Patent Publication Number: US-9430832-B2

Title: Differential phase contrast imaging with energy sensitive detection

Description:
CROSS-REFERENCE TO PRIOR APPLICATIONS 
     This application is the U.S. National Phase application under 35 U.S.C. §371 of International Application Serial No. PCT/IB2012/054032, filed on Aug. 8, 2012, which claims the benefit of U.S. Application Ser. No. 61/529,450, filed on Aug. 31, 2011. These applications are hereby incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to differential phase contrast imaging. In particular, the invention relates to a method for correcting differential phase image data, a method for generating corrected differential phase image data, a computer program, a computer readable medium and differential phase imaging system. 
     BACKGROUND OF THE INVENTION 
     X-ray radiography and tomography are important methods for a variety of applications, for example non-destructive investigation of bulk samples, quality inspection of industrial products and non-invasive examination of anatomical structures and tissue regions of interest in the interior of a patient&#39;s body. 
     X-ray imaging based on attenuation of X-rays may yield excellent results where highly absorbing anatomical structures such as bones are embedded in a tissue of relatively weakly absorbing material. This is due to the fact that the penetration depth of hard X-ray beams may be rather high, which allows for recording sharp projections of the attenuation coefficient. 
     When different kinds of tissue with similar absorption cross-sections are under examination (for example in mammography or angiography), the contrast of X-ray absorption may be relatively poor. In this case, phase contrast X-ray radiography and tomography may be employed, where the change of phase of the X-rays penetrating the object of interest is examined. One method to obtain phase contrast information is the so-call differential phase contrast imaging as described in the following. 
     In differential phase contrast X-ray radiography and tomography, phase wrapping may occur. If the gradient of the phase front is outside the range of [−π; π] per grating period, the gradient is wrapped into this interval. This situation may appear in particular at the rim of an object, for example because of a big jump of the refractive index between air and tissue. The phase wrapping may happen for reasonable grating pitches of 2 μm for the phase grid already for object sizes below 1 mm. In particular in differential phase contrast X-ray tomography, this may lead to a strong capping artifact in the reconstructed image. 
     SUMMARY OF THE INVENTION 
     It is an object of the invention to provide images recorded with differential phase contrast X-ray imaging, in particular tomography, that illustrate and represent the imaged object of interest clearly and exactly. 
     This object is achieved by the subject-matter of the independent claims. Further exemplary embodiments are evident from the dependent claims and the following description. 
     A first aspect of the invention relates to a method for correcting differential phase image data. For example, the method may be applied in X-ray phase contrast imaging, in particular mammography. 
     According to an embodiment of the invention, the method comprising the steps of: Receiving differential phase image data acquired with radiation at different energy levels, wherein the differential phase image data comprises pixels, each pixel having a phase gradient value for each energy level; Determining an energy dependent behavior of phase gradient values of a pixel; Determining a corrected phase gradient value for the pixel from the phase gradient values of the pixel and a model for the energy dependence of the phase gradient values. With the method, phase unwrapping in differential phase contrast CT by using energy sensitive detection is possible. The method may be performed pixel per pixel, i.e. locally. 
     For example, with the model for the energy dependence of the phase gradient value, a wrapping number at the pixel may be determined from the energy dependent behavior. The wrapping number, i.e. the number of complete shifting by 2π of the phase of the radiation at the position of the pixel, may be determined by using the differential phase image data acquired at different energy levels, i.e. at different wave lengths of the radiation. Every pixel of the differential phase image data is associated with phase gradient values, for example at least three values, from which the energy dependent behavior of the phase gradient at the pixel may be determined. Since the energy dependent behavior is characteristic for the wrapping number, the wrapping number at the pixel may be determined. With the wrapping number, a corrected phase gradient value at a selected reference energy may be determined and corrected differential phase image data may be generated from the corrected pixels. 
     In such a way, artifacts in the image data that are based on phase wrapping during the detection process may be reduced or eliminated from the image data. 
     A further aspect of the invention is a method for generating corrected differential phase image data. 
     According to an embodiment of the invention, the method comprises integrating the differential data to obtain either a plain phase contrast image (radiograph) of the object or the facilitate image fusion with the attenuation contrast image. 
     According to an embodiment of the invention, the method comprises the steps of: Generating radiation at different energy levels; detecting the generated radiation penetrating an object of interest; Acquiring or recording differential phase image data from the detected radiation; and executing the steps of the method for correcting differential phase image data as described in the above and in the following. 
     A further aspect of the invention relates to a differential phase imaging system, for example a CT system. 
     According to an embodiment of the invention, the system comprises a radiation source, a detector and a controller, wherein the radiation source is adapted to generate radiation of different energy levels, wherein the detector is adapted to detect differential phase image data of an object of interest penetrated by the radiation, wherein the controller is adapted to carry out the method as described in the above and in the following. 
     Further aspects of the invention are a computer program for correcting differential phase image data or for generating corrected differential phase image data and a computer-readable medium on which such a computer program is stored. 
     It has to be understood that features of the method as described in the above and in the following may be features of the system, the computer program and the computer readable medium as described in the above and in the following. 
     These and other aspects of the invention will be apparent from and elucidated with reference to the embodiments described hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Below, embodiments of the present invention are described in more detail with reference to the attached drawings. 
         FIG. 1  schematically shows a differential phase imaging system according to an embodiment of the invention. 
         FIG. 2  shows a diagram with absolute phase gradient values according to an embodiment of the invention. 
         FIG. 3  shows a diagram with wrapped phase gradient values according to an embodiment of the invention. 
         FIG. 4  schematically shows image data according to an embodiment of the invention. 
         FIG. 5  shows a flow diagram for a method for generating and correction differential phase image data according to an embodiment of the invention. 
     
