a 2D CT partial image of it by multiplication. In this way, the method is very simple to apply, in particular simpler as compared to correction methods acting on the 2D projections. Further, the inventive method is based on correction information obtained from one or more reconstructed 3D CT calibration images, which can be obtained easily by measurement. The inventive method does not require detailed knowledge about the measurement setup (such as the anode geometry or anode material absorption behaviour or the detector sensitivity) or the object to be measured (such as object geometry and object material absorption behaviour). For this reason, too, the inventive method is easy to perform and may be applied universally at basically every system. 
     The scaling factors are determined using at least one 3D CT calibration image. The at least one 3D CT calibration pictures similar or identical object structures of a calibration object at different regions (positions) in the field of view of the measurement setup with respect to the y direction, along which the heel effect is acting. 
     The similar object structures are similar in that they exhibit (at least approximately) the same x-ray absorption properties, in particular wherein the similar object structures are made from the material having (at least approximately) the same density (typically wherein the similar object structures are at least predominantly made from the same material) and have (at least approximately) the same dimensions within the different regions of the field of view, in particular within the xz plane at corresponding (i.e., compared) slices. If the compared object structures are identical, then they inherently have identical x-ray absorption properties. 
     However, due to the heel effect, the similar or identical object structures appear somewhat different in the at least one 3D CT calibration image in the different regions of the field of view. Generally speaking, the grey values of the similar or identical object structures at different positions in y direction are shifted (and possibly also stretched) with respect to each other. In other words, the similar or identical object structures have a different brightness (and possibly also a different contrast) as a function of the y position. The alteration of the grey values of the similar or identical structures at different y positions in the field of view represents the grey value contribution (or grey level contribution) attributable to the slice position in y direction. 
     The grey value contribution results from the heel effect, wherein depending on the y position, the generated x-rays at the angled anode of the reflection type target have a different wavelength distribution, and this different wavelength distribution leads to different x-ray absorption and thus different apparent grey values in the (similar or identical) object structures located at different y positions, before correction. 
     With the knowledge of the grey value contribution in the uncorrected 3D CT image resulting from the heel effect as a function of y position (i.e., the slice position in y direction), scaling factors for each slice compensating for this grey value contribution can be determined. The scaling factor for a particular slice reverses the heel effect in that slice, at least approximately. Most simply, a reference slice can be defined for each 3D CT image (typically in the center with respect to y direction) representing a standard grey value level; at the reference slice no correction is required by definition (i.e. the scaling factor is 1), and above and below the reference slice, the scaling factor increases or decreases the grey value, generally with the scaling factor deviating the more from 1 the farther the slice to be corrected is away from the reference slice. 
     Specific scaling factors are attributed to slices having the same y position. With typical distances between the x-ray source (or the beam spot) and the object to be measured, x-rays within a particular slice have roughly the same emission direction (“heel angle”) from the beam spot and so have travelled roughly through the same thickness (path length) of anode material, and therefore have roughly the same spectral composition; accordingly one scaling factor will be adequate for a complete slice. 
     It should be noted that the similar or identical object structures may be (at least a part of) the object to be measured (i.e., the object is also used as calibration object), or may be (at least a part of) a calibration object (“phantom”) generally resembling the object to be measured (i.e., a calibration object/phantom different from the object to be measured is used). 
     Within the inventive method, it is possible to use one or more 3D CT images also as one or more 3D CT calibration images (so no separate calibration image measurements are necessary), or alternatively one or more 3D CT calibration images are taken in addition to the one or more 3D CT image of the object. If the 3D CT image(s) of the object are also used as 3D CT calibration images (with the object acting as calibration object then), the inventive calibration can be called a “self calibration”. 
     For executing the rotation of the object relative to the measurement setup, either the object may be rotated with respect to a (as far as rotation is concerned) stationary measurement setup (x-ray source/2D x-ray detector), or the measurement setup (x-ray source/2D x-ray detector) may be rotated with respect to a (as far as rotation is concerned) stationary object. For executing a shift of the object relative to the measurement setup, either the object may be shifted with respect to a (as far as shift is concerned) stationary measurement setup (x-ray source/2D x-ray detector), or the measurement setup (x-ray source/2D x-ray detector) may be shifted with respect to a (as far as shift is concerned) stationary object. 
     Note that the inventive correction may be limited to a 2D CT partial image of a 3D CT image, instead of correcting the complete 3D CT image. A typical 2D CT partial image corrected with the inventive method is a voxel plane extended in y direction and another direction x or z (Note that if the 2D CT partial image was chosen in a xz plane, no heel effect correction would be necessary). 
     Note that x, y, z form a Cartesian coordinate system, typically with z representing the general propagation direction of the x-rays, and the target surface of the angled anode extending parallel to the x direction and being inclined with respect to the y direction. The direction of an electron beam hitting the angled anode typically corresponds to the y direction. 
     In a preferred variant of the inventive method, the grey value contribution is approximated with a linear or piecewise linear function in y direction. This simplifies the inventive procedure, and results in most applications in an adequate (approximate) heel effect compensation. In other words, for most applications, the heel effect is basically linear in y direction in the image space (in the reconstructed 3D CT image), such that this variant fits the physical behaviour very well. For a linear function, an analysis of similar or identical object structures at only two regions/positions (“base points”) with different y positions is sufficient. In case of a piecewise linear function, a number NB of base points is applied, typically with 4≤NB≤20, or 5≤NB≤15, or 6≤NB≤12. The linear approximation of the grey value contribution function is particularly well suited if the emission angle of x-rays used in the field of view of the measurement setup is relatively low, e.g., within a maximum range of +/−25°, or even +/−20°, with respect to the xz plane containing the electron spot on the target surface. 
     In another advantageous variant, for the 2D projections of all sets of 2D projections, a flat field correction is applied. In the course of the flat field correction, a 2D projection of the x-rays without any object is measured with the measurement setup, and the measured intensity values of this 2D flat field projection are inverted in order to obtain local flat field correction factors (per pixel) to be multiplied on the 2D projections measured with an object or calibration object; if desired, also a dark field correction may be included here (a 2D dark field projection is measured without x-rays (and without any object), and the 2D dark field projection may be subtracted from the 2D flat field projection without object and each 2D projection of the object/calibration object). The flat field correction compensates for amplitude distortions originating from different 2D x-ray detector regions having different distances to the electron spot on the target surface and different incidence angles of x-ray radiation, as well as possible variances in local detector sensitivity. Further, within the flat field correction, differences in amplitude over all wavelengths of the generated x-rays as a function of y position, originating from the different path length within the anode material, are intrinsically offset. In other words, the intensity effect aspect of the heel effect can be compensated for directly and in advance by the flat field compensation. It should be noted that the flat field correction (which addresses intrinsically the intensity effect aspect of the heel effect and other artefacts) is a procedure done in projection space on a per pixel basis, whereas the inventive heel effect correction (which addresses the spectrum aspect of the heel effect) is a procedure done in image space on a per y-position (slice) basis. 
