Abstract:
A method for the reduction of motion artifacts in a helical CT reconstruction, the method including the steps of: (a) reconstructing from the raw CT data an initial estimate of the 3D attenuation distribution of the object of interest; (b) estimating a pose parameter set of the object for each projection angle; (c) undertaking a motion corrected reconstruction from the measured projections, accounting for the pose changes estimated in (b); (d) iterating steps (b)-(c) until some convergence criterion is met; and (e) making a final reconstruction of diagnostic quality using the pose estimates obtained in the previous steps.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention provides for systems and methods for the automated correction of motion artefacts in CT images, and, in particular, helical CT images. 
       BACKGROUND OF THE INVENTION 
       [0002]    Any discussion of the background art throughout the specification should in no way be considered as an admission that such art is widely known or forms part of common general knowledge in the field. 
         [0003]    One of the major sources of image artifacts in computed tomography (CT) is patient motion, which creates inconsistencies between acquired projections, leading to distortion and blurring when images are reconstructed. These motion artifacts may lead to false diagnosis, or in extreme cases, render images uninterpretable. Head motion is a common problem in young patients who are often sedated or anesthetized to prevent motion. According to the latest available data, over 70 million CT scans are performed annually in the USA alone, of which approximately 10% are performed in children. Moreover, a recent survey of CT practice in developing countries revealed that about 75% of pediatric CT scans were of the head. Due to the relatively high radiation dose associated with CT scanning, it is undesirable to repeat the scan if motion occurs, particularly in children who have a much higher estimated lifetime risk of radiation-induced cancer than adults. In adults, head motion is a problem for patients suffering from claustrophobia or a mental or behavioral incapacity, and in patients with head trauma. In a recent study, head movements classified as moderate or severe were observed in 25% of 103 patients with acute ischemic stroke during CT brain perfusion scans. 
         [0004]    A slight movement of the patient will lead to a reduction of spatial resolution in CT imaging, in severe cases resulting in corrupted images unsuitable for diagnosis or further processing. To reduce the likelihood of motion artifacts, CT manufacturers have made scanners faster by increasing the number of detector rows and the rate of rotation of the x-ray source and detector, which is not a complete solution. Several motion correction methods have been suggested in the literature, the majority of which are intended for circular cone beam CT (CBCT), and few studies have addressed motion correction for helical CT. Motion correction is arguably simpler in CBCT since the entire object will normally be in the field of view at all times. In contrast, in helical CT, the object is always truncated in axial direction, which complicates the application of analytical motion correction algorithms. 
         [0005]    Assessment of the subject motion is of considerable general interest in tomography. A variety of methods for the estimation of motion in CT exist, including direct motion estimation using a camera system with visual markers or without markers. Artificial or anatomical landmarks can be also tracked in the image or projection domain. Indirect estimation methods have been proposed where motion is estimated through the minimization of errors in consistency conditions using projection moments, or by an iterative process using re-projected image information. Another approach has used similarity measures to quantify changes between projections to measure subject motion. Previously, there has been some progress in applying rigid motion correction techniques to helical brain scans, by measuring the head motion with a Polaris system (Spectra, Northern Digital Inc, Waterloo, Canada). 
         [0006]    Even with these techniques, errors are still visible in the resulting images. The possible reason is that there still is some residual unrecorded motion in each pose, caused by small errors in the pose measurements. This will result in ‘jagged’ artifacts which are most visible at the edge of the phantom. 
         [0007]    It would be desirable to provide for improved processing of CT imagery to reduce the level of motion artifacts. 
       SUMMARY OF THE INVENTION 
       [0008]    It is an object of the invention, in its preferred form to provide an improved form of processing of helical CT images. 
         [0009]    In accordance with a first aspect of the present invention, there is provided a method for the reduction of motion artifacts in a helical CT reconstruction, the method including the steps of: (a) reconstructing from the raw CT data an initial estimate of the 3D attenuation distribution of the object of interest; (b) estimating a pose parameter set of the object for each projection angle; (c) undertaking a motion corrected reconstruction from the measured projection, accounting for the pose changes estimated in (b); (d) iterating steps (b)-(c) until a convergence criterion is met; (e) determining a final reconstruction of diagnostic quality using the pose estimates obtained in the previous steps. 