    
    
     In principle, identical parts are provided with the same reference symbols in the figures. 
     DETAILED DESCRIPTION OF EMBODIMENTS 
       FIG. 1  schematically shows a differential phase imaging system  10  with a radiation source  10 , a detector  12  and a controller  14 . 
     The radiation source  10  may comprise an incoherent X-ray source  16 , for example an X-ray tube  18 , and a source grating  20  for achieving spatial beam coherence. The radiation source  10  may be adapted to generate a spatial coherent beam of radiation. 
     The detector  12  may comprise a phase grating  22 , an absorber grating  24  and an X-ray detector element  26  adapted for detecting image data from X-rays radiated from the radiation source through an object of interest  28 . 
     The source grating  20 , the phase grating  22  and the absorber grating  24  have a plurality of equidistant X-ray absorbing (source and absorber grating) or phase shifting (phase grating) strips which extend in parallel in a direction normal to an optical axis A of the imaging system  10 . 
     The phase grating  22  serves as a phase-shifting beam splitter, which transforms the variation of the phase front of the x-ray beam into an intensity modulation, i.e., an interference pattern with a typical length scale of half the pitch of the phase grating  22  at the location of the absorber grating  24 . The absorber grating  24  generates a Moire interference pattern from the X-rays leaving the phase grating  22 . The Moire interference pattern on the detector element  26  contains information about the phase shift of the deflected and phase-shifted X-rays after passing both the object  28  and the phase grating  22 . 
     The controller  16  comprises a processor  30  for recording or acquiring the image data from the detector element  26  and for commanding and/or controlling a motor  32  for moving the absorber grating  24  in a direction orthogonal to the extension of the strips of the gratings  22 ,  24 . Due to the movement of the absorber grating  26  different Moire patterns are generated on the detector element  26 . These different Moire patterns may be recorded by the processor  30  and transformed into differential phase image data, which may be stored into memory  34  of the controller  14 . 
     The controller  16  and in particular the processor  30  may be further adapted to control the incoherent X-ray source  16  in such a way that different energy levels of X-rays are generated. For example, the tube voltage of an X-ray tube  16  may be adjusted in such a way that different energy levels of X-rays are generated. 
     Summarized, the differential phase imaging system  10  may comprise a radiation source  12 , a detector  14  and a controller  16 . The radiation source  10  may be adapted to generate radiation of different energy levels, the detector  14  may be adapted to detect data of an object of interest  28  penetrated by the radiation. The data may be transformed to differential phase image data by the controller  16 . 
     Furthermore, the differential phase imaging system  10  is adapted for acquiring image data at different energy levels of the X-ray radiation. In other words, the acquisition of the image data may be energy sensitive. 
     On the one hand this may be achieved by controlling the radiation source  12  to generate X-rays at different energy levels at different time points, for example by altering the tube voltage with the controller  16  and with a detector that is sensitive to all generated energy levels. In this case several measurements (at different time points) with different settings for the tube voltage and/or beam filtration may be made for generating the image data. 
     On the other hand an energy sensitive acquisition may be achieved with an radiation source  12  that generates simultaneously different energy levels of radiation and a detector  14  that is adapted for differentiating between different energy levels, for example a spectroscopic detector. 
     It may be possible that the system  10  is an X-ray differential phase contrast radiography system  10  and/or an X-ray differential phase contrast tomography system  10 . In the later case, the system  10  may calculate slices or three dimensional representations from the object  18  from image data that has been acquired from different directions by moving the arrangement of radiation source  12  and detector  14  and the object  28  relative to each other. 
     The generated radiography or tomography images may be displayed on a display  36  of the system  10 . 
     Due to the spatial variation of the refractive index δ(x,y,z) of the object  28 , two different beams of x-rays through the object  28  may undergo different phase changes in its phase, which may be detected with the detector  12  and differential phase image data may be calculated from the detected raw data with the processor  30  and stored in the memory  34 . 
     However, with the detector  14  and the following transformation process of the raw data, the phase gradient cannot be determined unambiguously. In other words, when the real phase gradient is outside the interval [−π, π] the determined phase gradient is wrapped to this interval, i.e. only the real phase gradient modulo 2π may be determined. 
     The ambiguity of the determined phase gradient may be resolved by the use of energy sensitive acquisition in combination with a spectral model of the measurement process, which will be explained in the following. 
     The dependence of the real part of the refractive index δ can be described using a potential law 
               δ   ⁡     (   E   )       =         (       E   0     E     )     2     ⁢       δ   ⁡     (     E   0     )       .             
Since the phase of the wave-front φ of a beam of radiation along the z-direction is approximately
 