     In another preferred variant, at least two 3D CT calibration images are measured, wherein the at least two 3D CT calibration images form at least one calibration pair of 3D CT calibration images, wherein the two 3D CT calibration images of a respective calibration pair picture respective identical object structures of the calibration object placed within the beam path of the x-rays in different regions of the field of view of the measurement setup with respect to the y direction, wherein the two 3D CT calibration images of the calibration pair are measured at different shifting positions of the calibration object relative to the measurement setup, wherein the respective identical object structures located in different regions of the field of view of the measurement setup are pictured in different 3D CT calibration images of this calibration pair, in particular wherein the grey values of at least a part of the voxels within said different regions are compared for determining the grey value contribution. By using identical object structures for determining the grey value contribution, i.e., for compensating for the heel effect, the heel effect compensation can be particularly accurate. Possible differences in x-ray absorption behaviour of (only) similar object structures to base the correction upon are entirely excluded. 
     In a preferred further development of this variant, the calibration object is identical with the object. In this way, no separate calibration object in addition to the object to be measured (imaged) is needed, simplifying the method. Further, possible differences in material or structure between a calibration object and the object to be measured cannot distort the compensation, since the object to be measured is used as the calibration object. Note that generally, a part of the object that is representative for the object is preferably chosen as the identical object structure. 
     Particularly preferred is a subvariant of this further development, providing that the object is longer than the field of view of the measurement setup with respect to the y-direction, that at least two sets of 2D projections of parts of the object are recorded, with the at least two sets of 2D projections being recorded in at least two different shifting positions of the object relative to the measurement setup with respect to the y-direction, that at least two 3D CT images of the parts of the object are generated from the at least two sets of 2D projections, wherein the at least two 3D CT images form at least one pair of 3D CT images, with the respective parts of the object pictured in a respective pair of the 3D CT images overlapping in y-direction in a respective overlapping object section, and that at least one said pair of 3D CT images is used as at least one said calibration pair of 3D CT calibration images, wherein for a respective calibration pair of 3D CT calibration images the respective overlapping object section provides the respective identical object structures for this calibration pair, in particular wherein the entirety of the at least two 3D CT images pictures the complete object. This procedure can also be called “self calibration”, and can be applied in “connected scans”. In the connected scans, the 3D CT images recorded overlap pairwise, such that a respective overlapping object section is scanned and its volume is reconstructed twice, typically once with a small and once with a large heel angle (i.e. once with a low and once with a high y position in the field of view); here the heel effect is most pronounced and well observable, what can be used for determining the grey value contribution attributable to the slice position in y direction. So the 3D CT images of the object can be used as the 3D CT calibration images, with the object being used as calibration object. In this way, no additional scans (sets of 2D projections) are necessary, and the method is particularly fast and accurate. 
     In another preferred variant, a respective 3D CT calibration image pictures similar object structures of the calibration object placed within the beam path of the x-rays in different regions of the field of view of the measurement setup with respect to the y-direction, wherein the similar object structures located in different regions of the field of view of the measurement setup are included in the same respective 3D calibration image, and wherein the calibration object is different from the object to be measured. This variant is simple to do, and in particular does not require any shifting of an object or calibration object. The calibration object is chosen as to have at least two regions offering similar object structures for doing the calibration. Typically, in this variant, only one 3D CT calibration image from the calibration object is measured, independent from the at least one 3D CT image of the object (wherein typically only one 3D CT image of the complete object is measured, but there can also be multiple 3D CT images which piecewise picture the object too large of a single 3D CT image). However, it is also possible to measure multiple 3D CT calibration images of a calibration object, and to average the calibration information (information about the grey value distribution) from these multiple 3D CT calibration images for determining the scaling factors. 
     In a preferred further development of this variant, the calibration object is chosen at least approximately cylinder shaped and aligned with the y direction. In this way, the object can be used along its entire length for choosing similar object structures for calibration purposes; in particular a practically arbitrary number of base points may be chosen for a piecewise linear function for the grey value contribution, or even a continuous grey value contribution function may be applied. Note that the calibration object is generally made of a uniform material. 
     Another preferred further development provides that a calibration material density CMD of calibration material of the calibration object and a density D of a material or predominant material of the object are similar, in particular with 90%≤CMD/D≤110%, and/or that a geometry of the calibration object and the object are similar, in particular wherein for a height COH of the calibration object and a height H of the object in y direction 80%≤COH/H≤120% applies and/or for a largest diameter COLD of the calibration object and a largest diameter LD of the object in cross-section perpendicular to the y direction 80%≤COLD/LD≤120% applies for at least 80% of the respective height COH or H. In this way, the heel effect compensation by the invention can be highly accurate. 
     An advantageous variant of the inventive method provides that the slices of voxels of the at least one 3D CT image are consecutively numbered in regard to their y position, that one slice of voxels of the at least one 3D CT image is set as a reference slice with a scaling factor of 1, in particular wherein the reference slice is a central slice of the at least one 3D CT image with respect to the y direction; and that the scaling factor determined for a respective slice of voxels of the at least one 3D CT image is determined as follows: 
       sf( n   s )=1+sc*( n   s   −n   ref ) 
     with sf: scaling factor, sc: scaling coefficient, n s : position number of the respective slice, n ref : position number of the reference slice. In this variant, the scaling factor is determined by one linear function for the complete respective 3D CT image, what is particularly simple. Further, by choosing the scaling factor of 1 in the center of the 3D CT image, a good contrast can be achieved in most practical applications. 
     In a preferred further development of this variant, for determining the grey value contribution, the grey values of voxels belonging to the similar or identical object structures are compared in two different regions of the field of view of the measurement setup, and a total of K sets of pictured similar or identical object structures in the two different regions are analysed, wherein K is a natural number ≥1, in particular wherein K calibration pairs of 3D CT calibration images in two different regions are analysed, wherein the two different regions comprise a first region and a second region. Comparing the similar or identical structures in just two different regions of the field of view (per calibration pair) is particularly simple and fully sufficient for determining the scaling coefficient sc. When a plurality of calibration pairs are analysed (i.e. K≥2), then a single scaling coefficient ssc may be determined for each calibration pair, and the scaling coefficient sc may be calculated by averaging the single scaling coefficients ssc. In case of one or more calibration pairs being analysed, the different regions typically correspond to respective overlapping parts of the image volumes of the 3D CT calibration images of the calibration pair. However, the regions may be chosen smaller, though, if desired. 
     Advantageous is a subvariant of this further development, which provides that for each of the first and second region, or each of a respective section of the first and second region with the respective section including voxels belonging to the respective similar or identical object structures, a characteristic grey value and a characteristic position number in regard to the y direction is determined, and that the scaling coefficient is determined as follows: 
     