         [0010]    The step (b) preferably can include the minimization of a cost function including a difference measure of the measured projection image and an image produced by forward projection from a current pose estimate for the object. 
         [0011]    The step (c) preferably can include forming a motion corrected image from the pose parameter set, using a reconstruction algorithm that accounts for the motion. 
         [0012]    Preferably, the iterations (b)-(c) are performed across multiple resolutions of the helical CT image, with the multiple resolutions being of increasing fidelity. 
         [0013]    In some embodiments, the step (b) can further include estimating a pose parameter set including three dimensional rotation and translation of the object of interest. In some embodiments, one of said translations is perpendicular to the CT image sensor and need not be estimated as it has a negligible effect on the measured CT projection image. In some embodiments, the step (b) can include applying an ordered subset expectation maximization reconstruction algorithm to the measured CT projections and the pose parameter set. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]    Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which: 
           [0015]      FIG. 1  illustrates schematically the scanner and detector coordinate systems on which motion correction is based. 
           [0016]      FIG. 2  illustrates a flow diagram of the general motion estimation scheme. μ is the update of the attenuation image, S is the update of the rigid transform, n is the iteration number. A multi-resolution scheme was applied to the motion estimation, increasing the sampling as the iteration number. The motion update uses the current reconstruction image μ n , the image update uses the current motion update S n . In the figure the resolution is increased for every iteration, but in practice, multiple iterations can be typically carried at each resolution level. 
           [0017]      FIG. 3  illustrates schematically that the effect of object translation or rotation parallel to the detector can be well approximated as translation and rotation of the projection. For simplicity, the curvature of the detector was ignored. t is the magnification factor from the object to detector. However, for the two other rotations, more complicated calculations are required to compute their effect on the projection. 
           [0018]      FIG. 4  illustrates selected transaxial and coronal slices from reconstructions without and with motion correction from a simulation experiment with motion, and also from the true image. The simulated motions were based on recorded motions from human volunteers. 
           [0019]      FIG. 5  and  FIG. 6  are graphs of the estimated motion values as a function of the view angle for the two most prominent motion parameters, with  FIG. 5  showing φ x , and  FIG. 6  showing t x . 
           [0020]      FIG. 7  shows a selected transaxial plane, without (left) and with correction (right) for motion artifacts in a clinical CT scan. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0021]    The preferred embodiments provide a data-driven method to iteratively correct for motion errors together with the reconstruction of helical CT imagery. In every iteration, an estimate of the motion is undertaken, view-by-view, which then can be used to update a system matrix which is used during reconstruction. A multi-resolution scheme was used to speed up the convergence of the joint estimation of the motion and reconstruction. The quality of the reconstructed images can be improved substantially after the correction. 
         [0022]    The approach was found to reduce or eliminate the motion artifacts in helical-CT reconstruction. The correction method was found to only need the measured raw helical scan data, hence it is called a ‘data-driven’ approach. 
         [0023]    The Method 
         [0024]    Coordinate Systems: 
         [0025]    Turning initially to  FIG. 1 , there is shown the operational aspects  10  of a helical CT scan and associated coordinate systems. A clinical helical CT system usually has a curved detector  12 . It is possible to define a world coordinate system c′=(x, y, z)ε           3    14 , which is fixed with respect to the scanner, its z-axis coinciding with the rotation axis of the scanner. Detector coordinate system c=(u. v, z′)ε           3    13  is fixed with respect to the rotating source-detector system  12 : Its origin  13  is in the center of the detector  12 , z′ is parallel to z, u is tangent and v is orthogonal to the detector. 
         [0026]    For one projection view, a rigid motion in the world coordinate system may be written as: 
         [0000]        S   world =(φ x ,φ y ,φ z   ,t   x   ,t   y   ,t   z′ ) T   (1)
 