                 φ   ⁡     (     x   ,   y   ,   E     )       =       -       2   ⁢   π   ⁢           ⁢   E     hc       ⁢       ∫     -   ∞     0     ⁢       δ   ⁡     (     x   ,   y   ,     z   ′     ,   E     )       ⁢     ⅆ     z   ′               ,         
the absolute phase gradient g of the wave front has the following dependence on the energy, i.e. the wavelength
 
     
       
         
           
             
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     This may be viewed as a model of the energy behavior of the phase gradient. 
       FIG. 2  shows a diagram with true phase gradient values for different energies and shows the energy dependence of the phase gradient. The x-axis of the diagram depicts the energy in keV, the y-axis the absolute phase gradient in rad.  FIG. 2  illustrates the dependency of the phase gradient for different values. Five curves  40   a ,  40   b ,  40   c ,  40   d  and  40   d  are illustrated which are selected such that the gradient g(E 0 ) at the design energy E 0 =20 keV is
 
 g ( E   0 )=(1 +n π)/(grid spacing of the phase grid 22)
 
with n=0, 2, 4, 6, 8. The curve  40   a  belongs to n=0, the curve  40   b  to n=2 and so on. Due to the choice of the above formula for the curves  40   a ,  40   b ,  40   c ,  40   d ,  40   e , the phase gradient values g(E 0 ) at E 0 =20 keV differ by 2π.
 