       
         
           
             sc 
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 K 
               
               
                 
                   1 
                   K 
                 
                 * 
                 
                   ( 
                   
                     
                       gv 
                       1 
                       k 
                     
                     - 
                     
                       gv 
                       2 
                       k 
                     
                   
                   ) 
                 
                 / 
                 
                   ( 
                   
                     
                       i 
                       1 
                       k 
                     
                     - 
                     
                       i 
                       2 
                       k 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     with gv 1   k : characteristic grey value of the first region or section of it for set k, gv 2   k : characteristic grey value of second region or section of it for set k, i 1   k : characteristic position number of the first region or section of it for set k, i 2   k : characteristic position number of the second region or section of it for set k, k: index of sets of pictured similar or identical object structures analysed. In this way, the scaling factor can be calculated in a simple way. Averaging over K increases the liability of the determination of the scaling coefficient sc. A section is typically chosen such that only the object/calibration object (or a part of it) is pictured, and free space around the object/calibration object is excluded. Note that the different regions (here the first and second region) or the respective sections are generally chosen with an identical size. If identical object structures of the calibration object (or object) are used for calibration, then the different regions of the field of view relate to the same area within the calibration object (or object). The characteristic grey value may e.g., be chosen as a local extreme value, such as a local maximum value, or as a mean grey value. 
     In an advantageous subdevelopment of the above subvariant, it is provided that the characteristic grey value of the first region or section of it is chosen as a mean grey value, that the characteristic grey value of second region or section of it is chosen as a mean grey value, that the characteristic position number of the first region or section of it is chosen as a mean position number, and that the characteristic position number of the second region or section of it is chosen as a mean position number. The mean gray values and means position number can be determined easily and fast, in particular with automatic means. 
     Another preferred subvariant of the above further development provides that for each of the first and second region, or each of a respective section of the first and second region with the respective section including voxels belonging to the respective similar or identical object structures, 
     a histogram of the quantity of voxels as a function of the grey value of these voxels is generated, and the histograms of the first region and second region are overlaid, and the histogram of the second region is shifted with respect to the grey values with respect to the first histogram or vice versa until a deviation of the histograms is minimized, 
     a characteristic position number, in particular mean position number, in regard to the y direction is determined, 
     and that the scaling coefficient is determined as follows: 
     
       
         
           
             sc 
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 K 
               
               
                 
                   1 
                   K 
                 
                 * 
                 
                   ( 
                   
                     shift 
                     k 
                   
                   ) 
                 
                 / 
                 
                   ( 
                   
                     
                       i 
                       1 
                       k 
                     
                     - 
                     
                       i 
                       2 
                       k 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     with shift k : shift of histograms with respect to the grey values at which a deviation of the histograms is minimized for set k, i 1   k : characteristic position number, in particular mean position number, of the first region or section of it for set k, i 2   k : characteristic position number, in particular mean position number, of the second region or section of it for set k, k: index of sets of pictured similar or identical object structures analysed. In this way, the scaling coefficient can be determined with particularly high accuracy. Shifting the histograms and determining the shift of best match is very robust and in particular insensitive to numerous measurement artefacts such as single wrong detector readouts. The averaging over K increases the liability of the determination of the scaling factor. A section is typically chosen such that only the object/calibration object (or a part of it) is pictured, and free space around the object/calibration object is excluded. 
     In a preferred further development of the variant wherein a respective 3D CT calibration image pictures similar object structures of the calibration object in different regions of the field of view, it is provided that a curve plotting the mean grey value mgv for each slice of the at least one 3D CT calibration image as a function of the slice position j s  of the respective slice in y direction is determined, that a reference slice is chosen having a slice position j ref  in y direction, in particular wherein the reference slice is a central slice of the at least one 3D CT calibration image, that the curve is multiplied with a weighting factor wf, resulting in a weighted curve of weighted mean grey values wf*mgv, wherein the weighting factor wf is chosen such that the weighted mean grey value wf*mgv(j ref ) of the reference slice has a value of wf*mgv(j ref )=1, and that the scaling factor sf(j s ) for a respective slice at slice position j s  is chosen with sf(j s )=1/[wf*mgv(j s )]. In this way, a continuous heel effect compensation can be established, what is particularly accurate. The mean grey values for each slice can be determined by measuring a calibration object/phantom extending over the complete field of view, and calculating the mean grey value for each slice from the measured grey values of the voxels of that slice. Alternatively, the curve is approximated by a piecewise linear function, wherein the mean grey values are determined by measurement only for a few base points in the field of view, and the mean grey values between the base points are determined by linear interpolation. 
     Further advantages can be extracted from the description and the enclosed drawing. The features mentioned above and below can be used in accordance with the invention either individually or collectively in any combination. The embodiments mentioned are not to be understood as exhaustive enumeration but rather have exemplary character for the description of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    shows a schematic side view of a CT measurement setup for use with the present invention; 
         FIG.  2    shows a magnification of  FIG.  1    in the region of the angled anode, explaining the origin of the heel effect; 
         FIG.  3    illustrates schematically the intensity effect aspect of the heel effect and the flat field correction, to be applied in a variant of the invention; 
         FIG.  4    illustrates schematically the spectrum effect aspect of the heel effect, for a variant with single 3D CT image, and the effect of the inventive correction; 
         FIG.  5    illustrates schematically the spectrum effect aspect of the heel effect, for a variant with a connected scan of 3D CT images, and the effect of the inventive correction; 
         FIG.  6    illustrates schematically the inventive heel effect compensation for a 3D CT image in an exemplary variant, applying one separate 3D CT calibration image using a separate cylindrical calibration object and using two different regions of the field of view for calibration; 
         FIG.  7    illustrates schematically the inventive heel effect compensation for a 3D CT image in an exemplary variant, applying two connected 3D CT calibration images using the object as calibration object and using two different regions of the field of view for calibration; 
         FIG.  8    illustrates schematically the inventive heel effect compensation for a connected scan of three 3D CT images in an exemplary variant, applying the 3D CT images as 3D CT calibration images, using the object as calibration object, and using two different regions of the field of view for calibration; 
         FIG.  9    illustrates schematically the choice of two different regions in the field of view containing similar object structures of a cylindrical calibration object used for the inventive correction method in an exemplary variant; 
         FIG.  10    illustrates schematically the choice of two different regions in the field of view containing identical object structures of an object to be measured, used as calibration object, applying connected scans used in the inventive correction method in an exemplary variant; 
         FIG.  11    illustrates schematically the determination of a scanning coefficient for determining scaling factors in accordance with the invention in an exemplary variant, applying a histogram of grey values in sections of two different regions in the field of view, with part A before shifting, and part B after shifting for minimizing histogram deviation; 
         FIG.  12    shows a table used in the determination of a scanning coefficient for determining scanning factors in accordance with an exemplary variant of the invention, listing mean position numbers and mean grey values in sections of two different regions in the field of view contained in three overlapping images; 
         FIG.  13    illustrates schematically the determination of scanning factors according to an exemplary variant of the invention, calculating mean grey values for each slice of voxels (upper three diagrams A, B, C) or alternatively for six base points (lower diagram D); 
         FIG.  14    shows an uncorrected 3D CT image of a cylindrical object (left picture A) and a corrected 3D CT image of the cylindrical object (right picture B), applying scaling factors for each slice in accordance with a variant of the invention, based on the mean grey values for each slice; 
         FIG.  15    shows an uncorrected connected scan/overall 3D CT image of three 3D CT images of a mouse femur (left picture A) and the corrected overall 3D CT image of the mouse femur (right picture B), applying scaling factors obtained by self-calibration according to a variant of the invention; 
         FIG.  16   a    shows a diagram plotting the mean grey values and standard deviation intervals as a function if the slice index of an overall 3D CT image of an AI rod before a heel effect correction; 
         FIG.  16   b    shows the diagram of  FIG.  16   a    after heel effect correction according to a variant of the invention, applying self calibration. 
     
