Patent Publication Number: US-9412184-B2

Title: Regularized phase retrieval in differential phase-contrast imaging

Description:
FIELD OF THE INVENTION 
     The present invention relates to X-ray imaging technology in general. 
     More particularly, the present invention relates to differential phase-contrast imaging. 
     In particular, the present invention relates to a method for regularized phase retrieval in phase-contrast imaging, an apparatus for regularized phase retrieval in phase-contrast imaging, an X-ray system comprising an apparatus according to the present invention, the use of an apparatus according to the present invention in one of an X-ray system and a CT system, a computer-readable medium, and a program element. 
     BACKGROUND OF THE INVENTION 
     In transmission X-ray image acquisition, an object to be examined, e.g., a patient, is arranged between an X-ray generating device, for example an X-ray tube, and an X-ray detector. 
     X-ray radiation emanating from the X-ray generating device penetrates the object to be examined and subsequently arrives at the X-ray detector for acquiring image information, which may later then be reconstructed into an X-ray image for presentation. The inner structure of the object, e.g., the tissue structure, provides spatial attenuation of the X-radiation penetrating the object. Accordingly, the X-ray detector is registering the spatially attenuated X-ray radiation. 
     Certain objects may attenuate X-ray radiation only to a smaller extent or may rather uniformly attenuate the X-ray beam resulting in an acquired X-ray image having low contrast. 
     However, even objects imposing only a small amount of attenuation of an X-ray beam penetrating the object, a phase of a wave front of X-ray radiation may be influenced to a rather large extent by the same object. 
     Accordingly, phase-contrast imaging may be employed for obtaining image information of an object with enhanced contrast. 
     In phase-contrast imaging, an X-ray source together with a so-called source grating element arranged adjacent to the X-ray source generates at least partly spatially coherent X-ray radiation. Coherent X-ray radiation penetrating the object may allow a subsequent retrieval of phase information. 
     Since a phase of a wave may not be measured directly, a further grating element, a so-called phase grating, is employed, arranged between the object to be examined and the X-ray detector. The phase grating allows for a conversion of a phase-shift to an intensity modulation by interference of a plurality of waves, which intensity modulation may then be detectable by an X-ray detector. 
     However, an interference pattern generated by employing a phase grating only may be too small for a current X-ray detector to be precisely detectable, due to a lack of spatial resolution of the X-ray detector. Here, an additional grating element, a so-called analyzer grating, may be employed, which is arranged between the phase grating element and the X-ray detector in the vicinity of the X-ray detector. The analyzer grating provides an interference pattern, which is large enough to be detectable by a current X-ray detector. 
     To obtain appropriate phase-contrast image information, phase stepping is conducted for obtaining a plurality of phase-contrast projections. In phase stepping, one of the source grating element, the phase grating element and the analyzer grating element is displaced laterally with respect to the other gratings and the X-ray detector element by a fraction of the respective grating pitch, e.g., a fourth, sixth, eighth of the grating pitch of, e.g., the phase grating. Image acquisition and lateral displacement is repeated, e.g., four, six, or eight times, for acquiring a plurality of phase contrast projections, constituting together a phase stepping interval. 
     In differential phase-contrast imaging, the first derivative of a phase front perpendicular to the grating direction of a grating element, i.e., the extension of the barrier regions and the trench regions of the grating structure, is detected, thus measured, by the X-ray detector. Due to the grating structure, the acquired image information may be considered to be highly asymmetric with edges being enhanced in particular in one direction, i.e., the direction perpendicular to the grating direction. The direction parallel to the grating direction may not be enhanced. 
     For reconstructing image information, an integration procedure along the lines of differentiation, i.e., perpendicular to the grating structure, may result in image information having streak-like artefacts or streaks due to noise or other errors, which are arranged locally in the differential image information and which may propagate along the line of reminder of the image data. 