     However, the detected (measured) and determined differential phase image data does not comprise the absolute (true) phase gradient values, but the values that are wrapped to the interval [−π, π]. 
     This is depicted with respect to  FIG. 3 , which shows a diagram with wrapped phase gradient values. The x-axis of the diagram depicts the energy in keV, the y-axis the wrapped phase gradient in rad. The wrapped phase gradient values related to the same value for n are connected by lines for illustration purpose. 
     In  FIG. 3 , the true phase gradients  40   a ,  40   b ,  40   c ,  40   d ,  40   e  from  FIG. 2  are wrapped into the interval [−π; π] per grating period into relative phase gradients  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g . The curves  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g  belong to the numbers n=0, 2, 4, 6, 8, 10, 12, respectively. 
     In the diagram, the data points or curve  44   a  relate to the case where no wrapping occurred. The curve  44   a  shows the expected E 0 /E scaling around the phase gradient of 1 rad/grid period. The data points or curve  44   b  relate to a true phase gradient of (1+2π)/grid period at 20 keV, which is wrapped to 1 rad/grid period. Since the difference of the phase gradient between 20 keV and, e.g. 22 keV is based on the true gradient, the wrapped phase gradient changes quicker with energy, which makes an unwrapping possible. 
     Since the phase gradients g(E 0 ) were selected to differ by 2π, the wrapping causes that at E 0 , all wrapped (i.e. measured) phase gradients  42  are the same. Even though the wrapped curves  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g  (that represent measured values) suffer from phase wrapping at the other energies as well for n&gt;0, it is clearly visible that different numbers n of wrapping result in quite different distributions of wrapped phase gradients  46  within the energy range shown. 
     In particular, all the curves  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g  have different wrapped phase gradients  46  at different energy levels (for example 16 keV, 18 keV, 22 keV, 24 keV) for several gradient curves  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g  which are mapped all to the same value  42  at the design or reference energy (for example 20 keV). 
     Even three data points  42 ,  46  (and therefore three measurements at three different energy levels) may be sufficient to distinguish all cases (i.e. curves  44   a ,  44   b ,  44   c ,  44   d ,  44   e ,  44   f ,  44   g ) shown in  FIG. 3 , for instance the data points of the energy levels E −1 , E 0  and E 1  corresponding to 18, 20, and 22 keV. 
       FIG. 4  schematically shows image data  50  that may be received, processed, and stored in the memory  34  of the controller  16  and in particular by the processor  30 . 
     The image data  50  may comprise differential phase image data  52 , comprising differential phase image data  54   a ,  54   b ,  54   c  acquired at different energy levels E −1 , E 0 , E 1 , respectively and/or may comprise attenuation image data  58 . 
     The image data  50  is composed of pixels  60  and each pixel may have a phase gradient value  62   a ,  62   b ,  62   c  for each energy level E −1 , E 0  and E 1  and/or an attenuation value  64 . 
     According to an embodiment of the invention, a pixel  60  has phase gradient values  62   a ,  62   b ,  62   c  corresponding to at least three energy levels E −1 , E 0  and E 1 . 
     According to an embodiment of the invention, a pixel  60  has an attenuation value  64 . 
     The image data  50  may comprise corrected differential phase image data  66 , that may be generated from the image data  52 ,  58  by the method as described in the above and in the following. In particular, every pixel  60  may comprise a corrected phase gradient value  68  that may be determined from the values  62   a ,  62   b ,  62   c ,  64  by the method. 
     It has to be noted that the image data  50  need not represent two dimensional images. Other representations, for example, line data may be possible. In other words, the detector element  26  may be a two dimensional detector or a line detector. 
       FIG. 5  shows a flow diagram for a method for generating and correction differential phase image data  52 . 
     In step S 10 , the radiation source  12  is controlled by the controller  16  to generate a radiation at a first energy level E −1 , for example X-ray radiation of 18 keV. The radiation penetrates the object  28  and falls onto the detector  26 . 
     According to an embodiment of the invention, the radiation is electromagnetic radiation, for example, x-ray radiation. 
     In step S 12 , the controller  16  controls the detector  14 , and in particular the grating  24 , such that different Moire interference patterns are detected by the detector element  26 . 
     According to an embodiment of the invention, the method comprises the step of detecting the generated radiation penetrating an object of interest  28 . 
     In step S 14 , the Moire interference patterns are transformed into differential phase image data  52   a  and attenuation image data  58  by the processor  30 . These image data  52   a ,  58  may be stored in the memory  34 . 
     According to an embodiment of the invention, the method comprises the step of acquiring differential phase image data  52  from the detected radiation. 
