Abstract:
The present invention is intended to provide an image reconstruction method based on an iterative reconstruction technique and characterized by a short image reconstruction time, a small memory capacity, and high image quality. The image reconstruction method comprises the steps of: constructing a system model that has a plurality of concentric rings each segmented into a plurality of sectors; constructing a system matrix associated with one view, by using some of all the sectors constituting the system model, the some of all sectors being located on respective projection lines concerning the one view; creating a system matrix associated with another view, by transforming the system matrix associated with the one view; reconstructing an image through iterative reconstruction by using the system matrix and projection data; and transforming the image into an image composed of rectangular pixels.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of Chinese Application No. 200610073962.5 filed Mar. 10, 2006, which is hereby incorporated by reference in its entirety. 
   BACKGROUND OF THE INVENTION 
   The present invention relates to an image reconstruction method and an X-ray computed tomography (CT) system, or more particularly, an image reconstruction method based on an iterative reconstruction technique and an X-ray CT apparatus that performs the image reconstruction. 
   Iterative reconstruction is known as one of techniques for reconstructing an image on the basis of projection data produced by an X-ray CT apparatus or a positron emission tomography (PET) system. 
   The iterative reconstruction is such that an estimated distribution of coefficients in a system model is sequentially corrected until projection data produced from the estimated distribution of coefficients will agree with projection data produced from actual measurement. The iterative reconstruction may be referred to as statistical reconstruction. Typical examples of the image reconstruction technique include an ordered subsets expectation maximization (OSEM) technique (refer to, for example, Non-patent Document 1). 
   [Non-Patent Document 1] “Study of a Convergent Subsetized List-mode EM Reconstruction Algorithm” by A. Rahmin, T. J. Ruth, and V. Sossi (Nuclear Science Symposium Conference Record 2004 IEEE, Oct. 2004, Vol. 6, pp. 3978-3982) 
   According to the foregoing technique, a spatial array of coefficients formed like a rectangular lattice is adopted as a system model. A system matrix for use in reconstructing an image of the system model has a complex structure and includes an enormous amount of data. The calculation of the system matrix is a large load to be imposed on a computer. 
   As shown in  FIG. 15(   a ), when a stored matrix having been calculated in advance is used, a large memory capacity is required. If a system matrix is, as shown in  FIG. 15(   b ), calculated at every time of image reconstruction, an image reconstruction time gets longer. If the system matrix is simplified in order to shorten the image reconstruction time, image quality deteriorates. 
   SUMMARY OF THE INVENTION 
   Therefore, an object of the present invention is to provide an image reconstruction method based on an iterative reconstruction technique and characterized by a short image reconstruction time, a small memory capacity, and high image quality, and an X-ray CT apparatus that performs such image reconstruction. 
   For accomplishing the above object, according to the first aspect of the present invention, there is provided an image reconstruction method characterized in that: a system model has a plurality of concentric rings each segmented into a plurality of sectors; a system matrix to be associated with one view is constructed with some of all the sectors constituting the system model, the some of all the sectors being located on projection lines concerning the one view; the system matrix associated with the one view is transformed in order to produce a system matrix to be associated with another view; the system matrix and projection data are used to reconstruct an image through interactive reconstruction; and the image is transformed into an image composed of rectangular pixels. 
   For accomplishing the foregoing object, according to the second aspect of the present invention, there is provided an X-ray CT apparatus including data acquisition equipment that scans a subject with X-rays so as to acquire projection data, and image reconstruction equipment that reconstructs an image on the basis of the acquired projection data. The image reconstruction equipment includes: a system model construction means for constructing a system model that has a plurality of concentric rings each segmented into a plurality of sectors; a system matrix construction means for constructing a system matrix associated with one view, by using some of all the sectors constituting the system model, the some of all the sectors being located on respective projection lines concerning the one view; a system matrix creation means for creating a system matrix associated with another view, by transforming the system matrix associated with the one view; an image reconstruction means that reconstructs an image through iterative reconstruction by using the system matrix and projection data; and a transformation means for transforming the image into an image composed of rectangular pixels. 
   Preferably, each of elements constituting the system matrix is a length of a projection line falling within each sector so that the system matrix can be simplified. 