     Thus, it may be beneficial to provide means for reducing or even removal of said streak-like artefacts. 
     Phase-contrast imaging is described in both Weitkamp T., Diaz A., David C. et al.: “X-ray phase imaging with a grating interferometer”; Optics Express 6296, 8. August 2005/vol. 13, no. 16 as well as Bartl P., Durst J., Haas W. et al. “Simulation of X-ray phase-contrast computed tomography of a medical phantom comprising particle and wave contributions”, Proc of SPIE vol. 7622 76220Q-1. 
     Determining regularization parameter is described in Engl H. W. and Greyer W.: “Using the L-curve for determining optimal regularization parameters”; Numer. Math. 69: 25-31 (1994). 
     SUMMARY OF THE INVENTION 
     One aspect of the present invention may be seen in employing regularization methods known for the solution of ill-posed problems for reducing image artefacts. 
     Accordingly, a method for regularized phase retrieval in phase-contrast imaging, an apparatus for regularized phase retrieval in phase-contrast imaging, an X-ray system comprising an apparatus according to the present invention, the use of an apparatus according to the present invention in one of an X-ray system and a CT system, a computer-readable medium as well as a program element according to the independent claims are provided. 
     Preferred embodiments may be taken from the dependent claims. 
     These and other aspects of the present invention will become apparent from and elucidated with reference to the embodiments described hereinafter. 
     Exemplary embodiments of the present invention will be described below with reference to the following drawings. 
     The illustration in the drawings is schematic. In different drawings, similar or identical elements are provided with similar or identical reference numerals. 
     The figures are not drawn to scale, however may depict qualitative proportions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows an exemplary embodiment of an X-ray system according to the present invention; 
         FIG. 2  shows an exemplary embodiment of a differential phase-contrast imaging system according to the present invention; 
         FIG. 3  shows an exemplary embodiment of an apparatus for regularized phase retrieval in phase-contrast imaging according to the present invention; 
         FIG. 4  shows an exemplary embodiment of the method for regularized phase retrieval in phase-contrast imaging according to the present invention; and 
         FIGS. 5 a - c    show an exemplary reconstruction of image information employing differential phase-contrast image data according to the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     The present invention applies regularization methods for the solution of ill-posed problems to the problem of artefact suppression when reconstructing X-ray images from differential phase-contrast image data acquired employing a differential phase-contrast imaging system. 
     For the following explanations, without loss of generality, it is assumed that the acquired two-dimensional image information is image information comprising an N×N matrix image structure. In other words, the two-dimensional image comprises as many pixels in an image column as in an image row, thus resulting in a quadratic image. Accordingly, the X-ray detector element employed for acquiring said image data may, e.g., comprise 512×512 or 1024×1024 single individual pixel elements. 
     Further, it is assumed that the image information to be reconstructed comprises a matrix form of N×(N+1). 
     However, it is to be understood that the following explanations may also be applied to image data having a substantially different pixel structure and a person skilled in the art may readily adapt the following teachings to the actually occurring image data size or shape. 
     For the following explanations, we assume that streak-like artefacts occur as horizontal streaks resulting from a grating structure aligned vertically with regard to the image information. Furthermore, without loss of generality, it is assumed that the direction of differentiation is along image rows. Rows of the image data are referred to by index i while columns are referred to by index j. 
     The measured differential image data is referred to as two-dimensional image data d i,j  while the desired two-dimensional image information, i.e., the reconstructed two-dimensional image information, is denoted as g i,j . 
     For the application of regularization methods, the row by row integration problem is reformulated to constitute a linear reconstruction problem. The differentiation operation may be considered a forward problem, which may be expressed as a relation between g and d in accordance with equation 1.
 