         [0000]    where φ x , φ y , φ z , are the 3 rotations, t x , t y , t z , are the 3 translations in the world coordinate system. The transform can be mapped into the detector coordinate system c as: 
         [0000]        S   detector =(φ u ,φ v ,φ z   ,t   u   ,t   v   ,t   z′ ) T   (2)
 
         [0000]    where φ u , φ v , φ z , are the 3 rotations, t u , t v , t z ′, are the translations in the detector coordinate system. A small motion t v  in the direction perpendicular to the detector results in a very small magnification of the projection, which can usually be assumed negligible. In every projection view, then, one can set t v  to zero and only 5 parameters need to be estimated in the correction scheme: φ u , φ v , φ z , t u  and t z′ : 
         [0000]        S   detector =(φ u ,φ v ,φ z   ,t   u   t   z′ ) T   (3)
 
         [0027]    The estimated motion in the detector coordinate system can later be transformed to the motion in the world coordinate system, as the reconstruction requires a transform in the world coordinate system: 
         [0000]    
       
      
       S 
       detector 
       →T 
       detector  
      
     
         [0000]        T   world   =RT   detector   (4)
 
         [0000]    where T is the 4×4 homogenous matrix representation of the transform, and R is the 4×4 transformation matrix that maps the detector coordinate system to the world coordinate system. 
         [0028]    B. OSEM Reconstruction 
         [0029]    In the presence of object motion, the helical CT-orbit is distorted into an effective orbit with arbitrary shape. This necessitates the use of a reconstruction algorithm with the ability to reconstruct from projections acquired along an arbitrary orbit. One such algorithm is Ordered Subset Expectation Maximization (OSEM). The OSEM-algorithm can be used for convenience, but if the use of a better noise model would be required, it can be replaced with, for example, a dedicated iterative algorithm for transmission tomography. Analytical algorithms capable of dealing with the motion could also be considered. 
         [0030]    C. General Motion Correction Scheme 
         [0031]    Turning to  FIG. 2 , there is illustrated a flow chart  20  of the method of the data-driven motion correction algorithm of this embodiment. The motion-corrected reconstruction and motion were alternately updated  26  to increase the likelihood, and the iterations were stopped when the motion estimate appeared to have converged  22 . 
         [0032]    The implementation involves: (1) a 2D-3D image registration to update the motion estimate for each view at the current iteration; (2) an image update  24  with iterative reconstruction, incorporating the updated motion in the system matrix; (3) alternate updates of both image and motion  23  with a multi-resolution scheme that increases the sampling as the number of iterations increases; (4) final reconstruction  25  with a system matrix based on the last motion estimate. 
         [0033]    1) Motion Update  24   
         [0034]    From the measured raw data an initial image is reconstructed. This image can be a reconstruction produced with the system software (postprocessed to convert Hounsfield units back to attenuation integrals), or a first iterative reconstruction (Eq. 5) from the measured data. 
         [0035]    For one projection line i, we integrate along the projection line to define the forward projection of the current estimate μ. The index j indicates the voxel index, μ j  is the attenuation value at voxel j, and α ij  is the sensitivity of sinogram bin i for activity in voxel j: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                     i 
                   
                   = 
                   
                     
                       ∑ 
                       j 
                     
                      
                     
                         
                     
                      
                     
                       
                         a 
                         ij 
                       
                        
                       
                         μ 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0036]    In helical CT, the line integrals are organized in views, where view θ contains all line integrals associated with a single source position: 
         [0000]        f   θ   ={f   i }  (6)
 
         [0037]    For each view, the 5 (or 6) motion parameters are estimated one after the other. Suppose the general motion correction scheme ( FIG. 2 ) is at the iteration n, hence the current motion update is s n . Assuming that the motion represented by the (rotation or translation) parameter ŝ is small, the derivative of projection f with respect to ŝ can be approximated as a finite difference of the intensities: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         f 
                         θ 
                       