    
    
     DETAILED DESCRIPTION 
       FIG.  1    schematically illustrates an exemplary measurement setup  1  for obtaining 3D CT images, for use with the present invention. 
     The measurement setup  1  comprises an x-ray source  2 , which comprises a reflection type target with an angled anode  3 . Electrons  5  originating from a cathode  4  form an electron beam  5   a , which is directed towards the angled anode  3  as a target. The electrons  5  are accelerated towards the angled anode  3  by a high voltage, such as 5 kV or more (not illustrated in detail). The electron beam  5   a  is parallel to a y direction here. Directions x, y and z form a Cartesian coordinate system. 
     Also apparent from the magnification of  FIG.  2   , the angled anode  3  has a target surface  6  which is inclined by an angle α towards the propagation direction of the electron beam  5   a , i.e., towards the y direction. Said angle α is also referred to as the anode angle α. In the example shown, the anode angle α is about 45°; note that in general, the anode angle α is typically chosen with 25°≤α≤65°. The target surface  6  is flat, and extends parallel to the x direction here. 
     The electrons  5  of the electron beam  5   a  penetrate a little into the anode material of the angled anode  3 , and generate x-rays  8  roughly in an electron beam spot region  7  a little below the target surface  6  (note that this somewhat simplifies the physics here, but for understanding the basics of the heel effect, this presentation is sufficient). The x-rays  8  result from a deceleration of the electrons  5 , causing bremsstrahlung having a basically continuous wavelength spectrum, and from a filling up of electron shells in the anode material the electrons of which have been shot out by the electrons  5  of the electron beam  5   a , resulting in characteristic x-rays of the anode material. Note that the angled anode  3  may comprise a covering layer of a particular material in order to influence the emission spectrum (“target material”). In general, the bremsstrahlung is the major part of generated x-rays  8 , and dominates CT imaging. The x-rays  8  are emitted from the electron beam spot region  7  in a spatial angle area roughly centered about the z-direction, i.e., the x-rays propagate roughly in z-direction from the electron beam spot region  7 . 
     Depending on the exact emission direction, the x-rays  8  have to travel through the anode material differently far before reaching the target surface  6  and thus leaving the anode material. In the example shown, the x-ray fraction  8   a  close to the cathode  4  has travelled a path length PLa through the anode material, and the x-ray fraction  8   b  far from the cathode  4  has travelled a path length PLb through the anode material, wherein PLb is about twice as long as PLa here. Generally speaking, in the example shown, x-ray fractions reaching the target surface  6  at larger y position values (such as fraction  8   b ) have travelled farther in the anode material as compared to x-ray fractions reaching the target surface  6  at smaller y position values (such as fraction  8   a ). 
     The x-rays  8  generated in the electron beam spot region  7  interact with the anode material before exiting the anode material. Generally, the anode material will attenuate the passing x-rays of all wavelengths, wherein the longer the path length in the anode material, the stronger is the attenuation (compare Lambert-Beer&#39;s law). Further, in general, softer x-rays are attenuated to a higher degree as compared to harder x-rays in the anode material, what is known as beam hardening; this beam hardening is more pronounced for longer path lengths in the anode material. As a result, the overall x-ray intensity is higher for x-ray fractions reaching the target surface  6  at smaller y position values and decreases with higher y position values (causing the “intensity effect” aspect of the heel effect). Further, the spectrum of the x-rays is harder for x-ray fractions reaching the target surface  6  at larger y position values and becomes softer with smaller y position values (causing the “spectral effect” aspect of the heel effect). In the example shown, the y axis is vertical, what is a typical and preferred orientation. 
     The measurement setup  1  further comprises an object stage  9 , at which for a 3D CT image measurement an object to be measured  10  is held (alternatively, a separate calibration object may be held at the object stage  9  for a 3D CT calibration measurement, not shown here). In the example shown, the object  10  can be rotated about a turning axis  12  by means of the object stage  9 , which is parallel to the y direction here. Further, in the example shown, the object  10  can also be shifted in y direction with the object stage  9 . The x-rays  8  are directed towards the object  10  and, depending on the object geometry and the (local) object material, are attenuated by the object  10 . The x-rays having penetrated the object  10  are measured at a 2D x-ray detector  11  of the measurement setup  1 , i.e., the 2D x-ray detector may record 2D projections (sometimes also referred to as 2D images) of the object  10 . 
     It should be noted here that in an alternative design, instead of turning and/or shifting the object  10  by means of the object holder  9  with respect to a (remaining) stationary measurement setup  1  (i.e. stationary x-ray source  2  and stationary 2D x-ray detector  11 ), it is possible to turn the remaining measurement setup  1  about the object  10  and/or to shift the remaining measurement setup  1  with respect to the object  10  (i.e. stationary object  10 , not shown here). 
     By means of the measurement setup  1 , for obtaining a 3D CT image of the object  10 , a set of 2D projections are measured at different rotation positions of the object  10 , and from this set, a 3D CT image is calculated (reconstructed). 
     Referring to  FIG.  3   , when recording a 2D projection  30  with the measurement setup  1  of  FIG.  1    and  FIG.  2    without any object, then the 2D projection  30  will show a gradient of the grey values recorded with the 2D x-ray detector  11 . In  FIG.  3    (as well as the following figures), higher x-ray intensity at the detector  11  is illustrated with darker color and vice versa. For low y position values of pixels in the 2D projection  30 , higher x-ray intensity will be found due to the “intensity effect” aspect of the heel effect. However, this artefact can be overcome by means of the so called “flat field correction” (abbreviated here also as FFC). For FFC, the intensity values of the pixels of a 2D projection are multiplied with an FFC correction factor corresponding to the inverse of the measured intensity without any object. When, for example, applying FFC to the 2D projection  30 , a uniform grey 2D projection  38  can be obtained. Note that flat field correction also compensates for local differences in detector sensitivity and for varying local distances of the 2D x-ray detector  11  to the electron beam spot region (“beam spot”). It should be noted here that in the course of the inventive method, FFC correction is preferably applied generally on the 2D projection level (i.e., in projection space). 
     Note that equivalent compensation of the “intensity effect” aspect of the heel effect can be achieved by air intensity line per line correction at 3D image reconstruction (not further explained here). 
     When applying FFC on 2D projections recorded with the measurement setup  1  with an object  10  included in the beam path, the “spectrum effect” aspect of the heel effect becomes relevant. When, as shown in the example of  FIG.  4   , a basically cylindrical and uniform object  10  aligned with the y direction is imaged, then in a resulting 2D projection  31 , the object projection  32  shows a gradient of the grey values with respect to the y direction. The reason for this artefact is as follows: X-rays  8  (or respective fractions) having a position larger in the y direction have a harder x-ray spectrum, and for this reason are less absorbed in the object  10 . Accordingly, in the 2D projection, the corresponding areas of the object projection  32  that are high in y position have a higher x-ray intensity. In turn, x-rays  8  (or respective fractions) having a position smaller in the y direction have a softer x-ray spectrum, and for this reason are more absorbed in the object  10 . Accordingly, in the 2D projection, the corresponding areas of the object projection  32  that are small in y position have a lower x-ray intensity. 