   Preferably, each of elements constituting the system matrix is a partial area of a projection stripe falling within each sector so that the system matrix can be formed precisely. 
   Preferably, the partial area is a normalized area so that the element can be standardized. 
   Preferably, a system matrix to be associated with another view is created at each time of iteration so that the system matrix can be created timely. 
   Preferably, an ordered subsets expectation maximization (OSEM) technique is adopted as the iterative reconstruction technique so that image reconstruction can be achieved properly. 
   Preferably, transformation to an image composed of rectangular pixels is achieved through bilinear interpolation so that the transformation can be achieved properly. 
   Preferably, the projection lines express fan-beam X-rays so that a matrix will be well associated with projection data provided by the fan-beam X-rays. 
   Preferably, the projection lines express parallel-beam X-rays so that a matrix will be well associated with projection data provided by the parallel-beam X-rays. 
   The present invention can provide an image reconstruction method based on an iterative reconstruction technique and an X-ray CT apparatus that performs the image reconstruction. Herein, a system model is constructed to have a plurality of concentric rings each segmented into a plurality of sectors. A system matrix to be associated with one view is constructed using part of all the sectors constituting the system matrix which is located on respective projection lines concerning the one view. The system matrix associated with the one view is transformed in order to create a system matrix to be associated with another view. The system matrix and projection data are used to reconstruct an image through iterative reconstruction. The image is transformed into an image composed of rectangular pixels. This results in a short image reconstruction time, a small memory capacity, and high image quality. 
   Further objects and advantages of the present invention will be apparent from the following description of the preferred embodiments of the invention as illustrated in the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows the configuration of an X-ray CT apparatus that is an example of the best mode for implementing the present invention. 
       FIG. 2  shows the configuration of the X-ray CT apparatus that is an example of the best mode for implementing the present invention. 
       FIG. 3  shows the configuration of X-ray irradiation/detection equipment. 
       FIG. 4  shows the structure of an X-ray incidence surface of an X-ray detector. 
       FIG. 5  outlines a process that is an example of the best mode for implementing the present invention. 
       FIG. 6  shows a system model and projection lines. 
       FIG. 7  shows part of a projection stripe. 
       FIGS. 8   a  and  8   b  show a system model and a projection line in association with two successive views. 
       FIG. 9  describes a process that is an example of the best mode for implementing the present invention. 
       FIG. 10  shows transformation of an image through interpolation. 
       FIG. 11  lists a reconstruction time required by the method in accordance with the present invention in comparison with a reconstruction time required by a conventional method. 
       FIG. 12  includes halftone photographs showing a reconstructed image produced according to the method in accordance with the present invention in comparison with a reconstructed image produced according to the conventional method. 
       FIG. 13  shows a profile of the reconstructed image produced according to the method in accordance with the present invention in comparison with a profile of the reconstructed image produced according to the conventional method. 
       FIG. 14   a   1 ,  14   a   2 ,  14   a   3 ,  14   b   1 ,  14   b   2 ,  14   b   3 ,  14   c   1 ,  14   c   2 , and  14   c   3  are halftone photographs showing the reconstructed image produced according to the method in accordance with the present invention in comparison with a reconstructed image produced according to a filtered back projection (FBP) technique. 
       FIGS. 15   a  and  15   b  show processes employed in the conventional method. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Referring to drawings, the best mode for implementing the present invention will be described below. Noted is that the present invention is not limited to the best mode for implementing the present invention.  FIG. 1  illustrates the configuration of an X-ray CT apparatus. The X-ray CT apparatus is an example of the best mode for implementing the present invention. The configuration of the X-ray CT apparatus is an example of the best mode for implementing the present invention in an X-ray CT apparatus. 
   The X-ray CT apparatus includes a gantry  100 , a table  200 , and an operator console  300 . The gantry  100  uses X-irradiation/detection equipment  110  to scan a subject  10  who lies down on the table  200 , thus acquires transmitted X-ray signals carrying a plurality of views (projection data), and transfers the signals to the operator console  300 . The gantry  100  is an example of data acquisition equipment included in the present invention. 