     The steps S 10  to S 14  are repeated for every further energy level E 0 , E −1 . The attenuation image data  58  may be acquired for only one energy level, for example the basic energy level E 0 . 
     According to an embodiment of the invention, the method comprises the step of generating radiation at different energy levels E −1 , E 0 , E 1 . 
     According to an embodiment of the invention, the energy levels E −1 , E 0 , E 1  of the radiation comprise a reference energy level E 0  and two neighboring energy levels E −1 , E 1  differing about 8% to 12%, e.g. 10%, from the reference energy level E 0 , for example 18, 20 and 22 keV. 
     It may be possible that the radiation at one energy level is generated at different time points and the radiation is detected at these different time points, as explained above. However, it may also be possible, that radiation with different energy levels is simultaneously generated and detected, for example with a multi-chromatic radiation source  12  and a spectroscopic detector  14 . 
     In step S 16 , the image data  50  is received by the processor  30  from the memory  34  (for example pixel wise). 
     According to an embodiment of the invention, the method comprises the step of receiving differential phase image data  52  acquired with radiation at different energy levels E −1 , E 0 , E 1 , wherein the differential phase image data  52  comprises pixels  60 , each pixel  60  having a phase gradient value  62   a ,  62   b ,  62   c  for each energy level E −1 , E 0 , E 1  and optionally an attenuation value  64 . 
     The following steps S 18  to S 22  may then be repeated for every pixel  60  of the image data  50 . 
     In step S 18 , the energy dependent behavior of the phase gradient values  62   a ,  62   b ,  62   c  of the selected pixel  60  is determined with the aid of the wrapping curves  44   a  to  44   g  shown in  FIG. 3 . 
     According to an embodiment of the invention, the method comprises the step of determining an energy dependent behavior  44   a  to  44   g  of phase gradient values  62   a ,  62   b ,  62   c  of a pixel  60 . 
     The wrapping curves  44   a  to  44   g  may be represented in the memory  30  with data points  41 ,  46 . For example, for every curve  44   a  to  44   g  at least three precalculated data values  42 ,  46  may be stored. 
     According to an embodiment of the invention, a wrapping curve  44   a  to  44   g  is represented by phase gradient values  42 ,  46  associated with energy levels E −1 , E 0 , E 1 . 
     According to an embodiment of the invention, the phase gradient values  42 ,  46  of a wrapping curve  44   a  to  44   g  are precalculated and/or stored in the controller  16 , for example the memory  34 . 
     According to an embodiment of the invention, the phase gradient values  42 ,  46  of the wrapping curves  44   a  to  44   g  are determined such that phase gradient values  42  at a reference energy level E 0  are equal. The phase gradient values  62   a ,  62   b ,  62   c  may then, by the processor  30 , be mapped to every wrapping curve stored in the controller  16 . For example, for each wrapping curve  44   a  to  44   g , a best fitting value is calculated, that indicates how much the phase gradient values  62   a ,  62   b ,  62   c  of the pixel correspond to the wrapping curve  44   a  to  44   g . In particular, a phase gradient value  62   a ,  62   b ,  62   c  of the pixel  60  may be mapped to the corresponding phase gradient values  42 ,  46  of a wrapping curve. After that a numerical fit may be calculated from the mapped values  62   a ,  62   b ,  62   c ,  42 ,  46 . 
     According to an embodiment of the invention, the energy dependent behavior is determined by fitting the phase gradient values  62   a ,  62   b ,  62   c  of the pixel  60  to a plurality of phase wrapping curves  44   a  to  44   g.    
     According to an embodiment of the invention, a phase gradient value  42 ,  46  of an energy level E −1 , E 0 , E 1  of a wrapping curve  44   a  to  44   g  is fitted with a phase gradient value  62   a ,  62   b ,  62   c  of the pixel  60  at this energy level. 
     According to an embodiment of the invention, the fitting is a numerical fitting, for example a fitting with a root mean square measure. 
     In step S 18 , the processor  30  may determine a wrapping number n for the pixel  60  with the aid of the best fit of the phase gradient values  62   a ,  62   b ,  62   c  to the wrapping curves  44   a  to  44   g . Since each wrapping curve may be associated with a wrapping number, the wrapping number n for the pixel may be chosen as the wrapping number of the wrapping curve with the best fit. 
     According to an embodiment of the invention, the method comprises the step of determining a wrapping number n of the pixel  60  from the energy dependent behavior. 
     According to an embodiment of the invention, the wrapping number n of the pixel  60  is determined by determining a wrapping curve  44   a  to  44   g  with a best fit of the fitted phase gradient values  62   a ,  62   b ,  62   c  of the pixel  60  and by selecting the wrapping number associated with the determined wrapping curve  44   a  to  44   g.    
     Alternatively or additionally, the phase gradient value may be determined directly from the model of the energy behavior of the phase gradient. 
     According to an embodiment of the invention, the phase gradient value  42  at E 0  is denoted as g 0  and is calculated by the least squares fit 
     