                     
                     
                       ∂ 
                       s 
                     
                   
                   ≈ 
                   
                     
                       
                         
                           f 
                           θ 
                         
                          
                         
                           ( 
                           
                             s 
                             ^ 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           f 
                           θ 
                         
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                           ( 
                           
                             s 
                             n 
                           
                           ) 
                         
                       
                     
                     
                       s 
                       ^ 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where f θ (s n ) is the calculated re-projection (using the current estimates for the image and motion) and f θ (ŝ) is the measured projection for view θ. 
         [0038]    To estimate ŝ in Eq. 7, it is necessary to know the derivative on the left hand side, hence another equation is introduced which is very similar to Eq. 7: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         f 
                         θ 
                       
                     
                     
                       ∂ 
                       s 
                     
                   
                   ≈ 
                   
                     
                       
                         
                           f 
                           θ 
                         
                          
                         
                           ( 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             s 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           f 
                           θ 
                         
                          
                         
                           ( 
                           
                             s 
                             n 
                           
                           ) 
                         
                       
                     
                     
                       Δ 
                        
                       
                           
                       
                        
                       s 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Δ s  is a known small increment of the parameter to be estimated. When Δ s  represents a translation, f θ (Δs) can be approximated as a simple translation of the current reprojection f θ (s n ), for in-plane rotation, again f θ (Δs) can be computed as a simple rotation of the re-projection f θ , as shown in  FIG. 3 . For the two out-of-plane rotations (and for the motion towards the detector, in cases where it cannot be ignored), f θ (Δs) is calculated with a forward projection using a system matrix adjusted with Δ s . Eq. 7 and 8 assume that a small increment of one degree of freedom rigid motion only results in a linear change of the intensities in the projection. All the above lead to a least squares minimization problem for the current view at the current iteration: 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     incre 
                     n 
                   
                   = 
                   
                     
                       
                         arg 
                          
                         
                             
                         
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                         ^ 
                       
                     
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                          
                         
                           
                             
                               
                                 
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                                     · 
                                     
                                       [ 
                                       
                                         
                                           
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                                                 s 
                                                 n 
                                               
                                               + 
                                               
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                                       ] 
                                     
                                   
                                 
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                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0039]    To find S incre   n , Eq. 9 is solved analytically. Defining 
         [0000]        P   θ   =f   θ ( s   n   +ŝ )− f   θ ( s   n )
 
         [0000]        Q   θ   =f   θ ( s   n   +Δs )− f   θ ( s   n )  (10)
 
         [0000]    and setting the derivative of the Eq. 9 with respect to ŝ to zero, one obtains: 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     incre 
                     n 
                   
                   = 
                   
                     
                       
                         
                           
                             ∑ 
                             N 
                           
                            
                           
                               
                           
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                               P 
                               θ 
                             
                             · 
                             
                               
                                 ∑ 
                                 N 
                               
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                                
                               
                                 Q 
                                 θ 
                               
                             
                           
                         
                         
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                               N 
                             
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                       · 
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                      
                     s 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where N is total number of voxels in one projection view θ. 
         [0040]    This procedure can be applied to estimate all five parameters in Eq. 3 (or all 6 parameters from Eq. 2). The sequence of the estimation is translation first, then rotation. The newly estimated parameter values are used immediately when estimating the value of the next parameter. This sequential estimation of the five motion parameters for all projection views completes the update of the rigid motion in the current iteration. Then the rigid motion parameters for each projection view are transformed into a homogenous matrix in the detector coordinate system ( FIG. 1 ). 
         [0041]    Applying the Eq. 4, the matrix is mapped into the world coordinate system. The transformation matrix obtained in the current iteration (n) is then used to update the previous motion estimate for every view, which will be used in the next iteration (n+1): 
         [0000]    
       
      
       S 
       incre 
       n 
       →ΔT 
       θ 
       n  
      
     
         [0000]        T   θ   n+1   =T   θ   n   ·ΔT   θ   n   (12)
 
         [0042]    2) Image Update  24   
         [0043]    After obtaining the motion, the image representing the attenuation coefficients can be updated with a suitable reconstruction algorithm such as OSEM, a dedicated iterative algorithm for transmission tomography, or an analytical algorithm capable of dealing with motion. 
         [0044]    Instead of moving the reconstruction image, rigid motion correction is done by considering a coordinate system fixed to the object and incorporating the motion (now associated to the source-detector pair) into the system matrix. This corresponds to arbitrary 3-dimensional (3D) motion of the virtual gantry around the object being scanned given by the inverse of the object&#39;s motion. Motion correction is enabled by introducing a modified version of the standard reconstruction algorithm. As an example, using OSEM, we have: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           T 
                           ^ 
                         
                         i 
                         
                           n 
                           + 
                           1 
                         
                       
                       = 
                       
                         invert 
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             T 
                             i 
                             
                               n 
                               + 
                               1 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         μ 
                         j 
                         
                           n 
                           + 
                           1 
                         
                       
                       = 
                       
                         
                           
                             μ 
                             j 
                             n 
                           
                           
                             
                               ∑ 
                               
                                 i 
                                 ∈ 
                                 
                                   S 
                                   b 
                                 
                               
                             
                              
                             
                                 
                             
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                                   1 
                                 
                               
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                                   ij 
                                 
                                 ) 
                               
                             
                           
                         
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                               ∈ 
                               
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                                 b 
                               
                             
                           
                            
                           
                               
                           
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                                   ^ 
                                 
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                                   n 
                                   + 
                                   1 
                                 
                               
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                                 ( 
                                 
                                   a 
                                   ij 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 f 
                                 i 
                               
                               
                                 
                                   ∑ 
                                   k 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       
                                         T 
                                         ^ 
                                       
                                       i 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         a 
                                         ik 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     μ 
                                     k 
                                     n 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0045]    where i is the projection line index, j is the voxel index, a ij  is the effective intersection length of line i with voxel j, f is the log converted sinogram, S b  is one subset of projections of b subsets. {circumflex over (T)} i  is a 4×4 rigid transformation matrix applied to the projection line i. If T i  is the identity matrix for all projection lines, then Eq. 13 is the same as conventional OSEM. In helical CT, T i  is constant for all projection lines in one projection view. Because of the high rotation speed and the large number of views, the motion within a single view is negligible. In one example, the distance-driven projector was used for interpolation during the (back) projection. 
         [0046]    The new estimation of the attenuation from step 2 (the image update) can then be used for next motion update step (step 1). 
         [0047]    3) Multi-Resolution Alternate Updates 
         [0048]    By repeating steps 1 and 2, one can estimate the reconstruction (Eq. 13) and motion (Eq. 11) alternately. Because the correction of the image and the correction of the transform parameters are jointly estimated from the measured data, the problem of error propagation is mitigated. 
         [0049]    An approach to reduce the computation time is to apply multi-resolution techniques. The embodiments utilize this technique by running the algorithm with a coarse to fine representation of the raw data and the image. As in  FIG. 2 , the image update can be reconstructed at coarse resolution at early iterations, with the resolution increasing as the iteration numbers increase ( 27 ,  28 ). Similarly, the projections in Eq. 10 can be computed with gradually increased resolution. A possible additional advantage of the multi-resolution scheme is that it may help to avoid convergence to an undesired local maximum. Since the computations at the finest resolution are the most expensive ones in the multiresolution scheme, in one example, the iterations were stopped at the one but finest resolution. 
         [0050]    As proposed in an article published by J.-H. Kim et. al in the Journal of Physics in Medicine and Biology, the motion estimates can be smoothed (by filtering each component independently) to remove outliers. A Savitzky-Golay filter can be applied after every motion update. It is found that, with an appropriate selection of the kernel size, this filter can achieve satisfactory jitter suppression in simulations and patient scans. The optimal kernel size depends on the CT projection acquisition rate and the axial detector extent. 
         [0051]    It is not obvious how to define good stopping criteria for the motion estimation, especially considering that a ground truth image is unavailable for clinical data. In the motion estimation scheme, a maximum number of iterations is chosen for each resolution level. In addition, the summation of projection errors between the re-projected and measured data over all the views is computed, and at each resolution level, the iterations are stopped earlier when the relative change of this error measure is less than some fraction (e.g. 5%) of the summed error between the last re-projected and measured data. 
         [0052]    4) Final Reconstruction  25   
         [0053]    When the motion estimate has converged, a final reconstruction of diagnostic quality is produced. In simulation studies the final reconstruction is started with the last image update from the alternate updates. To achieve a similar speedup in clinical studies the final reconstruction can be started from an approximate helical Feldkamp-Davis-Kress reconstruction (motion correction was enabled in the backprojection step), provided the image is not affected too much by motion artifacts. 
         [0054]    To further accelerate the final reconstruction, the forward and backward projection operations may be implemented in a language supporting parallel processing, such as OpenCL, and run on a GPU (e.g. NIVIDIA Tesla C2075) or multi-core computer. 
         [0055]    Experiments and Results 
         [0056]    Simulation: 
         [0057]    In the simulation, a segment of measured motion from a volunteer was applied to a voxelized phantom to generate a simulated scan. Reconstructions from this scan were analyzed quantitatively to assess the performance of the motion correction algorithm. The phantom was a 3D voxelized phantom from the Visible Human Project. The unit was converted from Hounsfield (HU) to attenuation coefficient (cm −1 ) at peak kilovoltage of 70 kVp. The image size was 256×256×240, pixel size was 1×1×1 mm 3 . A helical scan with a Siemens Definition AS CT scanner (Siemens Medical Solutions USA, Inc., Malvern, Pa.) was simulated, with reduced angular sampling to reduce the computation times. The crucial parameters were: angles per rotation 250, pitch 1.0, collimation 32×1.2 mm. The motion was applied to the phantom for the simulated helical scan. To avoid cone-beam artifacts, all simulated helical scans covered a bit more than the entire object. 
         [0058]    OSEM was used for all reconstructions, with motion correction enabled (Eq. 13). During the joint estimation of the attenuation image and the motion, the attenuation image was updated using 1 OSEM iteration with 40 subsets. Reconstruction pixel size was 1×1×1 mm 3  at the finest resolution. Alternate updates of both image and motion were performed within a multi-resolution scheme to obtain the optimal motion. For the final reconstruction, 4 iterations and 60 subsets were applied.  FIG. 4  shows the estimated reconstruction (images inverted for reproduction).  FIG. 5  and  FIG. 6  are graphs of the estimated motion values as a function of the view angle for the two most prominent motion parameters, with  FIG. 5  showing φ x , and  FIG. 6  showing t x . 
         [0059]    Real Scan: 
         [0060]    The method was applied to clinical studies in which motion artifacts had been observed. The outcome was evaluated by assessing the image visually. The anonymized raw data of one patient who had previously undergone a head CT scan in the Department of Radiology at Westmead Hospital, Sydney, Australia, were collected with the approval of the Human Research Ethics Committee of the Western Sydney Local Health District. The scan was performed on a Siemens Force scanner (pitch 0.55, tube voltage 120 kVp, tube current 150 mAs, angles per rotation 1050, collimation 64×0.6 mm). Flying focus was turned on in both z and phi directions. 
         [0061]    Because of the huge size of the raw data, the data was read and the average of every 8 projections was used for the motion estimation. This accelerated both the motion updates and the image updates. OSEM was used as the reconstruction algorithm. The motion correction was enabled for all reconstructions. Unlike in the simulation, 2 OSEM iterations with 40 subsets were done for the image updates. The final reconstruction pixel size was 0.40039×0.40039×0.75 mm 3 , image size was 512×512×436. To accelerate the motion estimation, the multi-resolution scheme was applied as described above. Stopping criteria are also described above. For the final reconstruction with motion correction, the starting image was set as the image from a helical FDK reconstruction. Six OSEM iterations with 60 subsets were applied in combination with Nesterov&#39;s acceleration [ 21 ] on the GPU.  FIG. 7  shows (inverted views of) the original reconstructed image produced by the CT vendor software and the motion-corrected reconstructed image. 
         [0062]    The forgoing described embodiments provide a motion estimation and correction approach for helical X-ray CT of the head and other rigidly moving parts of a body, only based on the measured raw data. Since no additional measurements are needed, it can be applied retrospectively to standard helical CT data. 
       Interpretation 
       [0063]    Reference throughout this specification to “one embodiment”, “some embodiments” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment”, “in some embodiments” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment, but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments. 
         [0064]    As used herein, unless otherwise specified the use of the ordinal adjectives “first”, “second”, “third”, etc., to describe a common object, merely indicate that different instances of like objects are being referred to, and are not intended to imply that the objects so described must be in a given sequence, either temporally, spatially, in ranking, or in any other manner. 
         [0065]    In the claims below and the description herein, any one of the terms comprising, comprised of or which comprises is an open term that means including at least the elements/features that follow, but not excluding others. Thus, the term comprising, when used in the claims, should not be interpreted as being limitative to the means or elements or steps listed thereafter. For example, the scope of the expression a device comprising A and B should not be limited to devices consisting only of elements A and B. Any one of the terms including or which includes or that includes as used herein is also an open term that also means including at least the elements/features that follow the term, but not excluding others. Thus, including is synonymous with and means comprising. 
         [0066]    As used herein, the term “exemplary” is used in the sense of providing examples, as opposed to indicating quality. That is, an “exemplary embodiment” is an embodiment provided as an example, as opposed to necessarily being an embodiment of exemplary quality. 
         [0067]    It should be appreciated that in the above description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, FIG., or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed invention requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing disclosed embodiment. Thus, the claims following the Detailed Description are hereby expressly incorporated into this Detailed Description, with each claim standing on its own as a separate embodiment of this invention. 
         [0068]    Furthermore, while some embodiments described herein include some but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those skilled in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination. 
         [0069]    Furthermore, some of the embodiments are described herein as a method or combination of elements of a method that can be implemented by a processor of a computer system or by other means of carrying out the function. Thus, a processor with the necessary instructions for carrying out such a method or element of a method forms a means for carrying out the method or element of a method. Furthermore, an element described herein of an apparatus embodiment is an example of a means for carrying out the function performed by the element for the purpose of carrying out the invention. 
         [0070]    In the description provided herein, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description. 
         [0071]    Similarly, it is to be noticed that the term coupled, when used in the claims, should not be interpreted as being limited to direct connections only. The terms “coupled” and “connected,” along with their derivatives, may be used. It should be understood that these terms are not intended as synonyms for each other. Thus, the scope of the expression a device A coupled to a device B should not be limited to devices or systems wherein an output of device A is directly connected to an input of device B. It means that there exists a path between an output of A and an input of B which may be a path including other devices or means. “Coupled” may mean that two or more elements are either in direct physical or electrical contact, or that two or more elements are not in direct contact with each other but yet still co-operate or interact with each other. 
         [0072]    Thus, while there has been described what are believed to be the preferred embodiments of the invention, those skilled in the art will recognize that other and further modifications may be made thereto without departing from the spirit of the invention, and it is intended to claim all such changes and modifications as falling within the scope of the invention. For example, any formulas given above are merely representative of procedures that may be used. Functionality may be added or deleted from the block diagrams and operations may be interchanged among functional blocks. Steps may be added or deleted to methods described within the scope of the present invention.