 d   i,j   =g   i,j+1   −g   i,j    Equation 1
 
     In accordance with equation 1, a linear relationship between g and d is given. Thus, the two-dimensional image data g and d may be represented as one-dimensional vectors while the forward operation may be formulated as matrix A in accordance with equation 2.
 
 d=A·g    Equation 2
 
     Since image information g is the image information to be reconstructed from the measured image data d or the measured vector d, the reconstruction operation may be considered estimating g from the measured vector d. 
     With regard to the vector notation, single subscript indices to d and g, e.g., d k , may be employed to identify single elements of the respective vectors d and g. 
     The indices are mapped according to the relation k(i,j)=Ni+j or vice versa j(k)=k mod N, i(k)=integer part of (k/N). Matrix A is then a sparse matrix with 
     
       
         
           
             
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     For reducing streak-like artefacts, e.g., horizontal streak artefacts, a gradient operator may be employed, which penalizes, i.e., reduces, gradients in the second dimension of the two-dimensional image data different from the dimension of integration, i.e., different from the dimension in which the integration is conducted. 
     Accordingly, the gradient operator may be defined in accordance with equation 3.
 
(∇ z   g ) i,j   =g   z+1,j   −g   i,j    Equation 3
 
     Thus in case the integration operation is conducted within the detector row i of the two-dimensional image data, the gradient operator operates on each column j of the two-dimensional image data. 
     Like the forward operator matrix A, the gradient operator ∇ z  may be understood to be a linear operator operating on a one-dimensional vector g. The desired image information g may thus be reconstructed by solving the minimization problem in accordance with equation 4.
 
∥ A·g−d∥   2 +λ·∥∇ z   ·g∥   2 =min.   Equation 4
 
     A regularization parameter λ may be employed for controlling a degree of smoothness that shall be obtained, i.e., an amount of reduction of streak artefacts. 
     The regularization parameter λ may be required to be found empirically, e.g., by interactive adaptation of λ by a user, who is visually inspecting the reconstructed image data in dependence on λ. The operator may thus interactively control the regularization parameter until a preferred reconstructed image is obtained. Possible values of the regularization parameter λ are real positive values. λ may also be determined by employing the L-curve or by the discrepancy principle by Morozov as described in Engl H. W. 
     Equation 4 may be considered to be a penalized maximum likelihood reconstruction algorithm. For obtaining a more general form, two generalizations may be employed. 
     First, the measured data d k  may be considered to be Gaussian distributed random variables with variances σ k   2 . Even more generalized, a noise co-variance matrix C kl  may be employed with the diagonal elements C kk  corresponding to σ k   2 . 
     Secondly, gradients in z-direction may be penalized differently by a potential function Ψ((∇ z g) k ). 
     For obtaining a unique solution for the minimization problem, the potential function employed may be required to be convex. 
     According potential functions may be one of the quadratic penalty, the Huber penalty or Huber loss function according to equation 5 or the generalized Gaussian Markov random field according to equation 6 
     
       
         
           
             
               
                 
                   
                     
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     where σ&gt;0 in the Huber penalty as well as p, q, and c (with p≧q≧1 and c&gt;0) in the generalized Gaussian Markov random field are empirical parameters, which are selected upfront. 
     By employing according generalizations, a regularized retrieved phase front may be obtained by solving the minimization problem in accordance with equation 7. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     In equation 7, additional weighting factors w k  may locally emphasize or de-emphasize the smoothing in the z-direction due to the penalty and are real non-negative values. 
     Both equation 4 as well as equation 7 may be seen as seeking to reconstruct an additional constant of integration for each detector row, since N+1 values are estimated for each row from just N data samples, which constant may be considered to be responsible for the streak-like artefacts. 
     In case the object to be examined fits entirely into a single fan of X-ray radiation, the matrix g i,j  may also be an N×N size matrix with the integration constant may be fixed to be 0. Specifically, in order to obtain the differential data d, the measured data are referenced to a so-called blank-scan, i.e., a scan without an object. By referencing the scan with the object to the blank-scan, the processing becomes sensitive to just the changes of the wave-front that the object imposes rather than the phase of the wave-front itself. If the object fits entirely into the fan, the first pixel in each detector row is known to be not affected by the object, thus, the change of the phase compared with the blank-scan is zero. 
     Penalizing, i.e., reducing the gradients in a direction, e.g., the z-direction, may be considered to be very efficient for suppressing streak-like artefacts, in particular horizontal streaks. The streak-like artefacts may be considered to create a gradient over the entire image with a penalty adding up quickly to a substantial value. Accordingly, a streak artefact may be removed by adjusting only a single estimated measured value, i.e., the data penalty term may not change as quickly as the penalty term. This can be easily seen considering as an example the case, where the noise in the data d is dominated by an outlier, i.e., an unusually large noise value, in the data d 21 . By standard phase integration, i.e, by estimating g according to g i,j =Σ j′=1   j  d i,j , the single outlier is propagated by the sum throughout the entire row and thus this outlier will contribute to the gradient in z for a large number of detector pixels. 
     It is to be understood that the application of methods known from the field of inverse, ill-posed problems is neither known nor evident for reducing artefacts within image information, since the actual reconstruction is mathematically not an ill-posed problem but a well-posed problem. 
     Now referring to  FIG. 1 , an exemplary embodiment of an X-ray system according to the present invention is depicted. 
     An X-ray system  100 , exemplarily depicted as CT system  100 , comprises an X-ray generating device  102 , e.g. an X-ray tube, and an X-ray detector  104  comprising individual detector pixel elements  116 . X-ray detector element  104  is exemplarily depicted as a two-dimensional array of detector pixel elements  116 . 
     Both the X-ray generating device  102  and the X-ray detector  104  are arranged on a gantry  106  opposing one another. Gantry  106  allows rotation of both the X-ray generating device  102  and the X-ray detector  104  about an axis. X-ray radiation  114  is generated by X-ray generating device  102  and subsequently arriving at X-ray detector  104 . X-ray generating device  102  may further comprise a source grating element G 0    202 , not depicted in  FIG. 1 , for generating at least partly spatially coherent X-ray radiation  114  for differential phase-contrast imaging. Further grating elements like e.g. a phase grating element G 1    204  as well as an analyzer grating element G 2    206  are not depicted in  FIG. 1 . 
     An object  108  to be examined is arranged on a support  110  adapted to allow moving and placing the object  108  within X-ray beam  114 . 
     A processing system  112  is communicatively coupled to X-ray system  100  for controlling an image acquisition procedure, for reconstructing image data and/or for subsequent presentation, e.g. display, of reconstructed image information. 
     Now referring to  FIG. 2 , an exemplary embodiment of a differential phase-contrast imaging system according to the present invention is depicted. 
     A source grating element G 0    202  is schematically arranged in the vicinity of X-ray generating device  102  for generating at least partly spatially coherent X-ray radiation  114 . Source grating element G 0    202  is spaced apart from phase grating element G 1    204  by distance l. 
     Object  108  is arranged between the X-ray generating device  102  and phase grating element G 1    204 . Apparatus  200 , a phase-contrast imaging system  200 , further comprises an analyzer grating element G 2    206  spaced apart from phase grating element G 1    204  by distance D. Analyzer grating element G 2    206  is arranged in the vicinity of X-ray detector element  104  comprising individual detector pixel elements  116 . 
     A wave front  210   a  is arriving at object  108 , having a uniform phase relationship in accordance with coherent X-ray radiation. After passage of object  108 , the uniform phase relationship may have been influenced, as depicted by wave front  210   b . After passing through phase grating element G 1    204  and analyzer grating G 2    206 , an interference pattern is projected on the X-ray detector element  104  and its individual detector pixel element  116 . 
     An actuator element  208  is exemplarily arranged at analyzer grating element G 2    206  for displacement of analyzer grating element G 2    206  about a fraction of pitch p of phase grating element G 1    204  for acquisition of individual phase-contrast projections during phase stepping, i.e., having individual phase stepping states in a single phase stepping acquisition interval. 
     Actuator element  208  may also be arranged at one of the phase grating element G 1    204  and the source grating element G 0    202 . 