   The operator console  300  uses a built-in computer to reconstruct an image on the basis of the projection data received from the gantry  100 , and displays the image on a display  302 . The operator console  300  is an example of image reconstruction equipment included in the present invention. 
   The operator console  300  controls the operations of the gantry  100  and table  200  respectively. The gantry  100  scans a subject according to predetermined scanning conditions under the control of the operator console  300 . The table  200  positions the subject  10  so that a predetermined region of the subject will be scanned. The positioning is achieved when a built-in alignment mechanism adjusts the height of a tabletop  202  and a horizontal distance by which a cradle  204  on the tabletop is moved. 
   A scan is performed with the cradle  204  at a halt, whereby an axial scan is achieved. A plurality of scans is continuously performed with the cradle  204  continuously moved, whereby a helical scan is achieved. The cradle  204  is intermittently moved and a scan is performed every time the cradle  204  is halted, whereby a cluster scan is achieved. 
   The height of the tabletop  202  is adjusted by swinging a columnar support  206  with the root of the columnar support at a base  208  as a center. With the swing of the columnar support  206 , the tabletop  202  is displaced vertically and horizontally. The cradle  204  is moved horizontally on the tabletop  202 , whereby the horizontal displacement of the tabletop  202  is canceled out. Depending on scanning conditions, a scan is performed with the gantry  100  tilted. The gantry  100  is tilted by a built-in tilting mechanism. 
   The table  200  may be, as shown in  FIG. 2 , of a type having the tabletop  202  lifted or lowered relative to the base  208 . The tabletop  202  is lifted or lowered by a built-in lifting/lowering mechanism. In this type of table  200 , the lifting or lowering is not accompanied by the horizontal movement of the tabletop  202 . 
     FIG. 3  illustrates the configuration of the X-irradiation/detection equipment  110 . In the X-irradiation/detection equipment  110 , an X-ray detector  150  detects X-rays  134  irradiated from a focal spot  132  in an X-ray tube  130 . 
   The X-rays  134  are reshaped by a collimator that is not shown to be a laterally symmetrical cone beam or fan beam. The X-ray detector  150  has an X-ray incidence surface  152  that spread two-dimensionally along with the fanning out of the X-rays. The X-ray incidence surface  152  is curved in the form of part of a cylinder. The center axis of the cylinder passes through the focal spot  132 . 
   The X-irradiation/detection equipment  110  rotates about a center axis passing through a radiographic center, that is, an isocenter O. The center axis extends parallel to the center axis of the partial cylinder formed by the X-ray detector  150 . 
   Assume that the direction of the center axis of rotation is a z direction, the direction linking the isocenter O and the focal spot  132  is a y direction, and the direction perpendicular to the z and y directions is an x direction. The x, y, and z axes serve as three axes of a rotating coordinate system that rotates with the z axis as a center axis. 
     FIG. 4  is a plan view illustrating the X-ray incidence surface  152  of the X-ray detector  150 . The X-ray incidence surface  152  has detector cells  154  two-dimensionally arrayed in the x and z directions. In other words, the X-ray incidence surface  152  is formed with the two-dimensional array of the detector cells  154 . When fan-beam X-rays are employed, the X-ray incidence surface  152  may be formed with a one-dimensional array of the detector cells  154 . 
   Each of the detector cells  154  serves as a detection channel in the X-ray detector  150 . Consequently, the X-ray detector  150  is a multi-channel X-ray detector. Each of the detector cells  154  is composed of, for example, a scintillator and a photodiode. 
     FIG. 5  outlines an image reconstruction process to be executed in the operator console  300 . The process is an example of the best mode for implementing an image reconstruction method in accordance with the present invention. The process is executed by a built-in computer. 
   As mentioned in  FIG. 5 , iterative reconstruction is performed at step  101 . At step  103 , interpolation is performed in order to display an image. The iterative reconstruction of step  101  is performed based on, for example, the ordered subsets expectation maximization (OSEM) technique. Consequently, image reconstruction can be achieved properly. 
   The iterative reconstruction may not be performed based on the OSEM technique but may be performed based on a multiplicative algebraic reconstruction (MART) technique or a paraboloidal surrogate coordinate descent (PSCD) technique. The image reconstruction based on the OSEM technique will be described below. Noted is that the same applies to the image reconstruction based on the MART or PSCD technique. 
   For the iterative reconstruction, projection data and a system matrix calculated at step  105  are employed. A partial matrix stored in advance is used to calculate the system matrix. 
   The partial matrix will be described below. Beforehand, a system model with which the partial matrix is concerned will be described.  FIG. 6  illustrates the system model. As illustrated in  FIG. 6 , the system model has a plurality of concentric rings. 
   Each of the rings is segmented into equiangular sectors. An angle in units of which each ring is segmented is equal to an angle step of view, or an integral multiple thereof. The number of sectors into which each ring is segmented need not be the same among all rings. The number of sectors into which the outermost ring is segmented may be the largest. In this case, the lengths of respective sectors become nearly uniform over an entire system model. Moreover, the widths of respective rings need not be the same but may vary. 
   A plurality of projection lines extending radially from a source expresses X-rays that provide projection data constituting one view (i denotes a view number) of the system model. The projection lines express respective X-rays of, for example, fan-beam X-rays. A projection line expressing a center X-ray of the fan-beam X-rays shall pass through the center of the system model and be perpendicular to one of the radii of the rings. Using the projection lines, a matrix can be well associated with the projection data provided by the fan-beam X-rays. 
   When parallel-beam X-rays are substituted for the fan-beam X-rays, the plurality of projection lines extends parallel to one another, and is perpendicular to one of the radii of the rings. Using the projection lines, a matrix can be well associated with projection data provided by the parallel-beam X-rays. The description will proceed by taking a case, where the radial projection lines are employed, for instance. Noted is that the same applies to a case where the parallel projection lines are employed. 
   The projection lines pass through specific sectors of the system model. One sector (highlighted) through which the j-th projection line passes shall be focused. A partial matrix is defined as a set of such sectors through which all projection lines pass while expressing X-rays that provide one view. The partial matrix may be regarded as a system matrix associated with a view i. A set of partial matrices associated with all views is regarded as a system matrix. Hereinafter, the partial matrix may be simply called a matrix. 
   Each of elements constituting a system matrix shall be a length aij of a projection line j falling within each sector. This definition simplifies the system matrix. 
   Each of the elements may be defined as a partial area aik of a projection stripe k falling within each sector (highlighted). This definition realizes a precise system matrix. 
     FIG. 7  shows the partial area in detail. A hatched part indicates the partial area of a projection stripe falling within a sector. The size of the area is expressed with an area Sik normalized as follows:
   aik=Sik /(( l   top   +l   bottom )/2)  Formula 1 
   Herein, ltop and lbottom denote the lengths of the arcs in a stripe k on the side of the source and an opposite side which define a sector. Thus, each element is standardized. 
   Referring to  FIG. 8 , the property of a partial matrix will be described.  FIG. 8(   a ) and  FIG. 8(   b ) show scenes where a projection line j that passes through a system model expresses one of X-rays providing either of two successive views  1  and  2 . Herein, for brevity&#39;s sake, the number of views is eight in total. 
   As shown in  FIG. 8(   a ) and  FIG. 8(   b ), the projection line j passes through a sector  1  while expressing one of X-rays that provide the view  1 , while the projection line j passes through a sector  2  while expressing one of X-rays that provide the view  2 . Since an angle in units of which each ring of a system model is segmented into sectors is equal to the angle step of the view, the positions of the sectors  1  and  2  through which the projection line j passes while expressing one of X-rays that provide the view  1  or  2  are identical to each other. 
   Consequently, the scene where the projection line j passes through the sector  2  while expressing one of X-rays that provide the view  2  is identical to the scene where the projection line j passes through the sector  1  while expressing one of X-rays that provide the view  1 . Consequently, an element aij included in the partial matrix associated with the view  2  assumes the same value as an element aij included in the partial matrix associated with the view  1 . 