       
         
           
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     where σ 2  are the variances of the measured wrapped phase gradient values g i  at energies E i , w denotes the wrapping operation, and ∥∥ π  denotes a special distance operation namely 
     
       
         
           
             
               
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     Summarized, for every meaningful wrapping number n (for example 0, 2, 4, . . . , 12) a numerical fit may be made to get the best fitting g(E 0 ) assuming that the phase wrapped n/2 times at E 0 . Finally, among these results the one with the best overall fit (using for instance a root mean square measure) may be picked or chosen. 
     In step S 20 , the processor  30  determines the absolute or corrected phase gradient value  68  from the determined wrapping number n. For example to the phase gradient value  62   b  at the reference energy level E 0 , nπ may be added for calculating the value  68 . 
     According to an embodiment of the invention, the method comprises the step of determining a corrected phase gradient value  68  for the pixel  60  from the phase gradient values  62   a ,  62   b ,  62   c  of the pixel  60  and the wrapping number of the pixel  60 . 
     According to an embodiment of the invention, the corrected phase gradient value  68  is determined by shifting the phase gradient value  62   b  of the pixel  60  associated with a reference energy level E 0  with the determined wrapping number of the pixel  60 . 
     When not preselecting the plurality of wrapping curves  44   a  to  44   g , the method may only allows distinguishing between reasonably small values for possible wrapping values n. If n become excessively large, however, the results may become ambiguous again. In this situation, the attenuation image  58  may provide a rough estimate of the wrapping number n. The attenuation image  58  may then be used to resolve the remaining uncertainty. 
     Optionally, in step S 18 , the gradient value at the pixel  60  of the attenuation image data  58  may be determined with the processor  30 . From the gradient value, meaningful wrapping numbers (for example n=20 to 30) may be determined. For example, a function or table is stored in the memory  34 , with which a gradient value may be mapped to an estimated region of wrapping numbers. 
     The plurality of wrapping curves is then delimited to wrapping curves associated with wrapping numbers in that region. 
     According to an embodiment of the invention, the phase gradient is obtained by a least squares fit 
     
       
         
           
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     for a large value of m to approximate the discrete penalty with a smooth function. In the aforementioned case where the wrapping number is estimated from the attenuation image to be e.g. in the order of n 0 , which is equivalent for g to be in the order of nit, the penalty can be selected to be 
     
       
         
           
             
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     According to an embodiment of the invention, the method comprises the step of determining the gradient of the attenuation value  64  at the pixel  60 . 
     According to an embodiment of the invention, the method comprises the step of choosing an estimated region of wrapping numbers. 
     In step S 20  then, optionally only the chosen wrapping curves are used for the fitting of the phase gradient values  62   a ,  62   b ,  62   c  of the pixel  60 . 
     According to an embodiment of the invention, the method comprises the step of fitting the phase gradient values  62   a ,  62   b ,  62   c  of a pixel  60  to a plurality of wrapping curves  44   a  to  44   g  associated with wrapping numbers in the estimated region. 
     If the system  10  is a tomography system  10 , the system may generate image data showing slices or three dimensional views of the object  28 . The image data may be displayed on display  36  of the controller  16 . 
     In this case, the differential phase image data  52  may be acquired in different directions with respect to the object  18 . 
     Thus, in step S 24  the controller  16  may change the optical axis A with respect to the object, for example by rotating the arrangement of radiation source  12  and detector  14  around the object  18  and may repeat the acquisition of image data  50  with respect to the changed direction. 
     After enough image data  50  has been acquired and corrected, the controller  16  and in particular the processor  30  may generate tomography image data in step S 26 . 
     According to an embodiment of the invention, the method comprises the step of generating tomography image data from the corrected differential phase image data  66 . 
     It has to be understood that the steps of  FIG. 5  need not be performed in the order as described with respect to  FIG. 5 . 
     Note further that a distinction between different values  62   a ,  62   b ,  62   c ,  42 ,  46  may require that the signal-to-noise ratio must be roughly large enough to distinguish correctly between different data points. More specifically, for measurement at the design energy E 0 , and a further measurement at E 0 +ΔE, the phase gradients differ by 
     
       
         
           
             
               
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     Since the gradients to be distinguished by the proposed method may differ by 2π/per grating period, the signal-to-noise ratio may have to allow for a distinction of (wrapped) phase gradient differences of 
     
       
         
           
             
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     Depending on the object  28 , the energy spacing between the energy levels E −1 , E 0 , E 1  used in the method can be tuned to achieve this. 
     The method as described in the above and the following may be a computer program executed in the processor  30  and stored in the memory  34 . The computer program may be stored in a computer-readable medium like a floppy disk, a hard disk, an USB (Universal Serial Bus) storage device, a RAM (Random Access Memory), a ROM (Read Only memory) and an EPROM (Erasable Programmable Read Only Memory). A computer readable medium may also be a data communication network, e.g. the Internet, which allows downloading a program code. 
     While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the invention is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art and practising the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single processor or controller or other unit may fulfil the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope.