Patent Publication Number: US-2022218296-A1

Title: Application for x-ray dark-field and/or x-ray phase contrast imaging using stepping and moiré imaging

Description:
FIELD OF THE INVENTION 
     The present disclosure relates to an apparatus for X-ray dark-field and/or X-ray phase-contract imaging using phase stepping and Moiré fringe analysis. The present invention also relates to a method of generating X-ray dark-field and/or X-ray phase-contrast images using this apparatus. 
     BACKGROUND OF THE INVENTION 
     In conventional techniques of X-ray imaging, an object is exposed to X-rays to obtain an attenuation image of the object. However, many object features do not have sufficient attenuation contrast so that they can be efficiently studied using attenuation-contrast imaging. In order to overcome this limitation, there have been major developments in recent years to obtain dark-field images and phase-contrast images using X-rays. 
     Some of these techniques can be classified as grating interferometric methods or grating non-interferometric methods. In each of these techniques, a fringe pattern of X-rays is generated after the X-rays have traversed the object. The changes in the fringe pattern induced by the object can be used to extract X-ray dark-field and X-ray phase-contrast images. 
     In order to determine the changes in the fringe pattern which are induced by the object, phase-stepping procedures or Moiré fringe analysis have been applied. However, it has been shown that phase-stepping techniques typically require acquisition of ten or even more images in order to sufficiently suppress vibrational disturbances. This, however, leads to artefacts caused by patient motion. A long acquisition time also makes it difficult to apply this technique to X-ray computed tomography. Although, in principle, Moiré fringe analysis offers the opportunity to obtain phase-contrast and dark-field information based on a single image, this technique results in a loss of spatial resolution compared to the phase stepping procedure. 
     Therefore, there is a need for an improved method and apparatus for generating X-ray phase-contrast and X-ray dark-field images. 
     SUMMARY OF THE INVENTION 
     According to an aspect of the present disclosure, an apparatus is provided, which has an X-ray imaging system which comprises a first and a second X-ray optical grating and an X-ray sensitive image detector having an ordered array of X-ray sensitive pixels. The imaging system is configured to be usable with an X-ray beam which traverses the first grating, then the second grating before being incident on the X-ray sensitive detector for imaging an object, which is positioned in the beam path of the X-ray beam upstream of the first grating or between the first and the second grating. The imaging system is configured so that when no object is present, the first grating generates a fringe pattern in an entrance plane of the second grating. The second grating is configured to generate a Moiré pattern from the fringe pattern so that the Moiré pattern has a pitch which is less than 20 times or less than 10 times a pixel pitch of the pixels of the X-ray sensitive image detector. The imaging system further comprises a controller, which is configured to control a relative movement between the second grating and the fringe pattern to acquire a Moire-image at each of a plurality of different relative image acquisition positions. The apparatus may be configured for X-ray dark-field and/or X-ray phase-contrast imaging. The apparatus may be configured to generate data, from which X-ray dark-field and/or X-ray phase-contrast images are derivable (i.e. without using further data). Alternatively, the apparatus may be configured to generate data depending on which X-ray dark-field images and/or X-ray phase-contrast images are generatable. 
     The X-ray source may be a synchrotron or an X-ray source which generates X-rays using an electron beam which impinges on an anode. A diameter of a focal spot of the electron beam on the anode may be smaller than 2 mm or smaller than 1 mm. The diameter may be larger than 1 μm or larger than 10 μm. 
     The first grating may be configured as a phase grating and/or as an amplitude grating. Specifically, the first grating may be configured as a pure amplitude grating. The phase grating may be configured to cause a phase shift of π or substantially π between a portion of the X-rays which is transmitted through an aperture of the grating and a portion of the X-rays, which is transmitted through a partially opaque portion of the grating. 
     The image detector may be configured as a two-dimensional image detector. The ordered array of the X-ray sensitive pixels may be a two-dimensional ordered array. 
     A pitch of the first grating may be less than 100 μm or less than 40 μm. The pitch of the first grating may be greater than 1 μm or greater than 5 μm. 
     A pitch of the second grating may be less than 200 μm or less than 80 μm. The pitch of the second grating may be greater than 2 μm or greater than 10 μm. 
     A pixel pitch of the detector may be smaller than 2 mm or smaller than 0.5 mm. The pixel pitch may be greater than 10 μm or greater than 50 μm. 
     The fringes of the fringe pattern may have a linear shape and/or may be oriented parallel relative to each other. 
     The controller may be in signal communication with an actuator system of the X-ray imaging system which is configured to move a component of the X-ray imaging system based on signals received from the controller. By way of example, the actuator may include one or more piezo actuators. 
     The apparatus may include a data processing system for generating the dark-field and/or phase contrast image based on the acquired Moiré images. The data processing system may include a processor and a memory for storing instructions processable by the processor. The processor may execute an operating system. The data processing system may further include an input and/or output unit configured to allow a user to receive data from the data processing system and/or to provide data to the data processing system. The computer system may further include a data storage system and/or a user interface for receiving user input. 
     The imaging system may be configured to generate the relative movement between the second grating and the fringe pattern by moving the first and/or the second grating. For each of the gratings, the movement may be within a plane of the respective grating. The imaging system may be configured so that the first and/or the second grating is controllably movable. The controlled movement of the first and/or second grating may be performed depending on control signals generated by the controller. The imaging system may include an actuator system which may include one or more actuators (such as a piezo actuator), which are in operative connection with the first and/or the second grating. The actuator system may be in signal communication with the controller. The actuator may actuate the relative movement between the first and the second grating depending on control signals received from the controller. 
     The system may be configured to at least partially generate the relative movement between the second grating and the fringe pattern by controllably displacing the first and/or the second grating. The displacement of the first grating and/or the displacement of the second grating may be a displacement relative to an X-ray source of the apparatus. The X-ray source of the apparatus may be stationary during the displacement of the first grating and/or during the displacement of the second grating. The object may be stationary during the displacement of the first grating and/or during the displacement of the second grating. 
     The system may include an adjustment mechanism for adjusting a period of the Moiré pattern. The adjustment mechanism may include an actuator. The actuator may be in signal communication with the controller. The actuator may be in operative connection with the first and/or the second grating. The actuator may be configured to adjust a relative position and/or a relative orientation between the first and the second grating based on signals received by the controller. 
     According to an embodiment, a pixel pitch of the pixels of the detector is smaller than 0.5 times or smaller than 0.3 times, or smaller than 0.2 times or smaller than 0.1 times a pitch of the Moiré pattern. 
     According to a further embodiment, the system further comprises a data processing system, which is configured to generate an X-ray dark-field image and/or an X-ray phase-contrast image based on one or more analysis parameters. The system may be further configured to determine each of the one or more analysis parameters based on a plurality of pixel data values of one of the Moiré images, which sample a Moiré fringe pattern of the Moiré image over a period of the Moiré fringe pattern. 
     The one or more analysis parameters may further be generated based on a position parameter of the image acquisition positions. The position parameter may be defined to be measured in a direction perpendicular to an axis of the second grating and within the entrance plane of the second grating. The data processing system may further be configured to calculate the dark-field image and/or the phase contrast image based on the determined analysis parameter. 
     According to a further embodiment, the data processing system is further configured to determine each of the one or more analysis parameters further based on pixel data values of pixels of different ones of the Moiré images. The Moiré images may be acquired at different relative image acquisition positions. The pixels taken from the different Moiré images may relate to a same position (e.g. a same detector pixel) on an active surface of the image detector. 
     According to a further embodiment, the imaging system is configured so that the Moiré pattern is at least partially caused by an axis of fringes of the fringe pattern being rotated relative to an axis of the second grating as seen in the entrance plane of the second grating. Additionally or alternatively, the Moiré pattern may be at least partially caused by a difference between a grating pitch of the second grating and a pitch of the fringe pattern. 
     The Moiré pattern may be at least partially caused by a deviation of a relative position and/or a relative orientation between the first and the second grating from an ideal configuration of a Talbot interferometer or a Talbot-Lau interferometer. In Talbot interferometers and Talbot-Lau interferometers, the first and the second grating are separated by a fractional or integer Talbot distance and the grid axes of the first and the second grating are oriented parallel relative to each other. 
     According to a further embodiment, the X-ray imaging system is configured so that the first and the second grating form a grating interferometer. The apparatus may be configured so that the first grating acts as a diffraction grating. The diffraction grating may generate a diffraction pattern, which fills the space between the first and the second grating. The grating interferometer may be configured as a Talbot interferometer or as a Talbot-Lau interferometer. The fringe pattern may represent a cross-section through a spatial Talbot pattern (also denoted as Talbot carpet). The Talbot pattern may fill the space between the first and the second grating. A distance between the first grating and the second grating may be or may substantially be a fractional or integer Talbot distance. Through a distance, which deviates from an exact fractional or integer Talbot distance (and thereby only substantially represents a fractional or integer Talbot distance), it is possible to generate a fringe pattern in the entrance plane of the second grid, which has a pitch, which deviates from the grating pitch of the second grid. This allows generation of a Moiré pattern from the fringe pattern using the second grid. 
     Alternatively, the imaging system may be configured for X-ray edge illumination. The first grating may form a plurality of beamlets for X-ray edge illumination of the second grating. The imaging system may be configured so that downstream of the first grating, a plurality of separate beamlets are formed. The beamlets may be configured to be separate from each other. Therefore, the beamlets do not fill the space between the first and the second grating. The first and second grating, which are configured for edge illumination, may be further configured so that the fringe pattern in the entrance plane of the second grating represents a projection image of the apertures of the first grating. The projection image may be an unmagnified or magnified projection image. 
     According to a further embodiment, the system further comprises a third grating, which is arranged in the beam path of the X-ray beam upstream of the first grating. The imaging system may be configured to at least partially generate the relative movement between the second grating and the fringe pattern by controllably displacing the first, the second and/or the third grating. The displacement of the first, second and/or third grating may be a displacement relative to an X-ray source of the apparatus. The X-ray source of the apparatus may be stationary during the displacement of the first, second and/or third grating. The object may be stationary during the displacement of the first, second and/or third grating. 
     The apparatus may include an actuator system which includes one or more actuators. The actuators may include piezo actuators. The actuators may be in operative connection with the first, second and/or third grating. The actuator may be configured to displace a grating, which is in operative connection with the actuator, within a plane of the grating and in a direction perpendicular to a grating axis of the grating. The actuator system may be in signal communication with the controller. The actuator may displace the first, second and/or third grating in response to control signals received from the controller. 
     The third grating may be configured as a pure amplitude grating. A distance between the third and the first grating may be less than 2 m or less than 1 m. The distance may be greater than 2 cm or greater than 10 cm. A pitch of the third grating may be less than 200 μm or less than 20 μm. The pitch of the third grating may be greater than 1 μm or greater than 3 μm. A slit width of the third grating may be less than 60% or less than 50% of the pitch. The slit width may be greater than 5% or greater than 20% of the pitch. 
     According to a further embodiment, the X-ray imaging system is configured to determine, based on the Moiré images: (a) a parameter of a spatial position of the fringe pattern in the entrance plane, measured in a direction perpendicular to a fringe axis of the fringe pattern; and/or (b) a parameter of an amplitude of the fringe pattern. The parameter of the spatial position and/or the parameter of the amplitude may be local parameters, which represent local properties of the fringe pattern in the entrance plane of the second grating. Additionally or alternatively, the parameters may represent a change of a fringe pattern acquired from an object relative to a fringe pattern acquired when no object is present. Additionally, the X-ray imaging system may further be configured to determine, depending on the Moiré images, a parameter of an average pixel data value of the fringe pattern representing an average across at least a period of the fringe pattern. The parameter of the average may represent a local parameter, which represents a local property of the fringe pattern in the entrance plane of the second grating. Additionally or alternatively, the parameter of the average may represent a change of a fringe pattern acquired from an object relative to a fringe pattern acquired when no object is present. 
     According to a further aspect of the present disclosure, a method for generating X-ray dark-field and/or X-ray phase-contrast images using an apparatus having an X-ray imaging system is provided. The imaging system comprises an X-ray sensitive image detector having an ordered array, in particular a two-dimensional ordered array, of X-ray sensitive pixels, a first and a second X-ray optical grating. The imaging system is configured to be usable with an X-ray beam which traverses the first grating, then the second grating before being incident on the detector for imaging an object, which is positioned in the beam path of the X-ray beam upstream of the first grating or between the first and the second grating. The imaging system is further configured so that when no object is present, the first grating generates a fringe pattern in an entrance plane of the second grating. The method comprises generating a Moiré pattern from the fringe pattern using the second grating so that the Moiré pattern has a pitch which is less than 20 pixels or less than 10 pixels of the detector. The method further comprises controlling, using a controller, a relative movement between the second grating and the fringe pattern to acquire a Moire-image at each of a plurality of different relative image acquisition positions. 
     According to a further aspect of the present disclosure, a method for generating an object image based on at least two fringe pattern images is provided. Each of the fringe pattern images shows a different fringe pattern. In each of the fringe pattern images, the fringe pattern has a pitch which is less than 20 pixels or less than 10 pixels of the respective image and a pixel pitch of the pixels of the respective image is smaller than 0.5 times or smaller than 0.3 times, or smaller than 0.2 times or smaller than 0.1 times the pitch of the fringe pattern of the respective image. The method comprises reading and/or generating, using a data processing system, the fringe pattern images and one or more position parameters for each of the fringe pattern images. The method further comprises determining, using the data processing system, one or more analysis parameters, wherein each of the one or more analysis parameters is determined based on the position parameters and further based on a plurality of pixel data values of a plurality of pixels of the images which include: a) pixels of one of the images, which sample the fringe pattern of the respective image over a period of the fringe pattern; and b) pixels of different ones of the images. The method further comprises determining at least one pixel of the object image based on the one or more analysis parameters. 
     Each of the pixels, which are taken from different ones of the fringe pattern images, may have a same or substantially a same pixel position within the respective image. The object image may show an object to be imaged. The object image may represent a phase-contrast image or a dark-field image. 
     Each of the position parameters may represent a parameter of a relative image acquisition position. The relative image acquisition position may relate to a relative position between a grid and a fringe pattern. The grid and fringe pattern may be generated using a Talbot interferometer, a Talbot-Lau interferometer, and/or a system for edge illumination phase-contrast and/or dark-field imaging. 
     According to a further embodiment, the one or more analysis parameters are parameters of a model function. The model function may represent a fringe pattern, in particular a linear fringe pattern. The determining of the one or more analysis parameters may comprise fitting the model function to the plurality of pixel data values using the position parameters. The model function may include a trigonometric function, such as a cosine and/or a sinus function. 
     According to a further embodiment, the one or more analysis parameters comprise a parameter of a spatial position, in particular a parameter of a phase or a phase shift of the model function. Additionally or alternatively, the one or more analysis parameters may comprise a parameter of an amplitude of the model function. The spatial position may be a position within the image plane. The spatial position may be a spatial position in a direction perpendicular or substantially perpendicular to a fringe axis of a fringe pattern represented by the model function. The amplitude may be an amplitude of a periodic function such as the amplitude of a cosine function or the amplitude of a sine function. The amplitude may be a level of a local extremum relative to an average level of the model function. If the model function represents a periodic function, the average level may be determined by averaging values of the model function over one or more periods of the model function. 
     According to a further embodiment, the method further comprises determining, for each of a plurality of pixels of the object image, one or more analysis parameters. Each of the analysis parameters may be determined based on the position parameters and further based on a plurality of pixel data values of a plurality of pixels of the fringe pattern images and selected for the respective object image pixel. The pixels of the fringe pattern images may include: a) pixels of one of the fringe pattern images, which sample the fringe pattern of the respective fringe pattern image over a period of the fringe pattern. The fringe pattern image may be selected for the respective object pixel or may be the same for each of the object pixels. The pixels of the fringe pattern images for the respective object image pixel may further include b) pixels of different ones of the fringe pattern images. 
     For each of the object image pixels, the set of pixels of the fringe pattern images which are used for determining the object image pixel may be different. Specifically, the sets may be overlapping or non-overlapping. 
     According to a further aspect of the present disclosure, a program element for generating an object image based on at least two fringe pattern images is provided. Each of the fringe pattern images shows a different fringe pattern. In each of the fringe pattern images, the fringe pattern has a pitch which is less than 20 pixels or less than 10 pixels of the respective image and a pixel pitch of the pixels of the respective image is smaller than 0.5 times, or smaller than 0.3 times, or smaller than 0.2 times or smaller than 0.1 times the pitch of the fringe pattern of the respective image. The object image is generated using a data processing system, wherein the program element, when being executed by a processor of the data processing system, is adapted to carry out: reading and/or generating, using the data processing system, the fringe pattern images and one or more position parameters for each of the fringe pattern images; determining, using the data processing system, one or more analysis parameters, wherein each of the one or more analysis parameters is determined based on the position parameters and further based on a plurality of pixel data values of a plurality of pixels of the images, which include: a) pixels of one of the images, which sample the fringe pattern of the respective image over a period of the fringe pattern; and b) pixels of different ones of the images. The program element, when being executed by a processor of the data processing system, is further adapted to carry out determining at least one pixel of the object image based on the one or more analysis parameters. 
     According to a further aspect, a computer readable medium is provided having stored thereon the computer program element described in the preceding paragraph. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is as schematic perspective illustration of an apparatus for X-ray dark-field and X-ray phase-contrast imaging according to a first exemplary embodiment; 
         FIG. 2  is a schematic illustration of the imaging process in the apparatus according to the first exemplary embodiment, which is shown in  FIG. 1 ; 
         FIGS. 3A to 3C  are schematic illustrations of the change in the average signal, the change in the amplitude and the change in the phase of the fringe pattern measured using the apparatus according to the first exemplary embodiment which is shown in  FIG. 1 ; 
         FIG. 4  is a Moiré image acquired using the apparatus according to the first exemplary embodiment, which is shown in  FIG. 1 ; 
         FIG. 5  is a schematic illustration of an apparatus for X-ray phase-contrast imaging and X-ray dark-field imaging according to a second exemplary embodiment; 
         FIG. 6  is a flowchart of an exemplary method for analyzing the images which have been acquired using the apparatus according to the first exemplary embodiment, which is shown in  FIG. 1  or the apparatus according to the second exemplary embodiment, which is shown in  FIG. 5 . 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
       FIG. 1  is a schematic perspective illustration of an apparatus  10  according to a first exemplary embodiment, which is configured for X-ray dark-field imaging and X-ray phase-contrast imaging.  FIG. 2  schematically illustrates the imaging process in the apparatus  10 . The apparatus  10  has a grating interferometer, which includes a first grating  1 , a second grating  2  and a third grating  3 , which are arranged in a beam path of a divergent X-ray beam  11 . The second grating  2  is arranged in the beam path downstream of the first grating  1 . The first grating  1  and the second grating  2  are arranged so that their grating planes are parallel relative to each other or substantially parallel relative to each other. The first grating may be a pure amplitude grating or a phase grating, which shifts the phase of the portion of the X-ray beam  11 , which traverses a semi-transparent portion of the phase grating. By way of example, the phase shift amounts to π or π/2. 
     In the illustrated embodiment, the object  4  under inspection is disposed in the beam path of the X-ray beam  11  between the third grating  3  and the first grating  1 . However, it is also conceivable that the object  4  is in the beam path of the X-ray beam  11  between the first grating  1  and the second grating  2 . 
     The X-ray beam  11  is generated using an X-ray source  5  of the apparatus  10 . The X-ray source  5  may be configured as an X-ray tube, which generates a focus of an electron beam on a surface of an anode. The anode may be configured as a rotating anode disk. Alternatively, the X-ray source may be configured as a synchrotron. If a synchrotron is used as the X-ray source  5  or if the source size of an X-ray tube is small enough (such as smaller than 1 square millimeter), it is possible to operate the apparatus  10  without a third grating  3 . The third grating  3  is a pure amplitude grating and provides a plurality of apertures, each of which creating a sufficiently coherent virtual line source. 
     In the exemplary embodiment, the third grating  3  is placed close to the source  5 . By way of example, a distance k (shown in  FIG. 2 ) of the third grating  3  from the source may be smaller than 40 cm or smaller than 5 cm. 
     As it is illustrated in  FIG. 2 , it has been shown that improved dark-field and phase-contrast images can be obtained if the following equation is satisfied: 
     
       
         
           
             
               
                 p 
                 3 
               
               = 
               
                 
                   
                     p 
                     2 
                   
                   ⁢ 
                   l 
                 
                 d 
               
             
             , 
           
         
       
     
     with p 3  being the pitch (i.e. period) of the third grating  3 , p 2  being the pitch of the second grating  2 , l being the distance between the third grating  3  and the first grating  1  and d being the distance between the first grating  1  and the second grating  2 . The above equation ensures that each line source generated using the third grating  3  contributes constructively to the image-formation process. 
     The illumination of the first grating  1  using the X-ray beam  11 , which is generated using the source  5 , and which traverses the third grating  3 , generates a spatial Talbot pattern (also denoted as Talbot carpet), which fills the space  9  between the first grating  1  and the second grating  2 . The first grating  1  therefore acts as a diffraction grating. The distance between the first grating  1  and the second grating  2  corresponds or substantially corresponds to a fractional Talbot distance or to an integer Talbot distance. Thereby, in the entrance plane of the second grating  2 , a fringe pattern is generated formed by linear fringes. 
     It has been shown that a comparison between a first fringe pattern which is generated when the X-ray beam  11  is transmitted through the object  4  and a second fringe pattern, which is generated when no object is present, allows generation of X-ray dark-field and X-ray phase-contrast images of the object  4 . This is explained in more detail in relation to  FIGS. 2 and 3A to 3C . 
     Specifically, as is illustrated in  FIG. 2 , a phase object  4 , which is placed in the beam path of the X-ray beam  11  causes the X-ray beam  11 , which is transmitted through the object, to be slightly refracted, thereby generating a phase shift in the wavefront of the X-ray beam, which causes the fringes of the fringe pattern in the entrance plane of the second grating  2  to be displaced by an angle a. The angle a is directly proportional to a local directional derivative of the wavefront phase profile O. The directional derivative is also denoted as the differential phase contrast and expressed by the equation: 
     
       
         
           
             
               α 
               = 
               
                 
                   
                     λ 
                     ⁢ 
                     d 
                   
                   
                     p 
                     2 
                   
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       Φ 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                   
                   
                     ∂ 
                     x 
                   
                 
               
             
             , 
           
         
       
     
     where x and y are the cartesian coordinates in the entrance plane of the second grating (see the coordinate systems  38  shown in  FIGS. 1 and 2 ) and d (shown in  FIG. 2 ) denotes the distance between the first grating  1  (close to which the object is placed in this exemplary embodiment) and the second grating  2 . In configurations in which the object  4  is placed between the first grating  1  and the second grating  2 , d should be replaced by the distance between the object  4  and the third grating  3 . The x-axis is oriented perpendicular to the axis of the fringes. Therefore, by measuring, for each pixel of the detector  7 , a local displacement of the fringe pattern in the entrance plane of the second grating  2 , it is possible to generate a differential phase-contrast image. Based on this image, a phase-contrast image can be obtained by integration of the pixel data values which represent the differential phase contrast. 
     Further, specimens, which produce small-angle X-ray scattering contributions show a decrease of the amplitude of the fringe pattern. Therefore, by measuring, for each pixel of the detector  7 , the change in the amplitude of the fringe pattern which is caused by the object (i.e. by measuring the amplitude with and without the object), it is possible to generate a dark-field image. 
       FIGS. 3A to 3C  are plots of the intensity of the fringe pattern (y-axis), as measured in the entrance plane of the second grating versus position (x-axis). The position is measured along an axis in the entrance plane, which is oriented perpendicular to the axis of the fringes. 
     The dotted curves  16 ,  17  and  18  in  FIGS. 3A to 3C  indicate the intensity which is measured when the X-rays have traversed an object, whereas the solid curves  13 ,  14  and  15  in  FIGS. 3A to 3C  indicate the intensity, which is measured when no object is present. 
     For the sake of simplicity, the dotted curve in  FIG. 3A  illustrates the intensity of an object model, which attenuates the X-ray beam without causing a differential phase shift and without causing small-angle scattering. The dotted curve of  FIG. 3B  illustrates the intensity of an object model, which causes a differential phase shift without attenuating the X-ray beam and without causing small-angle scattering.  FIG. 3C  illustrates the intensity of an object model, which causes small-angle scattering without causing a differential phase shift and without attenuating the X-ray beam. 
     As can be seen from  FIG. 3A , the attenuation of the X-ray beam leads to a decrease of the average intensity from a first level  21  to a second level  22 , as is indicated by arrow  23 . The average is calculated over a period of the fringe pattern. As can be seen from  FIG. 3B , the differential phase shift generated by the object causes a spatial shift of the fringe pattern in a direction perpendicular to the fringe axis of the fringe pattern from a position  39  to a position  40 , as is indicated by arrow  25 . Further, as can be seen from  FIG. 3C , the small angle scattering by the object reduces the amplitude of the intensity of the fringe pattern from a level  25  to a level  26 , as is indicated by arrow  41 . The amplitude is measured as an intensity level at a local extremum (such as the extremum  28 ) relative to an average intensity level  27 , which represents an average over one period of the fringe pattern. 
     Returning back to  FIG. 1 , since the pixel pitch of the array of X-ray sensitive pixels of the detector  7  is greater than the pitch of the fringe pattern in the entrance plane of the second grating  2 , the apparatus  10  is configured to perform an electronically controllable relative movement between the second grating and the fringe pattern to sample the fringe pattern at a plurality of different relative image acquisition positions. In the exemplary embodiment, the pixel pitch of the detector  7  may be greater than 10 μm or greater than 50 μm. The pixel pitch may be smaller than 2 mm or smaller than 0.5 mm. 
     The direction vector of the relative movement is angled (such as perpendicular of substantially perpendicular) relative to the fringe axis of the fringe pattern and/or the grating axis of the second grating  2 . 
     The relative movement is controlled by a controller  30  of the imaging system, which is in signal communication with an actuator (not shown in  FIG. 1 ) of the imaging system. The actuator may be configured as a piezo actuator. The actuator is configured to actuate the relative movement between the fringe pattern and the second grating  2  based on control signals received from the controller  30 . 
     The circular dots on the curves of  FIGS. 3A to 3C  indicate the fringe pattern intensity which has been recorded using the relative movement between the second grating and the fringe pattern. Based on the fringe pattern images which have been acquired, it is possible to determine, for each of the detector pixels of the detector  7  (shown in  FIG. 1 ), a parameter D i  of the local amplitude of the fringe pattern (i being an index designating the pixel), a parameter ψ i  of the local phase of the fringe pattern and a parameter T i  of a local intensity of the fringe pattern. 
     The apparatus of the first exemplary embodiment may be configurable so that the grating pitch of the second grating is equal to or substantially equal to the pitch of the fringe pattern in the entrance plane of the second grating. In this configuration, the parameters D i , ψ i  and T i  can be obtained using the intensity values f ij  measured using the i-th pixel (j being an index designating the fringe pattern image). Specifically, the parameters D i , ψ i  and T i  can be obtained by varying these parameters to minimize Δ i   2  (T i , D i , ψ i ) which is defined by the following equation: 
     