   The same relationship is established between all projection lines expressing X-rays that provide the view  1  and all projection lines expressing X-rays that provide the view  2 , and also established among all other projection lines expressing X-rays that provide a view  3 , those expressing X-rays that provide a view  4 , those expressing X-rays that provide a view  5 , those expressing X-rays that provide a view  6 , those expressing X-rays that provide a view  7 , and those expressing X-rays that provide a view  8 . Consequently, the structures of partial matrices associated with all the views are the same as one another except view numbers. Once a partial matrix associated with one view is calculated and stored in advance, therefore, all the remaining partial matrices can be created through transformation of the stored partial matrix. The same applies to a case where a partial area of a projection stripe falling within each sector is regarded as an element. 
     FIG. 9  details an image reconstruction process. As described in  FIG. 9 , initialization and reading of projection data and a matrix are performed at step  501 . The matrix is calculated and stored in a memory in advance. The matrix is a partial matrix, or in other words, a system matrix associated with one view. The calculation is not a large load imposed on a computer, and the storage of the matrix does not consume a large memory capacity. 
   At step  503 , a subset of projection data is selected. At step  505 , a matrix to be associated with the subset is calculated. The matrix to be associated with the subset is calculated from the partial matrix read from the memory. The calculation includes simple transformation and therefore imposes only a small load on the computer. 
   At step  507 , ordered subsets expectation maximization (OSEM) is iterated once. At step  509 , whether the iteration is continued is determined. If the iteration is recognized to be necessary, control is returned to step  503 . A subset of projection data is selected at step  503 , a matrix is calculated at step  505 , and OSEM is iterated at step  507 . Thus, an image is reconstructed through iterative reconstruction. Since a matrix to be associated with a subset is calculated at every time of iteration of OSEM, a system matrix can be created timely. 
   If iteration is recognized to be unnecessary at step  509 , interpolation is performed at step  511 . Since a reconstructed image is, as shown in the left of  FIG. 10 , an image composed of sector pixels, the image is transformed into an image composed of, as shown in the right of  FIG. 10 , rectangular pixels through the interpolation. As the interpolation, for example, bilinear interpolation is adopted. Consequently, an image can be transformed appropriately. However, the interpolation is not limited to the bilinear interpolation but may be performed using an appropriate algorithm. 
     FIG. 11  lists a time required for the foregoing image reconstruction in comparison with a time required for image reconstruction to be performed according to a conventional method. As listed in  FIG. 11 , when the method in accordance with the present invention is employed, a speed of image reconstruction increases to be approximately four to twenty times higher than a conventional speed. 
     FIG. 12  shows a reconstructed image of a phantom produced according to a conventional method in comparison with a reconstructed image thereof produced according to the method in accordance with the present invention, (a) shows a real image of the phantom, (b) shows the reconstructed image produced according to the conventional method, and (c) shows the reconstructed image produced according to the method in accordance with the present invention. As apparent from  FIG. 12 , the method in accordance with the present invention provides the reconstructed image that is as good as the reconstructed image produced according to the conventional method. 
     FIG. 13  shows one-dimensional profiles indicating the real image of the phantom and the reconstructed images produced according to the conventional method and the method in accordance with the present invention respectively. A dashed line depicts the profile of the real image of the phantom. An alternate long and two short dashes line depicts the profile of the reconstructed image produced according to the conventional method. A solid line depicts the profile of the reconstructed image produced according to the method in accordance with the present invention. The profiles of the reconstructed imaged produced according to the conventional method and the method in accordance with the present invention respectively have no difference. The alternate long and two short dashes line and solid line are fully superposed on each other. 
     FIG. 14  shows reconstructed images of three phantoms produced according to the method in accordance with the present invention in comparison with reconstructed images thereof produced through filtered back projection (FBP). In  FIG. 14 , ( a   1 ), ( a   2 ), and ( a   3 ) show the reconstructed images produced through FBP, (b 1 ), (b 2 ), and (b 3 ) show the reconstructed images produced according to the method in accordance with the present invention, and (c 1 ), (c 2 ), and (c 3 ) show difference images. As apparent from  FIG. 14 , the method in accordance with the present invention provides images having nearly no difference from those produced through FBP. 
   Many widely different embodiments of the invention may be configured without departing from the spirit and the scope of the present invention. It should be understood that the present invention is not limited to the specific embodiments described in the specification, except as defined in the appended claims.