     The artefact appears in each 2D projection of a set  33  of 2D projections  31  (recorded at different rotation positions), and also translates into a generated (reconstructed) 3D CT image  34  from the set  33 . The object image  35  in the 3D CT image  34  shows accordingly a gradient in grey values. 
     In order to overcome this artefact, the 3D CT image  34  undergoes the inventive heel effect correction explained below, and a corrected 3D CT image  36  is obtained. In the illustrated example with a uniform object  10 , the object image  37  in the corrected 3D CT image  36  will show uniform grey values. 
     It should be noted that in accordance with the invention, also connected scans can be processed, as illustrated in  FIG.  5   . The object  10  in this example is again chosen as cylindrical and uniform object  10 , and aligned with they direction, but it is much longer than in  FIG.  4   , and cannot be imaged with a single 3D CT image any more with the measurement setup  1 . If, as shown in  FIG.  5   , the object  10  is larger than the field of view  50  of the 2D x-ray detector  11  (which is determined by the sensor area of the 2D x-ray detector  11  and the corresponding limiting x-ray beams  50   a  to the electron beam spot area on the angled anode), then a plurality of 3D CT images (or corresponding sets of 2D projections) of the object  10  may be recorded at different shifting positions of the object  10  with respect to the y direction relative to the measurement setup  1 , wherein the 3D CT images somewhat overlap, compare overlapping object sections  51   a ,  51   b . The overlapping object sections  51   a ,  51   b  are used for aligning the individual 3D CT images when combining them to an overall 3D CT image; in particular relative spatial shifts may be calculated “on the fly”. In  FIG.  5   , three different shifting positions are illustrated (one in full lines, two in dotted lines), which are subsequently visited (wherein here the object  10  is assumed fixed, and the measurement setup  1  is shifted), and a set of 2D projections is recorded at each shifting position. Note that the physical shifts applied in a connected scan between recording the individual 3D CT images (or respective sets of 2D projections) are in general chosen the same. The individual 3D CT images obtained this way may be combined to an overall 3D CT image  52  of the object  10 . 
     If only flat field correction is applied on the 2D projection level and no further measures are taken to compensate for the remaining heel effect (i.e., its spectral effect aspect), then the resulting overall 3D CT image  52  will show the so called “bamboo effect” artefact. The partial object images  35   a ,  35   b ,  35   c  (originating from the individual 3D CT images recorded at the different shifting positions) will each show their own gradient of grey values. Further, at the transitions between the individual 3D CT images that have been combined, a sharp jump in the grey values will be found in the imaged object. This sharp jump in contrast may be mistaken for a structure in the object, which in fact does not exist. 
     In order to overcome this artefact, the individual 3D CT images of the overall 3D CT image  52  undergo the inventive heel effect correction explained below, and a corrected overall 3D CT image  53  can be obtained then from the corrected individual 3D CT images. In the illustrated example, the partial object images  54   a ,  54   b ,  54   c  in the corrected 3D CT image  53  will show uniform grey values then, and the overall object image  55  has in total uniform grey values, without bamboo effect. 
     By means of  FIG.  6   , the general procedure of heel effect correction according to the invention is explained in a first variant, wherein only a single 3D CT image is obtained, and correction is done with a single 3D CT calibration image recorded with a separate calibration object. However, most of the description applies analogously to the other variants of the invention. 
     Similar to the process explained under  FIG.  4   , a set of 2D projections of the object  10  to be measured is recorded (with FFC) with the measurement setup  1 , and the corresponding 3D CT image  34  is reconstructed (compare  FIG.  4    above). In the example illustrated, the object  10  is generally extended in the y-direction with a complex outer shape, but is made of a basically uniform material. In the (uncorrected) 3D CT image  34 , the object image  35  shows a significant gradient of grey values in the y direction, due to the spectrum effect aspect of the heel effect. 
     Further, a set of 2D projections of a calibration object  61  is recorded with the measurement setup  1 , and a corresponding 3D CT calibration image  62  is reconstructed. The calibration object  61  is of cylindrical shape and made of a uniform material here. Generally, the calibration object  61  is chosen preferably such that the x-ray absorption properties resemble the x-ray absorption properties of the object  10  to be measured (in particular for continuous correction). In particular, the calibration object  61  is made from the same material as the object  10  here. Further, the height H of the object  10  and the height COH of the calibration object  61  are identical here. Even further, the largest diameter LD of the object  10  and the largest diameter COLD of the calibration object  61 , in the xz plane, is about the same at most y positions, namely with COLD deviating from LD by at maximum 20% for at least 80% of the y positions. For simplicity, only LD and COLD for the lowermost y position are shown here, where LD and COLD are about the same. The calibration object image  63  in the 3D CT calibration image  62  shows a significant grey value gradient in y direction, too, since the heel effect works basically the same on the object  10  and the calibration object  61 . 
     The 3D CT image  34  is made of a plurality of voxels, with each voxel having attributed a grey value indicating the x-ray absorption of the measured object in this voxel (note that a darker color here indicates a low x-ray absorption; for simplicity, the air area surrounding the object has not been marked with black color in the 3D CT image, so the object image is better visible). 
     The voxels may be attributed to a number of slices  64 , with each slice indicating the voxels having the same position in y direction. In other words, each slice  64  indicates a particular xz voxel plane. The 3D CT image  34  has a number of N slices, typically with N≥500. Note further each slice  64  typically includes a few thousand voxels, such as at least 250×250=62500 voxels. The slices  64  are numbered consecutively, with a slice number index n s , starting at n s =1 at the smallest y position, and ends at n s =N at the largest y position here. Note that analogue classification can be made for 3D CT calibration scans (see below). 
     For each slice  64 , a scaling factor sf, or sf(n s ) respectively, is determined for correcting the heel effect in the respective slice  64  of index number n s . This scaling factor sf(ns) is derived from the 3D CT calibration image  62 . By means of the scaling factor sf of a particular slice  64 , the uncorrected grey values of the voxels in this slice  64  can be corrected into corrected grey values, by multiplying the respective uncorrected grey value with the scaling factor sf for the particular slice  64 . The resulting corrected 3D CT image of the object  10  is shown as corrected 3D CT image  36 , wherein the object image  37  is now uniformly grey. In other words, the correction information from the scaling factor sf is obtained from an information in image space, here the 3D CT calibration image  62  (and not in projection space of 2D projections), and is applied in image space, here to 3D CT image  34  (and not in projection space of 2D projections); by this, the correction is decoupled from individual systems. 
     It should be noted here that instead of applying the scaling factor sf to the complete slice of the 3D CT image  34 , only a 2D CT partial image  65  of the 3D CT image  34  can be corrected, such as a section parallel to the xy plane at a particular z position as shown (or a section parallel to the zy plane at a particular x position, not shown); for correction of the heel effect, then a line of voxels in the section  65  belonging to a respective slice  64  is multiplied with the scaling factor sf for this respective slice  64 , resulting in the corrected 2D CT partial image  66 . 