       
         
           
             
               
                 Δ 
                 i 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   
                     T 
                     i 
                   
                   , 
                   
                     D 
                     i 
                   
                   , 
                   
                     ψ 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 j 
               
               ⁢ 
               
                 
                   1 
                   
                     Var 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         
                           i 
                           ⁢ 
                           j 
                         
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         f 
                         
                           i 
                           ⁢ 
                           j 
                         
                       
                       - 
                       
                         
                           T 
                           i 
                         
                         ⁢ 
                         
                           
                             
                               A 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     D 
                                     i 
                                   
                                   ⁢ 
                                   
                                     V 
                                     i 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ψ 
                                           i 
                                         
                                         + 
                                         
                                           ϕ 
                                           i 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                             x 
                                             j 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                     , 
                   
                 
               
             
           
         
       
     
     wherein x j  denotes the fraction of the period of the fringe pattern which corresponds to the image acquisition position of the j-th fringe pattern image. The parameters A i , V i , and ϕ i  for each pixel i in the above equation can be obtained by acquiring a plurality of fringe pattern images when no object is present. Var(f ij ) denotes a variance estimate for the measurement f ij . A simple estimate for the variance is Var(f ij )=f ij . Since a pixel of the detector receives X-ray radiation transmitted through a plurality of slits of the second grating, the parameters A i , V i , and ϕ i  for the i-th pixel represent the local behavior of the fringe pattern averaged over the area of the i-th pixel. 
     The minimum number of measurements which are necessary for acquiring intensity samples based on the above equation so that the phase shift and the change of the amplitude can be determined is three. However, it has been shown by the inventors that due to mechanical vibrations, it is typically necessary to acquire  10  or even more images to ensure that a sufficient image quality is obtained. This in turn, however, may cause image artefacts due to patient motion (such as respiratory movements or heart beat) so that the resulting images do not qualify as diagnostic images. 
     However, the inventors have found that it is possible to overcome these problems. Specifically, the inventors have shown that X-ray dark-field imaging as well as X-ray phase-contrast imaging is possible with a comparatively low number of images if the second grating is configured to generate a Moiré pattern from the fringe pattern in the entrance plane of the second grating. The Moiré pattern is generated so that it has a period which is less than  20  times or less than  10  times a pixel pitch of the X-ray sensitive image detector. In other words, a Moiré pattern is generated at an exit plane of the second grating (shown in  FIG. 1 ) from the fringe pattern, which is generated in the entry plane of the second grating  2 . 
     By way of example, the imaging system may be configured so that the Moiré pattern is caused by one or a combination of a deviation of a relative position and/or a relative orientation between the first and second gratings from an ideal Talbot configuration. By way of example, the distance between the first grating and the second grating may be selected so that the pitch of the fringe pattern in the entrance plane of the second grating, is different from the pitch of the second grating. Additionally or alternatively, the grating plane of the first grating may be inclined relative to the grating plane of the second grating. The inclination may be caused by a rotation of an axis of the first grating relative to an axis of the second grating. Additionally or alternatively, the grating axis of the first grating may be rotated relative to the grating axis of the second grating about an axis which is perpendicular to the grating plane of the first and/or second grating. These configurations allow switching between Moiré pattern imaging and non-Moiré pattern imaging by providing an actuator, which is configured for actuating a relative movement between the first and the second grating. Additionally or alternatively, it is also conceivable that the grating pitchp 1  (shown in  FIG. 2 ) of the first grating  1  and/or the grating pitch p 2  of the second grating  2  are adapted to generate Moiré images. 
       FIG. 4  shows an example of a Moiré image  34  acquired using the image detector  7  (shown in  FIG. 1 ). In the Moiré image  35  of  FIG. 4 , the fringe pattern has fringes  32  which are not exactly linear (but rather substantially linear) and which are not exactly parallel (but rather substantially parallel) relative to other fringes of the Moiré image  34 . This is caused by a deviation of the gratings from a perfect plane. 
     The Moiré pattern therefore represents an irregular sampling of the fringe pattern in the entrance plane of the second grating  2  (shown in  FIG. 1 ). The irregular sampling allows using a plurality of neighboring or substantially neighboring pixel data values acquired using the detector  7  (shown in  FIG. 1 ) to determine a parameter of a local phase shift of the fringe pattern, a parameter of a local average intensity of the fringe pattern and a parameter of a local average intensity of the fringe pattern. The spatial resolution achieved with this technique is limited by the spatial extent of the neighboring or substantially neighboring pixels, which are used to determine the parameters. Since the Moiré pattern has a period which is less than 20 times or less than 10 times the pixel pitch of the X-ray sensitive image detector, the neighboring or substantially neighboring pixels can be selected so that their spatial extent is comparatively small. 
     Further, in the exemplary embodiment, the pixel pitch of the pixels of the detector is smaller than 0.5 times or smaller than 0.3 times or smaller than 0.2 times or smaller than 0.1 times the period of the Moiré pattern. This allows resolving the Moiré pattern with a sufficient accuracy. 
     Specifically, based on the plurality of Moiré images acquired at different image acquisition positions provided by the relative movement between the second grating and the fringe pattern at the entrance plane of the second grating, for each of a plurality of pixels, a parameter D i  of the local amplitude of the fringe pattern (i being an index designating the pixel), a parameter ψ i  of a local phase shift of the fringe pattern and a parameter T i  of a local average intensity of the fringe pattern can be determined. These analysis parameters D i , ψ i  and T i  can be obtained by varying these parameters to minimize Δ i   2 (T i , D i , ψ i ) in the following equation: 
     
       
         
           
             
               
                 Δ 
                 i 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   
                     T 
                     i 
                   
                   , 
                   
                     D 
                     i 
                   
                   , 
                   
                     ψ 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 Σ 
                 
                   j 
                   , 
                   
                     
                       i 
                       ′ 
                     
                     ∈ 
                     
                       𝒩 
                       i 
                     
                   
                 
               
               ⁢ 
               
                 1 
                 
                   Var 
                   ⁡ 
                   
                     ( 
                     
                       f 
                       
                         
                           i 
                           ′ 
                         
                         ⁢ 
                         j 
                       
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       f 
                       
                         
                           i 
                           ′ 
                         
                         ⁢ 
                         j 
                       
                     
                     - 
                     
                       
                         T 
                         i 
                       
                       ⁢ 
                       
                         
                           
                             A 
                             
                               i 
                               ′ 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   D 
                                   i 
                                 
                                 ⁢ 
                                 
                                   V 
                                   
                                     i 
                                     ′ 
                                   
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         i 
                                       
                                       + 
                                       
                                         ϕ 
                                         
                                           i 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                           x 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
     wherein   is a set of pixel indices, which is formed by pixels in the neighborhood of pixel i. The i-th pixel may, but not necessarily, be an element of  . The set of pixels   may be chosen so that the pixel data values of the set of pixels sample an intensity fluctuation within a same period of the Moiré-pattern. 
     Accordingly, since the Moiré pattern allows using a plurality of pixel data values of different pixels from a same Moiré image to sample the fringe pattern at different phase positions, it is possible to determine the parameters D i , ψ i  and T i  with a sufficient accuracy using a small number of Moiré images, such as only two Moiré images. Since for each pixel i on the detector, the parameters D i , ψ i  and T i  are chosen based on the pixel set  , the resolution of the images, which can be obtained using these parameters is limited by the spatial extent of the pixel set  . However, since the proposed method uses a plurality of Moiré images which are acquired at different relative image acquisition positions between the second grating and the fringe pattern in its entrance plane, a sufficiently high image quality can be obtained, even if the number of pixels   is comparatively small and represent a small spatial extent. 
     In the foregoing equation, the parameters A i , V i , and ϕ i  for each pixel i can be obtained by acquiring a plurality of fringe pattern images when no object is present. Var(f ij ) denotes a variance estimate for the measurement f ij . 
     Therefore, using the proposed system and method, it is possible to generate dark-field images and phase-contrast images at a high spatial resolution based on only a small number of images acquired from the object. 
     The inventors have further shown that even better values for the analysis parameters D i  and T i  can be obtained if a second minimization step is performed which minimizes Δ i   2 (T i , D i ) in the equation: 
     
       
         
           
             
               
                 Δ 
                 i 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   
                     T 
                     i 
                   
                   , 
                   
                     D 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 Σ 
                 j 
               
               ⁢ 
               
                 1 
                 
                   Var 
                   ⁡ 
                   
                     ( 
                     
                       f 
                       ij 
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       f 
                       
                         i 
                         ⁢ 
                         j 
                       
                     
                     - 
                     
                       
                         T 
                         i 
                       
                       ⁢ 
                       
                         
                           
                             A 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   D 
                                   i 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         i 
                                       
                                       + 
                                       
                                         ϕ 
                                         i 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                           x 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
     wherein the parameters D i  and T i  are varied to minimize Δ i   2 (T i , D i ) and ψ i  is kept at the value retrieved from the first optimization step. It has been shown that the second minimization step according to the above equation leads to a higher accuracy for the analysis parameters D i  and T i , especially in cases in which the differential phase varies more slowly in space than the transmission and dark-field. 
       FIG. 5  schematically shows an apparatus  33  for generating X-ray phase-contrast images and X-ray dark-field images according to a second exemplary embodiment. In the second embodiment, the apparatus  33  is configured for edge illumination phase-contrast and dark-field imaging. Edge illumination is a non-interferometric and incoherent X-ray imaging technique. Like in the first exemplary embodiment described above, the images acquired using edge illumination contain a mixture of attenuation and refraction (or differential phase) contrast, the latter being proportional to the spatial derivative of the X-ray phase shift. 
     The apparatus  33  comprises a first grating  1  and a second grating  2 , each of which being configured as a pure absorption mask. The first grating  1  shapes the incident radiation into an array of beamlets (such as the beamlet  36 ) so that the openings in the first grating  1  form a plurality of incoherent X-ray line sources. Each of the beamlets traverses the object  4  and is incident on an entry plane of the second grating  2 . The beamlets form a fringe pattern in the entrance plane of the second grating  2 . Therefore, unlike the apparatus  10  of the first exemplary embodiment (shown in  FIG. 1 ), the first grating  1  of the apparatus  33  of the second exemplary embodiment does not function as a diffraction grating which generates a spatial interference pattern which fills the space  9  between the first grating  1  and the second grating  2 . 
     The apparatus  33  according to the second exemplary embodiment is configured for a controllable relative movement between the second grating  2  and the fringe pattern which is generated at the entrance plane of the second grating  2  for acquiring, using the image detector  7 , an image at each of a plurality of relative image acquisition positions. The relative movement may be actuated by an actuator (not shown in  FIG. 5 ) which is in operative connection with the first and/or the second grating for moving the first and/or second grating in a direction which has a component perpendicular to the grating axis of the movable respective grating. The actuator may be in signal communication with a controller  30  of the apparatus  33  and actuates the relative movement between the fringe pattern and the second grating  2  in response to a control signal received from the controller  30 . 
     Further, the apparatus  33  is also configured so that the second grating  2  generates a Moiré pattern from the fringe pattern which is generated in the entrance plane of the second grating  2 . The Moiré pattern may be generated by configuring the first grating  1 , the second grating  2  and a position and/or orientation between the first grating  1  and second grating  2  in a manner as has been described in connection with the first exemplary embodiment. 
     Thereby, the apparatus  33  is configured to acquire a Moiré image at each of a plurality of image acquisition positions provided by the relative movement between the fringe pattern and the second grating. The analysis parameters can then be determined in the same way as explained in connection with the first exemplary embodiment. 
       FIG. 6  is a flow-chart of a method  100  performed by the data processing system  37  (shown for the first exemplary embodiment in  FIG. 1  and for the second exemplary embodiment in  FIG. 5 ) for generating an object image (i.e. an X-ray phase-contrast image and/or an X-ray dark-field image) based on a plurality of fringe pattern images, each of which representing a Moiré image (such as the Moiré image  34  of  FIG. 4 ). 
     The data processing system  37  reads and/or generates  110  a plurality of fringe pattern images. By way of example, the data processing system  37  may read the fringe pattern images from the image detector  7  (shown in  FIGS. 1 and 5 ) and/or from a data storage system of the data processing system  37 . Additionally or alternatively, the data processing system  37  may receive signals from the image detector  7  and generates the fringe pattern images based on the received signals. 
     In each of the fringe pattern images, the fringe pattern has a period which is less than 20 pixels or less than 10 pixels of the respective image, wherein for each of the images a pixel pitch of the pixels of the respective image is smaller than 0.5 times or smaller than 0.3 times of the pitch of the linear fringe pattern of the respective image. A first group of the fringe pattern images are images acquired without an object being present, whereas a second group of the fringe pattern images are images acquired from an object. 
     The data processing system  37  then determines  120 , for each pixel i of the object image, one or more analysis parameters based on the fringe pattern images, which have been acquired without an object being present. By way of example, the analysis parameters include a parameter of a spatial position of a model function, such as a parameter of a phase or a phase shift of the model function (such as the analysis parameter ϕ i  in the first exemplary embodiment above). The model function may describe the fringe pattern in the entrance plane of the second grating. Additionally or alternatively, the analysis parameters may include a parameter of an amplitude of the model function (such as the parameter V i  in the first exemplary embodiment above) and/or a parameter of an average intensity of the model function (such as the parameter A i  in the first exemplary embodiment above). 
     The data processing system  37  further determines  130 , for each pixel i of the object image, one or more analysis parameters based on the fringe pattern images, which have been acquired from the object and further depending on the analysis parameters, which have been determined in the previous step  120  (i.e. based on the images acquired without the object). By way of example, the analysis parameters include a parameter of a phase of the model function (such as a parameter of a spatial shift of the model function generated by the object). An example for such a parameter is the analysis parameter ψ i  in the first exemplary embodiment above. Additionally or alternatively, the analysis parameters may include a parameter of an amplitude of the model function (such as a parameter of a change of the amplitude of the model function generated by the object). An example for such a parameter is the parameter D i  described in connection with the first exemplary embodiment above. Additionally or alternatively, the analysis parameters may include a parameter of an intensity of the model function (such as a parameter of a change of the intensity generated by the object). An example for such a parameter is the parameter T i  described above in connection with the first exemplary embodiment. 
     One or both of the steps  120  and  130  may be performed by fitting the model function to pixel data values of the fringe pattern images. The fitting procedure may include a regression algorithm. 
     In view of the foregoing, an improved apparatus and an improved method are provided for generating X-ray phase-contrast images and X-ray dark-field images. 
     The above embodiments as described are only illustrative, and not intended to limit the technique approaches of the present invention. Although the present invention is described in detail referring to the preferable embodiments, those skilled in the art will understand that the technique approaches of the present invention can be modified or equally displaced without departing from the protective scope of the claims of the present invention. 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. Any reference signs in the claims should not be construed as limiting the scope.