     Now referring to  FIG. 3 , an exemplary embodiment of an apparatus for regularized phase retrieval in phase-contrast imaging according to the present invention is depicted. 
     Processing system  112  exemplarily comprises processing element  300 , which is attached to a storage element  302  and a display element  304 . 
     Processing system  112  is communicatively coupled with X-ray system  100  for controlling the X-ray system, for reconstructing acquired image information and/or for presenting reconstructed image information, e.g., displaying reconstructed image information on display  304 . 
     However, presenting reconstructed image information may also be understood as providing said reconstructed image information to a further processing system or display system and/or storing reconstructed image information locally, e.g., within storage element  302 , or externally in a further storage system, e.g., for long-term storage and archiving. 
     Processing element  300  is adapted to carry out the method  400  for regularized phase retrieval in phase-contrast imaging, in particular for reconstructing acquired or received phase-contrast image data of an object in accordance with the method  400  of the present invention. 
     Now referring to  FIG. 4 , an exemplary embodiment of the method for regularized phase retrieval in phase-contrast imaging according to the present invention is depicted. 
       FIG. 4  shows the method  400  for regularized phase retrieval in phase-contrast imaging, comprising receiving  402  differential phase-contrast image data of an object, generating  404  reconstructed image data of the object and presenting  406  reconstructed image data of the object. 
     The generation of the reconstructed image data may comprise employing regularization methods in accordance with the present invention for recombinant image data in particular for removing or suppressing artefacts like e.g., streak-like artefacts in the reconstructed image data. 
     Presenting reconstructed image data of object  108  may thus be a display of image information to a user of the X-ray system  100 . However, presenting may also comprise storing reconstructed image data for a later display, archival of reconstructed image data, thus permanent or temporal storage of image data and may also comprise providing image data to a further processing system for, e.g., display, storage, or further processing. 
     Now referring to  FIGS. 5 a - c   , an exemplary reconstruction of image information employing differential phase-contrast image data according to the present invention is depicted. 
     In  FIG. 5 a   , two-dimensional differential phase-contrast image data is presented. The image data of  FIG. 5 a    depicts the differential image relationship of individual pixels of the image as acquired by the X-ray detector  104  in differential phase-contrast imaging. While individual columns are referred to with index j, individual rows are referred to with index i. 
     In  FIG. 5 a   , the grey values of the individual pixels substantially depict a difference in image value with respect to adjacent pixels. E.g., in the background of the circular-shaped object  108  of  FIG. 5 a   , a uniform grey value is present, exemplifying substantially no change of image information with regard to adjacent pixels. Accordingly, in  FIGS. 5 b  and 5 c   , the corresponding background of object  108  is depicted as black. Light pixel values at the left hemisphere of object  108  depict a rather steep change in image information, either positive or negative. Dark pixel values at the right hemisphere of object  108  exemplifies a substantially similar steep change in the other direction, thus either negative or positive. Image information in  FIGS. 5 b  and 5 c    are thus obtainable by integration, e.g., along individual rows (i=const), of  FIG. 5 a   . 
       FIG. 5 b    depicts a reconstructed image from the differential phase-contrast imaging information of  FIG. 5 a   , employing a common reconstruction algorithm, e.g., an integration operation. Streak-like artefacts  500 , in  FIG. 5 b    exemplarily depicted as horizontal streaks are present. Image information of  FIG. 5 b    was obtained from the image information of  FIG. 5 a    by row by row integration of image information of  FIG. 5   a.    
     Now when employing method  400  according to the present invention, while reconstructing image information, an X-ray image in accordance with  FIG. 5 c    may be obtained. With the image information of  FIG. 5 c   , in addition to the row by row integration, a gradient operator penalizing gradients in z-direction, i.e., with regard to columns (j=const) is employed. By employing a gradient operator in accordance with the present invention, streak-like artefacts  500  are substantially removed from the reconstructed image data as presented in  FIG. 5 c   . Accordingly by employing a gradient operator in accordance with the present invention, the streak-like artefacts  500  may be reduced or even removed. 
     It should be noted that the term “comprising” does not exclude other elements or steps and that “a” or “an” does not exclude a plurality. Also, elements described in association with different embodiments may be combined. 
     It should also be noted, that reference numerals in the claims shall not be construed as limiting the scope of the claims. 
     LIST OF REFERENCE SIGNS 
     
         
           100  X-ray system 
           102  X-ray generating device/X-ray source 
           104  X-ray detector 
           106  Gantry 
           108  Object 
           110  Support 
           112  Processing system 
           114  X-ray radiation 
           200  Differential phase-contrast imaging system 
           202  Source grating 
           204  Phase grating G 1    
           206  Analyzer grating G 2    
           208  Actuator element 
           210   a,b  Wave front 
           300  Microprocessor 
           302  Storage element 
           304  Display element 
           400  Method for regularized phase retrieval and phase-contrast imaging 
           402  Receiving differential phase-contrast image data 
           404  Generating reconstructed image data 
           406  Presenting reconstructed image data 
           500  Streak-like artefacts