     In the variant of  FIG.  6   , in the 3D CT calibration image  62 , two different regions, namely a first region R 1  and a second region R 2 , have been chosen for basing the inventive heel effect correction on (compare also  FIG.  11   ,  FIG.  12    in this respect); it should be noted that alternatively, a continuous analysis or an analysis based on more than two base points of the calibration object  61  or its 3D CT calibration image is also possible (compare  FIG.  13   ). The first region R 1  is at the bottom of the field of view  50  of the 2D x-ray detector  11  or the respective 3D CT calibration image  62  here, and the second region R 2  is at the top of the field of view  50  here. In this variant, it is above all important that the reference object  61  has similar object structures  64   a ,  64   b  in the two regions R 1 , R 2 , which behave similar to each other, and preferably also behave similar as compared to the object  10  to be measured, as far as x-ray absorption is concerned. Typically, the similar object structures  64   a ,  64   b  are made of the same material, and differ in size within their respective region at maximum by 20% (based on the volume of the possibly smaller object structures). 
     The heel effect or its spectral effect aspect, respectively, causes a grey value contribution to the grey values of voxels in similar object structures, i.e., having similar x-ray absorption behavior, located at different positions in the field of view  50 . With the cylindrical and uniform calibration object  61 , the similar object structures  64   a  of the object image  63  in first region R 1  should look the same as the similar object structures  64   b  of the object image  63  in the second region R 2 , but the object structures  64   b  show darker grey values than the object structures  64   a  due to the heel effect, which acts different on the calibration object  61  in different y positions. Accordingly, the shift in grey values between structures  64   a  and  64   b  represents a grey value contribution that can be traced back to the different y positions of the similar object structures  64   a ,  64   b  in the slices  64 , said shift being caused by the heel effect, what should be corrected for. 
     To determine said grey value contribution and compensating scaling factors, the grey values of the similar object structures  64   a ,  64   b  in regions R 1  and R 2  of the 3D CT image are analyzed (see further below) for determining the scaling factor sf(n s ). 
     If shifting of the object to be measured is possible with the measurement setup  1 , which is the case for most CT measurement setups, a separate calibration object can be done without. Instead, as illustrated in  FIG.  7   , the object  10  to be measured can be used as calibration object. In addition to the 3D CT image of the object catching the object altogether (upper part of  FIG.  7   , also acting as 3D CT calibration image here), an additional 3D CT calibration image is obtained with a shifted object position (see lower part of  FIG.  7   ), wherein the object  10  has been lowered relative to the remaining measurement setup  1  here. In each case, the top part of the object  10  is used here as the identical object structures  71  to base the inventive correction upon. These identical object structures  71  are positioned in the additional 3D CT calibration image (lower part of  FIG.  7   ) in a first region R 1  in the lower part of the field of view  50  of the 2D x-ray detector  11 , and in the 3D CT image (upper part of  FIG.  7   , used here also as 3D CT calibration image) in a second region R 2  in the upper part of the field of view  50 . 
     To determine the grey value contribution here, the grey levels of the identical object structures  71  in regions R 1  and R 2  of the two 3D CT calibration images (one of which is at the same time the 3D CT image of the object  10 ) are analyzed (see further below) for determining the scaling factor sf(n s ). Said two 3D CT calibration images form a so-called calibration pair CP here, which picture the identical object structures  71  at different regions R 1 , R 2  of the field of view  50  of the measurement setup  1  with respect to the y direction. 
     In case of a connected scan of an object  10  larger than the field of view  50  of the measurement setup  1 , in general multiple calibration pairs can be built as can be seen in  FIG.  8    in the course of a self-calibration. From the object  10 , here three 3D CT images are recorded (see upper, middle and lower part of  FIG.  8   ), which act at the same time as 3D CT calibration images with the object  10  as calibration object; note that different parts of the object  10  are recorded in each of the three 3D CT images, with a pairwise overlap of the parts of the object  10 . 
     The upper and middle 3D CT calibration images picture identical structures  71  in regions R 1  and R 2 , thus forming a first calibration pair CP 1 . Further, the middle and lower 3D CT calibration images picture identical structures  72  in regions R 1  and R 2 , thus forming a second calibration pair CP 2 . 
     From each calibration pair CP 1 , CP 2 , an information on the scaling factor SFI can be derived (such as a single scaling coefficient). By averaging the SFI of the different calibration pairs (and possibly some further calculation), the scaling factor sf(n s ) for each slice with slice index number n s  can be derived then. 
       FIG.  9    once more illustrates the similar object structures  64   a ,  64   b  of  FIG.  6    in regions R 1 , R 2  of the field of view  50  in 3D CT calibration image  62 . For obtaining the grey value contribution to be attributable to the slice position, the grey values in the similar object structures  64   a ,  64   b  of the two regions R 1 , R 2  are compared. For this purpose, one may select a respective section  91   a ,  91   b  (“region of interest”) in the similar object structures  64   a ,  64   b , safely excluding any “air area” or other non-representative material. The sections  91   a ,  91   b  are chosen to be of identical size. 
     Analogously, when a calibration pair of 3D CT calibration images picture identical object structures  71  in different regions R 1 , R 2  of the field of view  50 , with the object  10  shifted with respect to the field of view  50  as illustrated in  FIG.  10    (compare also  FIG.  7   ), then the grey values of the identical structures  71  in the different regions R 1 , R 2  are compared. Again, one may select a section  92  (“region of interest”) in the identical object structures  71  to safely exclude “air area” and other non-representative material. 
       FIG.  11    illustrates a way to determine the scaling factors sf(n s ) for the slices ns in a preferred variant. In a first region and a second region (or respective sections of them), the grey values of the similar or identical object structures are analyzed. For each region, the grey values are plotted in a histogram.  FIG.  11    on the left (part A) shows on the abscissa the grey value (here as bin number of a total of 256 bins), and on the ordinate the number of voxels having said grey value, with curve HR 1  for the voxels in the first region, and with curve HR 2  for the voxels in the second region. It should be noted that the first region and second region each contain similar or identical structures to be compared. The histograms have preferably undergone a boxcar filtering, and they may be treated to remove zero peaks, if any. Also, the first and/or last bin may be reset to zero to remove potential saturated regions. 
     As can be seen from  FIG.  11    part A, the two histograms HR 1 , HR 2  each show one strong maximum, at bin  208  for HR 1 , and at bin  191  for HR 2 . 
     For a simple evaluation, the bin of the respective maximum can be used as characteristic grey value gv 1 , gv 2  for the respective histogram HR 1 , HR 2  or the respective first and second region. Further, for each of the first and second region, a characteristic position number (of the slices consecutively numbered in y direction) i 1 , i 2  shall be determined; typically the position number of a central slice of the respective region is chosen. Then a scaling coefficient sc can be calculated with sc=(gv 1 −gv 2 )/(i 1 −i 2 ). For example, with the characteristic (average) position numbers i 1 =377 and i 2 =1225, a resulting scaling coefficient (or “slope”) sc would be: 
       sc=(208−191)/(377−1255)=−0.019
 
     The resulting scaling factor sf(n s ) would then be: 
       sf( n   s )=1+sc*( n   s   −n   ref ), 
     with n ref  being a reference (central) slice in the field of view, positioned, for example, at n ref =801, with sf=1−0.019*(n s −801). 
     Alternatively, a shift of the histograms RH 1 , RH 2  may be determined such that their deviation is minimized. This shift represents the global intensity change of the overlapping area in the two regions or sections of them. The determination of the shift can be done, e.g., by calculating a convolution of the two histograms HR 1 , HR 2 , and determining the value of the shift parameter where the convolution has its maximum. Preferably, for determining the shift, higher weight is given to high intensity voxels, what reduces the influence of artefacts and seems to improve heel effect correction in particular for strongly absorbing materials. 
     In  FIG.  11   , right hand side (part B) the histogram HR 1  has been shifted somewhat to the right until the best match was reached with HR 2 ; this was at a shift of 17 bins. The scaling factor sc then results with sc=shift/(i 1 −i 2 ), which results again in: 
       sc=17/(377−1225)=−0.019.
 
     It should be noted that in the case that multiple calibration pairs are available, a scaling coefficient should be determined for each calculation pair (“single scaling coefficient” ssc), and the final scaling factor sf should be calculated with an averaged scaling coefficient. 
     An example for such a procedure is illustrated in  FIG.  12   , showing a table belonging to a connected scan comprising three 3D CT images of an object, also used as 3D CT calibration images, with the object used as calibration object. The three 3D CT calibration images are noted as “part1”, “part2” and “part3” in the table of  FIG.  12   . From the three 3D CT calibration images, two calibration pairs CP 1 , CP 2  can be built. Calibration pair CP 1  uses the top part of the vertical field of view (“vFOW”) of part1 and the bottom part of the vFOV of part2 as different regions for grey value comparison (note that this calibration pair was also the basis for the histograms of  FIG.  11   ). Further, calibration pair CP 2  uses the top part of the vertical field of view (“vFOW”) of part2 and the bottom part of the vFOV of part3 as different regions for grey value comparison (compare also  FIG.  15    for attribution of the part1 P 1 , part2 P 2 , part3 P 3  at a mouse femur). The slice index or average/characteristic position number (in y direction), which is here simply the mean position number of the respective region, is noted in the second column. Further, the mean intensity (as characteristic grey value) in the respective region is noted in the third column; note that here an average grey value of the respective region containing the here identical object structures, limited to a section (“region of interest”) of the (first or second) region excluding “air area”, was determined. 
     From CP 1 , a “single” scaling coefficient: 
       ssc(CP1)=(gv 1 −gv 2 )/( i   1   −i   2 )=(188−171)/(337−1225)=−0.019
 
     results. From CP 2 , a “single” scaling coefficient: 
       ssc(CP2)=(181−166)/(337−1225)=−0.017
 
     results. In the example given, there is a total number of K=2 calibration pairs to average, so the scaling factor can be determined from the single scaling coefficients here with: 
       sc=(½)*[(−0.019)+(−0.017)]=−0.018.
 
     With this scaling coefficient sc, again the scaling factor sf(n s ) and n ref =801 can be calculated with: 
       sf( n   s )=1+sc*( n   s   −n   ref )=1−0.018*( n   s −801).
 
     In the examples of  FIG.  11    and  FIG.  12   , the scaling factor sf has been determined based on the assumption of a linear grey value contribution caused by the heel effect. However, it is also possible to determine a grey value contribution continuously, without such an assumption. The latter is particularly simple when having a 3D CT calibration scan of a cylindrical uniform calibration object available. As illustrated in  FIG.  13   , part A, in an example of such a variant, for each slice (with slice index j s , consecutively numbered in y direction) of the 3D CT calibration image an average mean grey value mgv (in bins) within that slice is determined and entered into the diagram. In the example shown, a total of 2500 slices are available in the 3D CT calibration image, with a central reference slice  130  at index j s =1251 having the reference slice index j ref .  FIG.  13   , part A shows the resulting curve  131 ; note that this curve  131  consists of a sequence of single points very close to each other. 
     Now a weighting factor wf is determined such that the product of the weighting factor wf and the mean grey value mgv of the reference slice  130  becomes exactly 1. In the illustrated example, since mgv at j s =1251 is 100 here, accordingly wf is 1/100=0.01 here.  FIG.  13   , part B illustrates the resulting weighted curve  132 , which corresponds to curve  131  of part A now multiplied with the weighting factor wf. This influences, in fact, only the label of the ordinate in the diagram of part B, as compared to the diagram of part A. 
     The scaling factor sf(j s ) for a particular slice j s  now can be determined by calculating the “inverse” of the curve  132  of part B. This inverse curve  133  is illustrated in the diagram of  FIG.  13   , part C, with: 
       sf( js )=1/[wf*mgv( js )] 
       and here: 
       sf( js )=1/[0.01*mgv( js )]. 
     It should be noted that curve  133  consists of a sequence of single points very close to each other. 
     In order to reduce the calculation efforts a little, instead of determining mean grey values mgv for each slice j s  as in part A of  FIG.  13   , it is alternatively possible to calculate mean grey values mgv only for a small number of slices, such as for six slices as illustrated in  FIG.  13   , part D. The mean grey values at these slices act as base points for a section-wise linear curve  134 , obtained by linear interpolation between the base points. With this curve  134 , the scaling factors sf(j s ) can be determined as shown in part B and part C of  FIG.  13    above. 
     It should be noted that the variants of  FIG.  13    are most often applied for preparation of single scans of objects small enough to fit into the field of view of one 3D CT image; however scaling factors sf obtained as described in  FIG.  13    can also be used to correct connected scans or its 3D CT images to be combined, respectively. 
       FIG.  14    shows as a first experimental example for the inventive correction of the heel effect. A cylindrical NbTi alloy rod, aligned with the y direction, was used as both object and calibration object. On the left (part A), a 2D CT partial image of the rod (taken as section in a yz plane at the x-position of the center of the rod) is shown before correcting for the heel effect. A gradient of the grey values along the y direction is visible, wherein the rod is brighter near the top and darker near the bottom, despite the rod being of uniform composition. Then a heel effect correction was done as explained in  FIG.  13    parts A/B/C. The resulting corrected 2D CT partial image, shown on the right of  FIG.  14    as part B, does not show any significant gradient of grey values along the y direction anymore. 
       FIG.  15    shows a second practical example for the inventive correction of the heel effect. Here, a mouse femur was used as object and calibration object in a connected scan of three 3D CT images/3D CT calibration images, here named P 1 , P 2  and P 3 , applying self-calibration in accordance with the invention. On the left (part A), an (overall) 2D CT partial image of the mouse femur (taken as section in a yz plane at the x-position roughly at the center of the femur) is shown before correcting for the heel effect. At the transitions of images P 1  to P 2  and P 2  to P 3 , a jump of the grey values is visible in the femur (“bamboo artefact”), compare arrows  150 ,  151 . Further, e.g., well visible in image P 2 , within the femur, a gradient of the grey values in y direction is visible, with the object image being brighter towards the top and being darker towards the bottom of P 2 . The individual 3D CT images then underwent a heel effect correction according to the invention and as illustrated in  FIG.  11   , wherein images P 1 /P 2  formed a first calibration pair, and images P 2 /P 3  formed a second calibration pair, and with shifts being determined for each calibration pair by shifting the histograms until their deviation was minimized. The shifts were used to determine single scaling coefficients for each calibration pair, and the single scaling coefficients were averaged to obtain an (overall) scaling coefficient and the scaling factors. In the corrected (overall) 2D CT image on the right (part B) of  FIG.  15   , the transitions between the images P 1 /P 2  and P 2 /P 3  are no more visible. Further, there is no more discernible gradient in the grey values within the femur part of the (corrected) images P 1 , P 2 , P 3  visible. 
       FIG.  16   a    and  FIG.  16   b    illustrate a third practical example for the inventive correction of the heel effect. A uniform aluminium rod, aligned along y direction, was used as object in a connected scan, and a self-calibration was applied as described for  FIG.  15   , but here five 3D CT images (and therefore four useable calibration pairs) that were combined into an overall 3D CT image (not shown).  FIG.  16   a    shows a diagram plotting on the abscissa the position in y direction (i.e., a slice index), and on the ordinate the mean grey value for the respective slice, together with its standard deviation interval, for the uncorrected overall 3D CT image. At the transitions of the combined 3D CT images, jumps (or steps) in the grey values are clearly visible, and between the jumps, significant gradients of the grey values are visible.  FIG.  16   b    shows the corresponding diagram for the overall 3D CT image, wherein the underlying 3D CT images underwent said self-calibration in accordance with the invention. There are no more jumps (or steps) of the grey values visible, and the gradients of the grey values between the jumps have disappeared. 
     In summary, the invention relates to a method for obtaining a CT image ( 36 ,  53 ) of an object ( 10 ), with the following steps: 
     a) generating x-rays ( 8 ) using an x-ray source ( 2 ) comprising an angled anode ( 3 ),
 
b) recording at least one set ( 33 ) of 2D projections ( 31 ) of the object ( 10 ) or a part of the object ( 10 ), and
 
c) generating at least one 3D CT image ( 34 ) of the object ( 10 ), characterized in that the method further comprises a step of
 
d) for each generated 3D CT image ( 34 ), correcting the 3D CT image ( 34 ), wherein scaling factors (sf) for slices ( 64 ) of voxels are determined with at least one 3D CT calibration image ( 62 ) measured with the measurement setup ( 1 ), wherein the at least one 3D CT calibration image ( 62 ) pictures similar or identical object structures ( 64   a ,  64   b ;  71 ,  72 ) of a calibration object ( 61 ) placed within the beam path of the x-rays ( 8 ) in different regions (R 1 , R 2 ) of a field of view ( 50 ) of the measurement setup ( 1 ) with respect to a y-direction, wherein for the at least one 3D CT calibration image ( 62 ) a grey value contribution to the grey values of voxels belonging to the similar or identical object structures ( 64   a ,  64   b ;  71 ,  72 ) in said different regions (R 1 , R 2 ) attributable to the slice position (n s , j s ) in y direction is determined at least approximately, and the scaling factor (sf) for a respective slice ( 64 ) of voxels is chosen such that it compensates for the determined grey value contribution for that slice ( 64 ). The method reduces heel effect artefacts in a simple way.